Sparse Signal Recovery Methods for Multiplexing PET Detector Readout

Abstract

Nuclear medicine imaging detectors are commonly multiplexed to reduce the number of readout channels. Because the underlying detector signals have a sparse representation, sparse recovery methods such as compressed sensing may be used to develop new multiplexing schemes. Random methods may be used to create sensing matrices that satisfy the restricted isometry property. However, the restricted isometry property provides little guidance for developing multiplexing networks with good signal-to-noise recovery capability. In this work, we describe compressed sensing using a maximum likelihood framework and develop a new method for constructing multiplexing (sensing) matrices that can recover signals more accurately in a mean square error sense compared to sensing matrices constructed by random construction methods. Signals can then be recovered by maximum likelihood estimation constrained to the support recovered by either greedy ℓ₀ iterative algorithms or ℓ₁-norm minimization techniques. We show that this new method for constructing and decoding sensing matrices recovers signals with 4%–110% higher SNR than random Gaussian sensing matrices, up to 100% higher SNR than partial DCT sensing matrices 50%–2400% higher SNR than cross-strip multiplexing, and 22%–210% higher SNR than Anger multiplexing for photoelectric events.

I. Introduction

The anger camera [1] and cross strip multiplexing [2] are two common methods for spatially multiplexing the readouts of positron emission tomography (PET) detectors where each readout channel is a linearly weighted sum of the signals from the detector pixels. However, there is no evidence that these multiplexing methods maximize the signal-to-noise ratio (SNR) of the decoded signals. Are there better choices for linearly-weighted multiplexing?

Sparse signal recovery theory can be used to develop an alternative linearly-weighted multiplexing method for PET detector For signals that have a sparse representation with respect to some basis, compressed sensing [3], [4] employs non-adaptive linear projections that allow a signal to be recovered at a lower sampling rate than suggested by Nyquist-Shannon theory. A sampling function or sensing matrix is used to sample the signal and the signal can then be recovered by ℓ₀-norm or ℓ₁-norm minimization.

PET systems are often constructed from “block” detectors, 2-D imaging arrays. To reduce cost, the electrical readouts of the block detectors typically employ spatial multiplexing to position events. The sensitivity and spatial resolution of PET systems can be improved by increasing the number of crystal elements and detector pixels. Consequently, it is important to develop better techniques for spatially multiplexing detector readouts such that these performance gains can be achieved at reasonable cost.

Because of the low gamma photon event rate in PET, the detector images recorded by each block detector are sparse both spatially and temporally. For a time sampled readout system, there are generally no more than one photon event recorded in any detector block in one readout sampling interval. When two photon events are recorded in a detector block in the same sampling interval, the resulting pile up event is typically lost. For intercrystal scatter, each photon can interact in multiple crystal elements. However, it is unlikely for a photon to interact with more than three detector crystal elements within the typical time window used for PET [5]. The sparseness of the signal is defined as the number of simultaneous crystal elements hit within one sampling period of the digitizer.

Gamma camera designs exploit the sparseness of the signal on the crystal array for any instance of time to spatially multiplex the readout of detectors by using the distance-weighted sum of the photodetectors signals and Anger logic [1] to decode the signal. In this approach, an array of crystal elements is read out using only four channels (e.g., m = 4). Each channel corresponds to a corner of the 2-D imaging array with each pixel contributing signal magnitude that is attenuated by its distance to the channel. The Anger decoder can be used to recover the position and magnitude of the crystal hit with a significant reduction in readout cost. For example, a large crystal array (such as 13 × 13) can be readout by four channels.

A schematic of a new proposed multiplexing method for crystal elements coupled one-to-one to photodetectors such as silicon photomultipliers (SIPM) is shown in Fig. 1. A pre-amplifier is used to provide gain for each photodetector before a multiplexing network takes linear combinations of the analog pixel signals and combines them into a fewer number of analog channels before digitization. Multiplexing may comprise resistive or capacitive networks that reduce the number of expensive analog-to-digital converters (ADC) that are needed to digitize the signals. Time is picked off using time-to-digital converters placed on each of the readout channels, allowing a lower sampling rate such as 100 MHz (100 ns sampling interval) to be used.

Fig. 1.

Schematic for multiplexing PET detector readout channels is shown. Scintillator crystal elements are coupled one-to-one to photodetector pixels that are in turn coupled to preamplifiers and an electronic multiplexing network linearly combines the output from the preamplifiers that are then sampled by ADC converters. The multiplexing network is an analog network that may be a resistive or capacitive network that allows data to be collected with fewer digitizing channels than detector pixels, reducing sampling readout costs.

The key development presented here are spatial multiplexing methods that have the ability to resolve the position and energies of interactions that occur in multiple crystal elements within the same time sampling interval. The detector signals are discretized in both the spatial and time domain. The number of spatial positions that can be estimated in the detector block cannot exceed the number of detector pixels.

In [6], a mathematical model describing readout spatial multiplexing for this type of PET detector was described as an undetermined linear system of equations mapping the crystal elements to a set of readout channels at one time point. When a photon interacts in the 2-D array of crystal elements, pulses (time-domain signals) are produced in one or more photodetector pixels. Let the noiseless signal at one instance of time or sampling interval for the individual pixels of the photodetectors in a block be denoted by the d by 1 vector x. Noise is also present at each pixel and is denoted by the d by 1 vector n_x.Let the detector readout, or multiplexed channels, be denoted by the m by 1 vector y. The detector readout can be described simply as

$$\mathbf{y}=\mathbf{C}\left(\mathbf{x}+{\mathbf{n}}_{\mathbf{x}}\right)+\mathbf{e}$$ (1)

where the matrix C describes the multiplexing network that maps the d detector pixels to the m readout channels where d > m and e is an additional random measurement noise vector produced by the multiplexing electronics. Each readout is a linearly weighted sum of the photodetector pixels with weights described by the matrix C. The matrix $\mathbf{C}=\left[\begin{array}{lll}{\mathbf{c}}_1& \cdots & {\mathbf{c}}_d\end{array}\right]$ will also be referred to as the dictionary with column vectors c_i that will be called the codes. A p-subdictionary is a matrix formed by using p codes of the dictionary. This system model is widely applicable to medical imaging applications besides PET such as SPECT and X-ray CT. The objective is then to recover the original position and magnitudes of detected events on the image array x by decoding the multiplexed readouts y at each time point or sampling interval. In this work, the decoder is implemented digitally, which requires that the multiplexed signals y be time sampled by ADCs and then decoded. Since the spatial multiplexing is an underdetermined system of equations, unique recovery is possible only if the underlying signal x is sparse.

In the Anger camera, each photon is positioned at the energy-weighted centroid of all the individual interactions resulting in a loss of spatial resolution when intercrystal scattering occurs. Recent advances in sparse recovery theory, such as compressed sensing, can be used to design alternatives to the weighing matrices C used by the Anger camera and cross strip multiplexing. Unlike an Anger camera, compressed sensing has the capability to resolve the energy and position of each interaction of a multiple interaction event that occur simultaneously due to intercrystal scattering. This capability can be utilized to fully exploit the potential of emerging 3-D PET detectors [7]–[9].

In this work, we develop a mathematical model for describing spatial multiplexing readouts in a sparse signal recovery framework. Within this framework, commonly used multiplexing methods such as Anger and cross-strip multiplexing can be viewed as just two of many possible linearly weighted coding schemes for compressing the spatial information from the photodetector pixels into a set of readout channels. However, compressed sensing methods such as ℓ₀-norm and ℓ₁-norm minimization of random sensing matrices are not sufficiently noise robust for PET. Therefore, we investigate a method for improving the ℓ₀-norm minimization decoder and for constructing sensing matrices with better SNR for PET detectors compared to other compressed sensing, Anger multiplexing, and cross-strip multiplexing. We performed a simulation study of a detector design that we are developing as a simple demonstration of the multiplexing methods presented in this work.

The new multiplexing method can be used to develop novel photodetector array devices. As such, it could be used in a wide range of imaging applications including PET, SPECT, and other modalities that require single photon counting capability such as lifetime fluorescence imaging. We also provide a brief description of how this multiplexing readout method can be practically integrated into a photodetector array.

II. Compressed Sensing Theory

In this section, we describe how compressed sensing can be applied to the spatial multiplexing problem. We describe how to handle light sharing, review how to design the weights of a linear multiplexing network, and review algorithms for decoding the positions and amplitudes of the signal.

For compressed sensing, a sparse basis must be found for the signal vector. For a gamma camera, the sparsifying function is the light-sharing function. Compressed sensing requires that the signals be discretized both spatially and temporally. If light sharing occurs between scintillation array crystal elements, the light-sharing function must be digitized as a light-sharing matrix $\mathbf{\Psi }$, which maps the light from each crystal element to the photodetector pixels. For PET, $\mathbf{z}=\mathbf{\Psi }\mathbf{x}$ is a sparse vector for each time sample interval, e.g., has many zero coefficients. If we let ${\mathbf{n}}_{\mathbf{z}}=\mathbf{\Psi }{\mathbf{n}}_{\mathbf{x}}$, then (1) becomes

$$\mathbf{y}=\mathbf{C}{\mathbf{\Psi }}^{-1}\left(\mathbf{z}+{\mathbf{n}}_{\mathbf{z}}\right)+\mathbf{e}.$$ (2)

The multiplexing matrix is then given by

$$\mathbf{\Phi }=\mathbf{C}{\mathbf{\Psi }}^{-1}$$

where C corresponds to the sensing matrix. Compressed sensing methods can be used if ${\mathbf{\Psi }}^{-1}$ exists.

We constructed a 4 × 4 array of 3 mm × 3 mm × 20 mm LYSO crystals coupled one-to-one to a photodetector array and measured the light sharing at each crystal position. Each crystal was wrapped with a reflector on all sides except one end of the crystal that was coupled by a thin layer of optical grease to the SIPM array of 3 mm × 3 mm pixels. Due to imperfect light reflectors, light sharing between adjacent crystals was measured. The measured light sharing at two crystal positions are shown in Fig. 2. When a photon interacts in one crystal element, light is detected in the corresponding detector pixel and the eight nearest neighbor pixels. The measured light-sharing matrix for the 4 × 4 array is shown on the left side of Fig. 3. This 16 × 16 matrix was found to well-conditioned and invertible. There was approximately 20% variance in the coefficients of the 16 measured PSFs.

Fig. 2.

Shown are experimental measurements of the light spread function at two positions of a 4 × 4 crystal array coupled one-to-one to SiPM photodetectors. The light spread was localized to the eight nearest neighbor detector pixels. Beyond this neighborhood, the amount of measured light was negligible.

Fig. 3.

(Left) Shown is the light sharing matrix corresponding to experimental measurements of a 4 × 4 crystal array coupled one-to-one to SiPM photodetectors. (Right) Shown is a light sharing matrix constructed for the simulation study.

A sufficient, but not necessary, condition for a sensing matrix C to be used to recover the signal is the “restricted isometry property” (RIP) [10]. The matrix C is an isometry if it preserves the length of a vector after matrix multiplication. If multiplication by C produces a vector with a length that is approximately the same (RIP condition), it has been shown that a sparse vector, one with only a few nonzero coefficients, can be recovered with high probability [4], [11]. For a sensing matrix that satisfies the RIP condition, ℓ₁-norm minimization can be used to decode the position and amplitudes of the crystal element interactions, that is

$${\stackrel{\mathbf{ˆ}}{\mathbf{x}}}_1=\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{i}\mathrm{n}{‖{\mathbf{x}}^{\prime }‖}_1,\mathbf{y}=\mathbf{C}{\mathbf{x}}^{\prime }.$$ (3)

Different variations of objective functions based on ℓ₁-norm minimization have also been proposed [11]–[15]. It has been shown that randomly sampling certain distributions can produce sensing matrices that satisfy the RIP condition. Random Gaussian spherical codes, partial Fourier matrices or partial discrete cosine transform (DCT) matrices are possible choices for sensing matrices for multiplexing PET detectors [11], [16].

The minimum ℓ₀-norm solution

$${\stackrel{\mathbf{ˆ}}{\mathbf{x}}}_0=\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{i}\mathrm{n}{‖{\mathbf{x}}^{\prime }‖}_0,\mathbf{y}=\mathbf{C}{\mathbf{x}}^{\prime }$$ (4)

is the solution with the fewest nonzero coefficients and this solution can recover signals from sensing matrices with a larger RIP bound than the minimum ℓ₁-norm solution. Computationally efficient greedy iterative algorithms for ℓ₀-norm minimization have been proposed that include compressed sampling matching pursuit (CoSaMP) [17], subspace pursuit (SP) [18], and iterative hard thresholding (IHT) [19]–[21]. These three algorithms have previously been compared [22]. These algorithms all introduce a regularization constraint to improve signal recovery under noisy conditions similar to that used by basis pursuit denoising [23] where

$${\stackrel{\mathbf{ˆ}}{\mathbf{x}}}_0=\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{i}\mathrm{n}\left(\mathrm{\lambda }{‖{\mathbf{x}}^{\prime }‖}_0+{‖\mathbf{y}-{\mathbf{C}\mathbf{x}}^{\prime }‖}_2\right)$$ (5)

is minimized instead of (4).

For an undetermined system of equations, the pseudoinverse of C does not provide a satisfactory solution. However, if the support can be constrained such that k = supp(x) ≤ m, then a pseudoinverse ${\mathbf{C}}_k^{+}$ can be applied to provide a good estimate of the signal $\stackrel{\mathbf{ˆ}}{\mathbf{x}}={\mathbf{C}}_k^{+}\mathbf{y}$. CoSaMP is a support recovery algorithm where the support is estimated iteratively. At each iteration, C^T is applied to the residual and the largest k = 2s ≤ m coefficients are added to the support from the previous iteration. The estimate $\stackrel{\mathbf{ˆ}}{\mathbf{x}}$ is then computed by retaining the $s$ largest coefficients after applying the pseudoinverse ${\mathbf{C}}_k^{+}$ while all other coefficient are set to zero. The residual is then updated by subtracting $\mathbf{C}\stackrel{\mathbf{ˆ}}{\mathbf{x}}$.

SP operates very similarly to CoSaMP. At each iteration, C^T is applied to the residual and the largest s coefficients are added to the support from the previous iteration. SP uses a pseudoinverse over a support that is generally smaller than CoSaMP. To estimate $\widehat{\mathbf{x}}$, the pseudoinverse on this support set is applied to y and then the support is constrained to the largest s coefficients before a second pseudoinverse operation is used to update the residual. Therefore, SP uses two pseudoinverse operations per iteration unlike CoSaMP, which uses one pseudoinverse operation per iteration. However, the pseudoinverse operations are smaller in dimensions that CoSaMP. Both CoSaMP and SP can be accelerated by replacing the pseudoinverse operations with a few iterations of a fast iterative algorithm such as the conjugate gradient algorithm.

IHT is an iterative operation that does not employ the pseudoinverse operation. It is essentially a gradient descent algorithm with all but the largest s coefficients of $\widehat{\mathbf{x}}$ set to zero after every iteration. This algorithm is a nonlinear operator applied to Landweber iterations [21]. The computational cost of an IHT iteration is significantly less than that of CoSaMP and SP. However, more iterations are required to reach an acceptable level of convergence.

III. Approach

In this section, we describe new methods for improving the noise robustness of compressed sensing. In Section III-A, we propose to use the existing decoders to position events but, not to recover the signal amplitude by maximum likelihood estimation or weighted least squares estimation. In Section III-B, we propose an alternative to the RIP condition for designing the weights of a linear multiplexing network.

A. Multiplexing Decoder

The standard compressed sensing decoders based on ℓ₀-norm minimization and ℓ₁-norm minimization perform poorly in the presence of noise. Therefore, instead of using ℓ₀-norm minimization (4) to estimate both the position and magnitudes of the signal, we propose using (4) by any of the ℓ₀-norm minimization algorithms such as CoSaMP, SP, or IHT to estimate the support of the signal, e.g., the position of each interaction, followed by maximum likelihood estimation (MLE) to recover the signal magnitude, e.g., the energy of each interaction. Once the interactions have been positioned, the multiplexing (1) reduces to an overdetermined system of equations that can be easily solved by MLE or weighted least squares (WLS).

Let $S=\left\{S_i\right\}=\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\left({\stackrel{\mathbf{ˆ}}{\mathbf{x}}}_1\right)$, for i = 1 … p. The support of x, the indices corresponding to the nonzero pixels of the detector image, can also be found by using ℓ₁-norm minimization instead of ℓ₀-norm minimization algorithms. Using the solution from (4), a permutation matrix that indicates the nonzero pixels is formed as

$$P=\left[e_{j(i)}\right],e_{j(i)}=\left\{\begin{array}{ll}j(i)=S_i,& i\le p\\ j(i)=S_{i-p}^C,& p<i\le d\end{array}\right.$$

where e_j(i) is the j(i)th elementary vector, a vector with the value 1 at the j(i)th entry if i ≤ p and zero everywhere else. The permutation matrix is an orthogonal matrix and therefore (1) can be rearranged into an overdetermined system of linear equations

$$\begin{gathered} \mathbf{y}=\left[\begin{array}{ll}\mathbf{A}& \mathbf{B}\end{array}\right]\left[\begin{array}{c}\mathbf{s}\\ \mathbf{0}\end{array}\right]+{\mathbf{C}\mathbf{n}}_{\mathbf{x}}+\mathbf{e} \\ C{P}^T=\left[\begin{array}{ll}\mathbf{A}& \mathbf{B}\end{array}\right],\mathbf{P}\mathbf{x}=\left[\begin{array}{c}\mathbf{s}\\ \mathbf{0}\end{array}\right] \end{gathered}$$ (6)

where s represents the p by 1 sparse signal vector with p ≤ s. We assume that the signal contribution within the components of S^C is zero. The pixel noise is denoted by the d by 1 detector noise pixel vector n_x. The m by P matrix A is the P-subdictionary that maps the signal vector s to the measurements y while the remainder of the dictionary is denoted by B, an m by d – p matrix.

Let n_x and e be Gaussian distributed random noise with covariance matrices R_n and R_e, respectively. Let C satisfy the restricted isometry property and ${\stackrel{\mathbf{ˆ}}{\mathbf{x}}}_1$ be the solution to (3) or (4). If $\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\mathbf{x})=\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\left({\stackrel{\mathbf{ˆ}}{\mathbf{x}}}_1\right)$, then it follows that

$$\begin{gathered} \mathbf{s}={\left({\mathbf{A}}^T{\mathbf{R}}^{-1}\mathbf{A}\right)}^{-1}{\mathbf{A}}^T\mathbf{y} \\ \mathbf{R}={\mathbf{C}\mathbf{R}}_n{\mathbf{C}}^T+{\mathbf{R}}_e \end{gathered}$$ (7)

yields the optimal minimum mean square error estimate [24] of the pixel magnitude. All pixels outside of the support S are then estimated to have a magnitude of zero.

For large dimensional matrices or non-Gaussian distributed noise, the conjugate gradient method could be used to iteratively approximate the maximum likelihood solution instead of the direct solution shown in (7).

It is also possible to directly incorporate MLE into ℓ₀-norm and ℓ₁-norm minimization. CoSaMP and SP are support recovery algorithms that use the pseudoinverse to estimate the support and the solution. The ML estimator could be used to replace the pseudoinverse operation in CoSaMP and SP. IHT uses iterations built around a least squares gradient descent. This could be replaced with a gradient descent that maximizes the likelihood function. However, this approach was not pursued in this study.

MLE can be used to recover the signal when $S=\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\mathbf{x})\subset \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\stackrel{\mathbf{ˆ}}{\mathbf{x}})$. In this case, [24]

$$E\left[(\stackrel{\mathbf{ˆ}}{\mathbf{x}}-\mathbf{x}{)}^T(\stackrel{\mathbf{ˆ}}{\mathbf{x}}-\mathbf{x})\mid S\right]=\mathrm{t}\mathrm{r}{\left({\mathbf{A}}_S^T{\mathbf{R}}_S^{-1}{\mathbf{A}}_S\right)}^{-1}$$ (8)

where A_S corresponds to the subdictionary for S and R_S is the noise covariance matrix of A_S.

B. Spherical Codes Sensing Matrix

Matrices that satisfy the RIP condition (Definition 2.2) are so common that they can be generated by random methods. The RIP condition is so broad, that it includes many matrices that perform poorly in the presence of noise and therefore, it does not provide much practical guidance for designing good sensing matrices. A better criterion is needed. A logical objective is to design a sensing matrix that minimizes the mean square error of the decoded signals. When $S=\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\mathbf{x})\subset \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\stackrel{\mathbf{ˆ}}{\mathbf{x}})$, the mean square error (8) can also be expressed as

$$E\left[(\stackrel{\mathbf{ˆ}}{\mathbf{x}}-\mathbf{x}{)}^T(\stackrel{\mathbf{ˆ}}{\mathbf{x}}-\mathbf{x})\mid S\right]=\mathrm{t}\mathrm{r}{\left({\mathbf{A}}_S^T{\mathbf{R}}_S^{-1}{\mathbf{A}}_S\right)}^{-1}=\sum {\mathrm{\lambda }}_{i,S}$$

where λ_i,S are the eigenvalues of ${\left({\mathbf{A}}_S^T{\mathbf{R}}_S^{-1}{\mathbf{A}}_S\right)}^{-1}$. Therefore, the best mean square error would be achieved by minimizing the condition number of the matrices ${\mathbf{A}}_S^T{\mathbf{R}}_S^{-1}{\mathbf{A}}_S$ over all possible subdictionaries A_S. Given that

$${‖{\mathbf{A}}_S^T{\mathbf{R}}_S^{-1}{\mathbf{A}}_S‖}_2\le {‖{\mathbf{A}}_S‖}_2^2{‖{\mathbf{R}}_S^{-1}‖}_2$$

the condition number can be minimized by minimizing ${‖{\mathbf{A}}_S‖}_2$ and ${‖{\mathbf{R}}_S^{-1}‖}_2$. If we then assume that ${\mathbf{R}}_e$ is negligible in ${\mathbf{R}}_s$ and the pixel noise covariance ${\mathbf{R}}_n={\sigma }_n\mathbf{I}$, then the condition number of $\mathbf{R}\approx {\sigma }_n{\mathbf{C}\mathbf{C}}^T$ can be minimized by choosing the codes of $\mathbf{C}$ that are as close to orthogonal as possible since the eigenvalues of an orthogonal matrix all have magnitude of 1. Since ${\mathbf{A}}_S$ is a submatrix of $\mathbf{C}$, then ${‖{\mathbf{A}}_S‖}_2^2\le \Vert \mathbf{C}{\Vert }_2^2$ and choosing the most orthogonal codes will also minimize the condition number of A_S. Therefore, C is chosen to minimize

$$g(\mathbf{C})=\underset{i,j}{\mathrm{m}\mathrm{a}\mathrm{x}}{\mathbf{c}}_i^T{\mathbf{c}}_j$$ (9)

which is equivalent to choosing codes that minimize the potential energy function

$$U(\mathbf{C})=\sum _i\sum _{j\ne i}\frac{1}{r(\mathbf{x})\left({\mathbf{c}}_i-{\mathbf{c}}_j\right)}$$ (10)

where r(x) is the length of the vector x. A dictionary that satisfies this criterion will henceforth be referred to as the minimum potential dictionary or minimum potential spherical codes (MPSC).

Minimizing the inner product between codes can also improve support estimation. Each code linearly maps a pixel to the readout channels. The magnitude of the inner product between two codes is proportional to the crosstalk between the corresponding pixels. With noise at each pixel, crosstalk can lead to support estimation errors, e.g., positioning errors from multiplexing.

A simple steepest descent algorithm can be used to produce the MPSC:

Algorithm 3.1.

MPSC Algorithm

Initialize: (Generate a random spherical code)
For each $i=1\dots d$
Set ci to a m-dimensional Gaussian random variable $${\mathbf{c}}_i=\frac{{\mathbf{c}}_i}{‖{\mathbf{c}}_i‖}.$$
Repeat:
For each i = 1 … d, $$\begin{gathered} {\mathbf{d}}_i=\sum _j\frac{1}{{\left({\mathbf{c}}_i-{\mathbf{c}}_j\right)}^T\left({\mathbf{c}}_i-{\mathbf{c}}_j\right)}\left({\mathbf{c}}_i-{\mathbf{c}}_j\right) \\ +\sum _j\frac{1}{{\left({\mathbf{c}}_i+{\mathbf{c}}_j\right)}^T\left({\mathbf{c}}_i+{\mathbf{c}}_j\right)}\left({\mathbf{c}}_i+{\mathbf{c}}_j\right) \\ {\mathbf{d}}_i=\frac{{\mathbf{d}}_i}{‖{\mathbf{d}}_i‖} \\ \mathrm{\Delta }{\mathbf{c}}_i=\gamma \left({\mathbf{d}}_i-{\mathbf{c}}_i^T{\mathbf{d}}_i{\mathbf{c}}_i\right) \end{gathered}$$
For each i = 1 … d, $$\begin{gathered} {\mathbf{c}}_i={\mathbf{c}}_i+\mathrm{\Delta }{\mathbf{c}}_i \\ {\mathbf{c}}_i=\frac{{\mathbf{c}}_i}{‖{\mathbf{c}}_i‖}. \end{gathered}$$

Until stop condition, such as when $\mathrm{m}\mathrm{a}\mathrm{x}\left(\mathrm{\Delta }{\mathbf{c}}_i\right)<\epsilon$

The value γ is a step-size parameter. This algorithm operates by simulating repulsive forces between the codes. Each code is treated as a dipole at position c_i and −c_i. If f_i is the net force applied to c_i and g_i is the net force applied to the opposite end of the dipole −c_i, the net force operating on c_i is f_i – g_i. The magnitude of the forces follows an inverse square law and the codes are constrained to lie on the unit m-sphere. We can easily impose a non-negativity constraint on the codes by choosing the end of the dipole that lies in the positive hemisphere, where all coordinates are nonnegative.

Better variations of this algorithm could be developed. However, for our application, the sizes of the matrices are reasonably small and the dictionary only needs to be computed once to determine the multiplexing topology. Therefore, the simple algorithm described here is sufficient for our needs.

IV. Numerical Studies

As an example of the application of the methods developed here, we performed simulations of one detector design using various multiplexing methods. The simulations performed here are intended only to demonstrate the potential and not to serve as a comprehensive evaluation of the multiplexing method. Partial Fourier matrices [3] and random spherical codes [11] have been suggested as useful sensing matrices. Therefore, we performed numerical studies to compare the MPSC sensing matrix against partial Fourier, random spherical codes, Anger multiplexing, and cross-strip sensing matrices for an 8 × 8 crystal array of 3 mm × 3 mm × 2 cm LYSO coupled one-to-one with 8 × 8 pixel array multiplexed to 16 channels (4:1 multiplexing ratio) that are digitized by analog-to-digital converters (ADC). We observed intercrystal scatter events in at least one detector for 50% of the simulated coincidence events.

We simulated a light-sharing matrix for the 8 × 8 crystal array using a simple method that approximates the PSF of measurements taken from the 4 × 4 crystal array described in Section II. Light produced in a crystal element was assumed to spread uniformly over a 4 mm × 4 mm square region. The light-sharing magnitude at each detector pixel corresponds to the intersection area between the 4 mm × 4 mm light region and the detector pixel. This produces a PSF similar to the measured PSF at all crystal positions including the boundary effects. Gaussian random noise was added to the PSF coefficients with variance equal to 20% of the coefficient value to create variation in the PSF coefficients that approximately matches the variation observed in the measured array described in Section II. The lightsharing matrix constructed by this method is shown on the right side of Fig. 3. This matrix was found to be invertible.

The positions and energies of photoelectric events and intercrystal scatter events on the detector was simulated using an in-house Monte Carlo simulation package called GRAY [25]. Like GATE (Geant4) [26], GRAY makes use of the NIST materials database and models the physics of high-energy photons interactions in different materials. GRAY has been compared with GATE (Geant4) and shown to produce statistically equivalent results for scattering and photoelectric interactions in common scintillator materials [25]. A point source was simulated at 200 cm from the detector to provide an approximately uniform distribution of 511 keV photons over the surface of the detector. We measured the signal-to-noise ratio for a 511 keV photon (SNR₅₁₁) by finding the amplitude of a 511 keV photon crystal interaction and the root mean square SIPM pixel noise for SIPM devices from SensL and Hamamatsu. We define SNR as

$${\mathrm{S}\mathrm{N}\mathrm{R}}_{511}=\frac{A_{511}^2}{A_{\text{noise}}^2}$$

where A₅₁₁ is the mean amplitude of a 511 keV photon and A_noise as the RMS amplitude of the pixel noise. Based on these measurements, we simulated white Gaussian noise at each SIPM pixel such that the SNR₅₁₁ was 30² (SensL SIPM), 100² (SNR between two devices), and 400² (Hamamatsu). ADC noise was added at the same noise level as the simulated SIPM device. Additional energy blurring was simulated by adding

$$\frac{0.12(.511)\sqrt{e+0.05}}{2.3548\sqrt{0.561}}n$$

where n is a Gaussian random variable with variance 1 and e corresponds to the energy of the photon in MeV. This simulates 12% FWHM energy blurring for 511 keV photons with square-root scaling at lower energies with the noise. A 250–770 keV and 450–570 keV energy windows were applied on the total detected energy to reject events that scattered in tissue.

The partial Fourier sensing matrix was constructed by repeatedly using a random uniform distribution until 16 unique rows were selected from a 64×64 discrete cosine transform (DCT) matrix. Random spherical codes were constructed by randomly generating Gaussian distributed vectors that were then normalized to lie on the unit sphere. Minimum potential spherical codes were constructed using 2000 iterations of the MPSC algorithm with a constant relaxation factor of 1/2000. For each sensing matrix, we calculated the average multiplexing SNR and positioning accuracy from 10,000 trials each for 1-, 2-, and 3-crystal element interactions and SNR₅₁₁ = 30², SNR₅₁₁ = 100², and SNR₅₁₁ = 400².

We compared CoSaMP, SP, and IHT to decode the multiplexing. We also used these same three algorithms for support recovery followed by MLE to decode the signals, henceforth referred to as ML-CoSaMP, ML-SP, and ML-IHT. A noise threshold γ was chosen such that the probability that the noise pixel magnitude would be equal to or greater than γ was 50% as determined by (7). This threshold was applied to further constrain the support produced by ℓ₀ -norm minimization before application of MLE.

To assess decoder/sensing matrix performance, the SNR was calculated as the ratio of the signal amplitude squared to the decoder RMS error squared.

V. Results

To assess compressed sensing decoder performance, CoSaMP, SP, IHT ML-CoSaMP, ML-SP, and MP-IHT were used to decode the imaging array multiplexed by different sensing matrices. Decoder comparisons for each sensing matrix (SNR₅₁₁ = 30²) are shown in Tables I–IV. For cross-strip multiplexing, the MLE improved the SNR of the CoSaMP decoder by 3.5 times for a single 511 keV interaction. The improvement was marginal for 2- and 3-crystal element interactions. Therefore, it was determined that ML-CoSaMP was the best overall given that the probability of single-pixel interaction events is higher than multiple-pixel interaction events.

TABLE I.

SNR Comparison of Decoders for Cross-Strip Mulitiplexing

	Number of Crystal Element Interactions
	1	2	3
CoSaMP	17.6±0.4	4.34±0.14	2.23±0.13
SP	20.7±0.6	3.99±0.16	2.03±0.13
IHT	29.6±0.6	5.14±0.16	2.50±0.13
ML-CoSaMP	62.0±0.6	5.15±0.17	2.27±0.14
ML-SP	23.1±0.7	4.07±0.16	2.05±0.13
ML-IHT	38.1±0.7	5.21±0.17	2.49±0.14

TABLE IV.

SNR Comparison of Decoders for MP Mulitiplexing

	Number of Crystal Element Interactions
	1	2	3
CoSaMP	35.6±0.6	7.39±0.15	3.56±0.12
SP	77.4±0.7	14.30±0.17	4.66±0.13
IHT	72.9±0.7	10.73±0.16	4.22±0.13
ML-CoSaMP	65.6±0.7	9.53±0.17	4.07±0.13
ML-SP	93.8±0.8	15.20±0.17	4.76±0.13
ML-IHT	89.8±0.7	11.28±0.17	4.31±0.13

For the DCT, RSC, and MPSC sensing matrices (Tables II–IV), Using the three ℓ₀-norm minimization techniques to position interactions followed by MLE improved the SNR of the recovered signals by up to 250%, 25%, and 14% for 1-, 2-, and 3-crystal element interaction cases. It was observed that IHT was significantly better than ML-IHT for 2-crystal element interactions for the RSC matrix. The improvement was most pronounced for a single 511 keV interaction. From these results, it was determined that ML-SP was the best for DCT, RSC, and MPSC multiplexing.

TABLE II.

SNR Comparison of Decoders for DCT Mulitiplexing

	Number of Crystal Element Interactions
	1	2	3
CoSaMP	16.8±0.4	4.66±0.17	2.38±0.13
SP	39.7±0.5	6.46±0.17	2.74±0.14
IHT	36.0±0.5	6.06±0.17	2.71±0.13
ML-CoSaMP	30.9±0.5	5.83±0.18	2.61±0.14
ML-SP	56.4±0.6	6.97±0.18	2.82±0.14
ML-IHT	56.3±0.6	6.59±0.17	2.79±0.14

To examine the behavior of the MPSC algorithm, sensing matrices were produced using varying number of iterations. Fig. 4 shows a plot of the maximum dot product g(C) versus iterations. The function g(C) generally improves with increasing number of iterations, however, g(C) is not monotonically decreasing. Fig. 5 is a plot of the MSE for the ML-SP decoder versus number of iterations for sensing matrices constructed using the MP algorithm for 1-, 2-, and 3-pixel interaction cases. The MSE generally improves with increasing number of iterations. The difference in MSE was not statistically significant between 1000 iterations up to 20 000 iterations (only the results up to 6000 iterations were plotted in the figure).

Fig. 4.

Shown is a plot of the maximum dot product g(C) vs. number of iterations. This indicates that the MPSC algorithm iterations are decreasing the function g(C) as intended.

Fig. 5.

Shown is a comparison of a mean square error (MSE) performance that have been normalized such that the MSE for a random spherical code (0 iterations) is 1.0 for 1-, 2-, and 3-crystal element interaction using the ML-SP decoder. The MSE generally improves with increasing number of MPSC iterations.

We randomly generated 100 RSC matrices and used these matrices as the initial matrix to generate 100 MPSC matrices using 2000 iterations of the MPSC algorithm. We also randomly generated 100 partial DCT and RSC matrices using a uniform distribution to select rows of the DCT matrix until 16 unique rows were selected. The best RSC and DCT matrices was found from the 100 randomly generated matrices was chosen for evaluation with the SNR for 1-, 2-, and 3-pixel interactions calculated by Monte Carlo simulation with 10 000 trials for each configuration using the ML-SP decoder. The XS sensing matrix was simulated with 10 000 trials for each interaction configuration using the ML-CoSaMP decoder. The SNR (signal amplitude squared/RMS noise squared) of the decoded signal are shown in Table V for the 450–570 keV energy window. For SNR₅₁₁ = 30², the SNR for MPSC was higher than RSC, DCT, and XS by 12%–42%, 50%–100%, and 57%–240%, respectively. For SNR₅₁₁ = 100², the improvement over RSC, DCT, and XS was 8%–47%, 17%–47%, and 70%–1600%, respectively. For SNR₅₁₁ = 400², the improvement over RSC, DCT, and XS was 4%–110%, up to 18%, and 65%–2400%, respectively. MPSC was 22%–210% better than Anger for photoelectric interactions.

TABLE V.

SNR Comparison of MPSC/RSC/DCT/XS/Anger (450–570 keV Energy Window)

		Number of Crystal Element Interactions
SNR511	Multiplexing	1	2	3
302	MPSC	97.6±0.8	17.5±0.2	5.34±0.13
	RSC	87.2±0.7	12.4±0.2	4.04±0.13
	DCT	57.0±0.6	8.61±0.17	3.54±0.14
	XS	62.0±0.6	5.15±0.17	2.27±0.14
	MPSC12	68.6±0.7	7.00±0.17	2.93±0.13
	Anger	31.2±0.5
1002	MPSC	306.2±2.1	103.1±0.5	26.2±0.3
	RSC	283.3±2.3	83.6±0.5	17.8±0.3
	DCT	252.1±2.1	70.1±0.5	19.4±0.3
	XS	177.3±1.9	6.01±0.43	2.83±0.33
	MPSC12	252.0±2.2	40.6±0.5	7.68±0.32
	Anger	231.1±1.7
4002	MPSC	383.6±6.6	146.5±1.9	42.9±1.2
	RSC	367.4±6.4	118.6±1.9	20.3±1.2
	DCT	367.8±6.7	123.7±1.9	40.7±1.2
	XS	232.9±7.0	5.69±1.6	2.87±1.3
	MPSC12	333.8±6.7	45.9±1.9	8.97±1.2
	Anger	314.9±5.9

The percentage of events that were positioned correctly for the various sensing matrices are shown in Table VI for 1-, 2-, and 3-crystal element interaction cases and the 450–570 keV energy window. All sensing matrices were nearly 100% at correctly positioning all single pixel interactions. MPSC was the most accurate at 82%–87% and 56%–65% for 2-, and 3-pixel interactions, respectively. For SNR₅₁₁ = 100² and SNR₅₁₁ = 400², there was not a statistically significant difference (to a 95% confidence interval, 1.96 standard deviations) in positioning accuracy among MPSC, RSC, and DCT sensing matrices. XS multiplexing was the least accurate at 48.3%–68.9%.

TABLE VI.

Positioning Accuracy MPSC/RSC/DCT/XS/Anger (450–570 keV Energy Window)

		Number of Crystal Element Irteractions
SNR511	Multiplexing	1	2	3
302	MPSC	99.6%	82.1%	56.0%
	RSC	99.8%	79.3%	52.3%
	DCT	99.6%	76.3%	52.0%
	XS	99.1%	68.9%	48.3%
	MPSC12	99.5%	72.2%	45.3%
	Anger	97.4%
1002	MPSC	100%	86.8%	64.2%
	RSC	100%	86.6%	62.8%
	DCT	100%	86.2%	64.0%
	XS	100%	67.9%	50.2%
	MPSC12	100%	85.1%	57.7%
	Anger	100%
4002	MPSC	100%	87.0%	65.0%
	RSC	100%	86.9%	62.9%
	DCT	100%	86.7%	65.2%
	XS	100%	66.5%	50.3%
	MPSC12	100%	85.5%	58.7%
	Anger	100%

^*standard error = ±0.1% for ail cases

The results for the 250–770 keV energy window are compiled in Tables VII and VIII. As with the narrower energy window, MPSC produced the best SNR in all cases. MPSC had 49%–116% higher SNR than XS and 45%–300% higher SNR than Anger logic for single interaction cases.

TABLE VII.

SNR Comparison of MPSC/RSC/DCT/XS/Anger (250–770 keV Energy Window)

		Number of Crystal Element Interactions
SNR511	Multiplexing	1	2	3
302	MPSC	95.8±0.4	16.9±0.2	5.29±0.13
	RSC	83.1±0.4	12.3±0.17	4.21±0.14
	DCT	57.5±0.4	8.3±0.17	3.24±0.14
	XS	44.2±0.5	5.26±0.16	2.28±0.14
	MPSC12	66.4±0.4	6.9±0.2	2.97±0.13
	Anger	23.8±0.5
1002	MPSC	288.8±1.9	93.5±0.5	30.0±0.3
	RSC	271.6±1.9	75.6±0.5	15.7±0.3
	DCT	235.3±1.9	61.2±0.5	19.6±0.3
	XS	183.3±1.8	6.3±0.4	2.96±0.33
	MPSC12	236.0±1.9	27.5±0.5	6.7±0.3
	Anger	166.9±1.8
4002	MPSC	339.2±8.1	134.9±1.8	42.5±1.2
	RSC	342.0±8.2	113.1±1.8	21.1±1.2
	DCT	315.6±8.0	104.0±1.8	32.1±1.2
	XS	227.8±7.9	5.93±1.6	2.97±1.3
	MPSC12	308.9±8.1	48.9±1.9	10.1±1.2
	Anger	232.7±7.8

TABLE VIII.

Positioning Accuracy MPSC/RSC/DCT/XS/Anger (250–770 keV Energy Window)

		Number of Crystal Element Interactions
SNR511	Multiplexing	1	2	3
302	MPSC	95.4%	78.0%	54.5%
	RSC	95.5%	75.7%	51.6%
	DCT	94.5%	72.3%	50.0%
	XS	91.4%	65.1%	46.3%
	MPSC12	94.7%	68.0%	44.1%
	Anger	90.1%
1002	MPSC	99.8%	84.1%	63.0%
	RSC	99.8%	83.8%	54.2%
	DCT	99.9%	83.6%	62.6%
	XS	99.6%	66.0%	49.1%
	MPSC12	99.7%	81.3%	54.2%
	Anger	99.7%
4002	MPSC	100%	84.8%	63.6%
	RSC	100%	84.7%	61.4%
	DCT	100%	84.6%	63.2%
	XS	100%	65.0%	49.2%
	MPSC12	100%	83.1%	58.2%
	Anger	100%

^*standard error = ±0.1% for all cases

Finally, we also evaluated the performance of the MPSC algorithm with only 12-channels (5.33:1 multiplexing ratio), which is listed as MPSC12 in Tables V–VIII. Only a single realization of the MPSC sensing matrix was created.

VI. Discussion and Conclusion

In this work, we have introduced models and methods for multiplexing imaging arrays using a sparse signal recovery framework. The noise at each element of the measurement channels of (1), the vector y, is proportional to the sum of the elements of the rows in the matrix C. The highest SNR at the measurements is therefore achieved with Nyquist-Shannon sampling—e.g., nonmultiplexed readout—and the more the sampling is compressed (increasing the detector channel multiplexing ratio), the more the measurement SNR is decreased. In this framework, commonly used nuclear medicine multiplexing methods such as the Anger camera and cross strip multiplexing specify just two possible choices of the matrix C.

In this work, we have shown how compressed sensing can be used to position and recover the energy of intercrystal scatter. We have also shown that the noise robustness of standard compressed sensing methods applied to decode the multiplexing and to specify the multiplexing weights can be improved.

Standard compressed sensing decoders such as greedy ℓ₀-norm minimization algorithms are used to position the interactions followed by least squares estimation to recover the magnitudes of the signal. In this work, a noise model was formulated from the estimated positions produced by a ℓ₀-norm decoder in for maximum likelihood estimation of the signal magnitudes. The various ℓ₀-norm decoders will produce different support estimated under noisy conditions. In the case when the estimated support does not match the true support, MLE does not necessarily produce a better solution than the LS approach. Because there is no analytic method for assessing the performance of the various decoders with different sensing matrices, it was necessary to use Monte Carlo simulations to compare the ML versus LS approach for the various decoders and sensing matrices as shown in Tables I–IV. The best decoder for RSC, DCT, and MPSC was the ML-SP decoder, approximately 1%–42% better SNR than the SP decoder. The largest gain in SNR was observed for single-pixel interaction cases, 21% for MPSC, 30% for RSC and 42% for DCT. For cross-strip (XS) multiplexing, ML-CoSaMP SNR was up to 3.5 times higher than CoSaMP. The addition ofML estimation is superior only when the sensing matrix and SNR is sufficient for accurate support estimation.

The RIP condition is not a practical method for finding sensing matrices that are robust under noisy conditions. In this work, we used the maximum likelihood estimation framework to create an objective function for designing sensing matrices by gradient methods. The resulting MPSC sensing matrix produced by minimizing the objective function of (10) is easily computed via gradient descent methods and was shown to outperform partial Fourier sensing matrices, random Gaussian-distributed spherical codes, and cross-strip multiplexing. Using the best decoders for each sensing matrix (ML-CoSaMP for XS multiplexing and ML-SP for DCT, RSC, and MPSC sensing matrices), MPSC yielded a 65%–2400% gain in SNR compared to XS at the same 64 to 16 multiplexing ratio. MPSC was able to position 2- and 3-pixel interaction events with 82%–87% and 56%–65% accuracy, respectively. Further, it was shown that with 25% fewer channels, 64 to 12 multiplexing with the ML-SP decoder and MPSC produced a 10%–43%, 35%–700%, and 29%–210% improvement in SNR than 64 to 16 channel cross-strip multiplexing for 1-, 2-, and 3-pixel interactions, respectively. We also showed that as the minimum distance between codes increased, the decoded SNR from multiplexing increased. MPSC produced a higher SNR by up to 110% and up to 100% than the RSC and DCT, respectively.

The focus of this paper was to present a new methodology. It was beyond the scope of this work to perform a thorough evaluation of the new multiplexing scheme. Using a simple and limited simulation study, we were able to demonstrate that gains in SNR are possible compared to existing multiplexing schemes. Such gains will generally translate to improvements in spatial resolution, energy resolution and time resolution. A full characterization of detector performance with the new multiplexing scheme is best done by constructing and evaluating a real detector under the full range of operating conditions, which is an area for future research.

The compressed sensing methods presented here require spatial discretization of the block detector such that the number of discrete positions equals the number of photodetector pixels. In a standard PMT-based PET detector, light sharing or light “multiplexing” is intentional and is required for event positioning. In such designs, light multiplexing is often used such that there are fewer PMTs than crystal elements. The approach presented in this paper is not suitable for such designs.

However, compressed sensing is amenable to detectors using SIPM photodetectors, which are available with small pixels and high packing fraction. In the detector array modeled in this work, there is one-to-one crystal to photodetector coupling and light sharing occurred between crystal elements due to imperfect reflector properties. In this case, we had to incorporate this inadvertent light sharing into our model. We showed that the measured light sharing function is invertible. This approach could also be used to multiplex pixelated CdTe and CdZnTe detectors.

The RSC, DCT, and MPSC sensing matrices are all high-weight codes. The weight of a code is equal to the number of nonzero coefficients. High-weight codes are difficult to implement with discrete components on a PCB because they require many interconnections. Due to the high weight, implementing the MPSC sensing matrix might only be practical with an ASIC or if it is integrated directly with the photodetector.

The ideal way to implement the proposed multiplexing scheme would be to integrate it directly into an SiPM array in an integrated circuit. The manufacturing cost of such a device would be comparable whether it used the new or conventional multiplexing methods.

A schematic of MPSC12 multiplexing implemented on a SiPM array device is shown in Fig. 6. A pixel comprises of an array of Geiger-mode APDs (G-APD) outlined by the dotted box. The signal for the pixel is proportional to the number of photons that hit the pixel. Amplifiers are used to multiply the pixel signal by a weight according to the MPSC12 sensing matrix. The signals are then added to the readout lines. On each of the four sides of the array are three traces corresponding to 3 of the 12 readout channels. This pattern is repeated for all pixels. This implementation requires three layers, one each for the amplifiers, horizontal readout channels, and vertical readout channels. This layout is inspired by the design of the interpolating SiPM device built by Fischer [27]. To increase the fill factor of the device, 3-D imaging sensors could also be used with the readout implemented behind the pixel similar to the 3-D single photon avalanche diode arrays proposed by Berube et al. [28] and Durini et al. [29]. The design, implementation, and evaluation of a SiPM array using the multiplexing methods described here is another area for further research.

Fig. 6.

Schematic of proposed MPSC-12 multiplexing on a SiPM-like device. In the center is a single pixel comprised of an array of Geiger mode APDs. Amplifiers multiply the pixel signal by a weight and adds the signal to a readout line. The device uses three layers, one each for the amplifiers, horizontal readout lines, and vertical readout lines.

The primary focus of this work was to establish a good criterion for multiplexing design and the potential gains from developing new multiplexing for nuclear medicine detectors. This work provides an estimate of the upper bound of performance for linear multiplexing weights. Future work can also investigate easier sensing matrices for multiplexing that can approach the performance of the MPSC matrix with fewer interconnections. The number of interconnections can be reduced in various ways. For example, the lowest magnitude coefficients of the MPSC matrix could be rounded down to zero, reducing the number of interconnections required to implement the multiplexing. Alternatively, the objective function of (10) could be modified with a maximum weight or a constant weight constraint to limit the number of nonzero coefficients. Another possibility is to factor the MPSC matrix into the product of two or more sparse matrices. Since sparse matrices correspond to low weight codes, they require fewer interconnections and are easier to implement. This factored sensing matrix would then be implemented by a hierarchical architecture with two or more levels of networking before sampling by ADCs. Implementation can be simplified further by limiting the codes to binary coefficients. Binary codes are easy to implement since a “1”’ indicates a connection from a pixel to an ADC channel while a “0” indicates no connection between a pixel and ADC channel. It should be noted that cross-strip multiplexing is a binary constant weight code that satisfies the RIP condition.

Formulating a mathematical model that predicts performance of this readout multiplexing design for any arbitrary detector design is beyond the scope of this current work. The performance of any readout multiplexing scheme is dependent on a number of parameters including detector pixel SNR, PSF coefficients, energy window, and energy resolution. Developing a mathematical model that can predict readout SNR, along with detector spatial resolution, energy resolution, and time resolution would be a valuable contribution for the evaluation of not only the multiplexing designs presented here but, for all multiplexing and detector designs. However, in this work we used a simulation study to characterize performance for one specific detector design with a limited range of detector parameters such as two energy windows and three different detector pixel SNRs. As such, we cannot assume that the gains observed from the simulation of this one detector design can be generalized to any other design.

The work presented here was for a one-to-one coupled photodetector to crystal element detector design, which is a commonly used approach for SiPM-based detectors. The method presented here could be easily translated to monolithic crystal detector designs. This is another area for further research.

For detector designs that employ more crystal elements than photo-detector channels, a different formulation of sparse signal recovery methods would need to be developed than is described in this work. This is another area for further research.

The multiplexing method here offers the greatest benefit with intercrystal scatter events. As such, the technique might be especially useful to cadmium telluride and cadmium zinc telluride detectors. Such detectors have a higher fraction of intercrystal scatter events compared to scintillation crystal detection elements of the same volume. This is another area for further research.

Another area of future work will be to extend the methods presented here to resolve pile up events. The same linear weights of the multiplexing network would still be used while the decoder would need to be modified to use temporal signal information. The development of this extension and the comparison with conventional multiplexing schemes such as Anger logic to resolve pile up events was beyond the scope of this work.

In summary, we have developed a new method for decoding multiplexed detector arrays and a criterion for designing good linearly-weighted multiplexing networks. We also showed that these new multiplexing methods could be used to reconstruct multiple interaction events.

TABLE III.

SNR Comparison of Decoders for RSC Mulitiplexing

	Number of Crystal Element Interactions
	1	2	3
CoSaMP	33.1±0.5	6.04±0.17	3.06±0.13
SP	63.2±0.6	10.98±0.17	3.95±0.13
IHT	54.3±0.6	11.28±0.17	3.61±0.13
ML-CoSaMP	57.8±0.6	7.58±0.18	3.44±0.14
ML-SP	82.3±0.7	12.03±0.17	4.12±0.13
ML-IHT	75.0±0.7	8.28±0.17	3.73±0.13