Reconstruction of difference in sequential CT studies using penalized likelihood estimation

A simplified flowchart for the RoD approach is presented in figure 1. The methodology presumes that a previously acquired image volume, μ_p, is available to serve as a prior image. A subsequent acquisition of tomographic projection data, y, is available where the current anatomy is presumed to share many similarities with the prior image. The current anatomy and the prior image are not necessarily registered. Thus, a 2D-to-3D registration is conducted to form an appropriately transformed prior image, $W(\lambda ){{\mu}_{\text{p}}}$, where the registration operator, W, is parameterized by the vector λ. As opposed to traditional reconstruction methods where one attempts to reconstruct the true current anatomy, ${{\mu}_{\text{true}}}$, from y, we instead propose to reconstruct only the difference, ${{\mu}_{\Delta }}$, between the current anatomy and the prior image. The RoD estimator takes the raw data and the transformed prior image as an input to produce an estimate of ${{\mu}_{\Delta }}$. The difference image itself is often of interest; however, if desired, one may compute μ from ${{\mu}_{\Delta }}$, and ${{\mu}_{\text{p}}}$. We allow the possibility of alternating optimizations between the 2D-to-3D registration and the RoD estimation to permit for possible refinement.

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A detailed discussion of the algorithm follows in section 2.2 where we introduce the general RoD framework and the likelihood-based joint reconstruction and registration objective function. Next, the optimization of the objective function and its numerical implementation is explained in 2.3. Finally, a series of cone-beam CT simulation and test-bench experiments are described in sections 2.4 and 2.5.

Consider the following model for the mean measurements of a transmission tomography system

$$\begin{eqnarray}&&{{\bar{y}}_{i~}}=~{{b}_{i}}\centerdot \exp \left(-{{\left[\mathbf{A}\mu \right]}_{i}}\right)\end{eqnarray} \tag{ 1 }$$

where ${{b}_{i}}$ is a gain term associated with the number of unattenuated photons (x-ray fluence) and detector sensitivities, μ is vector of attenuation coefficients representing the current anatomy, $\mathbf{A}$ is the system matrix, and ${{\left[\mathbf{A}\mu \right]}_{i}}$ is the line integral associated with the ith measurement. We presume that ${{y}_{i}}$ is independent and Poisson distributed.

We model the current image volume as the sum of a registered prior image, μ_p, and a difference image, μ_Δ, such that

$$\begin{eqnarray}&&\mu \,=W(\lambda ){{\mu}_{\text{p}}}+{{\mu}_{\Delta }},\end{eqnarray} \tag{ 2 }$$

where W is a general transformation operator with parameters $\lambda$ and could potentially represent deformable registration. In this work we parameterized $W$ as a rigid transform as explained in detail in Stayman et al (2012). With this object model, we can rewrite all the measurements from (1) in the following vector form

$$\begin{eqnarray}&&\bar{y}=b\centerdot \exp \left(-\mathbf{A}W(\lambda ){{\mu}_{p}}\right)\centerdot \exp \left(-\mathbf{A}{{\mu}_{\Delta }}\right)\end{eqnarray} \tag{ 3 }$$

where the · operator denotes an element-by-element vector multiplication. Furthermore, we may combine the first two terms of (3) into a single gain parameter, $~g$, such that

$$\begin{eqnarray}&&\bar{y}=~g(\lambda )\centerdot \exp \left(-\mathbf{A}{{\mu}_{\Delta }}\right).\end{eqnarray} \tag{ 4 }$$

Equation (4) is convenient since it reduces the difference forward model to the same form as the traditional forward model in (1). This formulation will permit the use of standard reconstruction algorithms with only a redefinition of the gain term. Similarly, the factorization in (4) separates the dependence of λ on ${{\mu}_{\Delta }}$, and suggests that the registration may also be easily decoupled from the reconstruction.

We choose the following PL objective for the reconstruction of the difference image:

$$\begin{eqnarray}&&\Phi\left({{\mu}_{\Delta }},\lambda ;y,{{\mu}_{\text{p}}}\right)=-L\left({{\mu}_{\Delta }},\lambda ;y,{{\mu}_{\text{p}}}\right)+{{\beta}_{\text{R}}}\parallel \Psi{{\mu}_{\Delta }}{{\parallel}_{1}}+~{{\beta}_{\text{M}}}\parallel {{\mu}_{\Delta }}{{\parallel}_{1}}\end{eqnarray} \tag{ 5 }$$

with the implicitly defined estimator:

$$\begin{eqnarray}&&\left\{{{\hat{\mu}}_{\Delta }},\hat{\lambda}\right\}=\arg \underset{{{\mu}_{\Delta }},\lambda}{\mathop{\min}}\,\Phi\left({{\mu}_{\Delta }},\lambda ;y,{{\mu}_{\text{p}}}\right)\end{eqnarray} \tag{ 6 }$$

The (Poisson) log-likelihood function is denoted with L. Our objective function uses two regularization terms leveraging sparsity in multiple domains. This is akin to other work that regularizes in multiple domains (Dutta et al 2012, Xu et al 2014). The second term in (5) is a traditional edge-preserving roughness penalty term that encourages smooth solutions and whose strength is controlled by the scalar regularization parameter ${{\beta}_{\text{R}}}$. In this work, we chose $\Psi$ to be a local pairwise voxel difference operator for a first-order neighborhood. To ensure a differentiable objective, the l₁ norm is approximated using a Huber penalty function (Huber 1964) with a small δ parameter. The parameter δ controls the location of the transition between the quadratic and linear portions of the Huber function. In this work we used δ  =  10⁻⁴ mm⁻¹ for all reconstructions. The third term in (5) is a magnitude penalty on μ_Δ with strength $\,{{\beta}_{\text{M}}}$ that encourages the difference image itself to be sparse (e.g. the change in anatomy is local and relatively small).

While the roughness penalty is fairly intuitive in its control of the noise-resolution tradeoff, the function of the magnitude penalty is more complex. This penalty controls the amount of prior image information used in image formation. A large $\,{{\beta}_{\text{M}}}$ forces the difference image to be closer to zero, enforcing smaller allowable differences from the prior image. A small $\,{{\beta}_{\text{M}}}$ permits larger differences from the prior image and therefore a greater reliance on the current projection data; however, this increased reliance on the measurement data can also lead to attenuation differences due to noise. Thus, a proper balance and control of prior information inclusion must be selected. This trade-off is investigated in following sections.

We propose to solve the optimization in (6) using a two-step alternating approach to jointly solve for $\hat{\lambda}$ and ${{\hat{\mu}}_{\Delta }}$. That is, the registration parameters $\lambda$ are updated using a traditional gradient-based approach with a fixed attenuation estimate, and the difference image ${{\mu}_{\Delta }}$ is estimated iteratively using a tomography-specific image update with fixed registration.

Mathematically, we may write the registration step of the problem as

$$\begin{eqnarray}&&{{\lambda}^{\left[n\right]}}=\arg \underset{\lambda \in {{\mathbb{R}}^{6}}}{\mathop{\min}}\,\Phi\left(\lambda ;y,{{\mu}_{\text{p}}},\mu _{\Delta }^{\left[n-1\right]}\right)=\arg \underset{\lambda \in {{\mathbb{R}}^{6}}}{\mathop{\min}}\,\left\{-L\left(\lambda ;y,{{\mu}_{\text{p}}},\mu _{\Delta }^{\left[n-1\right]}\right)\right\}\end{eqnarray} \tag{ 7 }$$

Equation (7) represents a 2D-to-3D likelihood-based rigid registration approach. This same form of registration was discussed in detail in Stayman et al (2012). In short, the W operator in (2) is parameterized using B-spline kernels to ensure its differentiability. This allows us to use a quasi-Newton update method using the Broyden–Fletcher–Goldfarb–Shanno (BFGS) (Liu and Nocedal 1989) updates to optimize the objective function (7). The required function and gradient evaluations are straightforward to compute and may be derived from (5) by eliminating factors dependent only on attenuation (including the regularization terms). The bracketed superscript, [n], denotes the nth estimate of the parameter vector, formalizing that the nth alternation of registration updates depends on the previous, (n  −  1)th alternation of image updates.

For the image volume updates, we may write the optimization part of the problem as:

$$\begin{eqnarray}&&\mu _{\Delta }^{\left[n\right]}=\arg \underset{{{\mu}_{\Delta }}\in {{\mathbb{R}}^{{{N}_{\mu}}}}}{\mathop{\min}}\,\Phi\left({{\mu}_{\Delta }};y,{{\mu}_{\text{p}}},{{\lambda}^{\left[n\right]}}\right)\\ &&\quad \ \;=\arg \underset{{{\mu}_{\Delta }}\in {{\mathbb{R}}^{{{N}_{\mu}}}}}{\mathop{\min}}\,\left\{-L\left({{\mu}_{\Delta }};y,{{\mu}_{\text{p}}},{{\lambda}^{\left[n\right]}}\right)+{{\beta}_{\text{R}}}\parallel \Psi{{\mu}_{\Delta }}{{\parallel}_{1}}+~{{\beta}_{\text{M}}}\parallel {{\mu}_{\Delta }}{{\parallel}_{1}}\right\}\end{eqnarray} \tag{ 8 }$$

which includes a transformed prior with fixed $\lambda$ from the previous set of registration updates. The roughness and magnitude penalty terms satisfy the five criteria in (Erdogan and Fessler 2002) for finding paraboloidal surrogates. Therefore, we may adopt the separable paraboloidal surrogates (SPS) approach introduced by Erdogan and Fessler in 1999 with ordered-subsets subiterations for improved convergence rates. The difference image ${{\mu}_{\Delta }}$ represents the change in attenuation coefficients between scans and could have positive or negative values. Consequently, traditional non-negativity constraints on the reconstruction are not applied. The SPS image update equation can be readily derived:

$$\begin{eqnarray}&&{{\left[\mu _{\Delta }^{\left[n+1\right]}\right]}_{j}}~=~{{\left[\mu _{\Delta }^{\left[n\right]}\right]}_{j}}+\frac{\underset{i=1}{\overset{N}{\mathop \sum}}\,{{\mathbf{A}}_{ij}}{{\overset{\centerdot}{\mathop{h}}\,}_{i}}\left({{\left[\mathbf{A}\mu _{\Delta }^{\left[n\right]}\right]}_{i}}\right)-~{{\beta}_{\text{R}}}\underset{k=1}{\overset{K}{\mathop \sum}}\,{{\Psi}_{kj}}~\overset{\centerdot}{\mathop{f}}\,\left({{\left[\Psi\mu _{\Delta }^{\left[n\right]}\right]}_{k}}\right)-{{\beta}_{\text{M}}}\underset{k=1}{\overset{K}{\mathop \sum}}\,\overset{\centerdot}{\mathop{f}}\,\left({{\left[\mu _{\Delta }^{\left[n\right]}\right]}_{k}}\right)}{\underset{i=1}{\overset{N}{\mathop \sum}}\,\mathbf{A}_{ij}^{2}{{c}_{i}}\left({{\left[\mathbf{A}\mu _{\Delta }^{\left[n\right]}\right]}_{i}}\right)+{{\beta}_{\text{R}}}\underset{k=1}{\overset{K}{\mathop \sum}}\,\Psi_{kj}^{2}~\omega \left({{\left[\Psi\mu _{\Delta }^{\left[n\right]}\right]}_{k}}\right)+{{\beta}_{\text{M}}}\underset{k=1}{\overset{K}{\mathop \sum}}\,\omega \left({{\left[\mu _{\Delta }^{\left[n\right]}\right]}_{k}}\right)}\end{eqnarray} \tag{ 9 }$$

where ${{c}_{i}}$ are optimal curvatures, $\overset{\centerdot}{\mathop{f}}\,$ is the derivative of the Huber penalty function, and $\omega (t)={\mathop{f(\overset{\centerdot}{\mathop{t}})/t}}\,$ calculated according to Erdogan and Fessler (2002). The derivatives of marginal log-likelihoods were defined as ${{\overset{\centerdot}{\mathop{h}}\,}_{i}}(l)={{g}_{i}}{{\text{e}}^{-{{l}_{i}}}}-{{y}_{i}}$, with ${{g}_{i}}=~{{b}_{i~}}{{\text{e}}^{{{\left[-\boldsymbol{A}W\left({{\lambda}^{\left[n\right]}}\right){{\mu}_{\text{p}}}\right]}_{i}}}}$.

Table 1 depicts pseudocode for the alternating joint registration and image update approach. The outer loop iterates over registration and image updates each with their own inner loops over BFGS and ordered subsets iterations, respectively.

Table 1. Pseudo code for the proposed reconstruction of difference approach.

$\mu _{\Delta }^{\left[0\right]}=$ Initial guess for difference image (zero or difference between FBPs)

${{\lambda}^{\left[0,0\right]}}=$ Initial guess for registration parameters

${{\mathbf{H}}^{\left[0,0\right]}}=\sigma \mathbf{I}$, Initial guess for inverse Hessian

FOR $n=0$ to ${{N}_{\text{iter}}}$

% registration update block

FOR $r=1$ to $R$

Compute ${{\nabla}_{\lambda}}\Phi\left({{\lambda}^{\left[n,r-1\right]}};y,{{\mu}_{p}},\mu _{\Delta }^{\left[n\right]}\right)$

${{\mathbf{H}}^{\left[n,r\right]}}=$ BFGS update based on $\left\{{{\nabla}_{\lambda}}\Phi\left({{\lambda}^{\left[n,r-1\right]}}\right),{{\mathbf{H}}^{\left[n,r-1\right]}}\right\}$

$\hat{\gamma}=$ line search in ${{\lambda}^{\left[n,r-1\right]}}+\gamma {{\mathbf{H}}^{\left[n,r\right]}}~{{\nabla}_{\lambda}}\Phi\left({{\lambda}^{\left[n,r-1\right]}}\right)$

${{\lambda}^{\left[n,r\right]}}={{\lambda}^{\left[n,r-1\right]}}+\hat{\gamma}{{\mathbf{H}}^{\left[n,r\right]}}~{{\nabla}_{\lambda}}\Phi\left({{\lambda}^{\left[n,r-1\right]}}\right)$

END

${{\lambda}^{\left[n+1,0\right]}}={{\lambda}^{\left[n,R\right]}};~{{\mathbf{H}}^{\left[n+1,0\right]}}={{\mathbf{H}}^{\left[n,R\right]}}$

% image update block

FOR $m=1$ to $M$ (number of ordered subsets)

${{g}_{i}}=~{{b}_{i~}}{{\text{e}}^{{{\left[-\mathbf{A}\text{W}\left({{\lambda}^{\left[n+1\right]}}\right){{\mu}_{\text{p}}}\right]}_{i}}}}$, $~{{l}_{i}}={{\left[\mathbf{A}\mu _{\Delta }^{\left[n\right]}\right]}_{i}}$, ${{\overset{\centerdot}{\mathop{h}}\,}_{i}}(l)={{g}_{i}}{{\text{e}}^{-{{l}_{i}}}}-{{y}_{i}}\,\forall i\in {{S}_{\text{m}}}$

FOR $j=1$ to ${{N}_{\mu}}$

${{\left[\mu _{\Delta }^{\left[n\right]}\right]}_{j}}~=~{{\left[\mu _{\Delta }^{\left[n\right]}\right]}_{j}}+\frac{M\underset{i\in {{S}_{\text{m}}}}{\mathop{\mathop{\sum}^{{}}}}\,{{\mathbf{A}}_{ij}}{{\overset{\centerdot}{\mathop{h}}\,}_{i}}\left(~{{l}_{i}}\right)-~{{\beta}_{\text{R}}}\sum_{k=1}^{K}{{\Psi}_{kj}}~\overset{\centerdot}{\mathop{f}}\,\left({{\left[\Psi\mu _{\Delta }^{\left[n\right]}\right]}_{k}}\right)-~{{\beta}_{\text{M}}}\sum_{k=1}^{K}\overset{\centerdot}{\mathop{f}}\,\left({{\left[\mu _{\Delta }^{\left[n\right]}\right]}_{k}}\right)}{M\underset{i\in {{S}_{\text{m}}}}{\mathop{\mathop{\sum}^{{}}}}\,\mathbf{A}_{ij}^{2}{{c}_{i}}\left(~{{l}_{i}}\right)~+~{{\beta}_{\text{R}}}\sum_{k=1}^{K}\Psi_{kj}^{2}~\omega \left({{\left[\Psi\mu _{\Delta }^{\left[n\right]}\right]}_{k}}\right)+~{{\beta}_{\text{M}}}\sum_{k=1}^{K}\omega \left({{\left[\mu _{\Delta }^{\left[n\right]}\right]}_{k}}\right)}$

END

END

END

The simultaneous image update in (9) can be easily parallelized for efficient computation on a graphical processing unit (GPU). We implemented all routines in Matlab (The Mathworks, Natick, MA) with calls to custom external libraries for the separable-footprint (Long et al 2010) projectors and back-projectors in C/C  +  + using CUDA libraries for execution on GPU. We have not directly compared the computational efficiency of this implementation to un-parallelized methods in this work; however, details of its computation efficiency and other possible speedup techniques are discussed in Wang et al (2014).

We simulated growth of a spherical lesion in the nasal cavity of an anthropomorphic head phantom. We formed a digital phantom from relatively low-noise cone-beam CT (CBCT) measurements (100 kVp, 453 mAs, 720 projections over 360°) using a CBCT test-bench (explained in detail in the next section) and a PL reconstruction with 0.5 mm isotropic voxel size. This image was used as the prior image ${{\mu}_{\text{p}}}$ for subsequent studies. A spherical lesion with 10.5 mm diameter and 0.02 mm⁻¹ attenuation (simulating a tumor, mucocele, or other abnormality to be detected in the RoD image) was digitally added to the nasal cavity, as seen in figure 4(A), to create the ground truth image for the current anatomy. Simulated new measurements were then created by projecting this 'with lesion' volume for 720 angles over 360°. Acquisitions with different levels of x-ray fluence were simulated by adding various levels of Poisson noise to the noiseless measurements. These data sets were used to investigate sensitivity to the regularization parameters (2.4.1), local versus global reconstruction (2.4.2), and performance of RoD with varying data fidelity (2.4.3). A separate dataset was simulated by rigid transformation of the prior image with a set of known $\lambda$ to investigate the performance of the registration step on image quality in section 2.4.4. In all experiments, the root-mean-square error (RMSE) between the RoD estimate and the ground truth difference image was used as a measure of image quality. The error was calculated over a large (100  ×  100 voxel) neighborhood around the spherical lesion in order to include the boney structures in the background as well as the air and soft tissue of the nasal cavity.

2.4.1. Regularization investigation.

2.4.2. Local versus global reconstruction of difference.

2.4.3. RoD performance versus data fidelity.

2.4.4. Performance of likelihood-based registration.

In addition to the previous simulation experiments, we acquired physical CBCT measurements using the test-bench setup in figure 2. The bench consisted of a Rad-94 x-ray source (Varian Medical Systems, Palo Alto, CA), a Varian PaxScan 4030CB flat-panel detector with 40  ×  30 cm² total size and 0.388 mm pixel pitch after 2  ×  2 binning, and a motion control system (Parker Hannifin, Mayfield Heights, OH). The bench geometry was chosen to simulate a C-arm system with a 118 cm source-to-detector distance and 77.4 cm source-to-axis distance.

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We placed an acrylic sphere with 12.5 mm diameter inside the nasal cavity of an anthropomorphic head phantom (The Phantom Laboratory, Salem, NY) to simulate a tumor growth (see inset in figure 2). Two scans were performed at 100 kVp and 453 mAs with 720 projections over 360° with and without the sphere inserted. The phantom was not moved during the removal of the sphere and between scans. We used a PL reconstruction of the scan with no sphere as the prior; similarly a PL reconstruction of the scan with the sphere inserted was used as the ground truth for the current anatomy.

We examined the performance of RoD in reconstructing a 3D volume, with an unregistered prior, to that of FBP and PL algorithms in a noisy scan. We created a rigid transformation matrix to form a misregistered prior image. The registration parameters ($\lambda$:  −4, 5, and  −10 mm shifts and 10°,−7°, and 30° rotations for x, y, and z axes respectively) were within the capture range of translations and rotations calculated from the results of the experiment explained in 2.4.4. Poisson noise was added to the projections of the 'with tumor' measurements to simulate fluence of 5000 photons and the penalty coefficients were chosen based on the results of the exhaustive search of the simulated data described in section 2.4.1.

We explored the effects of the penalty coefficients ${{\beta}_{\text{R}}},\,{{\beta}_{\text{M}}}$ on reconstructed image quality as explained in section 2.4.1. Figure 3(A) shows the RMSE (relative to truth) as a function of both regularization parameters, while figure 3(B) shows the zoomed ROI difference images that result from the (global) RoD approach. As expected, increasing ${{\beta}_{\text{R}}}$ resulted in less noisy difference images by forcing the reconstruction to be spatially smooth; coefficient values larger than 10² completely blurred the difference image. Similar to the roughness penalty coefficient, large values of ${{\beta}_{\text{M}}}$ also created less noisy images but by different means: the magnitude penalty forces the reconstructed image to be sparse. In the extreme, ${{\beta}_{\text{M}}}$ larger than 10^4.5 forced the reconstructed image to be completely sparse (i.e. zero everywhere). In this case ${{\beta}_{\text{R}}},\,{{\beta}_{\text{M}}}$  =  10^1.5 produced the best image quality in terms of RMSE. Exhaustive search of the 2D ${{\beta}_{\text{R}}},\,{{\beta}_{\text{M}}}$ space may be time-consuming especially in the case of large 3D volume reconstructions; however, the basic trend in RMSE map suggests that it is possible to find the optimum point by starting from low regularization levels and performing 1D searches first along the ${{\beta}_{\text{R}}}$, and next along the ${{\beta}_{\text{M}}}$ axes. This is somewhat similar to the results found and reported in Dang et al (2014b).

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The results of the global and local RoD experiment described in section 2.4.2 are summarized in figure 4. Both the summed current anatomy volume, ${{\mu}_{\text{p}}}+{{\hat{\mu}}_{\Delta }}$, and the difference volume, ${{\hat{\mu}}_{\Delta }}$ are shown for the local and global approaches. Images reconstructed were with RMSE-optimal penalty coefficients (though omitted for brevity, penalty coefficient RMSE maps for the local and global RoD showed similar trends.) RMSE was 4.02  ×  10⁻⁴ (mm⁻¹) for the local and 4.19  ×  10⁻⁴ (mm⁻¹) for the global protocols. The results indicate that the performance of local and global RoD were comparable, and there was slight improvement in RMSE when local RoD was used. This is possibly due to the fact that local RoD enforces zero difference image values outside the ROI, whereas global RoD must estimate those voxel values outside the ROI with some potential propagation of error/noise from the outside into the ROI where RMSE is computed. Since local RoD was faster to compute and at least as accurate as the global approach, we used the local approach for the remainder of the simulation experiments.

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We tested the performance of RoD under different levels of data fidelity. We changed the fidelity of the measurements by: (1) simulating noisy measurements, and (2) subsampling the number of projections. The results of the performance of RoD compared to PL in reconstructing measurements with decreasing photon fluence are presented in figure 5. The performance of both methods deteriorated as the photon fluence decreased to the point that at fluence of 100 (photons per pixel) both methods failed to reconstruct the difference; however, RoD performed consistently better than PL. The PL anatomical change images were calculated by subtracting the prior image from the PL estimate and show structural differences in the background.

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Note the significant amount of anatomical structure in the PL difference images (particularly due to bones near the sinus cavity) as opposed to RoD which does not exhibit such structure and yields an image much closer to the true anatomical difference. This performance difference is due, in part, to the differing regularization between the prior image and the PL reconstruction, whereas the regularization in RoD can be adjusted to mitigate the appearance of such differences. Figure 6 shows a similar trend in the performance of PL and RoD methods at different levels of data sparsity. The structural differences are also present in the background of subtracted PL images. Furthermore, the performance of RoD does not decline much compared to PL when the number of projections is reduced from 360 to 90.

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We tested the performance of the likelihood-based rigid registration of the global RoD algorithm under various random single translations and rotations in 3D as explained in 2.4.4. These results are summarized in figure 7. The RMSE in reconstruction for ensembles of random translation is shown in the top row, and for ensembles of random rotation in the bottom row. There is a fairly well-defined limit beyond which the RMSE rises dramatically. The capture range (defined as the range for which RMSE differs by  <  ±0.0005 mm⁻¹ of that in preregistered reconstruction) for translations along the $x,y,z$ axes were [−16 10], [−29 9], and [−11 10] (in mm), respectively. The translation capture ranges were limited by the prior image being cropped by moving outside the reconstructed volume by the transformation; theoretically the translation capture range can be improved by using a large enough reconstruction FOV. The capture range for a single random rotation about any axis was at least  ±50°. Overall, this is a broad capture range consistent with robust rigid registration and suggests that the method can handle large errors in initialization in the alternating optimization.

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For the multivariate registration experiment, where we tested the performance of the registration algorithm using 10 sets of random $\lambda$ with nonzero elements, the mean and standard deviation of the RMSE was 0.02  ±  0.0005 mm⁻¹. That is, all cases converged to essentially the same results in terms of registration and image quality (though there were minor differences due to noise).

Figure 8 shows the results of the 3D reconstruction of the physical CBCT test-bench data. Reconstructions using FBP, PL, and RoD with an unregistered prior image are presented. Both current anatomy volumes and (absolute) difference volumes are presented for each method. The RMSE was (1.27, 1.57, 1.97)  ×  10⁻³ mm⁻¹ for RoD, PL, and FBP reconstructions, respectively. The difference images for the PL and FBP change images included high levels of noise in the background. Moreover, the PL reconstruction exhibited highly structured noise showing anatomical boundaries (in particular the boundary between water and bone) as seen and described in previous studies. In contrast, the RoD does not show much noise outside the actual change and yields a much more accurate reconstruction of the difference image.

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