A stochastic analysis of distance estimation approaches in single molecule microscopy - quantifying the resolution limits of photon-limited imaging systems

Abstract

Optical microscopy is an invaluable tool to visualize biological processes at the cellular scale. In the recent past, there has been significant interest in studying these processes at the single molecule level. An important question that arises in single molecule experiments concerns the estimation of the distance of separation between two closely spaced molecules. Presently, there exists different experimental approaches to estimate the distance between two single molecules. However, it is not clear as to which of these approaches provides the best accuracy for estimating the distance. Here, we address this problem rigorously by using tools of statistical estimation theory. We derive formulations of the Fisher information matrix for the underlying estimation problem of determining the distance of separation from the acquired data for the different approaches. Through the Cramer-Rao inequality, we derive a lower bound to the accuracy with which the distance of separation can be estimated. We show through Monte-Carlo simulations that the bound can be attained by the maximum likelihood estimator. Our analysis shows that the distance estimation problem is in fact related to the localization accuracy problem, the latter being a distinct problem that deals with how accurately the location of an object can be determined. We have carried out a detailed investigation of the relationship between the Fisher information matrices of the two problems for the different experimental approaches considered here. The paper also addresses the issue of a singular Fisher information matrix, which presents a significant complication when calculating the Cramer-Rao lower bound. Here, we show how experimental design can overcome the singularity. Throughout the paper, we illustrate our results by considering a specific image profile that describe the image of a single molecule.

1 Introduction

The study of biomolecular interactions that occur within a cell is fundamental to all areas of basic biomedical research. The optical microscope is one of the most preferred tools to study biomolecular interactions, as it enables the direct visualization of these processes in real time. For instance, several technological advances in the past decade have made it possible to image individual biomolecules with an optical microscope even in live biological cells (Moerner (2007); Ober et al (2004a)). In many concrete applications, it is important to know the distance of separation between the biomolecules, as this has significant biological implications. The resolution limit of the optical microscope plays a crucial role in determining the ability to measure the distance of separation between biomolecules. Classical resolution criteria such as Rayleigh’s criterion, although extensively used, are well known to be based on heuristic notions that render them inadequate for present day microscopy systems. Therefore quantifying the resolution limit is a very important problem with significant implications on the nature and type of studies that can be carried out with an optical microscope.

Current experimental approaches to studying single molecule interactions can be broadly classified into two categories. In one set of approaches, which we refer to as the simultaneous detection approach (Figure 1A), photon emission from the point sources occurs simultaneously during image acquisition and hence the acquired images contain signal from both point sources (Santos and Young (2000); Ram et al (2006a); Chao et al (2009a, b)). In the other set of approaches, which we refer to as the separate detection approach (Figure 1C), photon emission from the point sources are temporally separated (e.g. stochastic photoactivation (Betzig et al (2006); Rust et al (2006); Hess et al (2006)) and blinking (Lidke et al (2005); Lagerholm et al (2006))). Hence the acquired images typically contain signal from only one of the point sources. For both types of approaches, the analysis of the acquired data is carried out using a parameter estimation framework. For example, in the case of the simultaneous detection approach the distance between the point sources is determined by fitting a pair of suitably parameterized image profiles to the acquired data. In the case of the separation detection approach, the analysis involves independently localizing the point sources and then deducing the distance. It has been reported that both approaches are capable of accurately measuring nanometer scale distances, well below the classical resolution criteria. However, an important question arises as to what are the fundamental performance limits of the two experimental approaches to measure the distance of separation.

Fig. 1.

Different experimental approaches to determine the distance of separation between two identical point sources. Panel A illustrates the simultaneous detection approach in which photon emission from both point sources occurs during image acquisition. In this approach, the data consists of a single image that contains signal from both point sources. Panel B illustrates the special case of the simultaneous detection approach, where the image of one of the point sources is additionally available. Here, the data consists of a pair of images where one of the images contains signal from only one point source, whereas the other image contains signal from both point sources. Panel C illustrates the separate detection approach, where photon emission from the point sources are temporally separated. Here, the data consists of a pair of images, where each image contains signal from either of the point sources.

In this paper, we use the tools of statistical signal processing to investigate this question in a rigorous manner. We formulate the resolution problem as a parameter estimation problem of determining the distance between two closely spaced point sources. The issue of resolvability of the two point sources then becomes a question of how accurately the distance can be estimated, i.e., how large is the standard deviation of the distance estimator. In this context, it is important to know what is the lowest possible standard deviation with which the distance can be estimated, as this can be used as a benchmark for the resolvability of the point sources. For this, we make use of the Cramer-Rao inequality (Rao (1965)) which, through the inverse Fisher information matrix, provides a lower bound to the variance of any unbiased estimator of an unknown parameter. Thus, in the present context we interpret the Cramer-Rao lower bound of the distance parameter as a measure of resolvability of the two point sources.

Here, we derive formulations of the Fisher informationmatrix for the parameter estimation problem that underlies the data analysis for the two approaches. Our analysis shows that the Fisher information matrices for the two techniques exhibit very distinct behaviors. For instance, in the simultaneous detection approach the Fisher information matrix depends on the distance of separation between the point sources. In contrast, for the separate detection approach the Fisher information matrix is independent of the distance of separation. As we will see, the distance dependence of the Fisher information matrix has several implications. In particular, for the simultaneous detection approach the Fisher information matrix becomes singular when the distance goes to zero assuming that the two point sources have identical image profiles and photon detection rates, which is typically the case in most imaging applications. An immediate implication is that for very small distances, the Cramer-Rao lower bound of the distance will be numerically very large, thereby predicting poor resolvability of the point sources. On the other hand, the Fisher information matrix for the separate detection approach is invertible for all values of the distance including when the distance is equal to zero.

Another problem that is of significance in the present context is the localization accuracy problem, which deals with how accurately the location of an object can be determined (Wong et al (2011); Ram et al (2006b); Ober et al (2004b); Rohr (2007)). For the separate detection approach, the localization accuracy problem naturally arises as part of the data analysis procedure. For the simultaneous detection approach, the localization accuracy problem arises as a special case where in some applications the image of one of the point sources is additionally available (e.g. photobleaching (Ram et al (2006a); Gordon et al (2004); Qu et al (2004)), which, in turn, can be used as a priori information (Figure 1B). Here, we investigate the relationship between the Fisher information matrix of the two approaches and that of the localization accuracy problem. Our analysis shows that for the separate detection approach, the expression for the Fisher information matrix is equivalent to that of the localization accuracy problem, whereas for the simultaneous detection approach the equivalence is attained only when the distance of separation between the point sources becomes very large (i.e., d → ∞). In this context, we also investigate the singularity of the Fisher information matrix for the simultaneous detection approach. In particular we show that the singularity can be removed when the location coordinates of one of the point sources is known a priori.

Previously, we have examined the distance estimation problem for optical microscopes, where we derived analytical expressions for the Fisher information matrix. In Ram et al (2006a), we investigated the 2D imaging scenario for the simultaneous detection approach, where the point sources were assumed to be located on the x axis of the plane of focus in the object space. In Chao et al (2009a,b), we considered the 3D imaging scenario for the simultaneous detection approach, where the point sources were assumed to be located anywhere in the object space. In Chao et al (2009c), we reported numerical calculations of the Cramer-Rao lower bound for the two detection approaches considered here. In the present work, we rigorously analyze the relationship between the Fisher information matrices of the two experimental approaches considered here and that of the localization accuracy problem.

In the past, other groups have investigated the distance estimation problem by adopting a simplified data model, where the acquired data is described as a deterministic signal corrupted by additive noise (Helstrom (1964); Smith (2005); Shahram and Milanfar (2004)). Because photon/light emission from a point source is inherently a random phenomenon (Young (1996)), it is important to take into account the stochastic nature of the signal (i.e., the photon statistics) from the point sources especially when dealing with photon-limited imaging systems (O’Sullivan et al (1998)). In our (prior and current) work, we have adopted a stochastic framework and model the acquired data as a spatio-temporal random process (marked Poisson process). In this way we explicitly take into account the photon statistics. Thus our results and analyses presented in this paper provide a broad framework to investigate the resolution limits for a wide variety of low light level imaging applications.

The paper is organized as follows. In Section 3, we derive general expressions of the Fisher information matrix for the estimation problem that underlies the simultaneous detection approach. We also derive the Fisher information matrix for a concrete scenario in optical microscopy where the image of an object is considered to be spatially invariant. In Section 4, we discuss the relationship between the Fisher information matrix for the simultaneous detection approach and that of the localization accuracy problem. In Section 5, we consider a special case of the simultaneous detection approach, where we assume that the location coordinates of one of the point sources is known and derive the Fisher information matrix. As we will see, the analysis of this special case provides important insights into the relationship between the two approaches considered here. In Section 6, we derive the Fisher information matrix for the separate detection approach. Finally, in Section 7 we validate our results by demonstrating that the maximum likelihood estimator of the distance attains the Cramer-Rao lower bound for the different experimental approaches considered here. Throughout the paper, we illustrate our results with examples relevant to single molecule microscopy.

2 Stochastic framework

We assume an acquired image to consist of the time points and the spatial coordinates of the detected photons and model it as a spatio-temporal random process. We refer to this process as the image detection process 𝒢 (see Ram et al (2006b) for details). The parameter space Θ is assumed to be an open subset of ℝⁿ and the detector that is used to capture the photons is denoted as 𝒞, where 𝒞 ⊆ ℝ² is open. The temporal part of 𝒢 is modeled as an inhomogeneous Poisson process with intensity Λ_θ called the photon detection rate and the spatial part of 𝒢 is modeled as a sequence of mutually independent random variables with densities {f_θ,τ}_τ≥t0 called the photon distribution profile. It is assumed that the spatial and temporal components are mutually independent of each other and that f_θ,τ satisfies the regularity conditions necessary for the calculation of the Fisher information matrix (Ram et al (2006b); Kay (1993)).

The general expression of the Fisher informationmatrix for the image detection process 𝒢 is given by (Ram et al (2006b))

$$\mathbf{I}(\theta )={\displaystyle {\int }_{t_0}^t}{\displaystyle {\int }_{𝒞}}\frac{1}{{\Lambda }_{\theta }(\tau )f_{\theta ,\tau }(r)}{\left(\frac{\partial [{\Lambda }_{\theta }(\tau )f_{\theta ,\tau }(r)]}{\partial \theta }\right)}^T\frac{\partial [{\Lambda }_{\theta }(\tau )f_{\theta ,\tau }(r)]}{\partial \theta }\mathit{\text{drd}}\tau ,\theta \in \Theta ,$$ (1)

where [t₀, t] denotes the time interval during which the data is acquired and the integration variable r denotes the 2D Cartesian coordinates (x,y). In the above equation, no specific assumptions have been made regarding the functional form of f_θ,τ or Λ_θ. Therefore, the above expression of I(θ) is applicable to a wide variety of imaging conditions, such as coherent/incoherent/partially-coherent light sources, polarized illumination and detection, etc. We note that the above equation is applicable to both stationary and moving objects, since we allow the density f_θ,τ, which describes the image profile of the object, to vary in time.

In order to quantify and compare the performance of the various experimental approaches considered in this paper, we make use of the Cramer-Rao inequality (Rao (1965)), which states that for any unbiased estimator θ̂ of a n × 1 vector parameter θ, Cov(θ̂) ≥ I⁻¹(θ), θ ∈ Θ, where I(θ) denotes the Fisher information matrix and it is assumed that the inverse exists. From this inequality, it immediately follows that the i^th leading diagonal entry of the inverse Fisher information matrix ([I⁻¹(θ)]_ii) provides a lower bound to the variance of the estimates of the i^th component of the parameter vector (θ_i), i = 1, …, n.

Throughout the paper, we adopt a parameterization in which the location of the two point sources are specified in terms of their Cartesian coordinates, i.e., (x₀₁, y₀₁) and (x₀₂, y₀₂). Hence the expressions for the Fisher information matrix will be given in terms of this parameterization. As we will see in subsequent sections, this parameterization not only simplifies the derivation of the Fisher information matrix for the different experimental approaches considered here, but it also helps in the analysis of the relationship between the distance estimation problem and the localization accuracy problem. To derive the Cramer-Rao lower bound for the distance parameter d, we require the analytical expression for the (inverse) Fisher information matrix of d. For this, we make use of the following coordinate transformation formula (Kay (1993))

$${\mathbf{I}}^{-1}(d)=\left(\frac{\partial d}{\partial \theta }\right){\mathbf{I}}^{-1}(\theta ){\left(\frac{\partial d}{\partial \theta }\right)}^T,d\in [0,\mathrm{\infty }),$$ (2)

where θ = (x₀₁, y₀₁, x₀₂, y₀₂), I⁻¹(θ) denotes the inverse Fisher information matrix corresponding to θ, and

$${\left(\frac{\partial d}{\partial \theta }\right)}^T≔\frac{1}{d}\left(\begin{array}{c}\hfill -(x_{02}-x_{01})\hfill \\ \hfill -(y_{02}-y_{01})\hfill \\ \hfill (x_{02}-x_{01})\hfill \\ \hfill (y_{02}-y_{01})\hfill \end{array}\right),\theta \in \Theta .$$

3 Fisher information matrix for the simultaneous detection approach

In the simultaneous detection approach, the acquired image is assumed to contain the signal from both objects. Hence the photon detection rate Λ_θ and the photon distribution profile f_θ,τ can be written as

$${\Lambda }_{\theta }(\tau )={\Lambda }_{\theta ,1}(\tau )+{\Lambda }_{\theta ,2}(\tau ),\theta \in \Theta ,\tau \ge t_0,$$ (3)

$$f_{\theta ,\tau }(r)={\epsilon }_{\theta ,1}(\tau )f_{\theta ,\tau ,1}(r)+{\epsilon }_{\theta ,2}(\tau )f_{\theta ,\tau ,2}(r),r=(x,y)\in 𝒞,\theta \in \Theta ,\tau \ge t_0,$$ (4)

where 𝒞 denotes the detector, Λ_θ,1, Λ_θ,2 and f_θ,τ,1, f_θ,τ,2 denote the photon detection rates and the photon distribution profiles of the two objects, respectively, and ε_θ,i(τ) ≔ Λ_θ,i(τ)/(Λ_θ,1 (τ) + Λ_{θ, 2} (τ)), θ ∈ Θ, τ ≥ t₀, i = 1, 2.

The results in this section are divided into two parts. In Section 3.1, we first derive general expressions of the Fisher information matrix for the simultaneous detection approach (Theorem 1). Here, we make no assumptions regarding the specific functional form of the photon detection rates Λ_θ,i or the photon distribution profiles f_θ,τ,i, i = 1, 2. Hence these results provide a general framework that is applicable to a wide variety of imaging scenarios.

In Section 3.2, we consider a concrete scenario (spatially invariant case) in optical microscopy where we assume a specific functional form for the photon distribution profiles f_θ,τ,i, i = 1, 2, which are expressed as a scaled and shifted version of the image of the objects. We then derive the Fisher information matrix for this functional form of f_θ,τ,i, i = 1, 2 (Theorem 2). As will be shown, the resulting Fisher information matrix can be expressed as a product decomposition of the form DCD^T, where D is an orthogonal matrix and C is a positive semidefinite matrix. Under weak assumptions of spatial symmetry for the image of the objects (which are typically satisfied in most situations), the product decomposition greatly simplifies the calculation of the Fisher information matrix and also facilitates the derivation of an analytical expression for the inverse Fisher information matrix (Corollary 1).

3.1 General expression of the Fisher information matrix

In many imaging applications, the unknown parameter vector θ can be expressed as θ = (θ_f, θ_Λ), where θ_f denotes the spatial component and θ_Λ denotes the temporal component. The spatial component θ_f typically consists of parameters that specify the location of one or more objects and the temporal component θ_Λ consists of parameters that specify the photon detection rates of the objects.

In the following theorem, we express the Fisher information matrix as a 2 × 2 block matrix. The terms in the leading diagonal (i.e., S_sim and T_sim) correspond to the Fisher information matrix of the spatial θ_f and temporal θ_Λ components while the terms in the off-diagonal (i.e., R_sim and ${\mathbf{R}}_{\mathit{\text{sim}}}^T$) correspond to the coupling between the spatial and temporal components. We derive expressions for three practical scenarios. In the first scenario, we derive a general expression for the Fisher information matrix. In the second scenario, we consider the case where the photon detection rates are related to one another by a known scalar function β, i.e., β(τ)Λ_θ,1 (τ) = Λ_θ,2 (τ) for τ ≥ t₀ and θ ∈ Θ, where β(τ) ≥ 0. In some applications, the photon detection rates of the objects are assumed to be the same, i.e., Λ_θ,1 (τ) = Λ_θ,2(τ), τ ≥ t₀. We note that this condition is a special case of the second scenario considered here with β(τ) = 1, τ ≥ t₀. For this scenario we show that the Fisher information matrix becomes block diagonal, which implies that the spatial θ_f and temporal θ_Λ components become decoupled. We note that this decoupling simplifies the subsequent analysis of the Fisher information matrix. In the third scenario, we assume that the photon distribution profiles of the objects are equal, i.e., f_θ,τ,1 (r) = f_θ,τ,2 (r) for r ∈ 𝒞, θ ∈ Θ and τ ≥ t₀. This scenario arises in many applications, where the image profiles of the objects are assumed to be identical. For this scenario also we show that the Fisher information matrix becomes block diagonal.

Theorem 1 Let Θ ⊆ ℝⁿ. For θ ≔ (θ_f, θ_Λ) ∈ Θ, let 𝒢(Λ_θ, {f_θ,τ}_τ≥t0, 𝒞) be an image detection process, where Λ_θ and f_θ,τ are defined in eqs. 3 and 4, respectively. Assume that for θ ∈ Θ, τ ≥ t₀ and i = 1, 2,

• A1 (∂f_θ,τ,i(r)/∂θ_Λ) = 0, r ∈ 𝒞,

• A2 (∂Λ_θ,i(τ)/∂θ_f) = 0.

  • 1.
    Then the Fisher information matrix of 𝒢 corresponding to the acquisition time interval [t₀, t] for the simultaneous detection approach is given by

    $${\mathbf{I}}_{\mathit{\text{sim}}}(\theta )=\left[\begin{array}{c}\hfill {\mathbf{S}}_{\mathit{\text{sim}}}(\theta ){\mathbf{R}}_{\mathit{\text{sim}}}(\theta )\hfill \\ \hfill {\mathbf{R}}_{\mathit{\text{sim}}}^T(\theta ){\mathbf{T}}_{\mathit{\text{sim}}}(\theta )\hfill \end{array}\right],\theta \in \Theta ,$$

    where for θ ∈ Θ,

    $${\mathbf{S}}_{\mathit{\text{sim}}}(\theta )≔{\displaystyle {\int }_{t_0}^t}{\displaystyle {\int }_{𝒞}}\frac{{\Lambda }_{\theta }(\tau )}{f_{\theta ,\tau }(r)}{\left(\frac{\partial f_{\theta ,\tau }(r)}{\partial {\theta }_f}\right)}^T\frac{\partial f_{\theta ,\tau }(r)}{\partial {\theta }_f}\mathit{\text{drd}}\tau ,$$ (5)

    $${\mathbf{R}}_{\mathit{\text{sim}}}(\theta )≔{\displaystyle {\int }_{t_0}^t}{\displaystyle {\int }_{𝒞}}\frac{{\Lambda }_{\theta }(\tau )}{f_{\theta ,\tau }(r)}{\left(\frac{\partial f_{\theta ,\tau }(r)}{\partial {\theta }_f}\right)}^T\frac{\partial f_{\theta ,\tau }(r)}{\partial {\theta }_{\Lambda }}\mathit{\text{drd}}\tau ,$$ (6)

    $${\mathbf{T}}_{\mathit{\text{sim}}}(\theta )≔{\displaystyle {\int }_{t_0}^t}\frac{1}{{\Lambda }_{\theta }(\tau )}{\left(\frac{\partial {\Lambda }_{\theta }(\tau )}{\partial {\theta }_{\Lambda }}\right)}^T\frac{\partial {\Lambda }_{\theta }(\tau )}{\partial {\theta }_{\Lambda }}d\tau +{\displaystyle {\int }_{t_0}^t}{\displaystyle {\int }_{𝒞}}\frac{{\Lambda }_{\theta }(\tau )}{f_{\theta ,\tau }(r)}{\left(\frac{\partial f_{\theta ,\tau }(r)}{\partial {\theta }_{\Lambda }}\right)}^T\frac{\partial f_{\theta ,\tau }(r)}{\partial {\theta }_{\Lambda }}\mathit{\text{drd}}\tau .$$ (7)
  • 2.For β(τ) ≥ 0, τ ≥ t₀, assume, in addition to A1 and A2, that

• A3 β(τ)Λ_θ,1(τ) = Λ_θ,2(τ), τ ≥ t₀ and θ ∈ Θ.

  Then the Fisher information matrix of 𝒢 corresponding to the acquisition time interval [t₀, t] for the simultaneous detection approach is given by

  $${\mathbf{I}}_{\mathit{\text{sim}}}(\theta )=\left[\begin{array}{cc}\hfill {\mathbf{S̃}}_{\mathit{\text{sim}}}(\theta )\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\mathbf{T̃}}_{\mathit{\text{sim}}}(\theta )\hfill \end{array}\right],\theta \in \Theta ,$$

  where for θ ∈ Θ,

  $${\mathbf{S̃}}_{\mathit{\text{sim}}}(\theta )≔{\displaystyle {\int }_{t_0}^t}{\displaystyle {\int }_{𝒞}}\frac{{\Lambda }_{\theta ,1}(\tau )}{f_{\theta ,\tau ,1}(r)+\beta (\tau )f_{\theta ,\tau ,2}(r)}{\left(\frac{\partial [f_{\theta ,\tau ,1}(r)+\beta (\tau )f_{\theta ,\tau ,2}(r)]}{\partial {\theta }_f}\right)}^T\times \frac{\partial [f_{\theta ,\tau ,1}(r)+\beta (\tau )f_{\theta ,\tau ,2}(r)]}{\partial {\theta }_f}\mathit{\text{drd}}\tau ,$$

  $${\mathbf{T̃}}_{\mathit{\text{sim}}}(\theta )≔{\displaystyle {\int }_{t_0}^t}\frac{1+\beta (\tau )}{{\Lambda }_{\theta ,1}(\tau )}{\left(\frac{\partial {\Lambda }_{\theta ,1}(\tau )}{\partial {\theta }_{\Lambda }}\right)}^T\frac{\partial {\Lambda }_{\theta ,1}(\tau )}{\partial {\theta }_{\Lambda }}d\tau .$$
  • 3.For θ ∈ Θ and τ ≥ t₀, assume, in addition to A1 and A2, that

• A4 f_θ,τ,1(r) = f_θ,τ,2(r) for r ∈ 𝒞.

Then the Fisher information matrix of 𝒢 corresponding to the acquisition time interval [t₀, t] for the simultaneous detection approach is given by

$${\mathbf{I}}_{\mathit{\text{sim}}}(\theta )=\left[\begin{array}{cc}\hfill {\mathbf{S̄}}_{\mathit{\text{sim}}}(\theta )\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\mathbf{T̄}}_{\mathit{\text{sim}}}(\theta )\hfill \end{array}\right],\theta \in \Theta ,$$

where for θ ∈ Θ,

$${\mathbf{S̄}}_{\mathit{\text{sim}}}(\theta )≔{\displaystyle {\int }_{t_0}^t}{\Lambda }_{\theta }(\tau )d\tau {\displaystyle {\int }_{𝒞}}\frac{1}{f_{\theta ,\tau ,1}(r)}{\left(\frac{\partial f_{\theta ,\tau ,1}(r)}{\partial {\theta }_f}\right)}^T\frac{\partial f_{\theta ,\tau ,1}(r)}{\partial {\theta }_f}\mathit{\text{dr}},$$

$${\mathbf{T̄}}_{\mathit{\text{sim}}}(\theta )≔{\displaystyle {\int }_{t_0}^t}\frac{1}{{\Lambda }_{\theta }(\tau )}{(\frac{\partial {\Lambda }_{\theta }(\tau )}{\partial {\theta }_{\Lambda }})}^T\frac{\partial {\Lambda }_{\theta }(\tau )}{\partial {\theta }_{\Lambda }}d\tau .$$

Proof See Section A.1 in Appendix for proof.

In many applications it is important to know whether the Fisher information matrix I(θ) is (block) diagonal. For instance, it is well known that under certain conditions the maximum likelihood estimator of a vector parameter θ is asymptotically Gaussian distributed with mean θ and covariance I⁻¹(θ) (see Van des Bos (2007)). From the above Theorem, we see that if the photon detection rates can be expressed as a scalar function of one another or if the photon distribution profiles are identical, then I(θ) becomes block diagonal. This implies that the maximum likelihood estimates of the spatial (θ_f) and temporal (θ_Λ) components of the unknown vector parameter θ are asymptotically independent. Moreover, if an efficient estimator of θ exists (i.e., an estimator whose covariance matrix is equal to I⁻¹(θ), θ ∈ Θ), then the estimates of θ_f and θ_Λ are uncorrelated. Another implication of block diagonality is that the Cramer-Rao lower bound of the spatial component θ_f is independent of the number of unknown parameters in the temporal component θ_Λ, and vice versa.

Remark 1 In result 2 of Theorem 1, we showed that the Fisher information matrix I_sim(θ) is block diagonal if β(τ)Λ_θ,1(τ) = Λ_θ,2(τ) for τ ≥ t₀ and θ ∈ Θ, where β(τ) ≥ 0, τ ≥ t₀, is a known scalar function. We note that I_sim(θ) will be block diagonal when Λ_θ,1(τ) = β(τ)Λ_θ,2(τ), τ ≥ t₀ and θ ∈ Θ for β(τ) ≥ 0, τ ≥ t₀.

3.2 Fisher information matrix for the spatially invariant case

We next investigate a concrete scenario in optical microscopy where the image of the objects is spatially invariant, and we derive the Fisher information matrix for the simultaneous detection approach. Here, we introduce a specific parameterization of the spatial component θ_f of the parameter vector θ given by θ_f = θ_c = (x₀₁, y₀₁, x₀₂, y₀₂) ∈ Θ_c, where (x₀₁, y₀₁) and (x₀₂, y₀₂) denote the Cartesian coordinates of the two objects, and Θ_c is the parameter space that is an open subset of ℝ⁴. We consider the infinitely large detector 𝒞 = ℝ². For any given imaging condition, this infinite detector provides the best case scenario, where all the photons that reach the detector plane are detected.

In many microscopy applications, the image of an object can be considered to be invariant with respect to shifts in the object location (Young (1996)). In the present context, the photon distribution profile f_θc,τ,i, i = 1, 2, can be expressed as a scaled and shifted version of the image of the object and is given by

$$f_{{\theta }_c,\tau ,i}(r)=\frac{1}{M^2}{q}_i\left(\frac{x}{M}-x_{0i},\frac{y}{M}-y_{0i}\right),r=(x,y)\in {ℝ}^2,$$ (8)

where θ_c ∈ Θ_c, τ ≥ t₀, i = 1, 2, M denotes the total lateral magnification of the optical system, and q_i denotes the image function of the i^th object, i = 1, 2. An image function q is defined as the image of an object at unit magnification when the object is located at the origin of the coordinate axes. By definition, f_θc,τ,i, i = 1, 2, is assumed to satisfy the regularity conditions that are necessary for the calculation of the Fisher information matrix. Hence we impose appropriate conditions on the image functions, which are given in Definition 6 (see Appendix).

In many imaging experiments, the temporal component θ_Λ of the vector parameter θ is either assumed to be known or the photon detection rates are unknown but assumed to be equal (Λ_θ,1(τ) = Λ_θ,2(τ), τ ≥ t₀). In the former case, the Fisher information matrix of the simultaneous detection approach I_sim(θ) trivially reduces to that of the spatial component θ_f i.e., I_sim(θ) = S_sim(θ), θ ∈ Θ. In the latter case, the Fisher information matrices of the spatial and temporal components are decoupled as shown in Result 2 of Theorem 1. Therefore in this section, we focus our analysis on the Fisher information matrix for the spatial component θ_f.

Without loss of generality, we assume that the photon detection rates of the objects are known, and hence we have

$${\Lambda }_{{\theta }_c}(\tau )={\Lambda }_1(\tau )+{\Lambda }_2(\tau ),\tau \ge t_0,{\theta }_c\in {\Theta }_c,$$ (9)

where Λ₁ and Λ₂ denote the photon detection rates of the two objects. Further, the photon distribution profile f_θ,τ is given by

$$f_{{\theta }_c,\tau }(r)≔{\epsilon }_1(\tau )f_{{\theta }_c,\tau ,1}(r)+{\epsilon }_2(\tau )f_{{\theta }_c,\tau ,2}(r),r\in {ℝ}^2,{\theta }_c\in {\Theta }_c,\tau \ge t_0.$$ (10)

where ε_i(τ) = Λ_i(τ)/(Λ₁(τ) + Λ₂(τ)), and f_θc,τ,i is given by eq. 8 for i = 1, 2, τ ≥ t₀ and θ_c ∈ Θ_c.

In the next Theorem we derive an analytical expression of the Fisher information matrix for the spatial component θ_c pertaining to the specific parameterization of the photon detection rate Λ_θc and the photon distribution profile f_θc,τ given in eqs. 9 and 10, respectively. Here, we express the Fisher information matrix S_sim(θ_c) as a 2 × 2 block matrix. As we shall see in Section 4, this expression will be used to analyze its relationship with the Fisher information matrix for the localization accuracy problem. We also derive a product decomposition for S_sim(θ_c). This decomposition simplifies the calculation of the inverse of S_sim(θ_c) and enables us to obtain an analytical expression for the same (Corollary 1).

Theorem 2 Let Θ_c ⊆ ℝ⁴. For θ_c = (x₀₁, y₀₁, x₀₂, y₀₂) ∈ Θ_c, let 𝒢(Λ_θc, {f_θc,τ}_τ≥t0, 𝒞) be an image detection process, where Λ_θ and f_θ,τ are given by eqs. eqs. 9 and 10, respectively.

• 1.
  For θ_c ∈ Θ_c, the Fisher information matrix of the spatial component corresponding to the acquisition time interval [t₀, t] for the simultaneous detection approach is given by

  $${\mathbf{S}}_{\mathit{\text{sim}}}({\theta }_c)=(\begin{array}{c}\hfill {\mathbf{K}}_{11}({\theta }_c){\mathbf{K}}_{12}({\theta }_c)\hfill \\ \hfill {\mathbf{K}}_{12}^T({\theta }_c){\mathbf{K}}_{22}({\theta }_c)\hfill \end{array}),$$ (11)

  where for θ_c ∈ Θ_c and i, j = 1, 2,

  $${\mathbf{K}}_{\mathit{\text{ij}}}({\theta }_c)≔{\displaystyle {\int }_{t_0}^t}{\displaystyle {\int }_{{ℝ}^2}}\frac{{\Lambda }_i(\tau ){\Lambda }_j(\tau )}{{\Lambda }_1(\tau )q_1(x-x_{01},y-y_{01})+{\Lambda }_2(\tau )q_2(x-x_{02},y-y_{02})}\times \left(\begin{array}{c}\hfill \frac{\partial q_i(x-x_{0i},y-y_{0i})}{\partial x}\frac{\partial q_j(x-x_{0j},y-y_{0j})}{\partial x}\frac{\partial q_i(x-x_{0i},y-y_{0i})}{\partial x}\frac{\partial q_j(x-x_{0j},y-y_{0j})}{\partial y}\hfill \\ \hfill \frac{\partial q_i(x-x_{0i},y-y_{0i})}{\partial y}\frac{\partial q_j(x-x_{0j},y-y_{0j})}{\partial x}\frac{\partial q_i(x-x_{0i},y-y_{0i})}{\partial y}\frac{\partial q_j(x-x_{0j},y-y_{0j})}{\partial y}\hfill \end{array}\right)\mathit{\text{ dxdyd}}\tau .$$ (12)
• 2.
  Let $d=\sqrt{{(x_{02}-x_{01})}^2+{(y_{02}-y_{01})}^2}$ and define ${\Theta }_c^0=\{(x_{01},y_{01},x_{02},y_{02})|(x_{01},y_{01})=(x_{02},y_{02})\}$. Then for ${\theta }_c\in {\Theta }_c\backslash {\Theta }_c^0$, the Fisher information matrix S_sim(θ_c) given in result 1 of this Theorem can be written as

  $${\mathbf{S}}_{\mathit{\text{sim}}}({\theta }_c)=\mathbf{D}({\theta }_c)\mathbf{C}({\theta }_c){\mathbf{D}}^T({\theta }_c),$$

  where for ${\theta }_c\in {\Theta }_c\backslash {\Theta }_c^0$

  $$\mathbf{D}({\theta }_c)≔\left(\begin{array}{cc}\hfill \mathbf{D̃}({\theta }_c)\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \mathbf{D̃}({\theta }_c)\hfill \end{array}\right),\mathbf{D̃}({\theta }_c)≔\frac{1}{d}\left(\begin{array}{cc}{x}_{02}-x_{01}\hfill & -(y_{02}-y_{01})\hfill \\ y_{02}-y_{01}\hfill & \hfill x_{02}-x_{01}\hfill \end{array}\right),$$ (13)

  $$\mathbf{C}({\theta }_c)≔\left(\begin{array}{c}\hfill {\mathbf{C}}_{11}({\theta }_c){\mathbf{C}}_{12}({\theta }_c)\hfill \\ \hfill {\mathbf{C}}_{12}^T({\theta }_c){\mathbf{C}}_{22}({\theta }_c)\hfill \end{array}\right),$$ (14)

  $${\mathbf{C}}_{\mathit{\text{ij}}}({\theta }_c)≔{\displaystyle {\int }_{t_0}^t}{\displaystyle {\int }_{{ℝ}^2}}\frac{{\Lambda }_i(\tau ){\Lambda }_j(\tau )}{{\Lambda }_1(\tau )q_1(x+\frac{d}{2},y)+{\Lambda }_2(\tau )q_2(x-\frac{d}{2},y)}\times (\begin{array}{c}\hfill q_{i,x}^{\prime }(x,y)q_{j,x}^{\prime }(x,y)q_{i,x}^{\prime }(x,y)q_{j,y}^{\prime }(x,y)\hfill \\ \hfill q_{i,x}^{\prime }(x,y)q_{j,y}^{\prime }(x,y)q_{i,y}^{\prime }(x,y)q_{j,y}^{\prime }(x,y)\hfill \end{array})\mathit{\text{ dxdyd}}\tau ,i,j=1,2,$$ (15)

  with

  $$q_{i,\zeta }^{\prime }(x,y)≔\{\begin{array}{c}\hfill \frac{\partial q_1(x+\frac{d}{2},y)}{\partial \zeta },i=1,(x,y)\in {ℝ}^2,\hfill \\ \hfill \frac{\partial q_2(x-\frac{d}{2},y)}{\partial \zeta },i=2,(x,y)\in {ℝ}^2,\hfill \end{array}\zeta \in \{x,y\}.$$ (16)
• 3.
  Assume that q₁ and q₂ are symmetric along the y axis with respect to y = 0, i.e., q_i(x, y) = q_i(x, −y), (x, y) ∈ ℝ² and i = 1, 2. Then for ${\theta }_c\in {\Theta }_c\backslash {\Theta }_c^0$ and i = 1, 2, C_ij(θ_c) is given by

  $${\mathbf{C}}_{\mathit{\text{ij}}}({\theta }_c)≔{\displaystyle {\int }_{t_0}^t}{\displaystyle {\int }_{{ℝ}^2}}\frac{{\Lambda }_i(\tau ){\Lambda }_j(\tau )}{{\Lambda }_1(\tau )q_1(x+\frac{d}{2},y)+{\Lambda }_2(\tau )q_2(x-\frac{d}{2},y)}\times \left(\begin{array}{cc}\hfill q_{i,x}^{\prime }(x,y)q_{j,x}^{\prime }(x,y)\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill q_{i,y}^{\prime }(x,y)q_{j,y}^{\prime }(x,y)\hfill \end{array}\right)\mathit{\text{ dxdyd}}\tau .$$ (17)

Proof Substituting for f_θc,τ and Λ_θc in the expression for I_ff (θ) given by eq. 5 (see result 1 of Theorem 1) and using Lemma 2, we obtain result 1. For proof of results 2 and 3, please see Section A.2 in Appendix.

In result 1 of the above Theorem, we obtained a block matrix representation of the Fisher information matrix S_sim(θ_c). The leading diagonal terms correspond to the individual contributions from the two objects and the off-diagonal terms correspond to the coupling between the two objects. As we will show in the next Section, the coupling plays an important role in the analysis of the relationship between the Fisher information matrix for the simultaneous detection approach and that for the localization accuracy problem.

The product decomposition D(θ_c)C(θ_c)D^T (θ_c) of S_sim(θ_c) that we obtained in result 2 of the above Theorem has an interesting structure. The matrix C(θ_c) is a special case of S_sim(θ_c) where the y coordinates of the two objects are assumed to be the same, i.e., y₀₂ = y₀₁, and the x coordinates of the two objects are equidistant from the origin. Note that the matrix D(θ_c) is orthogonal (i.e.,D⁻¹(θ_c) = D^T (θ_c)). It should be pointed out that the product decomposition holds only when (x₀₁, y₀₁) ≠ (x₀₂, y₀₂), i.e., when the distance d is not equal to zero, since at (x₀₁, y₀₁) = (x₀₂, y₀₂) the matrix D(θ_c) is not defined. An implication of this product decomposition is that for a given ${\theta }_c^s=(x_{01}^s,y_{01}^s,x_{02}^s,y_{02}^s)$ such that $(x_{01}^s,y_{01}^s)\ne (x_{02}^s,y_{02}^s)$, the Fisher information matrix for ${\theta }_c^s$ can be obtained by first computing the Fisher information matrix for $\left(-\frac{d}{2},0,\frac{d}{2},0\right)$ and then preand post-multiplying it with $\mathbf{D}({\theta }_c^s)\text{ and }{\mathbf{D}}^T({\theta }_c^s)$, respectively, where d denotes the distance between the two objects. In many practical situations, the image of the objects is symmetric along the y (and the x) axis. As shown in result 3 of Theorem 2, when this condition is satisfied, several entries of the matrix C(θ_c) become zero, which in turn simplifies the calculation of C(θ_c).

Remark 2 Consider the scenario when the distance between the two objects is zero, i.e. x₀₁ = x₀₂ and y₀₁ = y₀₂. For this scenario, the Fisher information matrix S_sim(θ_c) given in result 1 of Theorem 2 is singular, if the photon detection rates and the image functions of the two objects are identical, i.e., Λ₁ = Λ₂ and q₁ = q₂ (also see Section 4.1). However, for distinct photon detection rates and image functions, S_sim(θ_c) will, in general, be invertible even when the distance between the objects is zero.

In the following Corollary, we make use of the product decomposition of the Fisher information matrix S_sim(θ_c) and the orthogonality of D(θ_c) to obtain an analytical expression for the inverse of S_sim(θ_c) when the distance d between the objects is non-zero.

Corollary 1 Define ${\Theta }_c^0=\{(x_{01},y_{01},x_{02},y_{02})|(x_{01},y_{01})=(x_{02},y_{02})\}$. For ${\theta }_c\in {\Theta }_c\backslash {\Theta }_c^0$, let S_sim(θ_c) be given by result 2 of Theorem 2, D(θ_c) be given by eq. 13 and C_ij(θ_c), i = 1, 2, be given by eq. 17. Assume that q₁ and q₂ are symmetric along the y axis with respect to y = 0, i.e., q_i(x, y) = q_i(x, −y), (x, y) ∈ ℝ², i = 1, 2. Then for ${\theta }_c\in {\Theta }_c\backslash {\Theta }_c^0$, we have

$${\mathbf{S}}_{\mathit{\text{sim}}}^{-1}({\theta }_c)=\mathbf{D}({\theta }_c)\mathbf{H}({\theta }_c){\mathbf{D}}^T({\theta }_c),$$

where for ${\theta }_c\in {\Theta }_c\backslash {\Theta }_c^0$,

$$\mathbf{H}({\theta }_c)=\left(\begin{array}{cc}\hfill \Gamma ({\theta }_c)\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \Gamma ({\theta }_c)\hfill \end{array}\right)\left(\begin{array}{cc}\hfill {\mathbf{C}}_{22}({\theta }_c)\hfill & \hfill -{\mathbf{C}}_{12}({\theta }_c)\hfill \\ \hfill -{\mathbf{C}}_{12}({\theta }_c)\hfill & \hfill {\mathbf{C}}_{11}(\theta )\hfill \end{array}\right)\left(\begin{array}{cc}\hfill \Gamma ({\theta }_c)\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \Gamma ({\theta }_c)\hfill \end{array}\right),\Gamma ({\theta }_c)≔\left(\begin{array}{cc}\hfill \frac{1}{\sqrt{{\Sigma }_{11}({\theta }_c)}}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \frac{1}{\sqrt{{\Sigma }_{22}({\theta }_c)}}\hfill \end{array}\right),$$ (18)

with

$${\Sigma }_{\mathit{\text{ii}}}({\theta }_c)≔{[{\mathbf{C}}_{11}({\theta }_c)]}_{\mathit{\text{ii}}}{[{\mathbf{C}}_{22}({\theta }_c)]}_{\mathit{\text{ii}}}-{({[{\mathbf{C}}_{12}({\theta }_c)]}_{\mathit{\text{ii}}})}^2,i=1,2,{\theta }_c\in {\Theta }_c\backslash {\Theta }_c^0.$$ (19)

Proof The expression for ${\mathbf{S}}_{\mathit{\text{sim}}}^{-1}({\theta }_c)$ is obtained by making use of the product decomposition of S_sim(θ_c) and using the expression for the inverse of a block matrix (Zhang (1999)).

4 Simultaneous detection approach and the localization accuracy problem

In many optical microscopy applications, one of the central questions concerns the accuracy with which the location of a microscopic object (e.g., single molecule, biological sub-cellular structure such as a vesicle) can be determined, since this has several implications on the nature and type of studies that can be carried out (see Wong et al (2011); Ober et al (2004b)). The Fisher information matrix for the problem of estimating the location of the i^th object from its image is given by (see Ram et al (2006b); Ober et al (2004b))

$${\mathbf{Q}}_{\mathbf{i}}≔{\displaystyle {\int }_{t_0}^t}{\Lambda }_i(\tau )d\tau {\displaystyle {\int }_{{ℝ}^2}}\frac{1}{q_i(x,y)}\left(\begin{array}{cc}\hfill {\left(\frac{\partial q_i(x,y)}{\partial x}\right)}^2\hfill & \hfill \frac{\partial q_i(x,y)}{\partial x}\frac{\partial q_i(x,y)}{\partial y}\hfill \\ \hfill \frac{\partial q_i(x,y)}{\partial x}\frac{\partial q_i(x,y)}{\partial y}\hfill & \hfill {\left(\frac{\partial q_i(x,y)}{\partial y}\right)}^2\hfill \end{array}\right)\mathit{\text{dxdy}},$$ (20)

where i = 1, 2 and q_i and Λ_i denote the image function and the photon detection rate of the i^th object, respectively, for i = 1, 2. The above equation was derived using the same stochastic framework used in this paper and it is assumed that the image contains signal from only the i^th object, i = 1, 2.

In the following theorem we show how the Fisher information matrix S_sim(θ_c) for the spatially invariant case of the simultaneous detection approach (Theorem 2) is related to the Fisher information matrix for the localization accuracy problem. Specifically, we show that when the distance tends to infinity, the Fisher information matrix S_sim(θ_c) becomes equivalent to that of two independent localization accuracy problems.

Theorem 3 For θ_c = (x₀₁, y₀₁, x₀₂, y₀₂) ∈ Θ_c, let S_sim(θ_c) be given by result 1 of Theorem 2. For i = 1, 2, let Q_i be given by eq. 20. Let Λ₁ and Λ₂, and q₁ and q₂ denote the photon detection rates and the image functions of the two objects, respectively. Assume that for i = 1, 2, ζ ∈ {x, y} and y ∈ ℝ,

• A1 lim_x→±∞ q_i (x, y) = 0,

• A2 ${\text{lim}}_{x\to \pm \mathrm{\infty }}\frac{\partial q_2(x,y)}{\partial \zeta }=0.$

Then

$${\mathbf{S}}_{\mathit{\text{sim}}}^{\mathit{\text{inf}}}≔\underset{x_{02}\to \mathrm{\infty }}{\text{lim}}{\mathbf{S}}_{\mathit{\text{sim}}}({\theta }_c)=\underset{x_{02}\to \mathrm{\infty }}{\text{lim}}\left(\begin{array}{cc}\hfill {\mathbf{K}}_{11}({\theta }_c)\hfill & \hfill {\mathbf{K}}_{12}({\theta }_c)\hfill \\ \hfill {\mathbf{K}}_{12}^T({\theta }_c)\hfill & \hfill {\mathbf{K}}_{22}({\theta }_c)\hfill \end{array}\right)=\left(\begin{array}{cc}\hfill {\mathbf{Q}}_1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\mathbf{Q}}_2\hfill \end{array}\right),$$

where K_ij(θ_c), i, j = 1, 2 is given by eq. 12.

Proof See Section A.3 in Appendix for proof.

We would like to point out that in deriving the above result we assumed x₀₂ to go to infinity. In general, the above result will hold when any one of the coordinates i.e., x₀₁, y₀₁ or y₀₂ is assumed to go to infinity. From the above Theorem we see that as the distance of separation becomes sufficiently large, the leading diagonal terms (K₁₁(θ_c) and K₂₂(θ_c)) of the Fisher information matrix S_sim(θ_c) for the simultaneous detection approach reduce to that of the localization accuracy problem for the two point sources (i.e., Q₁ and Q₂), and the off-diagonal term K₁₂(θ_c) goes to zero. Note that the off-diagonal term represents the coupling between the two point sources.

From a practical standpoint, the knowledge of the behavior of the off-diagonal term as a function of the distance would enable the experimenter to determine whether it is necessary to calculate the full Fisher information matrix for the simultaneous detection approach or to only calculate the Fisher information matrix for the localization accuracy problem. As we will see in the next section the latter is typically much easier to calculate, since a closed form analytical expression can be obtained.

4.1 Example 1

We next illustrate the results derived in the prior sections by considering a specific image function and calculate the Fisher information matrix for the simultaneous detection approach and for the localization accuracy problem. Here, we make use of the Cramer-Rao inequality to obtain a lower limit to the accuracy (i.e., standard deviation) of the estimates of the parameters of interest (see below). We assume the photon detection rates to be constant and equal i.e., Λ₁(τ) = Λ₂(τ) = Λ₀, τ ≥ t₀. We also assume the image functions to be identical and be given by the Airy profile, which, according to optical diffraction theory describes the image of an in-focus point source that is illuminated by incoherent, unpolarized light (Born and Wolf (1999)). The analytical expression for the image functions can be written as

$$q_1(x,y)=q_2(x,y)≔\frac{J_1^2(\frac{2\pi n_a}{\lambda }\sqrt{x^2+y^2})}{\pi (x^2+y^2)},(x,y)\in {ℝ}^2,$$ (21)

where J₁ denotes the first order Bessel function of the first kind, n_a > 0 denotes the numerical aperture of the objective lens used to image the point source and λ > 0 denotes wavelength of the detected photons.

By making use of the Cramer-Rao inequality, we define three different quantities, namely the 2D fundamental resolution measure (FREM) for the simultaneous detection approach, the limit to the accuracy of the location coordinates for the simultaneous detection approach, and the fundamental limit to the localization accuracy. Then in Corollary 2, we consider two limiting cases of the distance parameter d, i.e., d → 0 and d → ∞, and derive analytical expressions of the 2D FREM for the simultaneous detection approach. In Section 4.1.1, we numerically calculate the above quantities for different values of d and discuss their implications.

Definition 1 The 2D FREM for the simultaneous detection approach is defined as ${\delta }_d^{\mathit{\text{sim}}}≔\sqrt{{\mathbf{I}}_{\mathit{\text{sim}}}^{-1}(d)},d\in [0,\mathrm{\infty }),\text{ where }{\mathbf{I}}_{\mathit{\text{sim}}}^{-1}(d)$, is obtained by substituting ${\mathbf{S}}_{\mathit{\text{sim}}}^{-1}({\theta }_c)$ (Corollary 1) in the transformation formula given by eq. 2.

Definition 2 The limit to the accuracy of the location coordinates x_0i and y_0j for the simultaneous detection approach are defined as ${\delta }_{x_{0i}}^{\mathit{\text{sim}}}≔\sqrt{[{\mathbf{S}}_{\mathit{\text{sim}}}^{-1}]{({\theta }_c)}_{(2i-1)(2i-1)}}\text{ and }{\delta }_{y_{0i}}^{\mathit{\text{sim}}}=\sqrt{{[{\mathbf{S}}_{\mathit{\text{sim}}}^{-1}({\theta }_c)]}_{(2j)(2j)}}$, respectively, where i, j = 1, 2 and ${\mathbf{S}}_{\mathit{\text{sim}}}^{-1}({\theta }_c)$ denotes the inverse Fisher information matrix given by Corollary 1 for θ_c = (x₀₁, y₀₁, x₀₂, y₀₂) ∈ Θ_c.

Definition 3 The fundamental limit to the localization accuracy of the x-coordinate of the i^th object is defined as ${\delta }_x^{\mathit{\text{loc}},i}≔\sqrt{{[{\mathbf{Q}}_i^{-1}]}_{11}}$, i = 1, 2, and for the y-coordinate it is defined as ${\delta }_y^{\mathit{\text{loc}},i}≔\sqrt{{[{\mathbf{Q}}_i^{-1}]}_{22}}$, i = 1, 2, where Q_i is given by eq. 20, for i = 1, 2.

For the specific image functions and photon detection rates considered in this example, it can be shown that (see Ober et al (2004b)).

$${\delta }^{\mathit{\text{loc}}}={\delta }_x^{\mathit{\text{loc}},i}={\delta }_y^{\mathit{\text{loc}},i}≔\frac{\lambda }{2\pi n_a\sqrt{{\Lambda }_0(t-t_0)}},i=1,2.$$ (22)

Corollary 2 For d ∈ [0, ∞), let ${\delta }_d^{\mathit{\text{sim}}}$ denote the 2D FREM for the simultaneous detection approach. For i = 1, 2, let Λ_i and q_i denote the photon detection rate and the image function of the i^th object, respectively.

• 1.Assume that q₁(x, y) = q₂(x, y), (x, y) ∈ ℝ² and Λ₁(τ) = λ₂(τ), τ ≥ t₀. Then ${\text{lim}}_{d\to 0}{\delta }_d^{\mathit{\text{sim}}}=\mathrm{\infty }$.
• 2.
  For i = 1, 2, assume that q_i is radially symmetric, i.e., there exists a q_i such that $q_i(x,y)≔q_i(\sqrt{(x^2+y^2)}$, (x, y) ∈ ℝ² and i = 1, 2. Then

  $$\underset{d\to \mathrm{\infty }}{\text{lim}}{\delta }_d^{\mathit{\text{sim}}}=\sqrt{{({\delta }_{\mathit{\text{rs}},1}^{\mathit{\text{loc}}})}^2+{({\delta }_{\mathit{\text{rs}},2}^{\mathit{\text{loc}}})}^2},$$

  where for i = 1, 2,

  $${\delta }_{\mathit{\text{rs}},i}^{\mathit{\text{loc}}}≔\frac{1}{\sqrt{\pi {\kappa }_i{\displaystyle {\int }_{t_0}^t}{\Lambda }_i(\tau )d\tau }}\mathit{\text{ with }}{\kappa }_i≔{\displaystyle {\int }_0^{\mathrm{\infty }}}\frac{1}{q_i(r)}{\left(\frac{\partial q_i(r)}{\partial r}\right)}^2\mathit{\text{rdr}}.$$ (23)
• 3.Let δ^loc be given by eq. 22. For i = 1, 2, let q_i be an Airy profile that is given by eq. 21 and Λ₁(τ) = Λ₂(τ) = Λ₀, τ ≥ t₀. Then ${\text{lim}}_{d\to \mathrm{\infty }}{\delta }_d^{\mathit{\text{sim}}}=\sqrt{2}{\delta }^{\mathit{\text{loc}}}$.

Proof 1. By definition ${\delta }_d^{\mathit{\text{sim}}}=\sqrt{{\mathbf{I}}_{\mathit{\text{sim}}}^{-1}(d)},\text{ where }{\mathbf{I}}_{\mathit{\text{sim}}}^{-1}(d)$, is obtained by substituting ${\mathbf{S}}_{\mathit{\text{sim}}}^{-1}({\theta }_c)$ (Corollary 1) in the transformation formula in eq. 2. When d → 0 then x₀₁ → x₀₂ and y₀₂ → y₀₂, and from Remark 2 it immediately follows that S_sim(θ_c) is singular, where θ_c = (x₀₁, y₀₁, x₀₂, y₀₂) ∈ Θ_c and S_sim(θ_c) is given by eq. 5. From this the result follows.

2. Without loss of generality, we assume that d → ∞ implies x₀₂ → ∞. For θ_c = (x₀₁, y₀₁, x₀₂, y₀₂) ∈ Θ_c, consider the term S_sim(θ_c) which is given by eq. 5. Using Theorem 3 and Lemma 3 (see Appendix), we have

$$\underset{x_{02}\to \mathrm{\infty }}{\text{lim}}{\mathbf{S}}_{\mathit{\text{sim}}}({\theta }_c)=\left[\begin{array}{cc}\hfill {\mathbf{Q}}_1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\mathbf{Q}}_2\hfill \end{array}\right]=\left[\begin{array}{cc}\hfill \frac{1}{{({\delta }_{\mathit{\text{rs}},1}^{\mathit{\text{loc}}})}^2}{\mathbf{1}}_{2\times 2}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \frac{1}{{({\delta }_{\mathit{\text{rs}},2}^{\mathit{\text{loc}}})}^2}{\mathbf{I}}_{2\times 2}\hfill \end{array}\right],$$ (24)

where Q_i, i = 1, 2, denotes the Fisher information matrix for the localization accuracy problem (eq. 20) and 1_2×2 denotes the 2 × 2 identity matrix. Define Δ_x ≔ x₀₂ − x₀₁ and Δ_y ≔ y₀₂ − y₀₁. Consider the term

$$\underset{x_{02}\to \mathrm{\infty }}{\text{lim}}{\left(\frac{\partial d}{\partial {\theta }_c}\right)}^T=\underset{x_{02}\to \mathrm{\infty }}{\text{lim}}\frac{1}{d}\left(\begin{array}{c}\hfill -(x_{02}-x_{01})\hfill \\ \hfill -(y_{02}-y_{01})\hfill \\ \hfill (x_{02}-x_{01})\hfill \\ \hfill (y_{02}-y_{01})\hfill \end{array}\right)=\underset{x_{02}\to \mathrm{\infty }}{\text{lim}}\frac{1}{\sqrt{{\Delta }_x^2+{\Delta }_y^2}}\left(\begin{array}{c}\hfill -{\Delta }_x\hfill \\ \hfill -{\Delta }_y\hfill \\ \hfill {\Delta }_x\hfill \\ \hfill {\Delta }_y\hfill \end{array}\right)=\underset{x_{02}\to \mathrm{\infty }}{\text{lim}}\left(\begin{array}{c}\hfill -\frac{1}{\sqrt{1+\frac{{\Delta }_y^2}{{\Delta }_x^2}}}\hfill \\ \hfill -\frac{1}{\sqrt{\frac{{\Delta }_x^2}{{\Delta }_y^2}+1}}\hfill \\ \hfill \frac{1}{\sqrt{1+\frac{{\Delta }_y^2}{{\Delta }_x^2}}}\hfill \\ \hfill \frac{1}{\sqrt{\frac{{\Delta }_x^2}{{\Delta }_y^2}+1}}\hfill \end{array}\right)=\left(\begin{array}{c}\hfill -1\hfill \\ \hfill 0\hfill \\ \hfill 1\hfill \\ \hfill 0\hfill \end{array}\right).$$ (25)

Using eqs. 24 and 25 in eq. 2 and taking the limit x₀₂ → ∞, we have

$$\underset{x_{02}\to \mathrm{\infty }}{\text{lim}}{\mathbf{I}}_{\mathit{\text{sim}}}^{-1}(d)=(\begin{array}{cccc}\hfill -1\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \end{array})\left[\begin{array}{cc}\hfill \frac{1}{{({\delta }_{\mathit{\text{rs}},1}^{\mathit{\text{loc}}})}^2}{\mathbf{1}}_{2\times 2}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \frac{1}{{({\delta }_{\mathit{\text{rs}},2}^{\mathit{\text{loc}}})}^2}{\mathbf{I}}_{2\times 2}\hfill \end{array}\right]\left(\begin{array}{c}\hfill -1\hfill \\ \hfill 0\hfill \\ \hfill 1\hfill \\ \hfill 0\hfill \end{array}\right)={({\delta }_{\mathit{\text{rs}},1}^{\mathit{\text{loc}}})}^2+{({\delta }_{\mathit{\text{rs}},2}^{\mathit{\text{loc}}})}^2.$$

From this the result follows.

3. The Airy profile given in eq. 21 is radially symmetric. Hence substituting for q_i and Λ_i, i = 1, 2, in eq. 23, we have ${\delta }_{\mathit{\text{rs}},1}^{\mathit{\text{loc}}}={\delta }_{\mathit{\text{rs}},2}^{\mathit{\text{loc}}}={\delta }^{\mathit{\text{loc}}}$ and from this the result immediately follows.

4.1.1 Results

Here we numerically calculate the various quantities defined in Definitions 1–3. For this purpose, we assume the two point sources to be equidistant from the origin and to lie on a line segment that passes through the origin and subtends an angle of 45° with respect to the x-axis. We choose this specific configuration, since some of the calculated values (i.e., particular ${\delta }_{x_{0i}}^{\mathit{\text{sim}}}\text{ and }{\delta }_{y_{0i}}^{\mathit{\text{sim}}}$, i = 1, 2) become equal, which simplifies the presentation of the results.

Fig. 2 shows the behavior of the 2D FREM ${\delta }_d^{\mathit{\text{sim}}}$ as a function of the distance of separation. The figure also shows the limit to the accuracy of x₀₁ and x₀₂ for the simultaneous detection approach, i.e., ${\delta }_{x_{01}}^{\mathit{\text{sim}}}\text{ and }{\delta }_{x_{02}}^{\mathit{\text{sim}}}$, respectively, (the result for y₀₁ and y₀₂ are analogous) as well as the fundamental limit to the localization accuracy δ^loc (eq. 22). According to Rayleigh’s resolution criterion, two identical point sources are said to be resolved in a microscope if their distance of separation is greater than or equal to 0.61λ/n_a, where n_a denotes the numerical aperture of the microscope and λ denotes the wavelength of light emitted by the point sources. For the specific numerical values considered in Figure 2, Rayleigh’s resolution limit is ≈ 219 nm, and according to this criterion distances below 219 nm cannot resolved. In contrast, in Figure 2 we see that the numerical value of the 2D FREM ${\delta }_d^{\mathit{\text{sim}}}$ is relatively small for a range of distances below the classical resolution limit of 219 nm. An immediate implication of this result is that if there exists an efficient estimator, then these distances can be determined with an accuracy as predicted by ${\delta }_d^{\mathit{\text{sim}}}$.

Fig. 2.

Behavior of the 2D FREM ${\delta }_d^{\mathit{\text{sim}}}$ and the limit to the accuracy of x₀₁ and x₀₂, i.e., ${\delta }_{x_{01}}^{\mathit{\text{sim}}}\text{ and }{\delta }_{x_{02}}^{\mathit{\text{sim}}}$, respectively, for the simultaneous detection approach. Panel A shows ${\delta }_d^{\mathit{\text{sim}}}(\diamond )\text{ and }{\delta }_{x_{01}}^{\mathit{\text{sim}}}(\circ )$ (the results for ${\delta }_{y_{01}}^{\mathit{\text{sim}}}$ are similar) for a distance range of 1–300 nm, while Panel B shows the same for a distance range of 1 – 50 nm. Panel C shows ${\delta }_d^{\mathit{\text{sim}}}(\diamond )\text{ and }{\delta }_{x_{02}}^{\mathit{\text{sim}}}(\circ )$ (the results for ${\delta }_{y_{02}}^{\mathit{\text{sim}}}$ are similar) for a distance range of 1–300 nm, while Panel D shows the same for a distance range of 1 – 50 nm. In all the panels, (—) denotes the fundamental limit to the localization accuracy δ^loc (eq. 22). In panels A and C, the vertical dashed line denotes the classical Rayleigh’s resolution limit, which is given by 0.61λ/n_a. For all the plots, the numerical aperture is set to n_a = 1.45, the wavelength of the detected photons is set to λ = 520 nm, the photon detection rate for both point sources is set to Λ₀ = 3000 photons/s and the acquisition time interval is set to [0,1] s. For each value of distance, the location coordinates are set to (x₀₁, y₀₁) = −(0.5d cos ϕ, 0.5d sin ϕ) and (x₀₂, y₀₂) = (0.5d cos ϕ, 0.5d sin ϕ), with ϕ = π/4 for all values of d. For the above numerical values, the Rayleigh’s resolution limit is ≈ 219 nm.

Note that as the distance of separation becomes very small, ${\delta }_d^{\mathit{\text{sim}}}$ becomes numerically large thereby predicting poor accuracy in estimating the distance of separation. This is expected since under the assumptions of identical photon detection rates and image functions, when the distance d goes to zero the corresponding Fisher information matrix becomes singular and the 2D FREM ${\delta }_d^{\mathit{\text{sim}}}$ becomes infinitely large (result 1 of Corollary 2). As the distance of separation increases, ${\delta }_d^{\mathit{\text{sim}}}$ becomes smaller thereby predicting a relatively high accuracy in determining the distance between the two point sources. In particular, for large distances ${\delta }_d^{\mathit{\text{sim}}}$ approaches the fundamental limit to the localization accuracy δ^loc. This is expected, as it was shown in Theorem 3 that when d → ∞, the Fisher information matrix for the simultaneous detection approach reduces to an expression that is equivalent to two independent localization accuracy problems. For the specific image functions considered here, ${\delta }_d^{\mathit{\text{sim}}}=\sqrt{2}{\delta }^{\mathit{\text{loc}}}$ in the limit d → ∞ (result 2 of Corollary 2).

The results for ${\delta }_{x_{01}}^{\mathit{\text{sim}}}\text{ and }{\delta }_{x_{02}}^{\mathit{\text{sim}}}$ are also analogous to that of ${\delta }_d^{\mathit{\text{sim}}}$. Note that although ${\delta }_{x_{01}}^{\mathit{\text{sim}}}({\delta }_{x_{02}}^{\mathit{\text{sim}}})$ and δ^loc provide lower bounds to the accuracy with which the x-coordinate of a point source can be determined, their behaviors are very different. In particular, ${\delta }_{x_{01}}^{\mathit{\text{sim}}}\text{ and }{\delta }_{x_{02}}^{\mathit{\text{sim}}}$ depend on the distance and become infinitely large in the limit d → 0 (see Remark 2), whereas δ^loc is independent of the distance and remains finite for all values of d.

The above discussion raises the question that under what conditions ${\delta }_{x_{01}}^{\mathit{\text{sim}}}\text{ and }{\delta }_{x_{02}}^{\mathit{\text{sim}}}$, and more importantly ${\delta }_d^{\mathit{\text{sim}}}$ will remain finite as the distance goes to zero. In the next Section, we investigate this problem by considering a specical case of the simultaneous detection approach where we assume that one of the location coordinates is known. As we will see in Section 5.1, for this special case the limit to the accuracy of the distance d remains finite as d → 0 for the specific image profiles and photon detection rates considered in the present example.

5 Special case of the simultaneous detection approach - location of one of the objects is known

It has been shown experimentally that distances well below the classical resolution criteria (e.g., Rayleigh’s resolution criterion) can be resolved in a regular optical microscope when the location coordinates of one of the point sources is known a priori (Ram et al (2006a); Gordon et al (2004); Qu et al (2004)). For example, in a concrete experimental setting such a scenario arises when one wishes to study the interaction between a stationary object and a slow moving object. In many cases, the location coordinates of the stationary object can be determined a priori (for instance from an image that only contains the stationary object) and therefore can be assumed to be known. Thus an important question then arises as to how accurately the distance between the two objects can be determined when the location of one of the objects is known. Here we address this problem by deriving the Fisher information matrix for this specific scenario.

For the present discussion, we assume that the acquired data consists of a pair of images, where one of the images contains photons from only one of the objects (for example, the stationary one) and the other image contains photons from both objects. Here, we assume that the location coordinates (x₀₁, y₀₁) of object 1 is determined from the first image and the location coordinates (x₀₂, y₀₂) of object 2 is determined from the second image. In the following Theorem, we derive the expression for the Fisher information matrix for the problem of estimating the location coordinates of the objects from such a pair of images. We assume that the photon detection rate of the objects is known. Further, we also assume the spatially invariant case (analogous to Section 3.2), where the photon distribution profile of the i^th object f_θ,τ,i, i = 1, 2, is expressed as a scaled and shifted version of the image of that object (see eq. 8).

As we will show, the Fisher information matrix reduces to that of two independent localization accuracy problems. We also show that the Fisher information matrix is invertible for all values of the location coordinates of the two objects including when the location coordinates are the same (i.e., when the distance equals zero).

Theorem 4 Let Θ_c ⊆ ℝ⁴ be open. For θ_c = (x₀₁, y₀₁, x₀₂, y₀₂) ∈ Θ_c, τ ≥ t₀ and i = 1,2, let f_{θc,τ, i} and Λ_i denote the photon distribution profile and the photon detection rate of the i^th object, respectively, where f_θ,τ,i is given by eq. 8. For θ_c ∈ Θ and τ ≥ t₀, let Λ(τ) ≔ Λ₁(τ) + Λ₂(τ), and f_θc,τ be given by eq. 10. For θ_c ∈ Θ_c, let 𝒢₁(Λ₁, {f_θc,τ,1}_τ≥t0, ℝ²) and 𝒢₂(Λ, {f_θc,τ}_τ≥t0, ℝ²) denote two independent image detection processes.

• 1.
  Then for the two independent image detection processes 𝒢₁ and 𝒢₂, the Fisher information matrix of the spatial component corresponding to the acquisition time interval [t₀, t] for the special case of the simultaneous detection approach is given by

  $${\mathbf{S}}_{\mathit{\text{sim}},\mathit{\text{sp}}}({\theta }_c)≔\left[\begin{array}{cc}\hfill {\mathbf{Q}}_1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\mathbf{K}}_{22}({\theta }_c)\hfill \end{array}\right],{\theta }_c\in {\Theta }_c,$$ (26)

  where Q₁ is given by eq. 20 and K₂₂(θ_c), θ_c ∈ Θ_c, is given by eq. 12.
• 2.For θ_c ∈ Θ_c, S_sim,sp(θ_c) is invertible including when (x₀₁, y₀₁) = (x₀₂, y₀₂).

Proof See Section A.4 in Appendix for proof.

From result 1 of the above Theorem we see that the Fisher information matrix for the special case of the simultaneous detection approach is a block diagonal matrix. The first term Q₁ (eq. 20) in the leading diagonal pertains to the Fisher information matrix for the localization accuracy problem corresponding to the location coordinates (x₀₁, y₀₁) of object 1. The second term K₂₂(θ_c) (eq. 12) in the leading diagonal is a component of the Fisher information matrix for the spatially invariant case of the simultaneous detection approach in which both location coordinates are unknown and are determined from a single image (Theorem 2). Importantly, this component K₂₂(θ_c) is equivalent to the Fisher information matrix of the localization accuracy problem for the location coordinates (x₀₂, y₀₂) of object 2 in the presence of an extraneous background signal given by Λ₁q₁, where Λ₁ and q₁ denote the photon detection rate and the image function of object 1, respectively. In this context, we would like to note that the effect of an extraneous background term on the localization accuracy problem has been extensively investigated before (Ram et al (2006b); Ober et al (2004b)).

In result 2 of the above Theorem, we showed that the Fisher information matrix is, in general, invertible for all values of the location coordinates of the two objects including when (x₀₁, y₀₁) = (x₀₂, y₀₂), i.e., when the distance between the two objects is zero. This is in contrast to the result obtained in Section 3.2, where we saw that the Fisher information matrix for the simultaneous detection approach becomes singular and therefore non-invertible when the distance is zero (assuming identical image profiles and photon detection rates; see Remark 2).

This brings out a very important aspect of the analyses carried out here. Specifically, the a priori knowledge of the location coordinates of one of the objects reduces the Fisher information matrix of the distance estimation problem to that of two independent localization accuracy problems. More importantly, it also removes the singularity of the Fisher information matrix when the distance is zero. The above result also explains the prior experimental observations of measuring nanometer scale distances well below the classical resolution criteria in a regular optical microscope when a priori information regarding the location coordinates of one of the objects is known (Gordon et al (2004); Qu et al (2004)). In the next section, we further illustrate this through a specific example where we show that the CRLB of the distance parameter remains finite when the distance goes to zero.

We note that in the derivation of the above theorem, the Fisher information matrix for the second image only depends on the location coordinates of object 2, since it is assumed that the location of object 1 is known. However, since the second image contains signal from both objects, it also provides information about the location of object 1. Hence this can be used to improve the location estimates of object 1. A detailed analysis of such a scenario has been previously carried out by us, where, analogous to Theorem 4, we derived the Fisher information matrix for a pair of images but considered the case where both location coordinates were estimated from the second image (Ram et al (2006a)).

Remark 3 The results derived in the above Theorem pertains to the Fisher information matrix for the spatial component θ_f (=θ_c) of the unknown parameter vector θ, and we have assumed the temporal component θ_Λ of θ (and in turn the the photon detection rates of the objects) to be known. The above results will hold even if the temporal component θ_Λ is unknown provided the photon detection rates of the objects are related to one another through a scalar function β, i.e. Λ_θ,1(τ) = β(τ)Λ_θ,2(τ), θ ∈ Θ and τ ≥ t₀. This is due to the fact that under this condition, the Fisher information matrix for the spatial θ_f and temporal components θ_Λ are decoupled (see result 2 of Theorem 1). It should be pointed out that the assumption Λ_θ,1 = βΛ_θ,2, θ ∈ Θ is satisfied in many practical situations since the photon detection rates of the objects are typically assumed to be the same (i.e., β = 1).

5.1 Example 2

We now illustrate the results derived in the previous section by considering a specific image profile. Analogous to Section 4.1, we assume the image functions q₁ and q₂ to be identical Airy profiles given by eq. 21 and set the photon detection rates to be constant and equal, i.e., Λ₁(τ) = Λ₂(τ) = Λ₀, τ ≥ t₀. We also define the 2D FREM for the special case of the simultaneous detection approach, which we denote as ${\delta }_d^{\mathit{\text{sim}},\mathit{\text{sp}}}$. In Corollary 3, we consider two limiting cases of the distance parameter d, i.e., d → 0 and d → ∞ and derive analytical expressions for ${\delta }_d^{\mathit{\text{sim}},\mathit{\text{sp}}}$ for the specific image functions and photon detection rates considered here.

Definition 4 The 2D FREM for the special case of the simultaneous detection approach is defined as ${\delta }_d^{\mathit{\text{sim}},\mathit{\text{sp}}}≔\sqrt{{\mathbf{I}}_{\mathit{\text{sim}},\mathit{\text{sp}}}^{-1}(d)},d\in [0,\mathrm{\infty })$, where I_sim,sp(d) is obtained by substituting ${\mathbf{S}}_{\mathit{\text{sim}},\mathit{\text{sp}}}^{-1}({\theta }_c)$ (result 2 of Theorem 4) in the transformation formula given by eq. 2.

Corollary 3 For d ∈ [0, ∞), let ${\delta }_d^{\mathit{\text{sim}},\mathit{\text{sp}}}$ denote the 2D FREM for the special case of the simultaneous detection approach. For i = 1, 2, let Λ_i and q_i denote the photon detection rate and the image function of the i^th object, respectively. Assume that Λ₁(τ) = Λ₂(τ), τ ≥ t₀, q₁(x, y) = q₂(x, y), (x, y) ∈ ℝ², and that q₁ is radially symmetric, i.e., there exists a q₁ such that $q_1(x,y)=q_1(\sqrt{x^2+y^2})$ for (x, y) ∈ ℝ².

Then

• 1.${\text{lim}}_{d\to 0}{\delta }_d^{\mathit{\text{sim}},\mathit{\text{sp}}}=\sqrt{3}{\delta }_{\mathit{\text{rs}},1}^{\mathit{\text{loc}}}$
• 2.${\text{lim}}_{d\to \mathrm{\infty }}{\delta }_d^{\mathit{\text{sim}},\mathit{\text{sp}}}=\sqrt{2}{\delta }_{\mathit{\text{rs}},1}^{\mathit{\text{loc}}}$, where ${\delta }_{\mathit{\text{rs}},1}^{\mathit{\text{loc}}}$ is given by eq. 23.
• 3.
  Let q₁ be an Airy profile that is given by eq. 21 and Λ₁(τ) = Λ₀, τ ≥ t₀. Then

  $$\underset{d\to 0}{\text{lim}}{\delta }_d^{\mathit{\text{sim}},\mathit{\text{sp}}}=\sqrt{3}{\delta }^{\mathit{\text{loc}}},\underset{d\to \mathrm{\infty }}{\text{lim}}{\delta }_d^{\mathit{\text{sim}},\mathit{\text{sp}}}=\sqrt{2}{\delta }^{\mathit{\text{loc}}},$$

  where δ^loc is given by eq. 22.

Proof 1. By definition, q₁ is radially symmetric and hence from Lemma 3 it follows that ${\mathbf{Q}}_1^{-1}={({\delta }_{\mathit{\text{rs}},1}^{\mathit{\text{loc}}})}^2{\mathbf{1}}_{2\times 2}$, where Q₁ denotes the Fisher information matrix for the localization accuracy problem of object 1 (eq. 20) and 1_2×2 denotes the 2 × 2 identity matrix. Using this and eq. 2, we get

$${({\delta }_d^{\mathit{\text{sim}},\mathit{\text{sp}}})}^2={\mathbf{I}}_{\mathit{\text{sim}},\mathit{\text{sp}}}^{-1}(d)≔\frac{\partial d}{\partial {\theta }_c}{\mathbf{S}}_{\mathit{\text{sim}},\mathit{\text{sp}}}^{-1}({\theta }_c){\left(\frac{\partial d}{\partial {\theta }_c}\right)}^T=\frac{1}{d^2}{\left(\begin{array}{c}\hfill -(x_{02}-x_{01})\hfill \\ \hfill -(y_{02}-y_{01})\hfill \\ \hfill (x_{02}-x_{01})\hfill \\ \hfill (y_{02}-y_{01})\hfill \end{array}\right)}^T\left(\begin{array}{cc}\hfill {\mathbf{Q}}_1^{-1}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\mathbf{K}}_{22}^{-1}({\theta }_c)\hfill \end{array}\right)\left(\begin{array}{c}\hfill -(x_{02}-x_{01})\hfill \\ \hfill -(y_{02}-y_{01})\hfill \\ \hfill (x_{02}-x_{01})\hfill \\ \hfill (y_{02}-y_{01})\hfill \end{array}\right)={({\delta }_{\mathit{\text{rs}},1}^{\mathit{\text{loc}}})}^2+\frac{1}{d^2}(\begin{array}{cc}\hfill x_{02}-x_{01}\hfill & \hfill y_{02}-y_{01}\hfill \end{array}){\mathbf{K}}_{22}^{-1}({\theta }_c)\left(\begin{array}{c}\hfill x_{02}-x_{01}\hfill \\ \hfill y_{02}-y_{01}\hfill \end{array}\right),d\in [0,\mathrm{\infty }),$$ (27)

where S_sim,sp(θ_c) is given by eq. 26 and K₂₂(θ_c) is given by eq. 12. Because the photon detection rates and the image functions of the objects are assumed to be identical, we have Q₁ = Q₂, where Q_i, i = 1, 2, is given by eq. 20. Using this, we have

$$\underset{x_{01}\to x_{02},y_{01}\to y_{02}}{\text{lim}}{\mathbf{K}}_{22}({\theta }_c)=\underset{x_{01}\to x_{02},y_{01}\to y_{02}}{\text{lim}}{\displaystyle {\int }_{t_0}^t}{\displaystyle {\int }_{{ℝ}^2}}\frac{{\Lambda }_2^2(\tau )}{{\Lambda }_1(\tau )q_1(x-x_{01},y-y_{01})+{\Lambda }_2(\tau )q_2(x-x_{02},y-y_{02})}\times \left(\begin{array}{cc}\hfill {\left(\frac{\partial q_2(x-x_{02},y-y_{02})}{\partial x}\right)}^2\hfill & \hfill \frac{\partial q_2(x-x_{02},y-y_{02})}{\partial x}\frac{\partial q_2(x-x_{02},y-y_{02})}{\partial y}\hfill \\ \hfill \frac{\partial q_2(x-x_{02},y-y_{02})}{\partial y}\frac{\partial q_2(x-x_{02},y-y_{02})}{\partial x}\hfill & \hfill {\left(\frac{\partial q_2(x-x_{02},y-y_{02})}{\partial y}\right)}^2\hfill \end{array}\right)\mathit{\text{ dxdyd}}\tau =\frac{{\Lambda }_0(t-t_0)}{2}{\displaystyle {\int }_{{ℝ}^2}}\frac{1}{q_2(x-x_{02},y-y_{02})}\times \left(\begin{array}{cc}\hfill {\left(\frac{\partial q_2(x-x_{02},y-y_{02})}{\partial x}\right)}^2\hfill & \hfill \frac{\partial q_2(x-x_{02},y-y_{02})}{\partial x}\frac{\partial q_2(x-x_{02},y-y_{02})}{\partial y}\hfill \\ \hfill \frac{\partial q_2(x-x_{02},y-y_{02})}{\partial y}\frac{\partial q_2(x-x_{02},y-y_{02})}{\partial x}\hfill & \hfill {\left(\frac{\partial q_2(x-x_{02},y-y_{02})}{\partial y}\right)}^2\hfill \end{array}\right)\mathit{\text{ dxdyd}}\tau =\frac{1}{2}{\mathbf{Q}}_2=\frac{1}{2}{\mathbf{Q}}_1=\frac{1}{2{({\delta }_{\mathit{\text{rs}},1}^{\mathit{\text{loc}}})}^2}{\mathbf{1}}_{2\times 2},$$ (28)

where we have used the shift-invariant property of Lebesgue integrals in the penultimate step. Define Δ_x ≔ x₀₂ − x₀₁ and Δ_y ≔ y₀₂ − y₀₁. Consider the term

$$\underset{x_{01}\to x_{02},y_{01}\to y_{02}}{\text{lim}}\frac{1}{d}\left(\begin{array}{c}\hfill x_{02}-x_{01}\hfill \\ \hfill y_{02}-y_{01}\hfill \end{array}\right)=\underset{x_{01}\to x_{02}}{\text{lim}}\underset{y_{01}\to y_{02}}{\text{lim}}\left(\begin{array}{c}\hfill \frac{1}{\sqrt{1+\frac{{\Delta }_y^2}{{\Delta }_x^2}}}\hfill \\ \hfill \frac{1}{\sqrt{\frac{{\Delta }_x^2}{{\Delta }_y^2}+1}}\hfill \end{array}\right)=\left(\begin{array}{c}\hfill 1\hfill \\ \hfill 0\hfill \end{array}\right).$$ (29)

Substituting eqs. 28 and 29 in eq. 27 and taking the limit d → 0, we get

$$\underset{d\to 0}{\text{lim}}{({\delta }_d^{\mathit{\text{sim}},\mathit{\text{sp}}})}^2=\underset{d\to 0}{\text{lim}}{\mathbf{I}}_{\mathit{\text{sim}},\mathit{\text{sp}}}^{-1}(d)=\underset{x_{01}\to x_{02},y_{01}\to y_{02}}{\text{lim}}{\mathbf{I}}_{\mathit{\text{sim}},\mathit{\text{sp}}}^{-1}(d)={({\delta }_{\mathit{\text{rs}},1}^{\mathit{\text{loc}}})}^2+\underset{x_{01}\to x_{02},y_{01}\to y_{02}}{\text{lim}}\frac{1}{d^2}(\begin{array}{cc}\hfill x_{02}-x_{01}\hfill & \hfill y_{02}-y_{01}\hfill \end{array}){\mathbf{K}}_{22}^{-1}({\theta }_c)\left(\begin{array}{c}\hfill x_{02}-x_{01}\hfill \\ \hfill y_{02}-y_{01}\hfill \end{array}\right)={({\delta }_{\mathit{\text{rs}},1}^{\mathit{\text{loc}}})}^2+2{({\delta }_{\mathit{\text{rs}},1}^{\mathit{\text{loc}}})}^2(\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \end{array}){\mathbf{1}}_{2\times 2}\left(\begin{array}{c}\hfill 1\hfill \\ \hfill 0\hfill \end{array}\right)=3{({\delta }_{\mathit{\text{rs}},1}^{\mathit{\text{loc}}})}^2.$$

From this the result immediately follows.

2. Proof is analogous to that of result 2 of Corollary 2.

3. The Airy profile given in eq. 21 is radially symmetric. Substituting for q₁ and Λ₁ in results 1 and 2 of this Corollary, we get the desired results.

Figure 3 shows the 2D FREM ${\delta }_d^{\mathit{\text{sim}},\mathit{\text{sp}}}$ as a function of the distance for the special case of the simultaneous detection approach when the location coordinates (x₀₁, y₀₁) of one of the objects is assumed to be known. The figure also shows the 2D FREM for the simultaneous detection approach ${\delta }_d^{\mathit{\text{sim}}}$ when both location coordinates are assumed to be unknown (Section 1), and as a reference the fundamental limit to the localization accuracy δ^loc (eq. 22). From the figure we see that as the distance of separation decreases, the ${\theta }_d^{\mathit{\text{sim}}}$ becomes infinitely large as d → 0. In contrast, ${\theta }_d^{\mathit{\text{sim}},\mathit{\text{sp}}}$ first increases but then decreases and then remains finite even when d = 0. In particular, for the specific image functions and photon detection rates considered here, ${\delta }_d^{\mathit{\text{sim}},\mathit{\text{sp}}}=\sqrt{3}{\delta }^{\mathit{\text{loc}}}$ when d = 0 (result 1 of Corollary 3). An immediate implication of this result is that if the location coordinates of one of the objects is known, then it is possible to determine very small (nanometer scale) distances with relatively very high accuracy in an optical microscope. As the distance of separation increases, the 2D FREM ${\delta }_d^{\mathit{\text{sim}},\mathit{\text{sp}}}$ behaves analogous to ${\delta }_d^{\mathit{\text{sim}}}$. In particular, ${\delta }_d^{\mathit{\text{sim}},\mathit{\text{sp}}}=\sqrt{2}{\delta }^{\mathit{\text{loc}}}$ when the distance becomes infinitely large. This implies that for very large distances of separation, the limit to the accuracy of estimating the distance is independent of the distance and is a constant.

Fig. 3.

Behavior of the 2D FREM ${\delta }_d^{\mathit{\text{sim}},\mathit{\text{sp}}}$ for the special case of the simultaneous detection approach. Panel A shows ${\delta }_d^{\mathit{\text{sim}},\mathit{\text{sp}}}$ for a distance range of 10 – 300 nm for the special case of the simultaneous detection approach when the location coordinates (x₀₁, y₀₁) of object 1 is known (⊳). The panel also shows the 2D FREM ${\delta }_d^{\mathit{\text{sim}}}$ for the simultaneous detection approach when both location coordinates are unknown (∘). Panel B shows the same as Panel A for a distance range of 1 – 50 nm. In all the panels, (—) denotes the fundamental limit to the localization accuracy δ^loc (eq. 22), and in Panel A the vertical dashed line denotes the Rayleigh’s resolution limit. The numerical values used to generate the above plots are identical to those used in Figure 2.

We would like to point out that the analyses carried out in this Section have implications in a broader context of dealing with a singular Fisher information matrix, which represents a significant complication in the analysis of parameter estimation problems (e.g., see Stoica and Marzetta (2001)). In particular our results illustrate how a priori information can be used to eliminate the singularity of the Fisher information matrix. It is important to note that the choice of a priori information intimately depends on the specifics of the experimental design, i.e., how the data is captured. This further underscores the importance of carrying out a rigorous analysis of the Fisher information matrix, as it provides the necessary insight into choosing the most appropriate experimental approach from the point of view of obtaining the best accuracy in estimating the parameters of interest.

6 Fisher information matrix for the separate detection approach

We next consider the case where the location coordinates of the two objects are independently estimated from two separate images. Such a scenario arises in a class of experimental techniques in which the photon emission from the objects are temporally separated (e.g., stochastic photoactivation (Betzig et al (2006); Rust et al (2006); Hess et al (2006)) and blinking (Lidke et al (2005); Lagerholm et al (2006))). In the following Theorem, we derive an analytical expression of the Fisher information matrix for the separate detection approach. Here, we assume the acquired data to consist of a pair of images, where the first image contains signal from only object 1 and the the second image contains signal from only object 2. As we will see, the Fisher information matrix for the separate detection approach will reduce to two independent localization accuracy problems.

Theorem 5 Let Θ_c ⊆ ℝ⁴ be open. For θ_c ∈ Θ_c, τ ≥ t₀ and i = 1, 2, let Λ_i and f_θc,τ,i denote the photon detection rate and the photon distribution profile of the i^th object, respectively, where f_θc,τ,i is given by eq. 8. For θ_c ∈ Θ_c, let 𝒢₁(Λ₁, {f_θc,τ,1}_τ≥t0, ℝ²) and 𝒢₂(Λ₂, {f_θc,τ,2}_τ≥t0, ℝ²) denote two independent image detection processes.

• 1.
  Then for the two independent image detection processes 𝒢₁ and 𝒢₂, the Fisher information matrix of the spatial component corresponding to the acquisition time interval [t₀, t] for the separate detection approach is given by

  $${\mathbf{S}}_{\mathit{\text{sep}}}({\theta }_c)=\left[\begin{array}{cc}\hfill {\mathbf{Q}}_1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\mathbf{Q}}_2\hfill \end{array}\right],{\theta }_c\in {\Theta }_c,$$ (30)

  where Q_i, i = 1,2, is given in eq. 20
• 2.For θ_c ∈ Θ_c, S_sep(θ_c) is invertible including when (x₀₁, y₀₁) =(x₀₂, y₀₂).

Proof Proof is analogous to that of Theorem 4.

From the above Theorem, we see that the Fisher information matrix for the separate detection approach is block diagonal and is equivalent to two independent localization accuracy problems in the absence of any extraneous background signal. Note that the Fisher information matrix for the separate detection approach is independent of the location coordinates of the objects. This is in contrast to the simultaneous detection approach, where we saw that the Fisher information matrix depended on the location coordinates of the two objects (Theorem 2). In addition to this, for the simultaneous detection approach when both object coordinates are unknown the Fisher information matrix becomes block diagonal and reduces to that of two independent localization accuracy problems (in the absence of any extraneous background signal) only when the distance becomes infinitely large i.e., d → ∞ (Theorem 3).

6.1 Example 3

To illustrate the result derived in this section, we consider a specific image function. Analogous to Sections 4.1 and 5.1, we assume the image functions q₁ and q₂ to be identical Airy profiles given by eq. 21 and set the photon detection rates to be constant and equal, i.e., Λ₁(τ) = Λ₂(τ) = Λ₀, τ ≥ t₀. We also define the 2D FREM for the separate detection approach ${\delta }_d^{\mathit{\text{sep}}}$. Then in Corollary 4, we derive an analytical expression for ${\delta }_d^{\mathit{\text{sep}}}$ for the specific image functions and photon detection rates considered here.

Definition 5 The 2D FREM for the separate detection approach is defined as ${\delta }_d^{\mathit{\text{sep}}}≔\sqrt{{\mathbf{I}}_{\mathit{\text{sep}}}^{-1}(d)},d\in [0,\mathrm{\infty }),\text{ where }{\mathbf{I}}_{\mathit{\text{sep}}}^{-1}(d)$ is obtained by substituting ${\mathbf{S}}_{\mathit{\text{sep}}}^{-1}({\theta }_c)$ (result 2 of Theorem 5) in the transformation formula given by eq. 2.

Corollary 4 For d ∈ [0, ∞), let ${\delta }_d^{\mathit{\text{sep}}}$ denote the 2D FREM for the separate detection approach. For i = 1, 2, let Λ_i and q_i denote the photon detection rate and the image function of the i^th object, respectively.

• 1.
  For i = 1, 2, assume that q_i is radially symmetric, i.e., there exists a q_i such that $q_i(x,y)≔q_i(\sqrt{(x^2+y^2)}$ (x, y) ∈ ℝ² and i = 1, 2. Then for d ∈ [0, ∞), we have

  $${\delta }_d^{\mathit{\text{sep}}}=\sqrt{{\left({\delta }_{\mathit{\text{rs}},1}^{\mathit{\text{loc}}}\right)}^2+{\left({\delta }_{\mathit{\text{rs}},2}^{\mathit{\text{loc}}}\right)}^2},$$

  where for i = 1, 2, ${\delta }_{\mathit{\text{rs}},i}^{\mathit{\text{loc}}}$ is given by eq. 23.
• 2.For i = 1, 2, let q_i be an Airy profile that is given by eq. 21 and Λ₁(τ) = Λ₂(τ) = Λ₀, τ ≥ t₀. Then for d ∈ [0, ∞), ${\delta }_d^{\mathit{\text{sep}}}=\sqrt{2}{\delta }^{\mathit{\text{loc}}}$, where δ^loc is given by eq. 22.

Proof 1. Using eq. 2 and Lemma 3, we have

$${({\delta }_d^{\mathit{\text{sep}}})}^2={\mathbf{I}}_{\mathit{\text{sep}}}^{-1}(d)≔\frac{\partial d}{\partial {\theta }_c}{\mathbf{S}}_{\mathit{\text{sep}}}^{-1}({\theta }_c){\left(\frac{\partial d}{\partial {\theta }_c}\right)}^T=\frac{1}{d^2}{\left(\begin{array}{c}\hfill -(x_{02}-x_{01})\hfill \\ \hfill -(y_{02}-y_{01})\hfill \\ \hfill (x_{02}-x_{01})\hfill \\ \hfill (y_{02}-y_{01})\hfill \end{array}\right)}^T\left(\begin{array}{cc}\hfill {\mathbf{Q}}_1^{-1}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\mathbf{Q}}_2^{-1}\hfill \end{array}\right)\left(\begin{array}{c}\hfill -(x_{02}-x_{01})\hfill \\ \hfill -(y_{02}-y_{01})\hfill \\ \hfill (x_{02}-x_{01})\hfill \\ \hfill (y_{02}-y_{01})\hfill \end{array}\right)=\frac{1}{d^2}{\left(\begin{array}{c}\hfill -{\Delta }_x\hfill \\ \hfill -\Delta y\hfill \\ \hfill {\Delta }_x\hfill \\ \hfill {\Delta }_y\hfill \end{array}\right)}^T\left(\begin{array}{cccc}\hfill {\left({\delta }_{\mathit{\text{rs}},1}^{\mathit{\text{loc}}}\right)}^2\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\left({\delta }_{\mathit{\text{rs}},1}^{\mathit{\text{loc}}}\right)}^2\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill {\left({\delta }_{\mathit{\text{rs}},2}^{\mathit{\text{loc}}}\right)}^2\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {\left({\delta }_{\mathit{\text{rs}},2}^{\mathit{\text{loc}}}\right)}^2\hfill \end{array}\right)\left(\begin{array}{c}\hfill -{\Delta }_x\hfill \\ \hfill -\Delta y\hfill \\ \hfill {\Delta }_x\hfill \\ \hfill {\Delta }_y\hfill \end{array}\right)=\frac{1}{d^2}(({\Delta }_x^2+{\Delta }_y^2){({\delta }_{\mathit{\text{rs}},1}^{\mathit{\text{loc}}})}^2+({\Delta }_x^2+{\Delta }_y^2){({\delta }_{\mathit{\text{rs}},2}^{\mathit{\text{loc}}})}^2)={({\delta }_{\mathit{\text{rs}},1}^{\mathit{\text{loc}}})}^2+{({\delta }_{\mathit{\text{rs}},2}^{\mathit{\text{loc}}})}^2,d\in [0,\mathrm{\infty }),$$

where Δ_x = x₀₂ − x₀₁ and Δ_y = y₀₂ − y₀₁. From this the result immediately follows. 2. The Airy profile given in eq. 21 is radially symmetric. Hence substituting for Λ_i and q_i, i = 1, 2, in result 1 of this Corollary, the result immediately follows.

From the above result we see that the 2D FREM for the separate detection approach ${\delta }_d^{\mathit{\text{sep}}}$ is a constant and is independent of the distance of separation, if the image functions of the objects are radially symmetric. More specifically, when the image functions are assumed to be Airy profiles (eq. 21), then the 2D FREM ${\delta }_d^{\mathit{\text{sep}}}\text{ is }\sqrt{2}$ times the fundamental limit to the localization accuracy δ^loc. This is in contrast to the simultaneous detection approach where the 2D FREM ${\delta }_d^{\mathit{\text{sim}}}$ (as well as ${\delta }_d^{\mathit{\text{sim}},\mathit{\text{sp}}}$) depends on the distance and only in the limiting case when d becomes infinitely large, ${\delta }_d^{\mathit{\text{sim}}}=\sqrt{2}{\delta }^{\mathit{\text{loc}}}$ (Corollary 2). An immediate implication of the above result is that, if there exists an efficient estimator of the distance for the separate detection approach, then all distances can be determined with the same level of accuracy when the image profiles are radially symmetric.

7 Simulations

In the previous sections we investigated the Fisher information matrix of the distance d and calculated the 2D FREM for different experimental approaches. An important question then arises as to whether for a given experimental approach there exists an unbiased estimator that can attain the corresponding 2D FREM. In this section we address this question, where we use the Maximum Likelihood (ML) estimator to determine the distance d from simulated data and compare its performance (i.e. standard deviation) to the 2D FREM for the different experimental approaches. We consider all three approaches, i.e., the simultaneous detection approach, the special case of the simultaneous detection approach when one of the object locations are known, and the separate detection approach. We generate the acquired data through Monte-Carlo simulations which are discussed below. Here, we consider the data generation process for an ideal (non-pixelated) detector, where the acquired data consists of the spatial coordinates of the detected photons. We then use the maximum likelihood estimation algorithm on the simulated data to estimate the location coordinates of the objects, and from this we deduce the distance. Table 1 lists the standard deviations of the distance estimates for the different experimental approaches considered here. As we will see the ML estimator is unbiased and attains the 2D FREM for a range of distances when the sample size is sufficiently large.

Table 1.

Results of the maximum likelihood estimator of the distance for the different experimental approaches considered here. Table A shows the results for the simultaneous detection approach. Table B shows the results for the special case of the simultaneous detection approach, where one of the location coordinates is independently determined and is assumed to be known. Table C shows the results of the separate detection approach. The numerical values used to generate the data are identical to those used in Fig. 2. For all the data sets, the mean and standard deviation are obtained from 2000 maximum likelihood estimates of the distance.

A. Simultaneous detection approach
Data set #	True value of distance nm	Mean distance estimates nm	Std. dev of distance estimates nm	Resolution measure ${\delta }_d^{\mathit{\text{sim}}}$ nm
1	10	10.22	5.87	5.89
2	20	20.01	4.14	4.23
3	50	49.99	2.67	2.65
4	100	99.99	2.14	2.12
5	200	200.06	1.97	1.93
6	500	500	1.64	1.68
B. Special case of the simultaneous detection approach
Data set #	True value of distance nm	Mean distance estimates nm	Std. dev of distance estimates nm	Resolution measure ${\delta }_d^{\mathit{\text{sim}}}$ nm
1	10	10.15	1.56	1.81
2	20	20.03	1.68	1.82
3	50	50.03	1.85	1.85
4	100	100.02	1.91	1.91
5	200	200.03	1.78	1.74
6	500	500.01	1.56	1.6
C. Separate detection approach
Data set #	True value of distance nm	Mean distance estimates nm	Std. dev of distance estimates nm	Resolution measure ${\delta }_d^{\mathit{\text{sim}}}$ nm
1	10	10.15	1.48	1.47
2	20	20.02	1.49	1.47
3	50	50.04	1.50	1.47
4	100	100.01	1.48	1.47
5	200	200.03	1.50	1.47
6	500	500.03	1.45	1.47

7.1 Data simulation

We consider the two objects to be identical point sources. We set the photon detection rates of the two objects to be equal and constant, i.e. Λ_θ,1(τ) ≔ Λ₀, τ ≥ t₀ and Λ_θ,2(τ) ≔ Λ₀, τ ≥ t₀ and assume the image functions q₁ and q₂ to be identical Airy profiles given by eq. 21. We generate a sequence of images {𝒥_θ,1, 𝒥_θ,2, …, 𝒥_θ,Nmax}, where N_max denotes the total number of images. For k = 1, …, N_max, the k^th image is given by 𝒥_θ,k ≔ {𝒥_θ,1,k, 𝒥_θ,2,k}, where

$${𝒥}_{\theta ,i,k}≔\{(x_1^{i,k},y_1^{i,k}),(x_2^{i,k},y_2^{i,k}),\dots ,(x_{N_{i,k}}^{i,k},y_{N_{i,k}}^{i,k})\},i=1,2,k=1,\dots ,N_{\mathit{\text{max}}},$$ (31)

denotes the signal from the i^th object in the k^th image for k = 1, …, N_max and i = 1, 2. In the above equation, N_i,k denotes the number of detected photons from the i^th object in the k^th image for i = 1, 2 and k = 1, …, N_max, and is a realization of the Poisson random variable with mean Λ₀(t − t₀). The sequence $\{(x_m^{i,k},y_m^{i,k});m=1,\dots ,N_{i,k}\}$ denotes the spatial coordinates of the detected photons from the i^th object in the k^th image for i = 1, 2 and k = 1, …, N_max, and is a realization of N_i,k random variables with density f_θc,τ,i given by eq. 8, which is generated by using a method described in Ober et al (2004b).

7.2 Maximum likelihood estimator

For a general parameter estimation problem, the maximum likelihood estimator can be written as argmax_θ ln(ℒ(θ | 𝒵) where 𝒵 denotes the data and ℒ(θ | ·) denotes the likelihood function. For the simultaneous detection approach, the acquired data pertaining to the k^th image is given by 𝒵 = 𝒥_θ,k = {𝒥_θ,1,k, 𝒥_θ,2,k}, k = 1, …, N_max where 𝒥_θ,i,k is defined in eq. 31 and θ = θ_c = (x₀₁, y₀₁, x₀₂, y₀₂) ∈ ℝ⁴.

For the special case of the simultaneous detection approach when one of the location coordinates are known, the acquired data consists of a pair of images {𝒵₁, 𝒵₂}. We assume 𝒵₁ to be the image that contains the signal from object 1, i.e., 𝒵₁ = 𝒥_θ1,1,k, and 𝒵₂ to be the image that contains the signal from both objects, i.e., 𝒵₁ = {𝒥_θ1,1,k, 𝒥_θ2,2,k}, where 𝒥 is defined in eq. 31, θ_i = (x_0i, y_0i) ∈ ℝ², i = 1, 2 and k = 1, … N_max. Here, we carry out two independent ML estimations on each image, i.e., argmax_θ1 ln(ℒ (θ₁ | 𝒵₁) and argmax_θ2 ln(ℒ(θ₂ | 𝒵₂, θ̂₁), where θ_i ≔ (x_0i, y_0i) ∈ ℝ², i = 1, 2. Note that while carrying out the maximum likelihood estimation with the second image 𝒵₂, we set the value of θ₁ to be equal to θ̂₁, where thêta₁ denotes the maximum likelihood estimate of θ₁, which is determined from the first image.

For the separate detection approach, the acquired data consists of a pair of images {𝒵 ₁, 𝒵₂} each of which contains the image of only one of the objects. Here, we have 𝒵₁ = 𝒥_θ1,1,k and 𝒵₂ = 𝒥_θ2,1,k, where 𝒥 is defined in eq. 31 for θ_i = (x_0i, y_0i) ∈ ℝ², i = 1, 2 and k = 1, …, N_max. For this approach, we carry out independent ML estimations on each image, i.e. argmax_θ1 ln(ℒ(θ₁ | 𝒵₁) and argmax_θ2 ln(ℒ (θ₂ | 𝒵₂), where θ_i ≔ (x_0i, y_0i) ∈ ℝ², i = 1, 2.

In all the three imaging scenarios, the ML estimates are determined computationally by using a gradient based optimization algorithm (fminunc) in the MATLAB programming language.

7.3 Comparison of ML estimator performance to the 2D FREM

Table 1 shows the results of the ML estimator for the different experimental approaches considered here. The table lists mean and standard deviation of the distance estimates as well as the 2D FREM of the distance. From the table we see that for all the experimental approaches considered here, the mean value of the distance estimates is very close to the true value suggesting that the ML estimator is unbiased. Moreover, for a range of distances, the standard deviation of the distance is also consistently close to the 2D FREM thereby suggesting that the ML estimator is capable of achieving the theoretically best possible accuracy provided the sample size is sufficiently large. Note that the standard deviation of the ML estimates for the separate detection approach is almost a constant for a range of distances in agreement with the 2D FREM, which in turn shows that different distances can be estimated with the same level of accuracy.

A comparison of the standard deviations of the distance estimates (as well as the 2D FREMs) for the three approaches shows that for a range of distances considered in Table 1, the separate detection approach provides the best accuracy (i.e., the smallest 2D FREM/standard deviation) for determining the distance, followed by the special case of the simultaneous detection approach, and then followed by the simultaneous detection approach.

Appendix

A Appendix

Definition 6 A function q : ℝ² → [0, ∞) is said to be an image function if the following properties are satisfied (see (Ram et al, 2006b, pg 37)).

• 1.∫_ℝ² q(x, y)dxdy = 1,
• 2.$\frac{\partial q(x,y)}{\partial x}\text{ and }\frac{\partial q(x,y)}{\partial y}$ exist for every (x, y) ∈ ℝ²,
• 3.${\int }_{{ℝ}^2}\frac{\partial q(x,y)}{\partial x}|\mathit{\text{dxdy}}<\mathrm{\infty },{\int }_{{ℝ}^2}|\frac{\partial q(x,y)}{\partial y}|\mathit{\text{dxdy}}<\mathrm{\infty },$ and
• 4.${\int }_{{ℝ}^2}\frac{1}{q(x,y)}{\left(\frac{\partial q(x,y)}{\partial x}\right)}^2\mathit{\text{dxdy}}<\infty ,{\int }_{{ℝ}^2}\frac{1}{q(x,y)}{\left(\frac{\partial q(x,y)}{\partial y}\right)}^2\mathit{\text{dxdy}}<\infty ,\text{ and }{\int }_{{ℝ}^2}\frac{1}{q(x,y)}\frac{\partial q(x,y)}{\partial x}\frac{\partial q(x,y)}{\partial y}\mathit{\text{dxdy}}<\infty .$

Lemma 1 For θ = (θ_f, θ_Λ) ∈ Θ, τ ≥ t₀ and i = 1, 2, let f_θ,τ,i and Λ_θ,i denote the photon distribution profile and the photon detection rate of the i^th object, respectively, and let Λ_θ and f_θ,τ be given by eqs. 3 and 4, respectively. Let 𝒞 denote the detector.

• 1.For θ ∈ Θ and τ ≥ t₀, if β(τ)Λ_θ,1 (τ) = Λ_θ,2 (τ) for some β(τ) ≥ 0 that is independent of θ, then $\frac{\partial f_{\theta ,\tau }(r)}{\partial {\theta }_{\Lambda }}=0$, θ ∈ Θ, τ ≥ t₀, r ∈ 𝒞.
• 2.For θ ∈ Θ and τ ≥ t₀, if f_θ,τ,1(r) = f_θ,τ,2(r), r ∈ 𝒞, then $\frac{\partial f_{\theta ,\tau }(r)}{\partial {\theta }_{\Lambda }}=0$, θ ∈ Θ, τ ≥ t₀, r ∈ 𝒞.

Proof 1. For θ ∈ Θ, τ ≥ t₀ and i = 1, 2, let ε_θ,i(τ) = Λ_θ,i(τ)/Λ_θ(τ). Consider the term

$$\frac{\partial {\epsilon }_{\theta ,1}(\tau )}{\partial {\theta }_{\Lambda }}+\frac{\partial {\epsilon }_{\theta ,2}(\tau )}{\partial {\theta }_{\Lambda }}=\frac{{\Lambda }_{\theta }(\tau )\frac{\partial {\Lambda }_{\theta ,1}(\tau )}{\partial {\theta }_{\Lambda }}-{\Lambda }_{\theta ,1}(\tau )\frac{\partial {\Lambda }_{\theta }(\tau )}{\partial {\theta }_{\Lambda }}}{{\Lambda }_{\theta }^2(\tau )}+\frac{{\Lambda }_{\theta }(\tau )\frac{\partial {\Lambda }_{\theta ,2}(\tau )}{\partial {\theta }_{\Lambda }}-{\Lambda }_{\theta ,2}(\tau )\frac{\partial {\Lambda }_{\theta }(\tau )}{\partial {\theta }_{\Lambda }}}{{\Lambda }_{\theta }^2(\tau )}=\frac{{\Lambda }_{\theta }(\tau )(\frac{\partial {\Lambda }_{\theta ,1}(\tau )}{\partial {\theta }_{\Lambda }}+\frac{\partial {\Lambda }_{\theta ,2}(\tau )}{\partial {\theta }_{\Lambda }})-({\Lambda }_{\theta ,1}(\tau )+({\Lambda }_{\theta ,2}(\tau ))\frac{\partial {\Lambda }_{\theta }(\tau )}{\partial {\theta }_{\Lambda }}}{{\Lambda }_{\theta }^2(\tau )}=\frac{{\Lambda }_{\theta }(\tau )\frac{\partial {\Lambda }_{\theta }(\tau )}{\partial {\theta }_{\Lambda }}-{\Lambda }_{\theta }(\tau )\frac{\partial {\Lambda }_{\theta }(\tau )}{\partial {\theta }_{\Lambda }})}{{\Lambda }_{\theta }^2(\tau )}=0,\theta \in \Theta ,\tau \ge t_0,$$ (32)

where we have used the fact that Λ_θ(τ) ≔ Λ_θ,1(τ) + Λ_θ,2(τ), τ ≥ t₀ and θ ∈ 𝒞. Consider the term

$$\frac{\partial f_{\theta ,\tau }(r)}{\partial {\theta }_{\Lambda }}=\frac{\partial {\epsilon }_{\theta ,1}(\tau )}{\partial {\theta }_{\Lambda }}{f}_{\theta ,\tau ,1}(r)+{\epsilon }_{\theta ,1}(\tau )\frac{\partial f_{\theta ,\tau ,1}(r)}{\partial {\theta }_{\Lambda }}+\frac{\partial {\epsilon }_{\theta ,2}(\tau )}{\partial {\theta }_{\Lambda }}{f}_{\theta ,\tau ,2}(r)+{\epsilon }_{\theta ,2}(\tau )\frac{\partial f_{\theta ,\tau ,2}(r)}{\partial {\theta }_{\Lambda }},$$ (33)

where θ ∈ Θ, τ ≥ t₀ and r ∈ 𝒞. Substituting A1 in eq. 33 and using eq. 32, we have for θ ∈ Θ, τ ≥ t₀ and r ∈ 𝒞

$$\frac{\partial f_{\theta ,\tau }(r)}{\partial {\theta }_{\Lambda }}=f_{\theta ,\tau ,1}(r)(\frac{\partial {\epsilon }_{\theta ,1}(\tau )}{\partial {\theta }_{\Lambda }}+\frac{\partial {\epsilon }_{\theta ,2}(\tau )}{\partial {\theta }_{\Lambda }})=0.$$

2. Using A2 we have, ${\epsilon }_{\theta ,1}(\tau )=\frac{1}{1+\beta (\tau )}$, θ ∈ Θ and τ ≥ t₀, and ${\epsilon }_{\theta ,2}(\tau )=\frac{\beta (\tau )}{1+\beta (\tau )}$, θ ∈ Θ and τ ≥ t₀. Since β(τ) is independent of θ for τ ≥ t₀, $\frac{\partial {\epsilon }_{\theta ,i}(\tau )}{\partial {\theta }_{\Lambda }}=0$, θ ∈ Θ, τ ≥ t₀ and i = 1, 2. Substituting this in eq. 33 the result follows.

Lemma 2 For θ_c = (x₀₁, y₀₁, x₀₂, y₀₂) ∈ Θ_c, τ ≥ t₀ and i = 1, 2, let f_θc,τ,i be given by eq. 8. Let M > 0. Then for θ_c ∈ Θ_c and τ ≥ t₀, we have

• 1.$\frac{\partial f_{{\theta }_c,\tau ,i}(r)}{\partial x_{0i}}=-M\frac{\partial f_{{\theta }_c,\tau ,i}(r)}{\partial x}$, r = (x, y) ∈ ℝ², i = 1, 2.
• 2.$\frac{\partial f_{{\theta }_c,\tau ,i}(r)}{\partial y_{0i}}=-M\frac{\partial f_{{\theta }_c,\tau ,i}(r)}{\partial y}$, r = (x, y) ∈ ℝ², i = 1, 2.

Proof 1. For _c = (x₀₁, x₀₂, y₀₁, y₀₂) ∈ Θ_c and i = 1, 2, define $u_i≔\frac{x}{M}-x_{0i}\text{ and }{v}_i≔\frac{y}{M}-y_{0i}$. Then for i = 1, 2, we have

$$\frac{\partial f_{{\theta }_c,\tau ,i}(r)}{\partial x_{0i}}=\frac{1}{M^2}\frac{\partial q_i(\frac{x}{M}-x_{0i},\frac{y}{M}-y_{0i})}{\partial x_{0i}}=\frac{1}{M^2}\frac{\partial q_i(u_i,v_i)}{\partial u_i}\frac{\partial u_i}{\partial x_{0i}}=-\frac{1}{M^2}\frac{\partial q_i(u_i,v_i)}{\partial u_i}=\frac{1}{M^2}\frac{\partial q_i(\frac{x}{M}-x_{0i},\frac{y}{M}-y_{0i})}{\partial x}\frac{\partial x}{\partial u_i}=-M\frac{1}{M^2}\frac{\partial q_i(\frac{x}{M}-x_{0i},\frac{y}{M}-y_{0i})}{\partial x}=-M\frac{\partial f_{{\theta }_c,\tau ,i}(r)}{\partial x},$$

for r = (x, y) ∈ ℝ², θ_c ∈ Θ_c and τ ≥ t₀.

2. Proof is similar to that of result 1.

Lemma 3 For i = 1, 2, let Q_i be given by eq. 20, and Λ_i and q_i denote the photon detection rate and the image function of the i^th object, respectively. For i = 1, 2, assume that q_i is radially symmetric with respect to the origin, i.e., there exists a q_isuch that $q_i(x,y)=q_i(\sqrt{x^2+y^2})$ for (x, y) ∈ ℝ² and i = 1, 2. Then for i = 1, 2,

$${\mathbf{Q}}_i=\frac{1}{{\left({\delta }_{\mathit{\text{rs}},i}^{\mathit{\text{loc}}}\right)}^2}{\mathbf{1}}_{2\times 2},$$

where 1_2×2 denotes the 2 × 2 identity matrix and ${\delta }_{\mathit{\text{rs}},i}^{\mathit{\text{loc}}}$, i = 1, 2, is given by eq. 23.

Proof By definition, q_i, i = 1, 2, is symmetric along the x and y axes with respect to the origin. Using this, it can be shown that (see (Ram et al, 2006b, pg 39))

$${\mathbf{Q}}_i=\left({\displaystyle {\int }_{t_0}^t}{\Lambda }_i(\tau )d\tau \right)\text{ Diag }\left[{\displaystyle {\int }_{{ℝ}^2}}\frac{1}{q_i(x,y)}{\left(\frac{\partial q_i(x,y)}{\partial x}\right)}^2\mathit{\text{ dxdy }}{\displaystyle {\int }_{{ℝ}^2}}\frac{1}{q_i(x,y)}{\left(\frac{\partial q_i(x,y)}{\partial y}\right)}^2\mathit{\text{ dxdy}}\right],$$

where diag denotes the diagonal matrix. Further, using the fact that q_i, i = 1, 2, is radially symmetric, we have

$${[{\mathbf{Q}}_i]}_{11}=\left({\displaystyle {\int }_{t_0}^t}{\Lambda }_i(\tau )d\tau \right){\displaystyle {\int }_{{ℝ}^2}}\frac{1}{q_i(x,y)}{\left(\frac{\partial q_i(x,y)}{\partial x}\right)}^2\mathit{\text{dxdy}}=({\displaystyle {\int }_{t_0}^t}{\Lambda }_i(\tau )d\tau ){\displaystyle {\int }_0^{2\pi }}{\displaystyle {\int }_0^{\mathrm{\infty }}}\frac{1}{q_i(r)}{\left(\frac{\partial q_i(r)}{\partial r}\frac{\partial r}{\partial x}\right)}^2\mathit{\text{rdrd}}\varphi =({\displaystyle {\int }_{t_0}^t}{\Lambda }_i(\tau )d\tau ){\displaystyle {\int }_0^{2\pi }}{\text{cos}}^2(\varphi )d\varphi {\displaystyle {\int }_0^{\mathrm{\infty }}}\frac{1}{q_i(r)}{\left(\frac{\partial q_i(r)}{\partial r}\right)}^2\mathit{\text{rdr}}=({\displaystyle {\int }_{t_0}^t}{\Lambda }_i(\tau )d\tau \left)\right({\displaystyle {\int }_0^{2\pi }}\frac{1+\text{cos}(2\varphi )}{2}d\varphi ){\kappa }_i=({\displaystyle {\int }_{t_0}^t}{\Lambda }_i(\tau )d\tau )\pi {\kappa }_i=\frac{1}{{\left({\delta }_{\mathit{\text{rs}},i}^{\mathit{\text{loc}}}\right)}^2},$$

where i = 1, 2, and κ_i is defined in eq. 23. Similarly, we can show that for i = 1, 2, ${[{\mathbf{Q}}_i]}_{22}=1/{({\delta }_{\mathit{\text{rs}},i}^{\mathit{\text{loc}}})}^2$.

A.1 Proof of Theorem 1

Proof 1. Substituting for Λ_θ and f_θ,τ in eq. 1, and using assumptions A1 – A2 we get

$${\mathbf{I}}_{\mathit{\text{sim}}}(\theta )={\displaystyle {\int }_{t_0}^t}{\displaystyle {\int }_{𝒞}}\frac{1}{{\Lambda }_{\theta }(\tau )f_{\theta ,\tau }(r)}\left(\begin{array}{c}\hfill {\Lambda }_{\theta }(\tau ){\left(\frac{\partial f_{\theta ,\tau }(r)}{\partial {\theta }_f}\right)}^T\hfill \\ \hfill {\Lambda }_{\theta }(\tau ){\left(\frac{\partial f_{\theta ,\tau }(r)}{\partial {\theta }_{\Lambda }}\right)}^T+f_{\theta ,\tau }(r){\left(\frac{\partial {\Lambda }_{\theta }(\tau )}{\partial {\theta }_{\Lambda }}\right)}^T\hfill \end{array}\right)\times \left({\Lambda }_{\theta }(\tau )\frac{\partial f_{\theta ,\tau }(r)}{\partial {\theta }_f}{\Lambda }_{\theta }(\tau )\frac{\partial f_{\theta ,\tau }(r)}{\partial {\theta }_{\Lambda }}+f_{\theta ,\tau }(r)\frac{\partial {\Lambda }_{\theta }(\tau )}{\partial {\theta }_{\Lambda }}\right)\mathit{\text{ drd}}\tau {\mathbf{S}}_{\mathit{\text{sim}}}(\theta )=\left[{\left({\displaystyle {\int }_{t_0}^t}{\displaystyle {\int }_{𝒞}}\frac{1}{f_{\theta ,\tau }(r)}{\left(\frac{\partial f_{\theta ,\tau }(r)}{\partial {\theta }_f}\right)}^T\right(f_{\theta ,\tau }(r)\frac{\partial {\Lambda }_{\theta }(\tau )}{\partial {\theta }_{\Lambda }}+{\Lambda }_{\theta }(\tau )\frac{\partial f_{\theta ,\tau }(r)}{\partial {\theta }_{\Lambda }}\left)\mathit{\text{ drd}}\tau \right)}^T{\displaystyle {\int }_{t_0}^t}{\displaystyle {\int }_{𝒞}}\frac{1}{f_{\theta ,\tau }(r)}{\left(\frac{\partial f_{\theta ,\tau }(r)}{\partial {\theta }_f}\right)}^T\left({\Lambda }_{\theta }(\tau )\frac{\partial f_{\theta ,\tau }(r)}{\partial {\theta }_{\Lambda }}+f_{\theta ,\tau }(r)\frac{\partial {\Lambda }_{\theta }(\tau )}{\partial {\theta }_{\Lambda }}\right)\mathit{\text{ drd}}\tau {\displaystyle {\int }_{t_0}^t}{\displaystyle {\int }_{𝒞}}\frac{1}{{\Lambda }_{\theta }(\tau )f_{\theta ,\tau }(r)}{\left({\Lambda }_{\theta }(\tau )\frac{\partial f_{\theta ,\tau }(r)}{\partial {\theta }_{\Lambda }}+f_{\theta ,\tau }(r)\frac{\partial {\Lambda }_{\theta }(\tau )}{\partial {\theta }_{\Lambda }}\right)}^T\left({\Lambda }_{\theta }(\tau )\frac{\partial f_{\theta ,\tau }(r)}{\partial {\theta }_{\Lambda }}+f_{\theta ,\tau }(r)\frac{\partial {\Lambda }_{\theta }(\tau )}{\partial {\theta }_{\Lambda }}\right)\mathit{\text{ drd}}\tau \right].$$ (34)

By definition, f_θ,τ is a probability density function, which satisfies the regularity conditions that are necessary for the calculation of the Fisher information matrix (Kay (1993)). Hence we have for θ ∈ Θ and τ ≥ t₀,

$${\displaystyle {\int }_{𝒞}}\frac{\partial f_{\theta ,\tau }(r)}{\partial \theta }\mathit{\text{dr}}=\left(\begin{array}{c}\hfill {\displaystyle {\int }_{𝒞}}\frac{\partial f_{\theta ,\tau }(r)}{\partial {\theta }_f}\mathit{\text{dr}}\hfill \\ \hfill {\displaystyle {\int }_{𝒞}}\frac{\partial f_{\theta ,\tau }(r)}{\partial {\theta }_{\Lambda }}\mathit{\text{dr}}\hfill \end{array}\right)=\left(\begin{array}{c}\hfill \frac{\partial }{\partial {\theta }_f}{\displaystyle {\int }_{𝒞}}{f}_{\theta ,\tau }(r)\mathit{\text{dr}}\hfill \\ \hfill \frac{\partial }{\partial {\theta }_{\Lambda }}{\displaystyle {\int }_{𝒞}}{f}_{\theta ,\tau }(r)\mathit{\text{dr}}\hfill \end{array}\right)=\left(\begin{array}{c}\hfill \frac{\partial }{\partial {\theta }_f}1\hfill \\ \hfill \frac{\partial }{\partial {\theta }_{\Lambda }}1\hfill \end{array}\right)=\left(\begin{array}{c}\hfill 0\hfill \\ \hfill 0\hfill \end{array}\right).$$ (35)

Using eq. 35, we have

$${[{\mathbf{I}}_{\mathit{\text{sim}}}(\theta )]}_{12}={[{\mathbf{I}}_{\mathit{\text{sim}}}(\theta )]}_{21}^T={\displaystyle {\int }_{t_0}^t}{\displaystyle {\int }_{𝒞}}\frac{1}{f_{\theta ,\tau }(r)}{\left(\frac{\partial f_{\theta ,\tau }(r)}{\partial {\theta }_f}\right)}^T\left({\Lambda }_{\theta }(\tau )\frac{\partial f_{\theta ,\tau }(r)}{\partial {\theta }_{\Lambda }}+f_{\theta ,\tau }(r)\frac{\partial {\Lambda }_{\theta }(\tau )}{\partial {\theta }_{\Lambda }}\right)\mathit{\text{ drd}}\tau ={\displaystyle {\int }_{t_0}^t}{\displaystyle {\int }_{𝒞}}\frac{{\Lambda }_{\theta }(\tau )}{f_{\theta ,\tau }(r)}{\left(\frac{\partial f_{\theta ,\tau }(r)}{\partial {\theta }_f}\right)}^T\frac{\partial f_{\theta ,\tau }(r)}{\partial {\theta }_{\Lambda }}\mathit{\text{ drd}}\tau ={\mathbf{R}}_{\mathit{\text{sim}}}(\theta ),\theta \in \Theta .$$ (36)

Using eq. 35 and the fact that ∫_𝒞 f_θ,τ (r)dr = 1 for θ ∈ Θ and τ ≥ t₀, we have

$${[{\mathbf{I}}_{\mathit{\text{sim}}}(\theta )]}_{22}={\displaystyle {\int }_{t_0}^t}{\displaystyle {\int }_{𝒞}}\frac{1}{{\Lambda }_{\theta }(\tau )f_{\theta ,\tau }(r)}{\left({\Lambda }_{\theta }(\tau )\frac{\partial f_{\theta ,\tau }(r)}{\partial {\theta }_{\Lambda }}+f_{\theta ,\tau }(r)\frac{\partial {\Lambda }_{\theta }(\tau )}{\partial {\theta }_{\Lambda }}\right)}^T\left({\Lambda }_{\theta }(\tau )\frac{\partial f_{\theta ,\tau }(r)}{\partial {\theta }_{\Lambda }}+f_{\theta ,\tau }(r)\frac{\partial {\Lambda }_{\theta }(\tau )}{\partial {\theta }_{\Lambda }}\right)\mathit{\text{ drd}}\tau ={\displaystyle {\int }_{t_0}^t}{\displaystyle {\int }_{𝒞}}\frac{{\Lambda }_{\theta }(\tau )}{f_{\theta ,\tau }(r)}{\left(\frac{\partial f_{\theta ,\tau }(r)}{\partial {\theta }_{\Lambda }}\right)}^T\frac{\partial f_{\theta ,\tau }(r)}{\partial {\theta }_{\Lambda }}\mathit{\text{drd}}\tau +{\displaystyle {\int }_{t_0}^t}\left({\displaystyle {\int }_{𝒞}}{\left(\frac{\partial f_{\theta ,\tau }(r)}{\partial {\theta }_{\Lambda }}\right)}^T\mathit{\text{ dr}}\right)\frac{\partial {\Lambda }_{\theta }(\tau )}{\partial {\theta }_{\Lambda }}d\tau +{\displaystyle {\int }_{t_0}^t}\frac{1}{{\Lambda }_{\theta }(\tau )}{\left(\frac{\partial {\Lambda }_{\theta }(\tau )}{\partial {\theta }_{\Lambda }}\right)}^T\frac{\partial {\Lambda }_{\theta }(\tau )}{\partial {\theta }_{\Lambda }}d\tau +{\displaystyle {\int }_{t_0}^t}{\left(\frac{\partial {\Lambda }_{\theta }(\tau )}{\partial {\theta }_{\Lambda }}\right)}^T{\displaystyle {\int }_{𝒞}}\frac{\partial f_{\theta ,\tau }(r)}{\partial {\theta }_{\Lambda }}\mathit{\text{drd}}\tau ={\mathbf{T}}_{\mathit{\text{sim}}}(\theta ),\theta \in \Theta .$$ (37)

Substituting eqs. 36 and 37 in eq. 34, the result immediately follows.

2. Using assumptions A1 and A3 it can be shown that (∂f_θ,τ (r)/∂θ_Λ) = 0, r ∈ 𝒞, θ ∈ Θ, τ ≥ t₀ (see result 3 of Lemma 1 in Appendix). Substituting this and using assumption A3 in eqs. 5, 6 and 7, we obtain the desired result.

3. Using assumptions A1 and A4 it can be shown that (∂f_θ,τ (r)/∂θ_Λ) = 0, r ∈ 𝒞, θ ∈ Θ, τ ≥ t₀ (see result 2 of Lemma 1 in Appendix). Further, by assumption A4 we have f_θ,τ(r) = f_θ,τ,1(r)(ε_θ,1(τ) + ε_θ,2(τ)) = f_θ,τ,1(r), r ∈ 𝒞 and τ ≥ t₀. Substituting these results in eqs. 5, 6 and 7, we obtain the desired result.

A.2 Proof of results 2 and 3 of Theorem 2

Proof 2. For ${\theta }_c\in {\Theta }_c\backslash {\Theta }_c^0$, define s_x ≔ (x₀₁ + x₀₂)/2, s_y ≔ (y₀₁ + y₀₂)/2 and ϕ = tan⁻¹((y₀₂ − y₀₁)/(x₀₂ − x₀₁)). Then we have $x_{01}≔s_x-\frac{d\text{ cos }\varphi }{2},y_{01}≔s_y-\frac{d\text{ sin }\varphi }{2},x_{02}≔s_x-\frac{d\text{ cos }\varphi }{2},y_{02}≔s_y-\frac{d\text{ sin }\varphi }{2}$. Substituting this in result 1 of the Theorem 2 and using the shift invariant property of Lebesgue intergrals, we get for ${\theta }_c\in {\Theta }_c\backslash {\Theta }_c^0$,

$${\mathbf{S}}_{\mathit{\text{sim}}}({\theta }_c)≔{\displaystyle {\int }_{t_0}^t}{\displaystyle {\int }_{{ℝ}^2}}\frac{1}{{\Lambda }_1(\tau )q_1(x+\frac{d}{2}\text{cos }\varphi ,y+\frac{d}{2}\text{sin }\varphi )+{\Lambda }_2(\tau )q_2(x-\frac{d}{2}\text{cos }\varphi ,y-\frac{d}{2}\text{sin }\varphi )}\times \left[\begin{array}{c}\hfill {\Lambda }_1(\tau )\frac{\partial q_1(x+\frac{d}{2}\text{cos }\varphi ,y+\frac{d}{2}\text{sin }\varphi )}{\partial x}\hfill \\ \hfill {\Lambda }_1(\tau )\frac{\partial q_1(x+\frac{d}{2}\text{ cos }\varphi ,y+\frac{d}{2}\text{ sin }\varphi )}{\partial y}\hfill \\ \hfill {\Lambda }_2(\tau )\frac{\partial q_2(x-\frac{d}{2}\text{ cos }\varphi ,y-\frac{d}{2}\text{ sin }\varphi )}{\partial x}\hfill \\ \hfill {\Lambda }_2(\tau )\frac{\partial q_2(x-\frac{d}{2}\text{ cos }\varphi ,y-\frac{d}{2}\text{ sin }\varphi )}{\partial y}\hfill \end{array}\right]{\left[\begin{array}{c}\hfill {\Lambda }_1(\tau )\frac{\partial q_1(x+\frac{d}{2}\text{ cos }\varphi ,y+\frac{d}{2}\text{ sin }\varphi )}{\partial x}\hfill \\ \hfill {\Lambda }_1(\tau )\frac{\partial q_1(x+\frac{d}{2}\text{ cos }\varphi ,y+\frac{d}{2}\text{ sin }\varphi )}{\partial y}\hfill \\ \hfill {\Lambda }_2(\tau )\frac{\partial q_2(x-\frac{d}{2}\text{ cos }\varphi ,y-\frac{d}{2}\text{ sin }\varphi )}{\partial x}\hfill \\ \hfill {\Lambda }_2(\tau )\frac{\partial q_2(x-\frac{d}{2}\text{ cos }\varphi ,y-\frac{d}{2}\text{ sin }\varphi )}{\partial y}\hfill \end{array}\right]}^T\mathit{\text{ dxdyd}}\tau .$$ (38)

For (x, y) ∈ ℝ², τ ≥ t₀ and ${\theta }_c\in {\Theta }_c\backslash {\Theta }_c^0$, let

$$Q_{{\theta }_c}^{+}(x,y,\tau )≔{\Lambda }_1(\tau )q_1(x+\frac{d}{2}\text{ cos }\varphi ,y+\frac{d}{2}\text{ sin }\varphi ),$$ (39)

$$Q_{{\theta }_c}^{-}(x,y,\tau )≔{\Lambda }_2(\tau )q_2(x-\frac{d}{2}\text{ cos }\varphi ,y-\frac{d}{2}\text{ sin }\varphi ).$$ (40)

For ϕ ∈ (0, 2π), define T_ϕ : ℝ² → ℝ²

$$\left(\begin{array}{c}\hfill x\hfill \\ \hfill y\hfill \end{array}\right)\mapsto \left(\begin{array}{c}\hfill u\hfill \\ \hfill v\hfill \end{array}\right)=\left(\begin{array}{c}\hfill x\text{ cos }\varphi +y\text{ sin }\varphi \hfill \\ \hfill -x\text{ sin }\varphi +y\text{ cos }\varphi \hfill \end{array}\right).$$

The transformation T_ϕ maps the coordinates of a point on the 2D plane when the coordinate axes is rotated by an angle ϕ. Let $P^{\pm }≔(x\pm \frac{d}{2}\text{ cos }\varphi ,y\pm \frac{d}{2}\text{ sin }\varphi )$. Then

$${\mathrm{P̃}}^{\pm }≔T_{\varphi }{P}^{\pm }=\left(\begin{array}{c}\hfill \text{cos }\varphi \text{ sin }\varphi \hfill \\ \hfill -\text{sin }\varphi \text{ cos }\varphi \hfill \end{array}\right)\left(\begin{array}{c}\hfill x\pm \frac{d}{2}\text{cos }\varphi \hfill \\ \hfill y\pm \frac{d}{2}\text{sin }\varphi \hfill \end{array}\right)=\left(\begin{array}{c}\hfill x\text{ cos }\varphi +y\text{ sin }\varphi \pm \frac{d}{2}\hfill \\ \hfill -x\text{ sin }\varphi +y\text{ cos }\varphi \hfill \end{array}\right).$$ (41)

Using eq. 41, we have for τ ≥ t₀ and ${\theta }_c\in {\Theta }_c\backslash {\Theta }_c^0$,

$$(Q_{{\theta }_c}^{+}◦T_{\varphi })(x,y,\tau )={\Lambda }_1(\tau )q_1(T_{\varphi }(x+\frac{d}{2}\text{cos }\varphi ,y+\frac{d}{2}\text{sin }\varphi ))={\Lambda }_1(\tau )q_1(T_{\varphi }(P^{+}))={\Lambda }_1(\tau )q_1({\mathrm{P̃}}^{+})={\Lambda }_1(\tau )q_1(x\text{ cos }\varphi +y\text{ sin }\varphi +\frac{d}{2},-x\text{ sin }\varphi +y\text{ cos }\varphi ),(x,y)\in {ℝ}^2,$$ (42)

$$(Q_{{\theta }_c}^{-}◦T_{\varphi })(x,y,\tau )={\Lambda }_2(\tau )q_2(x\text{ cos }\varphi +y\text{ sin }\varphi -\frac{d}{2},-x\text{ sin }\varphi +y\text{ cos }\varphi ),(x,y)\in {ℝ}^2.$$ (43)

Similarly, for ${\theta }_c\in {\Theta }_c\backslash {\Theta }_c^0$, τ ≥t₀ and ζ ∈ {x, y},

$$(\frac{\partial Q_{{\theta }_c}^{+}}{\partial \zeta }◦T_{\varphi })(x,y)={\Lambda }_1(\tau )\frac{\partial q_1(T_{\varphi }(P^{+}))}{\partial \zeta }={\Lambda }_1(\tau )\frac{\partial q_1({\mathrm{P̃}}^{+})}{\partial \zeta }={\Lambda }_1(\tau )\frac{\partial q_1(x\text{ cos }\varphi +y\text{ sin }\varphi +\frac{d}{2},-x\text{ sin }\varphi +y\text{ cos }\varphi )}{\partial \zeta },(x,y)\in {ℝ}^2,$$ (44)

$$(\frac{\partial Q_{{\theta }_c}^{-}}{\partial \zeta }◦T_{\varphi })(x,y)={\Lambda }_2(\tau )\frac{\partial q_2(x\text{ cos }\varphi +y\text{ sin }\varphi -\frac{d}{2},-x\text{ sin }\varphi +y\text{ cos }\varphi )}{\partial \zeta },(x,y)\in {ℝ}^2.$$ (45)

By definition, the determinant of the Jacobian of T_ϕ is given by

$$\text{Det}[T_{\varphi }^{\prime }]≔\text{Det }\left[\begin{array}{cc}\hfill \text{cos }\varphi \hfill & \hfill \text{sin }\varphi \hfill \\ \hfill -\text{sin }\varphi \hfill & \hfill \text{cos }\varphi \hfill \end{array}\right]=1,\varphi \in (0,2\pi ),$$ (46)

and for (u, v) ≔ T _ϕ(x, y),

$$\mathit{\text{dudv}}=|\text{Det}[T_{\varphi }^{\prime }]|\mathit{\text{dxdy}}=\mathit{\text{dxdy}}.$$ (47)

Substituting eqs. 42 – 47 in the expression for S_sim(θ_c) given in eq. 38 and making use of the change of variables Theorem (Rudin (1987)) we get,

$${\mathbf{S}}_{\mathit{\text{sim}}}({\theta }_c)={\displaystyle {\int }_{t_0}^t}{\displaystyle {\int }_{{ℝ}^2}}\frac{1}{Q_{{\theta }_c}^{+}(x,y,\tau )+Q_{{\theta }_c}^{-}(x,y,\tau )}\left(\begin{array}{c}\hfill \frac{\partial Q_{{\theta }_c}^{+}(x,y,\tau )}{\partial x}\hfill \\ \hfill \frac{\partial Q_{{\theta }_c}^{+}(x,y,\tau )}{\partial y}\hfill \\ \hfill \frac{\partial Q_{{\theta }_c}^{-}(x,y,\tau )}{\partial x}\hfill \\ \hfill \frac{\partial Q_{{\theta }_c}^{-}(x,y,\tau )}{\partial y}\hfill \end{array}\right){\left(\begin{array}{c}\hfill \frac{\partial Q_{{\theta }_c}^{+}(x,y,\tau )}{\partial x}\hfill \\ \hfill \frac{\partial Q_{{\theta }_c}^{+}(x,y,\tau )}{\partial y}\hfill \\ \hfill \frac{\partial Q_{{\theta }_c}^{-}(x,y,\tau )}{\partial x}\hfill \\ \hfill \frac{\partial Q_{{\theta }_c}^{-}(x,y,\tau )}{\partial y}\hfill \end{array}\right)}^T\mathit{\text{dxdyd}}\tau ={\displaystyle {\int }_{t_0}^t}{\displaystyle {\int }_{T_{\varphi }({ℝ}^2)}}\left(\right(\frac{1}{Q_{{\theta }_c}^{+}+Q_{{\theta }_c}^{-}}\left(\begin{array}{c}\hfill \frac{\partial Q_{{\theta }_c}^{+}}{\partial x}\hfill \\ \hfill \frac{\partial Q_{{\theta }_c}^{+}}{\partial y}\hfill \\ \hfill \frac{\partial Q_{{\theta }_c}^{-}}{\partial x}\hfill \\ \hfill \frac{\partial Q_{{\theta }_c}^{-}}{\partial y}\hfill \end{array}\right){\left(\begin{array}{c}\hfill \frac{\partial Q_{{\theta }_c}^{+}}{\partial x}\hfill \\ \hfill \frac{\partial Q_{{\theta }_c}^{+}}{\partial y}\hfill \\ \hfill \frac{\partial Q_{{\theta }_c}^{-}}{\partial x}\hfill \\ \hfill \frac{\partial Q_{{\theta }_c}^{-}}{\partial y}\hfill \end{array}\right)}^T)\circ T_{\varphi })(x,y,\tau )\text{Det}|T_{\varphi }^{\prime }|\mathit{\text{dxdyd}}\tau ={\displaystyle {\int }_{t_0}^t}{\displaystyle {\int }_{{ℝ}^2}}\frac{1}{Q_{{\theta }_c}^{+}(T_{\varphi }(P^{+}))+Q_{{\theta }_c}^{-}(T_{\varphi }(P^{-}))}\left(\begin{array}{c}\hfill \frac{\partial Q_{{\theta }_c}^{+}(T_{\varphi }(P^{+}))}{\partial x}\hfill \\ \hfill \frac{\partial Q_{{\theta }_c}^{+}(T_{\varphi }(P^{+}))}{\partial y}\hfill \\ \hfill \frac{\partial Q_{{\theta }_c}^{-}(T_{\varphi }(P^{-}))}{\partial x}\hfill \\ \hfill \frac{\partial Q_{{\theta }_c}^{-}(T_{\varphi }(P^{-}))}{\partial y}\hfill \end{array}\right){\left(\begin{array}{c}\hfill \frac{\partial Q_{{\theta }_c}^{+}(T_{\varphi }(P^{+}))}{\partial x}\hfill \\ \hfill \frac{\partial Q_{{\theta }_c}^{+}(T_{\varphi }(P^{+}))}{\partial y}\hfill \\ \hfill \frac{\partial Q_{{\theta }_c}^{-}(T_{\varphi }(P^{-}))}{\partial x}\hfill \\ \hfill \frac{\partial Q_{{\theta }_c}^{-}(T_{\varphi }(P^{-}))}{\partial y}\hfill \end{array}\right)}^T\mathit{\text{dxdyd}}\tau ={\displaystyle {\int }_{t_0}^t}{\displaystyle {\int }_{{ℝ}^2}}\frac{1}{Q_{{\theta }_c}^{+}({\mathrm{P̃}}^{+})+Q_{{\theta }_c}^{-}({\mathrm{P̃}}^{-})}\left(\begin{array}{c}\hfill \frac{\partial Q_{{\theta }_c}^{+}({\mathrm{P̃}}^{+})}{\partial x}\hfill \\ \hfill \frac{\partial Q_{{\theta }_c}^{+}({\mathrm{P̃}}^{+})}{\partial y}\hfill \\ \hfill \frac{\partial Q_{{\theta }_c}^{-}({\mathrm{P̃}}^{-})}{\partial x}\hfill \\ \hfill \frac{\partial Q_{{\theta }_c}^{-}({\mathrm{P̃}}^{-})}{\partial y}\hfill \end{array}\right){\left(\begin{array}{c}\hfill \frac{\partial Q_{{\theta }_c}^{+}({\mathrm{P̃}}^{+})}{\partial x}\hfill \\ \hfill \frac{\partial Q_{{\theta }_c}^{+}({\mathrm{P̃}}^{+})}{\partial y}\hfill \\ \hfill \frac{\partial Q_{{\theta }_c}^{-}({\mathrm{P̃}}^{-})}{\partial x}\hfill \\ \hfill \frac{\partial Q_{{\theta }_c}^{-}({\mathrm{P̃}}^{-})}{\partial y}\hfill \end{array}\right)}^T\mathit{\text{dxdyd}}\tau ={\displaystyle {\int }_{t_0}^t}{\displaystyle {\int }_{{ℝ}^2}}\frac{1}{{\Lambda }_1(\tau )q_1(x\text{ cos }\varphi +y\text{ sin }\varphi +\frac{d}{2},-x\text{ sin }\varphi +y\text{ cos }\varphi )+{\Lambda }_2(\tau )q_2(x\text{ cos }\varphi +y\text{ sin }\varphi -\frac{d}{2},-x\text{ sin }\varphi +y\text{ cos }\varphi )}\times \left(\begin{array}{c}\hfill {\Lambda }_1(\tau )\frac{\partial q_1(x\text{ cos }\varphi +y\text{ sin }\varphi +\frac{d}{2},-x\text{ sin }\varphi +y\text{ cos }\varphi )}{\partial x}\hfill \\ \hfill {\Lambda }_1(\tau )\frac{\partial q_1(x\text{ cos }\varphi +y\text{ sin }\varphi +\frac{d}{2},-x\text{ sin }\varphi +y\text{ cos }\varphi )}{\partial y}\hfill \\ \hfill {\Lambda }_2(\tau )\frac{\partial q_2(x\text{ cos }\varphi +y\text{ sin }\varphi +\frac{d}{2},-x\text{ sin }\varphi +y\text{ cos }\varphi )}{\partial x}\hfill \\ \hfill {\Lambda }_2(\tau )\frac{\partial q_2(x\text{ cos }\varphi +y\text{ sin }\varphi +\frac{d}{2},-x\text{ sin }\varphi +y\text{ cos }\varphi )}{\partial y}\hfill \end{array}\right){\left(\begin{array}{c}\hfill {\Lambda }_1(\tau )\frac{\partial q_1(x\text{ cos }\varphi +y\text{ sin }\varphi +\frac{d}{2},-x\text{ sin }\varphi +y\text{ cos }\varphi )}{\partial x}\hfill \\ \hfill {\Lambda }_1(\tau )\frac{\partial q_1(x\text{ cos }\varphi +y\text{ sin }\varphi +\frac{d}{2},-x\text{ sin }\varphi +y\text{ cos }\varphi )}{\partial y}\hfill \\ \hfill {\Lambda }_2(\tau )\frac{\partial q_2(x\text{ cos }\varphi +y\text{ sin }\varphi +\frac{d}{2},-x\text{ sin }\varphi +y\text{ cos }\varphi )}{\partial x}\hfill \\ \hfill {\Lambda }_2(\tau )\frac{\partial q_2(x\text{ cos }\varphi +y\text{ sin }\varphi +\frac{d}{2},-x\text{ sin }\varphi +y\text{ cos }\varphi )}{\partial y}\hfill \end{array}\right)}^T\mathit{\text{dxdyd}}\tau ={\displaystyle {\int }_{t_0}^t}{\displaystyle {\int }_{{ℝ}^2}}\frac{1}{{\Lambda }_1(\tau )q_1(u+\frac{d}{2},v)+{\Lambda }_2(\tau )q_2(u-\frac{d}{2},v)}\left(\begin{array}{c}\hfill {\Lambda }_1(\tau )(\text{cos }\varphi \frac{\partial q_1(u+\frac{d}{2},v)}{\partial u}-\text{ sin }\varphi \frac{\partial q_1(u+\frac{d}{2},v)}{\partial v})\hfill \\ \hfill {\Lambda }_1(\tau )(\text{sin }\varphi \frac{\partial q_1(u+\frac{d}{2},v)}{\partial u}+\text{ cos }\varphi \frac{\partial q_1(u+\frac{d}{2},v)}{\partial v})\hfill \\ \hfill {\Lambda }_2(\tau )(\text{cos }\varphi \frac{\partial q_2(u-\frac{d}{2},v)}{\partial u}-\text{ sin }\varphi \frac{\partial q_2(u-\frac{d}{2},v)}{\partial v})\hfill \\ \hfill {\Lambda }_2(\tau )(\text{sin }\varphi \frac{\partial q_2(u-\frac{d}{2},v)}{\partial u}+\text{ cos }\varphi \frac{{\partial }_{q_2}(u-\frac{d}{2},v)}{\partial v})\hfill \end{array}\right)\times {\left(\begin{array}{c}\hfill {\Lambda }_1(\tau )(\text{cos }\varphi \frac{\partial q_1(u+\frac{d}{2},v)}{\partial u}-\text{ sin }\varphi \frac{\partial q_1(u+\frac{d}{2},v)}{\partial v})\hfill \\ \hfill {\Lambda }_1(\tau )(\text{sin }\varphi \frac{\partial q_1(u+\frac{d}{2},v)}{\partial u}+\text{ cos }\varphi \frac{\partial q_1(u+\frac{d}{2},v)}{\partial v})\hfill \\ \hfill {\Lambda }_2(\tau )(\text{cos }\varphi \frac{\partial q_2(u-\frac{d}{2},v)}{\partial u}-\text{ sin }\varphi \frac{\partial q_2(u-\frac{d}{2},v)}{\partial v})\hfill \\ \hfill {\Lambda }_2(\tau )(\text{sin }\varphi \frac{\partial q_2(u-\frac{d}{2},v)}{\partial u}+\text{ cos }\varphi \frac{\partial q_2(u-\frac{d}{2},v)}{\partial v})\hfill \end{array}\right)}^T\mathit{\text{dudvd}}\tau ,{\theta }_c\in {\Theta }_c,$$ (48)

where u ≔ x cos ϕ + y sin ϕ and v ≔ −x sin ϕ + y cos ϕ. Further, for ${\theta }_c\in {\Theta }_c\backslash {\Theta }_c^0$, τ ≥ t₀ and (x, y) ∈ ℝ², we have

$$\left(\begin{array}{c}\hfill {\Lambda }_1(\tau )(\text{cos }\varphi \frac{\partial q_1(u+\frac{d}{2},v)}{\partial u}-\text{sin }\varphi \frac{\partial q_1(u+\frac{d}{2},v)}{\partial v})\hfill \\ \hfill {\Lambda }_1(\tau )(\text{sin }\varphi \frac{\partial q_1(u+\frac{d}{2},v)}{\partial u}+\text{cos }\varphi \frac{\partial q_1(u+\frac{d}{2},v)}{\partial v})\hfill \\ \hfill {\Lambda }_2(\tau )(\text{cos }\varphi \frac{\partial q_2(u-\frac{d}{2},v)}{\partial u}-\text{sin }\varphi \frac{\partial q_2(u-\frac{d}{2},v)}{\partial v})\hfill \\ \hfill {\Lambda }_2(\tau )(\text{sin }\varphi \frac{\partial q_2(u-\frac{d}{2},v)}{\partial u}+\text{cos }\varphi \frac{\partial q_2(u-\frac{d}{2},v)}{\partial v})\hfill \end{array}\right)=\left[\begin{array}{cccc}\hfill \text{cos }\varphi \hfill & \hfill -\text{sin }\varphi \hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill \text{sin }\varphi \hfill & \hfill \text{cos }\varphi \hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \text{cos }\varphi \hfill & \hfill -\text{sin }\varphi \hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \text{sin }\varphi \hfill & \hfill \text{cos }\varphi \hfill \end{array}\right]\left[\begin{array}{c}\hfill {\Lambda }_1(\tau )\frac{\partial q_1(u+\frac{d}{2},v)}{\partial u}\hfill \\ \hfill {\Lambda }_1(\tau )\frac{\partial q_1(u+\frac{d}{2},v)}{\partial v}\hfill \\ \hfill {\Lambda }_2(\tau )\frac{\partial q_2(u-\frac{d}{2},v)}{\partial u}\hfill \\ \hfill {\Lambda }_2(\tau )\frac{\partial q_2(u-\frac{d}{2},v)}{\partial v}\hfill \end{array}\right]=\frac{1}{d}\left[\begin{array}{cccc}\hfill x_{02}-x_{01}\hfill & \hfill -(y_{02}-y_{01})\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill y_{02}-y_{01}\hfill & \hfill x_{02}-x_{01}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill x_{02}-x_{01}\hfill & \hfill -(y_{02}-y_{01})\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill y_{02}-y_{01}\hfill & \hfill x_{02}-x_{01}\hfill \end{array}\right]\left[\begin{array}{c}\hfill {\Lambda }_1(\tau )\frac{\partial q_1(u+\frac{d}{2},v)}{\partial u}\hfill \\ \hfill {\Lambda }_1(\tau )\frac{\partial q_1(u+\frac{d}{2},v)}{\partial v}\hfill \\ \hfill {\Lambda }_2(\tau )\frac{\partial q_2(u-\frac{d}{2},v)}{\partial u}\hfill \\ \hfill {\Lambda }_2(\tau )\frac{\partial q_2(u-\frac{d}{2},v)}{\partial v}\hfill \end{array}\right]=\mathbf{D}({\theta }_c)\left(\begin{array}{c}\hfill {\Lambda }_1(\tau )q_{1,x}^{\prime }(x,y)\hfill \\ \hfill {\Lambda }_1(\tau )q_{1,y}^{\prime }(x,y)\hfill \\ \hfill {\Lambda }_2(\tau )q_{2,x}^{\prime }(x,y)\hfill \\ \hfill {\Lambda }_2(\tau )q_{2,y}^{\prime }(x,y)\hfill \end{array}\right),$$

where D(θ_c) is defined in eq. 13, $q_{i,\zeta }^{\prime },i=1,2$, ζ ∈ {x, y} is given by eq. 16 and we have used the fact that cos ϕ ≔ (x₀₂ − x₀₁)/d and sin ϕ ≔ (y₀₂ − y₀₁)/d. Substituting the above expression in eq. 48, the result immediately follows.

3. To prove this result we need to show that the off-diagonal terms of C_ij(θ_c) are zero, for i, j = 1, 2 and ${\theta }_c\in {\Theta }_c\backslash {\Theta }_c^0$. For ${\theta }_c\in {\Theta }_c\backslash {\Theta }_c^0$, τ ≥ t₀ and (x, y) ∈ ℝ², let

$$W_{{\theta }_c}^1(x,y,\tau )≔{\Lambda }_1(\tau )q_1(x+\frac{d}{2},y),W_{{\theta }_c}^2(x,y,\tau )≔{\Lambda }_2(\tau )q_2(x-\frac{d}{2},y).$$ (49)

Define T_Y : ℝ² × [t₀, ∞) → ℝ² × [t₀, ∞), (x, y, τ) ↦ (x, −y, τ). Since q₁ and q₂ are symmetric along the y axis with respect to y = 0, we have $W_{{\theta }_c}^1(x,y,\tau )=(W_{{\theta }_c}^1\circ T_Y)(x,,y,\tau )\text{ and }{W}_{{\theta }_c}^2(x,y,\tau )=(W_{{\theta }_c}^2\circ T_Y)(x,,y,\tau )\text{ for }{\theta }_c\in {\Theta }_c\backslash {\Theta }_c^0,(x,y)\in {ℝ}^2$ and τ ≥ t₀. This implies that for ${\theta }_c\in {\Theta }_c\backslash {\Theta }_c^0$, (x, y) ∈ ℝ² and τ ≥ t₀, we have

$$U_{{\theta }_c}^{\pm }(x,y,\tau )={\Lambda }_1(\tau )q_1(x-\frac{d}{2},y)\pm {\Lambda }_2(\tau )q_2(x+\frac{d}{2},y)=(U_{{\theta }_c}^{\pm }\circ T_Y)(x,y,\tau ),$$ (50)

$$\frac{\partial W_{{\theta }_c}^i(x,y,\tau )}{\partial x}=(\frac{\partial W_{{\theta }_c}^i}{\partial x}\circ T_Y)(x,y,\tau ),i=1,2,$$ (51)

$$\frac{\partial W_{{\theta }_c}^i(x,y,\tau )}{\partial y}=-(\frac{\partial W_{{\theta }_c}^i}{\partial y}\circ T_Y)(x,y,\tau ),i=1,2.$$ (52)

Consider the term [C₁₁(θ_c)]₁₂, where C₁₁(θ_c) is given by eq. 15. Using eqs. 50, 51 and 52 we have

$${[{\mathbf{C}}_{11}({\theta }_c)]}_{12}={\displaystyle {\int }_{t_0}^t}{\displaystyle {\int }_{{ℝ}^2}}\frac{1}{{\Lambda }_1(\tau )q_1(x+\frac{d}{2},y)+{\Lambda }_2(\tau )q_2(x-\frac{d}{2},y)}\times ({\Lambda }_1(\tau )\frac{\partial q_1(x+\frac{d}{2},y)}{\partial x})({\Lambda }_2(\tau )\frac{\partial q_1(x+\frac{d}{2},y)}{\partial y})\mathit{\text{ dxdyd}}\tau ={\displaystyle {\int }_{t_0}^t}{\displaystyle {\int }_{{ℝ}^2}}\frac{1}{U_{{\theta }_c}^{+}(x,y,\tau )}\frac{\partial W^1(x,y,\tau )}{\partial x}\frac{\partial W_{{\theta }_c}^1(x,y,\tau )}{\partial y}\mathit{\text{dxdyd}}\tau =-{\displaystyle {\int }_{t_0}^t}{\displaystyle {\int }_{{ℝ}^2}}\frac{1}{(U_{{\theta }_c}^{+}\circ T_Y)(x,y,\tau )}(\frac{\partial W_{{\theta }_c}^1}{\partial x}\circ T_Y)(x,y,\tau )(\frac{\partial W_{{\theta }_c}^1}{\partial y}\circ T_Y)(x,y,\tau )\mathit{\text{dxdyd}}\tau =-{\displaystyle {\int }_{t_0}^t}{\displaystyle {\int }_{{ℝ}^2}}\left(\right(\frac{1}{U_{{\theta }_c}^{+}}\frac{\partial W_{{\theta }_c}^1}{\partial x}\frac{\partial W_{{\theta }_c}^1}{\partial y})\circ T_Y)(x,y,\tau )\mathit{\text{dxdyd}}\tau ={\displaystyle {\int }_{t_0}^t}{\displaystyle {\int }_{{ℝ}^2}}\frac{1}{U_{{\theta }_c}^{+}(x,y,\tau )}\frac{\partial W_{{\theta }_c}^1(x,y,\tau )}{\partial x}\frac{\partial W_{{\theta }_c}^1(x,y,\tau )}{\partial y}\mathit{\text{dxdyd}}\tau =-{[{\mathbf{C}}_{11}({\theta }_c)]}_{12},{\theta }_c\in {\Theta }_c\backslash {\Theta }_c^0,$$

where we have used the change of variables theorem in the final step. From the above equation it follows that [C₁₁(θ_c)]₁₂ = [C₁₁(θ_c)]₂₁ = 0, ${\theta }_c\in {\Theta }_c\backslash {\Theta }_c^0$. Similarly, by using eqs. 50, 51 and 52, we can show that [C₁₂(θ_c)]₁₂ = [C₁₂(θ_c)]₂₁ = 0, and [C₂₂(θ_c)]₁₂ = [C (θ_c)]₂₁ = 0 for ${\theta }_c\in {\Theta }_c\backslash {\Theta }_c^0$. From this the result follows.

Lemma 4 For θ_c = (x₀₁, y₀₁, x₀₂, y₀₂) ∈ Θ_c, let K₁₂(θ_c) be given by eq. 12 and for i = 1, 2 let Q_i be given by eq. 20. Then for θ_c ∈ Θ_c and i, j = 1, 2, we have

$${[{\mathbf{K}}_{12}({\theta }_c)]}_{\mathit{\text{ij}}}\le \sqrt{{[{\mathbf{Q}}_1]}_{\mathit{\text{ii}}}{[{\mathbf{Q}}_2]}_{\mathit{\text{jj}}}}<\mathrm{\infty }.$$

Proof Define Δ_x = x₀₂ − x₀₁ and Δ_y = y₀₂ − y₀₁. Applying the Cauchy-Schwarz inequality to the term [K₁₂(θ_c)]₁₁ and using the fact that Λ₁,Λ₂, q₁, q₂ ≥ 0, we have for θ_c ∈ Θ_c

$${[{\mathbf{K}}_{12}({\theta }_c)]}_{11}={\displaystyle {\int }_{t_0}^t}{\displaystyle {\int }_{{ℝ}^2}}\frac{{\Lambda }_1(\tau ){\Lambda }_2(\tau )}{{\Lambda }_1(\tau )q_1(x,y)+{\Lambda }_2(\tau )q_2(x-{\Delta }_x,y-{\Delta }_y)}\times \frac{\partial q_1(x,y)}{\partial x}\frac{\partial q_2(x-{\Delta }_x,y-{\Delta }_y)}{\partial x}\mathit{\text{dxdyd}}\tau \le {\left({\displaystyle {\int }_{t_0}^t}{\displaystyle {\int }_{{ℝ}^2}}\frac{{\Lambda }_1(\tau ){\Lambda }_2(\tau )}{{\Lambda }_1(\tau )q_1(x,y)+{\Lambda }_2(\tau )q_2(x-{\Delta }_x,y-{\Delta }_y)}{\left(\frac{\partial q_1(x,y)}{\partial x}\right)}^2\mathit{\text{dxdyd}}\tau \right)}^{\frac{1}{2}}\times {\left({\displaystyle {\int }_{t_0}^t}{\displaystyle {\int }_{{ℝ}^2}}\frac{{\Lambda }_1(\tau ){\Lambda }_2(\tau )}{{\Lambda }_1(\tau )q_1(x,y)+{\Lambda }_2(\tau )q_2(x-{\Delta }_x,y-{\Delta }_y)}{\left(\frac{\partial q_2(x-{\Delta }_x,y-{\Delta }_y)}{\partial x}\right)}^2\mathit{\text{dxdyd}}\tau \right)}^{\frac{1}{2}}\le {({\displaystyle {\int }_{t_0}^t}{\Lambda }_2(\tau )d\tau )}^{\frac{1}{2}}{\left({\displaystyle {\int }_{{ℝ}^2}}\frac{1}{q_1(x,y)}{\left(\frac{\partial q_1(x,y)}{\partial x}\right)}^2\mathit{\text{dxdyd}}\tau \right)}^{\frac{1}{2}}\times {({\displaystyle {\int }_{t_0}^t}{\Lambda }_1(\tau )d\tau )}^{\frac{1}{2}}{\left({\displaystyle {\int }_{{ℝ}^2}}\frac{1}{q_2(x-{\Delta }_x,y-{\Delta }_y)}{\left(\frac{\partial q_2(x-{\Delta }_x,y-{\Delta }_y)}{\partial x}\right)}^2\mathit{\text{dxdy}}\right)}^{\frac{1}{2}}={({\displaystyle {\int }_{t_0}^t}{\Lambda }_1(\tau )d\tau {\displaystyle {\int }_{{ℝ}^2}}\frac{1}{q_1(x,y)}{\left(\frac{\partial q_1(x,y)}{\partial x}\right)}^2\mathit{\text{dxdy}})}^{\frac{1}{2}}\times {({\displaystyle {\int }_{t_0}^t}{\Lambda }_2(\tau )d\tau {\displaystyle {\int }_{{ℝ}^2}}\frac{1}{q_2(x,y)}{\left(\frac{\partial q_2(x,y)}{\partial x}\right)}^2\mathit{\text{dxdy}})}^{\frac{1}{2}}=\sqrt{{[{\mathbf{Q}}_1]}_{11}{[{\mathbf{Q}}_2]}_{22}}<\mathrm{\infty },$$

where we have used the shift invariant property of Lebesgue integrals in the penultimate step, and we have used the properties of image functions (see definition 6) in the last step. Similarly, we can prove the other results.

A.3 Proof of Theorem 3

Proof Consider the term K₁₁(θ_c) given in eq. 12. By definition, the integral expression of K₁₁(θ_c) is measurable for every θ_c ∈ Θ_c. Define Δ_x ≔ x₀₂ − x₀₁ and Δ_y ≔ y₀₂ − y₀₁. Using the shift invariant property of Lebesgue integrals, and the fact that q_i(x, y) ≥ 0 and Λ_i(τ) ≥ 0 for i = 1, 2, (x, y) ∈ ℝ² and τ ≥ t₀, we have for θ_c ∈ Θ_c

$${\mathbf{K}}_{11}({\theta }_c)≔{\displaystyle {\int }_{t_0}^t}{\displaystyle {\int }_{{ℝ}^2}}\frac{{\Lambda }_1^2(\tau )}{{\Lambda }_1(\tau )q_1(x,y)+{\Lambda }_2(\tau )q_2(x-{\Delta }_x,y-{\Delta }_y)}\times \left(\begin{array}{cc}\hfill {\left(\frac{\partial q_1(x,y)}{\partial x}\right)}^2\hfill & \hfill \frac{\partial q_1(x,y)}{\partial x}\frac{\partial q_1(x,y)}{\partial y}\hfill \\ \hfill \frac{\partial q_1(x,y)}{\partial x}\frac{\partial q_1(x,y)}{\partial y}\hfill & \hfill {\left(\frac{\partial q_1(x,y)}{\partial y}\right)}^2\hfill \end{array}\right)\mathit{\text{ dxdyd}}\tau \le {\displaystyle {\int }_{t_0}^t}{\displaystyle {\int }_{{ℝ}^2}}\frac{{\Lambda }_1^2(\tau )}{{\Lambda }_1(\tau )q_1(x,y)}\left(\begin{array}{cc}\hfill {\left(\frac{\partial q_1(x,y)}{\partial x}\right)}^2\hfill & \hfill \frac{\partial q_1(x,y)}{\partial x}\frac{\partial q_1(x,y)}{\partial y}\hfill \\ \hfill \frac{\partial q_1(x,y)}{\partial x}\frac{\partial q_1(x,y)}{\partial y}\hfill & \hfill {\left(\frac{\partial q_1(x,y)}{\partial y}\right)}^2\hfill \end{array}\right)\mathit{\text{ dxdyd}}\tau ={\mathbf{Q}}_1.$$ (53)

By definition of the image function (see Definition 6), we have for ζ₁ = x and ζ₂ = y, ${\displaystyle {\int }_{{ℝ}^2}}\frac{1}{q_1(x,y)}\frac{\partial q_1(x,y)}{\partial {\zeta }_i}\frac{\partial q_1(x,y)}{\partial {\zeta }_j}\mathit{\text{dxdy}}<\mathrm{\infty }$ for i, j = 1, 2. This implies that K₁₁(θ_c) is dominated by the expression given in eq. 53 for every θ_c ∈ Θ_c. By definition of the image function (see Definition 6), q₁(x, y) and $\frac{\partial q_1(x,y)}{\partial x}$ are continuous for every x ∈ ℝ. Hence the integrand of K₁₁(θ_c) is continuous for every x ∈ ℝ. Hence by using the Theorem on changing integration and limits for Lebesgue integrals (see (Apostol, 1974, pg 281)), we have

$$\underset{x_{02}\to \mathrm{\infty }}{\text{lim}}{\mathbf{ K}}_{11}({\theta }_c)=\underset{x_{02}\to \mathrm{\infty }}{\text{lim}}{\displaystyle {\int }_{t_0}^t}{\displaystyle {\int }_{{ℝ}^2}}\frac{{\Lambda }_1^2(\tau )}{{\Lambda }_1(\tau )q_1(x,y)+{\Lambda }_2(\tau )q_2(x-{\Delta }_x,y-{\Delta }_y)}\times \left(\begin{array}{cc}\hfill {\left(\frac{\partial q_1(x,y)}{\partial x}\right)}^2\hfill & \hfill \frac{\partial q_1(x,y)}{\partial x}\frac{\partial q_1(x,y)}{\partial y}\hfill \\ \hfill \frac{\partial q_1(x,y)}{\partial x}\frac{\partial q_1(x,y)}{\partial y}\hfill & \hfill {\left(\frac{\partial q_1(x,y)}{\partial y}\right)}^2\hfill \end{array}\right)\mathit{\text{ dxdyd}}\tau ={\displaystyle {\int }_{t_0}^t}{\displaystyle {\int }_{{ℝ}^2}}\underset{x_{02}\to \mathrm{\infty }}{\text{lim}}\frac{{\Lambda }_1^2(\tau )}{{\Lambda }_1(\tau )q_1(x,y)+{\Lambda }_2(\tau )q_2(x-{\Delta }_{x,}y-\Delta y)}\left(\begin{array}{cc}\hfill {\left(\frac{\partial q_1(x,y)}{\partial x}\right)}^2\hfill & \hfill \frac{\partial q_1(x,y)}{\partial x}\frac{\partial q_1(x,y)}{\partial y}\hfill \\ \hfill \frac{\partial q_1(x,y)}{\partial x}\frac{\partial q_1(x,y)}{\partial y}\hfill & \hfill {\left(\frac{\partial q_1(x,y)}{\partial y}\right)}^2\hfill \end{array}\right)\mathit{\text{ dxdyd}}\tau ={\displaystyle {\int }_{t_0}^t}{\Lambda }_1(\tau )d\tau {\displaystyle {\int }_{{ℝ}^2}}\frac{1}{q_1(x,y)}\left(\begin{array}{cc}\hfill {\left(\frac{\partial q_1(x,y)}{\partial x}\right)}^2\hfill & \hfill \frac{\partial q_1(x,y)}{\partial x}\frac{\partial q_1(x,y)}{\partial y}\hfill \\ \hfill \frac{\partial q_1(x,y)}{\partial x}\frac{\partial q_1(x,y)}{\partial y}\hfill & \hfill {\left(\frac{\partial q_1(x,y)}{\partial y}\right)}^2\hfill \end{array}\right)\mathit{\text{ dxdy}}={\mathbf{Q}}_1,$$

where we have used assumption A1 in the next to last step. Similarly, we can show that lim_x02→∞ K₂₂(θ_c) = Q₂. For the term K₁₂(θ_c), by definition, the integrand is measurable. Further by definition of the image function, the integrand of K₁₂(θ_c) is continuous for every x ∈ ℝ. From Lemma 4 (see Appendix) we know that the entries of K₁₂(θ_c) are dominated by integral expressions that are independent of θ_c ∈ θ_c and are bounded. Hence using the above results pertaining to K₁₂(θ_c) and assumptions A1 and A2, we apply the Theorem on changing integration and limits for Lebesgue integrals (see (Apostol, 1974, pg 281)) to obtain

$$\underset{x_{02}\to \mathrm{\infty }}{\text{lim}}{\mathbf{ K}}_{12}({\theta }_c)={\displaystyle {\int }_{t_0}^t}{\displaystyle {\int }_{{ℝ}^2}}\underset{x_{02}\to \mathrm{\infty }}{\text{lim}}\frac{{\Lambda }_1(\tau ){\Lambda }_2(\tau )}{{\Lambda }_1(\tau )q_1(x,y)+{\Lambda }_2(\tau )q_2(x-{\Delta }_x,y-{\Delta }_y)}\times \left(\begin{array}{cc}\hfill \frac{\partial q_1(x,y)}{\partial x}\frac{\partial q_2(x-{\Delta }_x,y-{\Delta }_y)}{\partial x}\hfill & \hfill \frac{\partial q_1(x,y)}{\partial x}\frac{\partial q_2(x-{\Delta }_x,y-{\Delta }_y)}{\partial y}\hfill \\ \hfill \frac{\partial q_1(x,y)}{\partial y}\frac{\partial q_2(x-{\Delta }_x,y-{\Delta }_y)}{\partial x}\hfill & \hfill \frac{\partial q_1(x,y)}{\partial y}\frac{\partial q_2(x-{\Delta }_x,y-{\Delta }_y)}{\partial y}\hfill \end{array}\right)\mathit{\text{ dxdyd}}\tau ={\displaystyle {\int }_{t_0}^t}{\displaystyle {\int }_{{ℝ}^2}}\frac{{\Lambda }_1(\tau ){\Lambda }_2(\tau )}{{\Lambda }_1(\tau )q_1(x,y)+0}\left(\begin{array}{cc}\hfill \frac{\partial q_1(x,y)}{\partial x}0\hfill & \hfill \frac{\partial q_1(x,y)}{\partial x}0\hfill \\ \hfill \frac{\partial q_1(x,y)}{\partial y}0\hfill & \hfill \frac{\partial q_1(x,y)}{\partial y}0\hfill \end{array}\right)\mathit{\text{ dxdyd}}\tau =0.$$

A.4 Proof of Theorem 4

Proof 1. The image detection processes 𝒢₁ and 𝒢₂, which describe the first and second images, respectively, are assumed to be statistically independent of each other. Hence the general expression for the Fisher information matrix can be written as

$${\mathbf{S}}_{\mathit{\text{sim}},\mathit{\text{sp}}}({\theta }_c)={\mathbf{S}}_{\mathit{\text{sim}},\mathit{\text{sp}},1}({\theta }_c)+{\mathbf{S}}_{\mathit{\text{sim}},\mathit{\text{sp}},2}({\theta }_c),{\theta }_c\in {\Theta }_c,$$

where S_sim,sp,1 and S_sim,sp,2 denote the Fisher information matrices corresponding to the image detection processes 𝒢₁ and 𝒢₂, respectively. In the present case, we assume without loss of generality that (x₀₁, y₀₁) to be the location coordinates that is determined from the first image. Then it immediately follows that S_sim,sp,1(θ_c) = Q₁ for θ_c ∈ Θ_c, where Q₁ denotes the Fisher information matrix for the localization accuracy problem corresponding to object 1 and is given by eq. 20.

To derive an expression for S_sim,sp,2(θ_c), we make use of the fact that for the second image the location coordinates (x₀₁, y₀₁) of object 1 can be assumed to be known a priori, since it is already determined from the first image. Hence for the second image only the location coordinates (x₀₂, y₀₂) of the second object are the unknown parameters. Hence from this it immediately follows that the expression for S_{sim, sp,2} (θ_c) will be identical to K₂₂(θ_c) which is a component of the Fisher information matrix S_sim(θ_c) for the problem of estimating θ_c when the location coordinates of both objects are unknown (Theorem 2).

2. To show that S_sim,sp(θ_c) is invertible, we require that ${\mathbf{Q}}_1^{-1}\text{ and }{\mathbf{K}}_{22}^{-1}({\theta }_c)$ exist for every θ_c ∈ Θ_c. We prove the result by contradiction. Define Δ_x ≔ x₀₁ − x₀₂ and Δ_y = y₀₁ − y₀₂. For θ_c ∈ Θ_c, consider the term

$${\mathbf{K}}_{22}({\theta }_c)={\displaystyle {\int }_{t_0}^t}{\displaystyle {\int }_{{ℝ}^2}}\frac{{\Lambda }_2^2(\tau )}{{\Lambda }_1(\tau )q_1(x-x_{01},y-y_{01})+{\Lambda }_2(\tau )q_2(x-x_{02},y-y_{02})}\times \left(\begin{array}{cc}\hfill {\left(\frac{\partial q_2(x-x_{02},y-y_{02})}{\partial x}\right)}^2\hfill & \hfill \frac{\partial q_2(x-x_{02},y-y_{02})}{\partial x}\frac{\partial q_2(x-x_{02},y-y_{02})}{\partial y}\hfill \\ \hfill \frac{\partial q_2(x-x_{02},y-y_{02})}{\partial x}\frac{\partial q_2(x-x_{02},y-y_{02})}{\partial y}\hfill & \hfill {\left(\frac{\partial q_2(x-x_{02},y-y_{02})}{\partial y}\right)}^2\hfill \end{array}\right)\mathit{\text{ dxdyd}}\tau ={\displaystyle {\int }_{t_0}^t}{\displaystyle {\int }_{{ℝ}^2}}\frac{{\Lambda }_2^2(\tau )}{{\Lambda }_1(\tau )q_1(x-{\Delta }_{x,y}-{\Delta }_y)+{\Lambda }_2(\tau )q_2(x,y)}\left(\begin{array}{cc}\hfill {\left(\frac{\partial q_2(x,y)}{\partial x}\right)}^2\hfill & \hfill \frac{\partial q_2(x,y)}{\partial x}\frac{\partial q_2(x,y)}{\partial y}\hfill \\ \hfill \frac{\partial q_2(x,y)}{\partial x}\frac{\partial q_2(x,y)}{\partial y}\hfill & \hfill {\left(\frac{\partial q_2(x,y)}{\partial y}\right)}^2\hfill \end{array}\right)\mathit{\text{ dxdyd}}\tau ={\displaystyle {\int }_{t_0}^t}{\displaystyle {\int }_{{ℝ}^2}}{h}_{{\theta }_c}(x,y,\tau )\left(\begin{array}{cc}\hfill {\left(\frac{\partial q_2(x,y)}{\partial x}\right)}^2\hfill & \hfill \frac{\partial q_2(x,y)}{\partial x}\frac{\partial q_2(x,y)}{\partial y}\hfill \\ \hfill \frac{\partial q_2(x,y)}{\partial x}\frac{\partial q_2(x,y)}{\partial y}\hfill & \hfill {\left(\frac{\partial q_2(x,y)}{\partial y}\right)}^2\hfill \end{array}\right)\mathit{\text{ dxdyd}}\tau ,$$ (54)

where for θ_c ∈ Θ_c,

$$h_{{\theta }_c}(x,y,\tau )≔\frac{{\Lambda }_2^2(\tau )}{{\Lambda }_1(\tau )q_1(x-{\Delta }_x,y-{\Delta }_y)+{\Lambda }_2(\tau )q_2(x,y)},(x,y)\in {ℝ}^2,\tau \ge t_0.$$

Assume that there exists an image function q₂ such that the Fisher information matrix K₂₂(θ_c) is singular for θ_c ∈ Θ_c. Hence by eq. 54, it immediately follow that

$$\text{Det}[{\mathbf{K}}_{22}({\theta }_c)]={\displaystyle {\int }_{t_0}^t}{\displaystyle {\int }_{{ℝ}^2}}{h}_{{\theta }_c}(x,y,\tau ){\left(\frac{\partial q_2(x,y)}{\partial x}\right)}^2\mathit{\text{ dxdy}}{\displaystyle {\int }_{t_0}^t}{\displaystyle {\int }_{{ℝ}^2}}{h}_{{\theta }_c}(x,y,\tau ){\left(\frac{\partial q_2(x,y)}{\partial y}\right)}^2\mathit{\text{ dxdy}}-{\left(\underset{T_2}{\underbrace{{\displaystyle {\int }_{t_0}^t}{\displaystyle {\int }_{{ℝ}^2}}{h}_{{\theta }_c}(x,y,\tau )\frac{\partial q_2(x,y)}{\partial x}\frac{\partial q_2(x,y)}{\partial y}\mathit{\text{dxdy}}}}\right)}^2=0,{\theta }_c\in {\Theta }_c.$$

Note that the above expression pertains to the limiting case of equality of the Cauchy-Schwarz inequality applied to the term T₂. Hence by applying the condition for equality, we have for k ≠ 0

$$\frac{\partial q_2(x,y)}{\partial x}-k\frac{\partial q_2(x,y)}{\partial y}=0,(x,y)\in {ℝ}^2.$$ (55)

The above equation is analogous to the classical one-dimensional transport equation whose solutions are given by ((Strauss, 1992, pg 6–7))

$$q_2(x,y)=F(x+\frac{y}{k}),(x,y)\in {ℝ}^2,$$

where F is defined on ℝ. As q₂ is an image function satisfying the regularity conditions, we know that q₂ is continuous on ℝ². Hence it follows that F is also continuous on ℝ. Further, q₂(x, y) ≥ 0, (x, y) ∈ ℝ² and hence F (x) ≥ 0, x ∈ ℝ. This implies that there exists a constant K > 0 and a finite interval ℐ = (a, b) ⊂ ℝ such that F (x) ≥ K, x ∈ ℐ. Making use of the fact that ∫ℝ² q₂(x, y)dxdy = 1 (since q₂ is an image function) and substituting for q₂ in terms of F, we have

$$1={\displaystyle {\int }_{{ℝ}^2}}{q}_2(x,y)\mathit{\text{dxdy}}={\displaystyle {\int }_{{ℝ}^2}}F(x+\frac{y}{k})\mathit{\text{dxdy}}={\displaystyle {\int }_{ℝ}}({\displaystyle {\int }_{ℝ}}F(x+\frac{y}{k})\mathit{\text{dx}})\mathit{\text{ dy}}={\displaystyle {\int }_{ℝ}}({\displaystyle {\int }_{ℐ}}F(x+\frac{y}{k})\mathit{\text{dx}}+{\displaystyle {\int }_{ℐ\backslash ℝ}}F(x+\frac{y}{k})\mathit{\text{dx}})\mathit{\text{ dy}}\ge {\displaystyle {\int }_{ℝ}}({\displaystyle {\int }_{ℐ}}\mathit{\text{Kdx}}+{\displaystyle {\int }_{ℐ\backslash ℝ}}F(x+\frac{y}{k})\mathit{\text{dx}})\mathit{\text{ dy}}=K(b-a){\displaystyle {\int }_{ℝ}}\mathit{\text{ dy}}+{\displaystyle {\int }_{ℝ}}({\displaystyle {\int }_{ℐ\backslash ℝ}}F(x+\frac{y}{k})\mathit{\text{dx}})\mathit{\text{ dy}}=\mathrm{\infty },$$

which is a contradiction. Hence K₂₂(θ_c) is invertible for θ_c ∈ θ_c. Similarly we can show that Q₁ is also invertible. From this the result follows.