Transmission-less attenuation estimation from time-of-flight PET histo-images using consistency equations

Abstract

In positron emission tomography (PET) imaging, attenuation correction with accurate attenuation estimation is crucial for quantitative patient studies. Recent research showed that the attenuation sinogram can be determined up to a scaling constant utilizing the time-of-flight information. The TOF-PET data can be naturally and efficiently stored in a histo-image without information loss, and the radioactive tracer distribution can be efficiently reconstructed using the DIRECT approaches. In this paper, we explore transmission-less attenuation estimation from TOF-PET histo-images. We first present the TOF-PET histo-image formation and the consistency equations in the histo-image parameterization, then we derive a least-squares solution for estimating the directional derivatives of the attenuation factors from the measured emission histo-images. Finally, we present a fast solver to estimate the attenuation factors from their directional derivatives using the discrete sine transform and fast Fourier transform while considering the boundary conditions. We find that the attenuation histo-images can be uniquely determined from the TOF-PET histo-images by considering boundary conditions. Since the estimate of the attenuation directional derivatives can be inaccurate for LORs tangent to the patient boundary, external sources, e.g., a ring or annulus source, might be needed to give an accurate estimate of the attenuation gradient for such LORs. The attenuation estimation from TOF-PET emission histo-images is demonstrated using simulated 2D TOF-PET data.

1. Introduction

In positron emission tomography (PET) imaging, attenuation correction with accurate attenuation estimation is crucial for quantitative patient studies. For a modern PET scanner combined with an x-ray computed tomography (CT) scanner, the attenuation image is usually estimated from the x-ray CT scan, scaled to the PET energy of 511 keV, and forward projected to obtain the attenuation sinogram (Kinahan et al. 1998). However there are situations where the CT-based attenuation is incomplete or inaccurate, due to, e.g., the mismatch between PET and CT scans because of patient motion or different respiratory pattern during and between the two scans. PET images reconstructed without attenuation correction, or with an incomplete or mis-registered attenuation image can produce severe attenuation artifacts and lead to diagnostic errors (Gould et al. 2007). The newly emerged PET/MR scanners allow simultaneous data acquisitions of both PET functional imaging and high resolution MR anatomical imaging. However, the relation between the MR measurements and the linear attenuation coefficient at 511 keV is complex. Despite limited success in segmenting the registered MR images with attenuation assignment (Martinez-Möller et al. 2009, Hofmann et al. 2011, Keereman et al. 2013, Burgos et al. 2014), attenuation correction for PET/MR scanners remains more prone to errors than for PET-CT scanners (Wagenknecht et al. 2013).

TOF-PET data contain substantially more information about attenuation than non-TOF-PET data. Defrise et al. (2012) showed that TOF-PET data can determine the attenuation sinogram up to a constant, which results in a multiplicative factor in the activity image. Maximum likelihood methods were also developed to allow simultaneous reconstructions of attenuation and activity for TOF PET (Rezaei et al. 2012, Defrise et al. 2014, Rezaei et al. 2014).

An efficient partitioning scheme for TOF-PET data is the view-grouped histo-images (Matej et al. 2009, Daube-Witherspoon et al. 2012). The histo-images can be obtained by depositing TOF-PET events into the image space at the most likely annihilation (MLA) position or the confidence-weighted (CW) positions (Snyder et al. 1981, Watson 2007, Matej et al. 2009). The goal of this paper is to explore transmissionless attenuation estimation from the consistent TOF-PET histo-images. We first present the TOF-PET histo-image formation and the consistency equations in the histo-image parameterization. We then derive a least-squares (LS) solution for estimating the directional derivatives of the attenuation factors, which was parameterized in the histo-images format. Finally, we present a fast solver to estimate the attenuation factors from their directional derivatives using the discrete sine transform and fast Fourier transform.

2. Attenuation estimation from histo-image

2.1. TOF-PET histo-image formulation

TOF-PET data are generally parameterized as

$$p(t,s,\varphi ,z,\theta )={\int }_{-\infty }^{\infty }\mathrm{d}l{h}_F(t-l)\times f(scos\varphi -lcos\theta sin\varphi ,ssin\varphi +lcos\theta cos\varphi ,z+lsin\theta ),$$ (1)

where f is a 3D tracer distribution and the h_F(t − l) is a TOF profile centered at position l = t along the line-of-response (LOR), t is the TOF parameter, s and ϕ are the usual sinogram coordinates, z is the axial coordinate of the midpoint of the LOR, and θ is the co-polar angle between the LOR and a transaxial plane. The TOF profile is modeled as a Gaussian distribution with standard deviation σ_F,

$$h_F(t)=\frac{1}{\sqrt{2\pi }{\sigma }_F}exp(-\frac{t^2}{2{\sigma }_F^2}).$$ (2)

The TOF parameter t is related with the TOF time difference ΔT between the two arrival times of the two gammas by t = cΔT/2 where c denotes the speed of light. The standard deviation ${\sigma }_F={cT}_{\text{FWHM}}/(4\sqrt{2log2})$, and T_FWHM is the full width at half maximum (FWHM) of the measured time difference, which is on the order of 500 ps in current clinical scanners (Karp et al. 2008, Surti et al. 2007, Zaidi et al. 2011). A LaBr₃-based PET scanner developed at Penn has a timing resolution of 375 ps (Daube-Witherspoon et al. 2010), and a newly emerged scanner, Philips Vereos PET/CT based on digital photon counting technology, has a timing resolution of 345 ps.

The histo-image can be obtained by first grouping the TOF-PET events by the TOF direction n̂ depending on the azimuthal angle ϕ and co-polar angle θ. Then each event is deposited/backprojected into the image space at the most likely annihilation (MLA) position or the confidence-weighted (CW) positions (Snyder et al. 1981, Watson 2007, Matej et al. 2009). As shown in figure 1, each measured event associated with a single positron annihilation can be determined by the two detectors A and B and the difference of arrival time t of the gammas at the two locations. The MLA position x⃗ and the direction n̂ of a TOF event are

Figure 1.

Data parameterization of histo-image for a multi-ring TOF-PET scanner. A coincidence event between detectors A and B with TOF difference t can be parameterized in histo-image format by the most likely position x⃗ and the TOF direction n̂, the TOF profile is centered at position x⃗ along n̂, and x⃗₀ is the midpoint between A and B. The event can also be parameterized in the sinogram format by the variables t, s, z, ϕ and θ.

$$\begin{array}{l}\overrightarrow{x}={(scos\varphi -tcos\theta sin\varphi ,ssin\varphi +tcos\theta cos\varphi ,z+tsin\theta )}^T\\ \widehat{n}={(-cos\theta sin\varphi ,cos\theta cos\varphi ,sin\theta )}^T,\end{array}$$ (3)

where T denotes the vector or matrix transpose. We can use the MLA position x⃗ ∈ ℝ³ and the direction n̂ ∈ S² to parameterize and formulate the histo-image. An individual measured event determines a point (x⃗, n̂) in a 5D measurement space = ℝ³ × S², where ℝ³ and S² denote the 3D Euclidean space and the unit sphere in 3D space. The measurements obtained over an entire PET experiment form a Poisson point process on (Snyder et al. 1981, Snyder and Miller 1991). The expectation histo-image q(x⃗, n̂) can be modeled as a convolution

$$q(\overrightarrow{x},\widehat{n})={\int }_{-\infty }^{\infty }\mathrm{d}l{h}_B(t-l)p(l,s,\varphi ,z,\theta ),$$ (4)

where h_B(t) is the backprojection TOF profile and it can also be selected as a Gaussian distribution with standard deviation σ_B. One can use h_B(t) = δ(t) with σ_B = 0 and h_B(t) = h_F(t) with σ_B = σ_F for MLA and CW depositions, respectively.

Putting (1) into (4), we obtain

$$q(\overrightarrow{x},\widehat{n})={\int }_{-\infty }^{\infty }\mathrm{d}\mathit{lh}(l)f(\overrightarrow{x}-l\widehat{n}),$$ (5)

where h(t) is defined by the convolution of h_B(t) and h_F(t) as

$$h(t)=h_B\ast h_F(t)=\frac{1}{\sqrt{2\pi }\sigma }exp(-\frac{t^2}{2{\sigma }^2}),\sigma =\sqrt{{\sigma }_B^2+{\sigma }_F^2}.$$ (6)

Equation (5) is the histo-image formulation with parameters x⃗ and n̂. The standard deviation σ in the CW histo-image is increased by a factor of $\sqrt{2}$ compared to that in the MLA histo-image.

Some of the derivations below use two unit vectors û and v̂ that are orthogonal to n̂. All results in the paper are valid independently of the choice of these vectors. Some results will be made explicit, using the usual convention, which corresponds to the sinogram parameterization in (1). The two auxiliary unit vectors û, v̂ are given by

$$\widehat{u}={(cos\varphi ,sin\varphi ,0)}^T,$$ (7)

$$\widehat{v}=\widehat{n}\times \widehat{u}={(-sin\theta sin\varphi ,sin\theta cos\varphi ,-cos\theta )}^T.$$ (8)

Using these vectors, we can rewrite (5) as a 3D convolution

$$q(\overrightarrow{x},\widehat{n})=f(\overrightarrow{x})\ast \ast \ast \kappa (\overrightarrow{x},\widehat{n}),\kappa (\overrightarrow{x},\widehat{n})=h(\overrightarrow{x}·\widehat{n})\delta (\overrightarrow{x}·\widehat{u})\delta (\overrightarrow{x}·\widehat{v}).$$ (9)

Equation (9) states that the expectation q(x⃗, n̂) is a 3D convolution of the activity distribution f(x⃗) and the TOF kernel κ(x⃗, n̂).

2.2. Histo-image consistency equations

The histo-image q(x⃗, n̂) has five degrees of freedom (depends on five parameters) and the object f(x⃗) has only three—the two degrees of redundancy can be expressed as two independent consistency equations (Defrise et al. 2008, Defrise et al. 2013). For TOF-PET histo-images expressed in (4), (5) or (9), the vector calculus form of the consistency equations is (Defrise et al. 2015, Li et al. 2015)

$$\diamond q(\overrightarrow{x},\widehat{n})={\nabla }_{\widehat{n}}q(\overrightarrow{x},\widehat{n})-{\sigma }^2\widehat{n}·\nabla \nabla q(\overrightarrow{x},\widehat{n})=0.$$ (10)

Here we define ⋄ = ∇_n̂ − σ²n̂ · ∇∇ which maps q(x⃗, n̂) onto a vector field, the k-th component ${\diamond }_k=\frac{\partial }{\partial {\widehat{n}}_k}-{\sigma }^2{\sum }_{i=1}^3{\widehat{n}}_i\frac{{\partial }^2}{\partial x_i\partial x_k}$, k = 1, 2, 3, ∇ is the gradient with respect to position x⃗ and ∇_n̂ is the gradient with respect to TOF direction n̂. An alternative proof of (10) is given in Appendix A. Since the TOF direction n̂ is normalized, i.e., ||n̂|| = 1, ∇_n̂ is only meaningful in the directions perpendicular to n̂, e.g, û and v̂. All directions perpendicular to n̂ can be represented as a combination of û and v̂, so it is sufficient to consider the following two basis consistency equations

$$\widehat{u}·{\nabla }_{\widehat{n}}q(\overrightarrow{x},\widehat{n})-{\sigma }^2(\widehat{n}·\nabla )(\widehat{u}·\nabla )q(\overrightarrow{x},\widehat{n})=0,$$ (11)

$$\widehat{v}·{\nabla }_{\widehat{n}}q(\overrightarrow{x},\widehat{n})-{\sigma }^2(\widehat{n}·\nabla )(\widehat{v}·\nabla )q(\overrightarrow{x},\widehat{n})=0.$$ (12)

The consistency equations (10), (11) and (12) are independent of the coordinate system. For the specific parameterization (7) and (8), one has the following link with angular derivatives:

$$\widehat{u}·{\nabla }_{\widehat{n}}=-\frac{1}{cos\theta }\frac{\partial \widehat{n}}{\partial \varphi }·{\nabla }_{\widehat{n}}=-sec\theta \frac{d}{d\varphi },$$ (13)

$$\widehat{v}·{\nabla }_{\widehat{n}}=-\frac{\partial \widehat{n}}{\partial \theta }·{\nabla }_{\widehat{n}}=-\frac{d}{d\theta }.$$ (14)

Here, we used $\frac{\partial \widehat{n}}{\partial \varphi }=-cos\theta \widehat{u}$ and $\frac{\partial \widehat{n}}{\partial \theta }=-\widehat{v}$. Another useful consistency equation can be derived by adding v̂ · ∇ (11) and −û · ∇ (12):

$$(\widehat{v}·\nabla )(\widehat{u}·{\nabla }_{\widehat{n}})q(\overrightarrow{x},\widehat{n})-(\widehat{u}·\nabla )(\widehat{v}·{\nabla }_{\widehat{n}})q(\overrightarrow{x},\widehat{n})=0.$$ (15)

Equation (15) is John’s equation for TOF-PET histo-images (John 1938, John 1982, Defrise and Liu 1999, Defrise et al. 2013). The two consistency equations (11), (12) and John’s equation (15) for histo-images can be converted into the corresponding equations in the sinogram parameterization of the data in (1) (Defrise et al. 2013), and the details are shown in Appendix B.

The histo-image has an essentially bounded support because the object f has a bounded support and the kernel κ in (9) decays exponentially for a Gaussian TOF profile. So we can always use a Dirichlet/zero spatial boundary by selecting an image field-of-view larger than the object. From figure 1, we obtain the same TOF LOR by changing angles ϕ → ϕ + π, θ → −θ and time t → −t. So the consistent histo-images also satisfy the symmetry property

$$q(\overrightarrow{x},\widehat{n}(\varphi ,\theta ))=q(\overrightarrow{x},\widehat{n}(\varphi +\pi ,-\theta ))=q(\overrightarrow{x},-\widehat{n}(\varphi ,\theta )).$$ (16)

2.3. An analytical solution for attenuation gradients

After correcting for scattered and random coincidences, we can model the expectation histo-image m(x⃗, n̂) as

$$m(\overrightarrow{x},\widehat{n})=a(\overrightarrow{x},\widehat{n})q(\overrightarrow{x},\widehat{n}),$$ (17)

where a(x⃗, n̂) is the attenuation factor. Based on Beer-Lambert law, we can write the attenuation factor as

$$a(\overrightarrow{x},\widehat{n})=exp\left[-{\int }_{-\infty }^{\infty }\mathrm{d}l\mu (\overrightarrow{x}+l\widehat{n})\right],$$ (18)

where μ(x⃗) is the linear attenuation coefficients. To simplify notations and derivations below, we use here the same histo-image parameterization for the attenuation factors and for the emission data. Note however that the attenuation factor is independent of the TOF variable, i.e., it is constant along the TOF direction n̂, a(x⃗, n̂) = a(x⃗ + ln̂, n̂). So its directional derivative along n̂ is

$$\widehat{n}·\nabla a(\overrightarrow{x},\widehat{n})={\frac{d}{dl}a(\overrightarrow{x}+l\widehat{n},\widehat{n})|}_{l=0}=0.$$ (19)

The attenuation corrected emission histo-image q(x⃗, n̂) satisfies the consistency equations. Putting q(x⃗, n̂) = m(x⃗, n̂)/a(x⃗, n̂) into (10), applying the chain rule and using (19), we obtain

$$\begin{array}{l}0=\diamond \left(\frac{m(\overrightarrow{x},\widehat{n})}{a(\overrightarrow{x},\widehat{n})}\right)=\frac{\diamond m(\overrightarrow{x},\widehat{n})}{a(\overrightarrow{x},\widehat{n})}+m(\overrightarrow{x},\widehat{n})\diamond \frac{1}{a(\overrightarrow{x},\widehat{n})}-{\sigma }^2(\widehat{n}·\nabla m(\overrightarrow{x},\widehat{n}))\left(\nabla \frac{1}{a(\overrightarrow{x},\widehat{n})}\right)\\ =\frac{\diamond m(\overrightarrow{x},\widehat{n})}{a(\overrightarrow{x},\widehat{n})}-\frac{m(\overrightarrow{x},\widehat{n})}{a^2(\overrightarrow{x},\widehat{n})}{\nabla }_{\widehat{n}}a(\overrightarrow{x},\widehat{n})+{\sigma }^2\frac{\widehat{n}·\nabla m(\overrightarrow{x},\widehat{n})}{a^2(\overrightarrow{x},\widehat{n})}\nabla a(\overrightarrow{x},\widehat{n})\\ =\frac{\diamond m(\overrightarrow{x},\widehat{n})}{a(\overrightarrow{x},\widehat{n})}-\frac{m(\overrightarrow{x},\widehat{n})}{a(\overrightarrow{x},\widehat{n})}{\nabla }_{\widehat{n}}loga(\overrightarrow{x},\widehat{n})+{\sigma }^2\frac{\widehat{n}·\nabla m(\overrightarrow{x},\widehat{n})}{a(\overrightarrow{x},\widehat{n})}\nabla loga(\overrightarrow{x},\widehat{n}).\end{array}$$ (20)

Multiplying the above equation by a(x⃗, n̂), we obtain

$$m(\overrightarrow{x},\widehat{n}){\nabla }_{\widehat{n}}loga(\overrightarrow{x},\widehat{n})-{\sigma }^2\widehat{n}·\nabla m(\overrightarrow{x},\widehat{n})\nabla loga(\overrightarrow{x},\widehat{n})=\diamond m(\overrightarrow{x},\widehat{n}),$$ (21)

where ⋄m(x⃗, n̂) = ∇_n̂m(x⃗, n̂) − σ²n̂ · ∇∇m(x⃗, n̂). We would like to estimate the angular gradient ∇_n̂log a and spatial gradient ∇log a for a fixed LOR defined by a point x⃗₀ and a direction n̂, from the relevant data set {m(x⃗₀ + ln̂, n̂)|l ∈ ℝ}, which contains all TOF bins for this LOR. Along the TOF direction n̂, the attenuation factor a(x⃗₀, n̂) is unchanged, so its spatial gradient with respect to x⃗₀ is also unchanged,

$$\nabla a({\overrightarrow{x}}_0+l\widehat{n},\widehat{n})=\nabla a({\overrightarrow{x}}_0,\widehat{n}).$$ (22)

To proceed, we define the angular gradient of log a(x⃗₀ + ln̂, n̂) as

$${\nabla }_{\widehat{n}}loga({\overrightarrow{x}}_0+l\widehat{n},\widehat{n})={\mathcal{D}}_{\widehat{n}}loga({\overrightarrow{x}}_0+l\widehat{c},\widehat{n}){\mid }_{\widehat{c}=\widehat{n}},$$ (23)

where is the total angular gradient. Using the chain rule, we can then calculate the angular gradient ∇_n̂ log a(x⃗₀ + ln̂, n̂) as (Defrise et al. 2015)

$$\begin{array}{l}{\nabla }_{\widehat{n}}loga({\overrightarrow{x}}_0+l\widehat{n},\widehat{n})={\mathcal{D}}_{\widehat{n}}loga({\overrightarrow{x}}_0+l\widehat{n},\widehat{n})-{\mathcal{D}}_{\widehat{n}}loga({\overrightarrow{x}}_0+l\widehat{n},\widehat{c}){\mid }_{\widehat{c}=\widehat{n}}\\ ={\mathcal{D}}_{\widehat{n}}loga({\overrightarrow{x}}_0,\widehat{n})-l\nabla loga({\overrightarrow{x}}_0+l\widehat{n},\widehat{n})\\ ={\nabla }_{\widehat{n}}loga({\overrightarrow{x}}_0,\widehat{n})-l\nabla loga({\overrightarrow{x}}_0,\widehat{n}).\end{array}$$ (24)

Here we used a(x⃗₀ + ln̂, n̂) = a(x⃗₀, n̂) since a(x⃗₀ + ln̂, n̂) = a(x⃗₀, n̂). Using (22) and (24), we can rewrite (21) at x⃗ = x⃗₀ + ln̂ as

$$-[{\sigma }^2\widehat{n}·\nabla m+lm]\nabla loga({\overrightarrow{x}}_0,\widehat{n})+m{\nabla }_{\widehat{n}}loga({\overrightarrow{x}}_0,\widehat{n})=\diamond m.$$ (25)

Here we have dropped the arguments (x⃗₀ + ln̂, n̂) of m for conciseness. Again, (25) can be interpreted as two basis equations along directions û and v̂

$$-({\sigma }^2\widehat{n}·\nabla m+lm)\widehat{u}·\nabla loga({\overrightarrow{x}}_0,\widehat{n})+m\widehat{u}·{\nabla }_{\widehat{n}}loga({\overrightarrow{x}}_0,\widehat{n})=\widehat{u}·\diamond m,$$ (26)

$$-({\sigma }^2\widehat{n}·\nabla m+lm)\widehat{v}·\nabla loga({\overrightarrow{x}}_0,\widehat{n})+m\widehat{v}·{\nabla }_{\widehat{n}}loga({\overrightarrow{x}}_0,\widehat{n})=\widehat{v}·\diamond m.$$ (27)

For fixed (x⃗₀, n̂), the above equations hold for all measurements m(x⃗₀ + ln̂, n̂) with x⃗₀ · n̂ + l ∈ T, with T denoting the TOF interval where the data are measured. So one can solve (26) and (27) for the spatial/angular gradients in the least square sense (Defrise et al. 2012). For instance, the least square solution for û · ∇log a(x⃗₀, n̂) and û · ∇_n̂ log a(x⃗₀, n̂) in (26) can be obtained by minimizing the Euclidean norm

$$\mathcal{E}={\int }_{T}dl{\mid \widehat{u}·\diamond m+({\sigma }^2\widehat{n}·\nabla m+lm)\widehat{u}·\nabla loga({\overrightarrow{x}}_0,\widehat{n})-m\widehat{u}·{\nabla }_{\widehat{n}}loga({\overrightarrow{x}}_0,\widehat{n})\mid }^2.$$ (28)

After equating the derivatives with respect to the unknowns to zero, we obtain

$$\left[\begin{array}{cc}{H}_{11}& H_{12}\\ H_{12}& H_{22}\end{array}\right]\left[\begin{array}{c}-\widehat{u}·\nabla loga({\overrightarrow{x}}_0,\widehat{n})\\ \widehat{u}·{\nabla }_{\widehat{n}}loga({\overrightarrow{x}}_0,\widehat{n})\end{array}\right]=\left[\begin{array}{c}{J}_1\\ J_2\end{array}\right]$$ (29)

where

$$\begin{array}{l}{H}_{11}={\int }_{T}dl{({\sigma }^2\widehat{n}·\nabla m+lm)}^2,H_{12}={\int }_{T}dl({\sigma }^2\widehat{n}·\nabla m+lm)m,H_{22}={\int }_T{\mathit{dlm}}^2,\\ J_1={\int }_{T}dl({\sigma }^2\widehat{n}·\nabla m+lm)\widehat{u}·\diamond m,J_2={\int }_T\mathit{dlm}\widehat{u}·\diamond m.\end{array}$$ (30)

Solving (29), we obtain the least-squares (LS) estimate

$$-\widehat{u}·\nabla log{a}^{(\text{est})}({\overrightarrow{x}}_0,\widehat{n})=\frac{H_{22}{J}_1-H_{12}{J}_2}{H_{11}{H}_{22}-H_{12}^2},$$ (31)

$$\widehat{u}·{\nabla }_{\widehat{n}}log{a}^{(\text{est})}({\overrightarrow{x}}_0,\widehat{n})=\frac{H_{11}{J}_2-H_{12}{J}_1}{H_{11}{H}_{22}-H_{12}^2}.$$ (32)

The above LS solution for histo-images is equivalent to equation (24) for the sinogram parameterization in (Defrise et al. 2012). Similar to the attenuation histo-image, the directional derivatives in the LS solution are also spatially invariant along the TOF direction, and the detailed verification is shown in Appendix C. Based on the Schwarz inequality, the matrix in (29) is positive-semidefinite for a fixed (x⃗₀, n̂); the denominator $H_{11}{H}_{22}-H_{12}^2$ is nonnegative and it becomes zero if and only if there is only one point source along the line {x⃗₀ + ln̂| l ∈ ℝ} (Defrise et al. 2012). In this case, the two equations in (29) are dependent and there are infinitely many solutions. For an LOR tangent to the patient boundary, the solution is unstable since there is only one point having activity along the LOR. In such case, one can use regularization to stabilize the solution by adding a regularization term in (28). For instance, one can use the Tikhonov regularization to obtain solutions with smaller norms by replacing H₁₁ with H₁₁ + β₁ and H₂₂ with H₂₂ + β₂, with hyperparameters β₁, β₂ to control the amount of regularization (Tikhonov and Arsenin 1977). It is worth noting that the directional derivatives v̂ · ∇log a^(est)(x⃗₀, n̂) and v̂ · ∇_n̂ log a^(est)(x⃗₀, n̂) can also be solved using (30), (31) and (32) by replacing û with v̂.

2.4. Estimate attenuation from its directional derivatives

Methods to reconstruct shapes from gradient data (Southwell 1980, Ettl et al. 2008, Huang et al. 2015), can be used to reconstruct attenuation histo-images from their directional derivatives. However, the boundary conditions are not considered in these methods and the attenuation can only be determined up to an additive constant for log a(x⃗, n̂), which amounts to a multiplicative constant for the attenuation factors and activity image. This constant can be estimated using boundary conditions dictated by the finite support of the object. Here we consider the 2D case with θ = 0. We derive a fast dedicated method to uniquely determine attenuation histo-images (or sinograms) from their directional derivatives while explicitly considering the boundary conditions. For conciseness, we consider an azimuthal angular range ϕ ∈ [0, 2π). In practice, histo-images only have range ϕ ∈ [0, π); however, one can stack two replicates with one flipped along s by applying the symmetric property (16). Since attenuation is constant along the TOF direction n̂, and we can formulate the attenuation as μ(s, ϕ) = −log a(x⃗, n̂) with s = x⃗ · û, x⃗ · n̂ = 0, n̂ = (−sin ϕ, cos ϕ) and û = (cos ϕ, sin ϕ). Applying (13) and (B.3), we have the following partial derivatives

$$\frac{\partial }{\partial s}\mathcal{R}\mu (s,\varphi )=-\widehat{u}·\nabla loga(\overrightarrow{x},\widehat{n}),\frac{\partial }{\partial \varphi }\mathcal{R}\mu (s,\varphi )=\widehat{u}·{\nabla }_{\widehat{n}}loga(\overrightarrow{x},\widehat{n}).$$ (33)

The above equation is always implemented numerically to estimate the digitized version of μ(s, ϕ). We use N_s and N_ϕ to denote the number of samples, Δ_s and Δ_ϕ to denote the sampling intervals, along s and ϕ, respectively. We use vectorization vec(A) for a matrix A to denote a column vector obtained by stacking the columns of A, e.g. we use vec( μ) ∈ ℝ^N_sN_ϕ to denote the digitized version of attenuation μ(s, ϕ). The finite difference can be used to approximate the partial derivatives in (33) with zero boundary along s and periodic boundary along ϕ. We use higher order terms in the Taylor series (exact up to the 4-th order) to approximate the difference using the directional derivatives (Li et al. 2013a). We use the vectorized forms for the attenuation sinogram and the derivatives and rewrite (33) in matrix form

$$\left[\begin{array}{c}{\mathit{I}}_{\varphi }\otimes {\mathit{D}}_s\\ {\mathit{D}}_{\varphi }\otimes {\mathit{I}}_s\end{array}\right]\text{vec}(\mathcal{R}\mathit{\mu })=\left[\begin{array}{c}{\mathrm{\Delta }}_s{\mathit{I}}_{\varphi }\otimes {\mathit{M}}_s\text{vec}(-\widehat{u}·\nabla log\mathit{a})\\ {\mathrm{\Delta }}_{\varphi }{\mathit{M}}_{\varphi }\otimes {\mathit{I}}_s\text{vec}(\widehat{u}·{\nabla }_{\widehat{n}}log\mathit{a})\end{array}\right],$$ (34)

where I_s, I_ϕ are N_s × N_s and N_ϕ × N_ϕ identity matrices, D_s and D_ϕ are (N_s + 1) × N_s and N_ϕ × N_ϕ difference matrices given by

$${\mathit{D}}_s=\left[\begin{array}{rrrr}\hfill 1& \hfill & \hfill & \hfill \\ \hfill -1& \hfill 1& \hfill & \hfill \\ \hfill & \hfill -1& \hfill \ddots & \hfill \\ \hfill & \hfill & \hfill \ddots & \hfill 1\\ \hfill & \hfill & \hfill & \hfill -1\end{array}\right],{\mathit{D}}_{\varphi }=\left[\begin{array}{rrrr}\hfill -1& \hfill 1& \hfill & \hfill \\ \hfill & \hfill -1& \hfill \ddots & \hfill \\ \hfill & \hfill & \hfill \ddots & \hfill 1\\ \hfill 1& \hfill & \hfill & \hfill -1\end{array}\right],$$ (35)

and (N_s + 1) × N_s matrix M_s and N_ϕ × N_ϕ matrix M_ϕ are given by (Adapted from equation (9) in (Li et al. 2013a) after considering boundary condition)

$${\mathit{M}}_s=\frac{1}{24}\left[\begin{array}{rrrr}\hfill 13& \hfill -1& \hfill & \hfill \\ \hfill 13& \hfill 13& \hfill \ddots & \hfill \\ \hfill -1& \hfill 13& \hfill \ddots & \hfill -1\\ \hfill & \hfill \ddots & \hfill \ddots & \hfill 13\\ \hfill & \hfill & \hfill -1& \hfill 13\end{array}\right],{\mathit{M}}_{\varphi }=\frac{1}{24}\left[\begin{array}{rrrrr}\hfill 13& \hfill 13& \hfill -1& \hfill & \hfill -1\\ \hfill -1& \hfill 13& \hfill 13& \hfill \ddots & \hfill \\ \hfill & \hfill -1& \hfill 13& \hfill \ddots & \hfill -1\\ \hfill -1& \hfill & \hfill \ddots & \hfill \ddots & \hfill 13\\ \hfill 13& \hfill -1& \hfill & \hfill -1& \hfill 13\end{array}\right].$$ (36)

Here we used the Kronecker product ⊗ for block matrix operations. The Dirichlet boundary condition μ = 0 at the edge of the s range, is reflected by the first and last rows of D_s and M_s; the periodic boundary along ϕ is reflected by the circulant matrices D_ϕ and M_ϕ. Equation (34) is overdetermined since there are (2N_s + 1)N_ϕ equations but N_sN_ϕ unknowns in attenuation μ. The attenuation can be estimated in the least-squares sense by solving

$$({\mathit{I}}_{\varphi }\otimes {\mathit{K}}_s+C_{\varphi }\otimes {\mathit{I}}_s)\text{vec}(\mathcal{R}\mathit{\mu })=\text{vec}(\mathit{G}),$$ (37)

where

$$\mathit{G}={\mathrm{\Delta }}_s{\mathit{D}}_s^T{\mathit{M}}_s(-\widehat{u}·\nabla log\mathit{a})+{\mathrm{\Delta }}_{\varphi }(\widehat{u}·{\nabla }_{\widehat{n}}log\mathit{a}){\mathit{M}}_{\varphi }^T{\mathit{D}}_{\varphi },$$ (38)

and the symmetric tridiagonal matrix K_s and circulant matrix C_ϕ are given by

$${\mathit{K}}_s={\mathit{D}}_s^T{\mathit{D}}_s=\left[\begin{array}{rrrr}\hfill 2& \hfill -1& \hfill & \hfill \\ \hfill -1& \hfill 2& \hfill \ddots & \hfill \\ \hfill & \hfill \ddots & \hfill \ddots & \hfill -1\\ \hfill & \hfill & \hfill -1& \hfill 2\end{array}\right],{\mathit{C}}_{\varphi }={\mathit{D}}_{\varphi }^T{\mathit{D}}_{\varphi }=\left[\begin{array}{rrrr}\hfill 2& \hfill -1& \hfill & \hfill -1\\ \hfill -1& \hfill 2& \hfill \ddots & \hfill \\ \hfill & \hfill \ddots & \hfill \ddots & \hfill -1\\ \hfill -1& \hfill & \hfill -1& \hfill 2\end{array}\right].$$ (39)

Here we used the matrix equation vec(ABC) = (C^T ⊗ A)vec(B) in (38) (Horn and Johnson 1994). The matrix (I_ϕ ⊗ K_s + C_ϕ ⊗ I_s) in (37) is positive definite due to the positive-definite matrix K_s and positive semidefinite matrix C_ϕ (Strang 2007), so (37) has a unique solution for attenuation estimation. One can use the Gauss-Seidel method or Jacobi method to solve (37), and the diagonal elements of (I_ϕ ⊗ K_s + C_ϕ ⊗ I_s) are all equal to 4 (Southwell 1980). Noting that (37) is the vectorization form of Sylvester equation ${\mathit{K}}_s\mathcal{R}\mathit{\mu }+\mathcal{R}\mathit{\mu }{\mathit{C}}_{\varphi }^T=\mathit{G}$, one can also use the Sylvester equation solver to solve (37). Here, we use the special property of sparse matrices K_s and C_ϕ to derive a dedicated fast solver for (37). The matrix K_s can be diagonalized by the discrete sine transform and C_ϕ can be diagonalized by the fast Fourier transform (Strang 2007)

$${\mathit{K}}_s={\mathcal{S}}^{-1}\text{diag}({\mathit{\lambda }}_s)\mathcal{S},{\mathit{C}}_{\varphi }={\mathcal{F}}^{-1}\text{diag}({\mathit{\lambda }}_{\varphi })\mathcal{F},$$ (40)

where S is the discrete sine transform matrix/operator, and is the fast Fourier transform (FFT) matrix/operator, and the eigenvalues are

$${[{\mathit{\lambda }}_s]}_k=2-2cos\frac{k\pi }{N_s+1},k=1,2,\dots ,N_s,$$ (41)

$${[{\mathit{\lambda }}_{\varphi }]}_{\ell }=2-2cos\frac{2\pi \ell }{N_{\varphi }},\ell =0,1,\dots ,N_{\varphi }-1.$$ (42)

Applying (40)–(42), we obtain following fast attenuation solver/estimator

$$\mathcal{R}{\mathit{\mu }}^{(\text{est})}={\mathcal{S}}^{-1}\left(\frac{\mathcal{S}\mathit{G}{\mathcal{F}}^{-1}}{{\mathit{\lambda }}_s\oplus {\mathit{\lambda }}_{\varphi }}\right)\mathcal{F}.$$ (43)

We used Hadamard element-wise division and Kronecker sum ⊕ for convenience, the kℓ-th element [λ_s ⊕ λ_ϕ]_kℓ = [λ_s]_k + [λ_ϕ]_ℓ. We can apply the discrete sine transform along s and fast Fourier transform along ϕ in the implementation of (43), and the computational complexity is O(N_sN_ϕ log(N_sN_ϕ)). The fast dedicated solver (43) shares the same spirit as the fast Poisson solver in solving the Poisson equation (Strang 2007).

3. Numerical example

3.1. Simulation setup and XCAT phantom

To evaluate the attenuation estimation from TOF-PET data, we simulated a generic 2D TOF-PET system with 4mm crystals. A 2D TOF-PET projector was implemented using a strip-integral model with a Gaussian TOF profile with spatial FWHM of 75 mm, which corresponds to a timing resolution of about 500 ps FWHM. We used 41 TOF bins with bin size of 24 mm. The histo-images were generated by back-projecting the TOF sinograms for each angle using (4). The histo-image has 180 azimuthal angles uniformly spaced over 180°. To reduce the truncation in the convolution of the histo-image with the elongated TOF kernel, we used a large histo-image of 176 × 176 with 4mm pixel size. We used the XCAT phantom (formerly known as NCAT) (Segars and Tsui 2009, Li 2011), and the activity and attenuation images are shown in figure 2. Since the estimate of derivatives for LORs tangent to the object boundary is inaccurate, we added an external ring source containing 6.9% of body activity with diameter of 57.6 cm (Mollet et al. 2012, Mollet et al. 2014, Panin et al. 2013). The ring source allows an accurate estimate of derivatives for these LORs because it adds two activity points along each LOR passing through the scanner’s FOV.

Figure 2.

Activity and attenuation images of the simulated XCAT phantom. The image size is 70.4 cm × 70.4 cm, and the simulated TOF resolution was 7.5 cm FWHM, corresponding to time resolution of about 500 ps.

Figure 3 shows the histo-image generation along the direction n̂ at ϕ = 135°. The expectation TOF sinogram p(t, s, ϕ) (z = 0, θ = 0 for this 2D simulation and these arguments are omitted) was obtained using the TOF-PET projector. The attenuation factors are obtained by taking the exponential of the negative of the integral of the attenuation image μ(x⃗) along n̂, as formulated in (18). The attenuated TOF sinogram in figure 3(d) was obtained by taking the product of p(t, s, ϕ) and the corresponding attenuation factor. The noisy TOF sinogram was generated by applying a Poisson random generator to the attenuated TOF sinogram, and then the noisy histo-image m(x⃗, n̂) was obtained from the noisy TOF sinogram by convolving it with the kernel h_B according to (4). We simulated three different noise levels—noise-free and two noise levels. The total numbers of expected events in the histo-image are 4 × 10⁶ and 1 × 10⁶ for the moderate and the high noise cases, respectively. The attenuation derivatives estimated from noisy CW histo-images were always found to be more accurate than those estimated from noisy MLA histo-images. Therefore, we selected CW histo-images instead of MLA histo-images in the rest of this section.

Figure 3.

Histo-image generation along direction n̂ at ϕ =135°. (a) is a TOF sinogram p(t, s, ϕ) at the fixed direction, (b) is the CW activity histo-image obtained from (a), (c) is an attenuation factors in histo-image format, (d) is the attenuated TOF sinogram, (e) is obtained from (d) by adding Poisson noise, (f) is the CW histo-image obtained from (e). For TOF sinograms in (a), (d) and (e), the horizontal and vertical axes represent radial variable s and TOF variable t, respectively. For the histo-images in (b), (c,) and (f), the axes are the cartesian coordinates in image space. Note that for this 2D example the axes in the sinograms are rotated by ϕ compared with the histo-images.

3.2. Estimate directional derivatives of attenuation histo-image

The directional derivatives can be estimated using the LS method in (31) and (32), and we have θ = 0 in the 2D case considered for our simulation. The derivatives are invariant along TOF direction n̂, so we can estimate the coefficients in (30) for all x⃗₀ = sû. The line integrations over l in (30) can be naturally implemented as forward projections of the integrands formatted as histo-images along each direction n̂. The directional derivatives n̂ · ∇m(x⃗, n̂) was approximated as a convolution with [1/2, 0, −1/2]/Δ with bin size Δ. Applying (13), we can approximate û · ∇_n̂m(x⃗, n̂) using the central difference along ϕ. The least-square estimate of the gradients of the attenuation can be unstable for LORs having weak activity, which is a concern in particular for the LORs tangent to the ring source. To improve robustness to noise we regularized the gradient calculation in (31) and (32), as explained at the end of Section 2.3. The regularization parameters β₁ and β₂ were set equal to 10% of the minimum values of H₁₁ and H₂₂ in a central region of the histo-image (or sinogram). To avoid introducing bias for the LORs containing sufficient activity, regularization was applied only for LORs such that H₁₁ < β₁ or H₂₂ < β₂. The estimated directional derivatives −û · ∇log a and û · ∇_n̂ log a are shown in figure 4. The approximation of the true derivatives using the central difference is also shown for comparison.

Figure 4.

Estimation of directional derivatives −û · ∇log a (first row) and û · ∇_n̂ log a (second row), in sinogram format. The approximation of the true derivatives using central difference is also shown in the first column for comparison. The second to the fourth columns are the estimated derivatives from the CW histo-images of noise-free, moderate-noise and high-noise cases, respectively.

3.3. Estimate attenuation sinograms and reconstructed attenuation images

We used the central region of 54.4 cm (fully covering patient or “the activity phantom”) in estimating the attenuation sinogram to remove the unstable values in the estimated directional derivatives near the ring source in figure 4. The attenuation sinograms estimated from the directional derivatives in figure 4 using the fast solver (43) are shown in figure 5. Figure 6 shows a horizontal profile through the four sinograms at ϕ = 135° of figure 5. The underestimation for the noisy cases can be attributed to the nonlinearity in the gradient estimation with respect to the noisy histo-image m(x⃗, n̂). The bias can be reduced by direct attenuation estimation, e.g. (Li et al. 2013b).

Figure 5.

The true and estimated attenuation in sinogram format for the noise-free, moderate and high noise cases.

Figure 6.

Horizontal profiles through the sinogram shown in figure 5 at ϕ = 135°. The dotted curve denotes the true profile; the solid, dash-dotted and dashed curves denote the profiles through the noise-free and two noisy estimated sinograms.

We also present the reconstructed attenuation images μ(x⃗) to demonstrate the performance of the attenuation estimation in figure 7. The reconstruction of attenuation image is not required for PET attenuation correction. For comparison, we also show the reference images obtained by assuming that the true activity distribution in figure 2(a) was known. In this case, the attenuation reconstruction reduces to that in transmission tomography with a virtual blank scan equal to the true activity sinogram. The reference attenuation images were reconstructed using filtered back projection (FBP) and MLTR, a maximum-likelihood algorithm dedicated to transmission tomography (Nuyts et al. 1998). We used 12 iterations with 20 ordered-subsets in the MLTR reconstructions. We also used the FBP to reconstruct the attenuation image from the estimated sinograms. We smoothed the attenuation sinogram with a moving average kernel [1/3, 1/3, 1/3]^T applied along s to reduce noise in both the estimated and FBP reconstructions. There is a small crosstalk artifact (barely visible) from the activity in the heart region in the noise-free reconstructed attenuation images and the two reference images reconstructed using FBP and MLTR in figure 7. This artifact is attributed to the leaking of activity signal into the attenuation sinogram.

Figure 7.

Reconstructed attenuation images from simulated data without noise (left column), with moderate noise (middle column) and with high noise (right column). The attenuation images estimated using the LS solution (31,32) and fast solver (43) are shown in the first row. The reference images reconstructed using MLTR and FBP are shown in the second and third rows, respectively. The reference images were reconstructed from the TOF-integrated attenuated emission histo-image with the assumption that the true activity distribution was known.

4. Discussion

TOF-PET data can be naturally stored in histo-image format without information loss, and the DIRECT approach can be used for very efficient 3D TOF PET Reconstruction (Matej et al. 2009, Daube-Witherspoon et al. 2012). The attenuation histo-image formation allows estimating the attenuation directly from the selected data parameterization (histo-images) without having to first rebin to sinograms. The TOF-PET histo-image (or sinogram) has two degrees of redundancy; we formulated two consistency equations to characterize the redundancy. Thanks to the histo-image parameterization, the consistency equations in histo-image format are more concise and elegant than the consistency equations in the sinogram format, and they provide a better insight into the rich structure of TOF-PET data.

Consistency equations can also be derived in Fourier space, and again the histo-image parameterization leads to an elegant vector formulation independent of the coordinate system used. These equations are given in Appendix A. Equation (A.4) can be used to develop fast Fourier-based forward- and back-projectors for iterative image reconstruction (Matej et al. 2004, Cho et al. 2007), and to develop Fourier rebinning for TOF-PET histo-images, e.g., mapping the 3D TOF histo-images to 2D TOF histo-images (Defrise et al. 2005, Cho et al. 2009). The consistency equations for histo-image can also be applied to estimate the missing data in TOF PET due to incomplete linear or angular sampling due to, e.g., detector gaps (Karp et al. 1988).

Previously, it was showed that TOF-PET data determine attenuation sinograms up to a constant (Defrise et al. 2012). We developed a transmission-less attenuation estimation algorithm based on the consistency equations and showed from theory and an example that the attenuation histo-image (or sinogram) can be uniquely estimated from the TOF-PET histo-images when the time-of-flight (TOF) information is available. In principle, the attenuation can be fully determined from its directional derivatives, and the scaling constant is naturally solved by considering the boundary conditions. However, since the estimate of the directional derivatives of the attenuation is inaccurate for LORs tangent to object boundary, an external source might be needed to give accurate estimate for such LORs (Mollet et al. 2012, Mollet et al. 2014). The comparison of the directional derivatives estimated with and without ring source in figure 8, shows that the estimated derivatives are inaccurate for LORs tangent to the patient boundary when no ring source is used. After adding the ring source (considered as part of the entire activity during reconstruction), the inaccurate values move from the LORs tangent to patient to the LORs tangent to ring source. The LORs tangent to the ring source do not intersect the patient, and these errors do not affect the patient activity reconstruction. The ring or annulus source is not needed during the entire scan, so one can use a rotating point or line source (Panin et al. 2013). One can also use the source rotation with cardiac and respiratory gating for attenuation and activity estimation across gating frames. It is worth noting that one can also use the proposed attenuation estimation method to estimate the normalization factors for TOF PET with known attenuation (Rezaei et al. 2014). In this case, the external source can be removed, and the inaccurate values can be obtained using a model-based extrapolation method.

Figure 8.

Comparison of the directional derivatives estimated from the noise-free data with and without ring source. The derivatives −û · ∇log a and û · ∇_n̂ log a in the first and second columns are the same images as in the second column of figure 4.

How to optimally extract all the attenuation-related information from TOF-PET histo-images is still an open question, and it is beyond of the scope of this paper. The reconstruction incorporating all physics effects and modeling of Poisson noise is likely to produce superior image quality in the reconstructed attenuation image, e.g., the maximum-likelihood reconstruction for TOF-PET with simultaneous estimation of activity and the attenuation factors (MLACF) (Defrise et al. 2014, Rezaei et al. 2014). However, the joint maximum-likelihood estimation of activity and attenuation is not a convex optimization, and the estimation may not converge to the desired solution but to a saddle point or a local maximum. In this case, the fast analytic attenuation estimation can be used as initial sinogram/histo-image for MLACF which allows the MLACF converge to the desired solution. For 2D TOF PET, there is only one degree of redundancy, and we used only (26) to calculate the 2D directional derivatives in (31) and (32). For 3D TOF PET, there are two degrees of redundancy, and we can use both (26) and (27) to calculate the 4D directional derivatives—û · ∇ log a^(est), û · ∇ _n̂ log a^(est), v̂ · ∇ log a^(est) and v̂ · ∇_n̂ log a^(est), and then estimate attenuation from these derivatives. An alternative method is to estimate attenuation directly from (26) and (27) for 3D TOF PET. An efficient implementation of attenuation estimation for 3D TOF PET utilizing all measurement data needs to be investigated, which is our future work.

5. Conclusion

We developed an attenuation-estimation method for fully 3D TOF-PET emission histo-images with pre-correction of scattered and random coincidences. The attenuation estimation from histo-images is not only a theoretical result, but has a practical impact. It can provide better insight into the problem of simultaneous activity and attenuation estimation in TOF PET. The numerical example showed that the attenuation histo-images or sinograms can be uniquely estimated from the 2D TOF-PET emission histo-images with an external ring source. Future work will extend this work to 3D TOF-PET to efficiently estimate attenuation histo-images (or sinograms) utilizing all measured TOF-PET data.

Appendix A. An alternative proof of consistency equation (10)

We first prove that the TOF kernel κ(x⃗, n̂) satisfies (10). The TOF kernel is just the histo-image obtained from a point source f(x⃗) = δ(x⃗). The TOF kernel κ(x⃗, n̂) is given by the following inverse Fourier transform (Watson 2007)

$$\kappa (\overrightarrow{x},\widehat{n})=\frac{1}{{(2\pi )}^3}{\int }_{{ℝ}^3}{e}^{-{\sigma }^2{(\overrightarrow{\omega }·\widehat{n})}^2/2}{e}^{j\overrightarrow{\omega }·\overrightarrow{x}}{\mathrm{d}}^3\overrightarrow{\omega }={\mathcal{F}}^{-1}\left[e^{-{\sigma }^2{(\overrightarrow{\omega }·\widehat{n})}^2/2}\right].$$ (A.1)

Applying the derivative property of the Fourier transform (strictly speaking, in the distribution sense), we have

$${\nabla }_{\widehat{n}}\kappa (\overrightarrow{x},\widehat{n})={\mathcal{F}}^{-1}[-{\sigma }^2\widehat{n}·\overrightarrow{\omega }\overrightarrow{\omega }{e}^{-{\sigma }^2{(\overrightarrow{\omega }·\widehat{n})}^2/2}],$$ (A.2)

$$\widehat{n}·\nabla \nabla \kappa (\overrightarrow{x},\widehat{n})={\mathcal{F}}^{-1}[-\widehat{n}·\overrightarrow{\omega }\overrightarrow{\omega }{e}^{-{\sigma }^2{(\overrightarrow{\omega }·\widehat{n})}^2/2}].$$ (A.3)

Using (A.2) and (A.3), we prove the TOF kernel κ(x⃗, n̂) satisfies (10). Then we can prove any histo-image satisfies (10) using the fact that convolution commutes with differentiation.

In addition, we can write the Fourier transform of histo-image q(x⃗, n̂) using the convolution theorem as

$$\widehat{q}(\overrightarrow{\omega },\widehat{n})={\int }_{{ℝ}^3}q(\overrightarrow{x},\widehat{n})e^{-j\overrightarrow{\omega }·\overrightarrow{x}}{\mathrm{d}}^3\overrightarrow{x}=\widehat{f}(\overrightarrow{\omega })e^{-{\sigma }^2{(\overrightarrow{\omega }·\widehat{n})}^2/2},$$ (A.4)

where q̂(ω⃗) is the Fourier transform of the object f(x⃗). Equation (A.4) is the generalized projection slice theorem for 3D TOF-PET histo-image (Liu et al. 1999, Cho et al. 2009). By taking the Fourier transform of (10) with respect to x⃗, we can obtain the Fourier consistency equation in vector calculus form as

$$\widehat{\diamond }\widehat{q}(\overrightarrow{\omega },\widehat{n})={\nabla }_{\widehat{n}}\widehat{q}(\overrightarrow{\omega },\widehat{n})+{\sigma }^2\widehat{n}·\overrightarrow{\omega }\overrightarrow{\omega }\widehat{q}(\overrightarrow{\omega },\widehat{n})=0,$$ (A.5)

where ⋄̂ = ∇_n̂ + σ²n̂ · ω⃗ω⃗. One can verify that (A.4) is just the solution to (A.5).

Appendix B. Link with sinogram consistency equations

Histo-images and sinograms are connected by (4). Here we show that the histo-image consistency equations can be converted to consistency equations in sinogram format. By taking the inner product of x⃗ in (3) with the unit vectors n̂, û, v̂, we can derive

$$t=\overrightarrow{x}·\widehat{n}+tan\theta \overrightarrow{x}·\widehat{v},s=\overrightarrow{x}·\widehat{u},z=-sec\theta \overrightarrow{x}·\widehat{v}.$$ (B.1)

We can use (B.1) and rewrite the gradient ∇ as

$$\begin{array}{l}\nabla =\nabla (\overrightarrow{x}·\widehat{n}+tan\theta \overrightarrow{x}·\widehat{v})\frac{\partial }{\partial t}+\nabla (\overrightarrow{x}·\widehat{u})\frac{\partial }{\partial s}+\nabla (-sec\theta \overrightarrow{x}·\widehat{v})\frac{\partial }{\partial z}\\ =(\widehat{n}+tan\theta \widehat{v})\frac{\partial }{\partial t}+\widehat{u}\frac{\partial }{\partial s}-sec\theta \widehat{v}\frac{\partial }{\partial z}.\end{array}$$ (B.2)

From (B.2), we can obtain the directional derivatives

$$\widehat{n}·\nabla =\frac{\partial }{\partial t},\widehat{u}·\nabla =\frac{\partial }{\partial s},\widehat{v}·\nabla =tan\theta \frac{\partial }{\partial t}-sec\theta \frac{\partial }{\partial z}.$$ (B.3)

Using (13) and (B.1), we can rewrite the first term of (11) with a −cos θ factor as

$$\begin{array}{l}-cos\theta \widehat{u}·{\nabla }_{\widehat{n}}q(\overrightarrow{x},\widehat{n})={\frac{d}{d\varphi }p(t,s,\varphi ,z,\theta )|}_{\overrightarrow{x}}\\ =\frac{\partial p}{\partial \varphi }+{\frac{\partial (\overrightarrow{x}·\widehat{u})}{\partial \varphi }|}_{\overrightarrow{x}}\frac{\partial p}{\partial s}+{\frac{\partial (\overrightarrow{x}·\widehat{n}+tan\theta \overrightarrow{x}·\widehat{v})}{\partial \varphi }|}_{\overrightarrow{x}}\frac{\partial p}{\partial t}+{\frac{\partial (-sec\theta \overrightarrow{x}·\widehat{v})}{\partial \varphi }|}_{\overrightarrow{x}}\frac{\partial p}{\partial z}\\ =\frac{\partial p}{\partial \varphi }+\overrightarrow{x}·\frac{\partial \widehat{u}}{\partial \varphi }\frac{\partial p}{\partial s}+\left(\overrightarrow{x}·\frac{\partial \widehat{n}}{\partial \varphi }+tan\theta \overrightarrow{x}·\frac{\partial \widehat{v}}{\partial \varphi }\right)\frac{\partial p}{\partial t}-sec\theta \overrightarrow{x}·\frac{\partial \widehat{v}}{\partial \varphi }\frac{\partial p}{\partial z}\\ =\frac{\partial p}{\partial \varphi }+(cos\theta \overrightarrow{x}·\widehat{n}+sin\theta \overrightarrow{x}·\widehat{v})\frac{\partial p}{\partial s}-sec\theta \overrightarrow{x}·\widehat{u}\frac{\partial p}{\partial t}+tan\theta \overrightarrow{x}·\widehat{u}\frac{\partial p}{\partial z}\\ =\frac{\partial p}{\partial \varphi }+tcos\theta \frac{\partial p}{\partial s}-ssec\theta \frac{\partial p}{\partial t}+stan\theta \frac{\partial p}{\partial z}.\end{array}$$ (B.4)

Here we used ∂n̂/∂ϕ = −cos θû, ∂û/∂ϕ = cos θn̂ + sin θv̂ and ∂v̂/∂ϕ = −sin θû. Using (B.3), we can rewrite the second term of (11) with a −cos θ factor as

$$cos\theta {\sigma }^2(\widehat{n}·\nabla )(\widehat{u}·\nabla )q(\overrightarrow{x},\widehat{n})=cos\theta {\sigma }^2\frac{{\partial }^{2}p}{\partial s\partial t}.$$ (B.5)

Adding (B.4) and (B.5), we convert the first histo-image consistency equation (11) into the following sinogram consistency equation

$$\frac{\partial p}{\partial \varphi }+tcos\theta \frac{\partial p}{\partial s}-ssec\theta \frac{\partial p}{\partial t}+stan\theta \frac{\partial p}{\partial z}+{\sigma }^{2}cos\theta \frac{{\partial }^{2}p}{\partial s\partial t}=0.$$ (B.6)

Using (14) and (B.1), we can rewrite the first term of (12) with a negative sign as

$$\begin{array}{l}-\widehat{v}·{\nabla }_{\widehat{n}}q(\overrightarrow{x},\widehat{n})={\frac{d}{d\theta }p(t,s,\varphi ,z,\theta )|}_{\overrightarrow{x}}\\ =\frac{\partial p}{\partial \theta }+{\frac{\partial (\overrightarrow{x}·\widehat{u})}{\partial \theta }|}_{\overrightarrow{x}}\frac{\partial p}{\partial s}+{\frac{\partial (\overrightarrow{x}·\widehat{n}+tan\theta \overrightarrow{x}·\widehat{v})}{\partial \theta }|}_{\overrightarrow{x}}\frac{\partial p}{\partial t}+{\frac{\partial (-sec\theta \overrightarrow{x}·\widehat{v})}{\partial \theta }|}_{\overrightarrow{x}}\frac{\partial p}{\partial z}\\ =\frac{\partial p}{\partial \theta }+\overrightarrow{x}·\frac{\partial \widehat{u}}{\partial \theta }\frac{\partial p}{\partial s}+\left(\overrightarrow{x}·\frac{\partial \widehat{n}}{\partial \theta }+tan\theta \overrightarrow{x}·\frac{\partial \widehat{v}}{\partial \theta }+\frac{\partial tan\theta }{\partial \theta }\overrightarrow{x}·\widehat{v}\right)\frac{\partial p}{\partial t}-\left(sec\theta \overrightarrow{x}·\frac{\partial \widehat{v}}{\partial \theta }+\frac{\partial sec\theta }{\partial \theta }\overrightarrow{x}·\widehat{v}\right)\frac{\partial p}{\partial z}\\ =\frac{\partial p}{\partial \theta }+(-\overrightarrow{x}·\widehat{v}+tan\theta \overrightarrow{x}·\widehat{n}+{sec}^2\theta \overrightarrow{x}·\widehat{v})\frac{\partial p}{\partial t}-(sec\theta \overrightarrow{x}·\widehat{n}+sec\theta tan\theta \overrightarrow{x}·\widehat{v})\frac{\partial p}{\partial z}\\ =\frac{\partial p}{\partial \theta }+tan\theta (\overrightarrow{x}·\widehat{n}+tan\theta \overrightarrow{x}·\widehat{v})\frac{\partial p}{\partial t}-sec\theta (\overrightarrow{x}·\widehat{n}+tan\theta \overrightarrow{x}·\widehat{v})\frac{\partial p}{\partial z}\\ =\frac{\partial p}{\partial \theta }+ttan\theta \frac{\partial p}{\partial t}-tsec\theta \frac{\partial p}{\partial z}.\end{array}$$ (B.7)

Here we used ∂n̂/∂θ = −v̂, ∂û/∂θ = 0 and ∂v̂/∂θ = n̂. Using (B.3), we can rewrite the second term of (12) with a negative sign as

$${\sigma }^2(\widehat{n}·\nabla )(\widehat{v}·\nabla )q(\overrightarrow{x},\widehat{n})={\sigma }^{2}tan\theta \frac{{\partial }^{2}p}{{\partial }^{2}t}-{\sigma }^{2}sec\theta \frac{{\partial }^{2}p}{\partial z\partial t}.$$ (B.8)

Adding (B.7) and (B.8), we convert the second histo-image consistency equation (12) into the following sinogram consistency equation

$$\frac{\partial p}{\partial \theta }+ttan\theta \frac{\partial p}{\partial t}-tsec\theta \frac{\partial p}{\partial z}+{\sigma }^{2}tan\theta \frac{{\partial }^{2}p}{\partial t^2}-{\sigma }^{2}sec\theta \frac{{\partial }^{2}p}{\partial z\partial t}=0.$$ (B.9)

Equations (B.6) and (B.9) are identical to equations (4) and (2) with tan θ = δ in (Defrise et al. 2013). After removing the partial derivative with respect to z and using θ = 0, (B.6) becomes equation (8) in (Defrise et al. 2012), which was used to determine the attenuation sinogram from 2D TOF-PET data. Similar to (B.6) and (B.9), we can convert (15) to

$$sec\theta \frac{{\partial }^{2}p}{\partial z\partial \varphi }-tan\theta \frac{{\partial }^{2}p}{\partial \varphi \partial t}-s({sec}^2\theta +{tan}^2\theta )\frac{{\partial }^{2}p}{\partial z\partial t}+stan\theta sec\theta \frac{{\partial }^{2}p}{\partial z^2}+stan\theta sec\theta \frac{{\partial }^{2}p}{\partial t^2}-tan\theta cos\theta \frac{\partial p}{\partial s}+cos\theta \frac{{\partial }^{2}p}{\partial s\partial \theta }=0.$$ (B.10)

Equation (B.10) is just v̂ · ∇(11) −û · ∇(12), and it is equivalent to equation (6) in (Defrise et al. 2008) and equation (3) in (Defrise et al. 2013). After removing the three terms with the partial derivative of t, (B.10) becomes John’s equation for non-TOF PET data (John 1938, John 1982, Defrise and Liu 1999, Defrise et al. 2013).

Appendix C. Spatially invariant attenuation gradients along TOF direction

The attenuation histo-image a(x⃗, n̂) is invariant along the TOF direction n̂. To verify the internal consistency of the derivations in Section 2.4, we check below that the LS solution, (31) and (32), also has spatially invariant property along the TOF direction. Recalling that the argument of m and its derivatives in (30) is (x⃗₀ + ln̂, n̂), we can change the integration variable l + l₀ → l and rewrite the quantities in (30) at x⃗₀ + l₀n̂ for some l₀ as

$$\begin{array}{ll}{H}_{11}({\overrightarrow{x}}_0+l_0\widehat{n},\widehat{n})\hfill & =H_{11}({\overrightarrow{x}}_0,\widehat{n})+l_0^2{H}_{22}({\overrightarrow{x}}_0,\widehat{n})-2l_0{H}_{12}({\overrightarrow{x}}_0,\widehat{n}),\hfill \\ H_{12}({\overrightarrow{x}}_0+l_0\widehat{n},\widehat{n})\hfill & =H_{12}({\overrightarrow{x}}_0,\widehat{n})-l_0{H}_{22}({\overrightarrow{x}}_0,\widehat{n}),\hfill \\ H_{22}({\overrightarrow{x}}_0+l_0\widehat{n},\widehat{n})\hfill & =H_{22}({\overrightarrow{x}}_0,\widehat{n}),\hfill \\ J_1({\overrightarrow{x}}_0+l_0\widehat{n},\widehat{n})\hfill & =J_1({\overrightarrow{x}}_0,\widehat{n})-l_0{J}_2({\overrightarrow{x}}_0,\widehat{n}),\hfill \\ J_2({\overrightarrow{x}}_0+l_0\widehat{n},n)\hfill & =J_2({\overrightarrow{x}}_0,\widehat{n}).\hfill \end{array}$$ (C.1)

Then (29) at x⃗₀ + l₀n̂ becomes

$$\left[\begin{array}{ll}{H}_{11}+l_0^2{H}_{22}-2l_0{H}_{12}\hfill & H_{12}-l_0{H}_{22}\hfill \\ H_{12}-l_0{H}_{22}\hfill & H_{22}\hfill \end{array}\right]\left[\begin{array}{r}\hfill -(\widehat{u}·\nabla )loga({\overrightarrow{x}}_0+l_0\widehat{n},\widehat{n})\\ \hfill (\widehat{u}·{\nabla }_{\widehat{n}})loga({\overrightarrow{x}}_0+l_0\widehat{n},\widehat{n})\end{array}\right]=\left[\begin{array}{c}{J}_1-l_0{J}_2\\ J_2\end{array}\right].$$ (C.2)

Subtracting −l₀ times the second row from the first row, we obtain

$$\left[\begin{array}{ll}{H}_{11}\hfill & H_{12}\hfill \\ H_{12}\hfill & H_{22}\hfill \end{array}\right]\left[\begin{array}{cc}1& 0\\ -l_0& 1\end{array}\right]\left[\begin{array}{r}\hfill -(\widehat{u}·\nabla )loga({\overrightarrow{x}}_0+l_0\widehat{n},\widehat{n})\\ \hfill (\widehat{u}·{\nabla }_{\widehat{n}})loga({\overrightarrow{x}}_0+l_0\widehat{n},\widehat{n})\end{array}\right]=\left[\begin{array}{c}{J}_1\\ J_2\end{array}\right].$$ (C.3)

From (C.3), we obtain the following solution

$$-\widehat{u}·\nabla loga({\overrightarrow{x}}_0+l_0\widehat{n},\widehat{n})=\frac{H_{22}{J}_1-H_{12}{J}_2}{H_{11}{H}_{22}-H_{12}^2},$$ (C.4)

$$\widehat{u}·{\nabla }_{\widehat{n}}loga({\overrightarrow{x}}_0+l_0\widehat{n},\widehat{n})+l_0\widehat{u}·\nabla loga({\overrightarrow{x}}_0+l_0\widehat{n},\widehat{n})=\frac{H_{11}{J}_2-H_{12}{J}_1}{H_{11}{H}_{22}-H_{12}^2}.$$ (C.5)

After comparing (31, 32) and (C.4, C.5), we return to (22) and (24).