Anatomy-guided multi-resolution image reconstruction in PET

Objective. In this paper, we propose positron emission tomography image reconstruction using a multi-resolution triangular mesh. The mesh can be adapted based on patient specific anatomical information that can be in the form of a computed tomography or magnetic resonance imaging image in the hybrid imaging systems. The triangular mesh can be adapted to high resolution in localized anatomical regions of interest (ROI) and made coarser in other regions, leading to an imaging model with high resolution in the ROI with clearly reduced number of degrees of freedom compared to a conventional uniformly dense imaging model. Approach. We compare maximum likelihood expectation maximization reconstructions with the multi-resolution model to reconstructions using a uniformly dense mesh, a sparse mesh and regular rectangular pixel mesh. Two simulated cases are used in the comparison, with the first one using the NEMA image quality phantom and the second the XCAT human phantom. Main results. When compared to the results with the uniform imaging models, the locally refined multi-resolution mesh retains the accuracy of the dense mesh reconstruction in the ROI while being faster to compute than the reconstructions with the uniformly dense mesh. The locally dense multi-resolution model leads also to more accurate reconstruction than the pixel-based mesh or the sparse triangular mesh. Significance. The findings suggest that triangular multi-resolution mesh, which can be made patient and application specific, is a potential alternative for pixel-based reconstruction.


Introduction
Positron emission tomography (PET) is a medical imaging modality that provides information on the distribution of a biologically active radioactive tracer molecules in a patient.The radioactive tracer molecules undergo β + -decay where a positron is emitted.The positron eventually collides with an electron, resulting in annihilation of both particles, and releases two 511 keV gamma photons in opposite directions, which are detected by the PET scanner.When the two photons are detected by the PET scanner in a certain time window, it is called a coincidence event.These coincidence events are stored to form measurement data that needs to be reconstructed into an image, in order to obtain the information of the tracer distribution.The current standard in PET is to use iterative image reconstruction methods, where either maximum likelihood (ML) or maximum a posteriori (MAP) estimate is solved, with a wide range of algorithms already developed (Iriarte et al 2016, Wettenhovi et al 2021).
In most medical imaging modalities, including PET, the unknown images are discretized in a 2D pixel or 3D voxel grids.The problem with these pixel-based methods is the trade-off between computation time and accuracy of the reconstruction.For instance, if high accuracy is desired, the use of these pixel-based grids can lead to large discrete systems with long computation times.To address this issue, there has been interest to develop content adaptive, irregular triangular mesh methods (Brankov et al 2004, Sitek et al 2006, Pereira and Sitek 2010, Boutchko et al 2013, Massanes and Brankov 2014, Chen et al 2020).The advantage of the triangular mesh methods over the pixel-based methods is that these meshes are flexible with respect to how coarse or fine detail is wanted in different parts of the reconstructed image.This is due to the fact that in triangular meshes the location of vertices can be freely chosen contrary to pixel-based grids where vertices have to be at equal distances.
This flexibility can lead to a decreased size of the discrete system and, consequently, improved computation time.Additionally, the irregular triangular meshes are especially useful because they can utilize anatomical data provided by the hybrid PET scanners, since the current clinical scanners have either CT or MRI built-in.The previous works (Brankov et al 2004, Sitek et al 2006, Pereira and Sitek 2010, Boutchko et al 2013, Massanes and Brankov 2014, Chen et al 2020) are mainly focused on developing algorithms for obtaining an irregular mesh, either triangular or tetrahedral, which are then compared to pixel-or voxel-based methods.This work is focused on the utilization of an anatomical image for construction of a content-adaptive and locally dense multiresolution mesh, comparison of results against generic uniform imaging models, and the validation of such models.
In this work, we have compared the standard pixel model to the triangular mesh model in two different cases for PET imaging.For triangular mesh model, three different meshes are generated and two different algorithms are used for the reconstruction in both cases.The three meshes are a locally dense multi-resolution triangular mesh, uniformly dense mesh and sparse triangular mesh.The results in the sparse mesh are computed as a comparison for reconstruction time whereas the uniform dense mesh is computed as a comparison for accuracy.As for the reconstruction algorithms, the reason that two different algorithms were chosen, is the comparison of the different iterative methods with the triangular mesh model.Additionally, we incorporated a total variation prior to evaluate the effect of spatial regularization in the triangular mesh reconstructions.We have demonstrated that PET reconstructions in a locally dense multi-resolution triangular mesh will yield at least the same accuracy to the industry standard pixel model but with significantly smaller number of unknowns, and are able to conform to the anatomical outlines better.In this work, two different shapes for field-of-view are presented, a circular and a square shape.In the square case, triangular mesh was formed based on anatomical data, which demonstrates easy application of anatomical outlines in the mesh constructions.
This paper is organized as follows.In section 2 the algorithms used in this work are presented.Section 3 describes the data and meshes used.In section 4 the results of this work are presented.Finally, in section 5 we discuss the results and future work.

Theory
The mathematical problem in PET imaging can be written as where m is a vector containing the obtained measurements, A is a system matrix that models the relationship between the measurements and reconstructed image and f is a vector containing the reconstructed image.
In PET, the measurements m are often presented in the form of sinograms.In the sinograms, coincidence events are assigned to a line-of-response (LOR), which is a line between the two detectors that detected the photons of a coincidence event.Each LOR is presented in the sinogram with respect to the angle of the LOR and the orthogonal distance from the center of the field-of-view (FOV) of the PET system.Thus, each element of a sinogram contains the amount of detected coincidences for a specific LOR.
The system matrix, in its simplest form, portrays the probability of an event originating from a pixel j is detected in LOR i.In this form the system matrix is equal to a geometric projection matrix.The system matrix can also include for example correction matrices, but in this paper these were not considered.
In this paper, the system matrix is also calculated in a triangular mesh.The triangular mesh partitions the FOV into a collection of nonoverlapping triangles, called elements, that are defined by vertices.In the triangular mesh, the direction of each LOR is defined by the coordinates of the two detectors.A projection of a ray is formed to find from which boundary element and triangle edge the ray is coming from.The ray is then propagated through the mesh, in each element the triangle edge and distance in the element to the next triangle edge is determined.Further, the direction of the ray in each element is compared with the direction of the LOR.If the direction of the ray is aligned with the direction of the LOR, the distance that the ray travels in the element is calculated and each of the vertices connected to the element that the ray is going through are given values based on the distance traveled in the element.Finally, the distance traveled for each LOR divided by the total length of the LOR is allocated to the system matrix to the corresponding row and column.
The measurements in PET can be modeled as Poisson distributed events, since the coincidence events are measured individually as a counting process.A well known iterative algorithm to utilize this connection is the maximum likelihood expectation maximization (MLEM) algorithm (Dempster et al 1977, Shepp 1982).The MLEM algorithm solves the Poisson log-likelihood which is of the form is the expected value of coincidence events along LOR i where M is the number of pixels or vertices in the estimated image.The expected value can include estimated random and scattered coincidences, but these are not included in this paper.
From (2) we can obtain the iterative MLEM algorithm in matrix form as (Wettenhovi et al 2021) where A T 1 = ∑ i a ij is the sensitivity image and k is the iteration number.The operation Af k corresponds to the forward projection and the operation A T m Af k corresponds to the backprojection.A known disadvantage of MLEM is its slow convergence (Qi andLeahy 2006, Wettenhovi et al 2021).This has led to the development of variations of the MLEM algorithm, such as the current standard in clinical use, ordered subsets expectation maximization (OSEM) algorithm (Hudson andLarkin 1994, Qi andLeahy 2006).In the OSEM algorithm, the measurements and system matrix are divided into N S subsets, which accelerates the convergence based on the number of subsets.The OSEM algorithm image update is obtained by , 1 contain the LORs in the subset S. The downside of OSEM is that it cannot guarantee convergence to the ML estimate (Qi andLeahy 2006, Wettenhovi et al 2021).
Besides the ML methods, estimates can be obtained by maximizing the log-posterior instead.This has been implemented to the MLEM algorithm in one-step late (OSL) method (Green 1990).In the OSL method, the image update is obtained by In PET, a number of different priors have been developed (Ehrhardt et al 2019, Kang andLee 2021).These priors have been mainly applied to the pixel model, however, a Gibbs prior (Brankov et al 2004) has been applied to a triangular mesh.In single photon emission computed tomography (SPECT) a total variation (TV) (Chen et al 2020) prior has been utilized in the reconstruction of a triangular mesh.Some of the priors are computationally more difficult to apply to triangular meshes because the vertices, and elements, are distributed irregularly unlike with pixel meshes where the pixels are distributed uniformly.However, TV has been widely utilized in both pixel and triangular meshes (Tian et al 2011, Wang et al 2014, González et al 2017, Chen et al 2020).TV is a regularizer used in image reconstruction to preserve edges, given the assumption that the image is piecewise constant.In this work, TV is included in the OSL algorithm in the form based on (Arridge et al 2014, Harhanen et al 2015) where β is a small rounding parameter that allows R(f) to be differentiable.

Materials and methods
For this paper, we simulated two data sets in GATE (Jan et al 2004).In the NEMA case, an image of the number of photons detected in a GATE simulation was used in COMSOL to form the meshes and subsequently to form the simulated data by using the image as the true radioactivity distribution.Each of the three meshes had a diameter of 100 mm that corresponds to the length of one side of the square FOV in the scanner which is also the square pixel mesh FOV size.Since the triangular mesh FOV is circular, the total area is smaller than in the pixel mesh case.The dense mesh contained 4229 vertices with element size between 1 and 2 mm.The sparse mesh contained 1922 vertices with element size between 2 and 3 mm.Lastly, the locally dense mesh contained a sparse background region and a dense region around five cylinders of varying size.The number of vertices was 1251 with element sizes between 1 and 4 mm.The dense area in the locally dense mesh covered a slightly larger region than the size of the cylinders in the true radioactivity distribution.The meshes and the true radioactivity distribution are shown in figure 2. For the NEMA case alone, a fourth, denser, mesh was obtained with 26 637 vertices and an element size between 0.5 and 0.8 mm.This denser mesh was used to compute a system matrix that was then used in the forward problem of (1) with the true radioactivity distribution as f to form new simulated data.Inverse crime was avoided by inputting the resulting vector m as the expected value for Poisson noise and then using the output as the measurement data in the reconstructions.In short, the GATE simulated data itself was not used for reconstructions, but rather the true radioactivity distribution obtained from the GATE simulations was used as the image from which the final simulated measurements were obtained by using the dense mesh system matrix.
The second data set using the XCAT phantom used the measurement data obtained from the GATE simulations directly as the input measurement data for the reconstructions.From the XCAT phantom, five transverse thorax slices were selected with 1 mm thickness each.The duration of the simulated measurement was 10 minutes using back-to-back photons in GATE.Half-life corresponding to F-18 was used.Unlike in the NEMA case, all the meshes were created using a CT-based attenuation image.This was achieved by first obtaining the contour lines for the anatomical regions and then using them as a threshold value to generate a geometric model with mphimage2geom COMSOL Livelink function.For the dense and sparse mesh, only the body outline was used in the mesh creation.The locally dense mesh took into account also the other anatomical regions and contained a denser region in the heart.For the XCAT case, the mesh regions strictly followed the anatomical contours.The diameter of the meshes was 498 mm, which is slightly larger than the FOV size of 497.8 mm used in the scanner, and with the pixel mesh as well.The dense mesh contained 10 199 vertices with element sizes between 4 and 6 mm.The sparse mesh contained 4572 vertices with element sizes between 7 and 9 mm.Lastly, the locally dense mesh contained 5641 vertices with element sizes between 4 and 9 mm.Unlike the NEMA case, a square FOV was used in the XCAT case both for the pixel and triangular meshes.
In this work, we used the MLEM algorithm in the reconstructions of both the pixel and triangular meshes.For the pixel reconstructions, a 128 × 128 image was used.The OSL algorithm with TV (OSL-TV) was additionally used in the triangular mesh reconstructions.The TV prior, however, is not compared with the pixel mesh, but only to test its feasibility with the triangular mesh.MLEM algorithm was chosen instead of the clinical standard OSEM due to the 2D nature of the simulations.Use of subsets in the 2D case would only slow down the reconstructions rather than accelerate them.Furthermore, OSEM algorithm does not guarantee convergence unlike MLEM and the convergence of the log-likelihood of (2) was used as the stopping criteria for the reconstructions.The optimal number of iterations was selected from the log-likelihood curve by selecting the iteration such that the difference in log-likelihood values between subsequent iterations was less than 3 for the NEMA case and less than 120 for the XCAT case.For the NEMA case, this resulted in the number of iterations of 21 for the triangular meshes and 19 for the pixel mesh.For the XCAT case, this resulted in 23 iterations for all meshes.In all cases, only rays intersecting the FOV are computed.Finally, the quality of the reconstructed images were evaluated with root-mean-squared error (RMS), structural similarity index (SSIM) (Wang et al 2004) and peak signal-to-noise ratio (PSNR) (Wang and Bovik 2009) compared with the true radioactive distribution (i.e.ground truth, GT) in both cases.For the NEMA case, this is an image of the number of emitted photons and also served as the source for the simulated data, while for XCAT it is an image showing the number of detected photons per pixel during the simulation, excluding randoms and scattered photons.For comparison purposes, each reconstruction formed in the Table 1.RMS, SSIM and PSNR values for each reconstruction presented in figures 2 and 3, compared to the ground truth image and respective ROI.Each of the triangular mesh reconstructions had 21 iterations and the pixel mesh reconstruction had 19 iterations, respectively.The total computation times for each reconstruction is presented in the last column.The best value for the non-regularized reconstruction is shown in bold.triangular mesh was also interpolated to the pixel mesh because the ground truth image is available in the pixel mesh only.

Results
In figure 2, reconstructions for simulated NEMA data for the whole image domain is presented along with the ground truth image.The reconstructions formed in the triangular meshes are presented in both the corresponding triangular mesh and in the interpolated pixel mesh.Visually, in figure 2, the need for dense mesh in the ROI can be seen, as the reconstructions are noticeably more accurate in shape compared with the sparse mesh reconstruction in the triangular mesh and interpolated pixel mesh in both algorithms.The utilization of the TV prior with the OSL algorithm results in a smoother distribution when compared with the MLEM.
In table 1 each of the reconstructions in figure 2 are compared with the ground truth image with RMS, SSIM and PSNR.The RMS error is the largest in the sparse triangular mesh reconstruction in both algorithms, as expected.The rest of the values are closer to each other, though notably the locally dense triangular mesh yields the lowest RMS error in both algorithms.Similar behavior can be seen in the SSIM values in table 1.In the PSNR values in table 1, the only notable difference, compared with RMS and SSIM values is that the OSL-TV reconstruction in locally dense triangular mesh gives clearly the highest value.Additionally, in table 1, computation times for each reconstruction is presented.Every reconstruction with MLEM algorithm in the triangular mesh is consistently faster than the pixel reconstruction.However, the OSL-TV algorithm reconstructions are slower, which is due to the added TV regularization.
In figure 3, ROI areas for each reconstruction of the simulated NEMA data is presented.In (a), the chosen ROI area is highlighted in a white box of the ground truth image and close up of the ROI area is indicated with white arrows.In (b), the close up of ROI areas for each reconstruction is presented.
The same observations that were made from the whole image domain, also apply to the ROI areas as the values in table 1 show similar behavior.The only noteworthy difference in values compared to figure 2 is that the RMS error in the ROI of OSL-TV reconstruction in the locally dense triangular mesh is smaller compared with MLEM reconstruction in the same mesh.
In figure 4, reconstructions for the GATE simulated XCAT heart data for the whole image domain is presented.As in the NEMA case, the reconstructions formed in the triangular meshes are presented in both the corresponding triangular mesh and in the interpolated pixel mesh.In figure 4, it is harder to visually see a lot of difference between the reconstructions, since the anatomical area is small compared with the size of the FOV.However, in the sparse mesh values of the distribution are lower on average compared with the ground truth image and the other reconstructions with both of the algorithms.
In table 2, RMS, SSIM and PSNR values compared with the ground truth image are presented for each of the reconstructions in figure 4.There is not much difference in the RMS error between the reconstructions, which is also due to the small size of the area of interest, or the origin of the radiation, compared with the size of the FOV.Similar behavior can be noticed in the PSNR values.Notably, each reconstruction in the triangular mesh has better SSIM value than the pixel reconstruction.Further, in table 2, the computation times of the reconstructions show similar behavior as in the NEMA case.However, the dense triangular mesh reconstruction is slower.This due to the fact, that the amount of vertices is closer to the amount of the pixel case, oppositely to the NEMA case.
In the figure 5, ROI areas for each reconstruction of the GATE simulated heart data are presented.As in the NEMA case, in (a) the chosen ROI area is highlighted in a white box of the ground truth image and close up of the ROI area is indicated with white arrows.In (b), the close up of ROI areas for each reconstruction is presented.
Here, the differences between each reconstruction is more evident than in figure 4. Both of the sparse mesh In the first row first column: ground truth image and second column: MLEM reconstruction in pixel mesh.In the first column from second to last row: sparse, dense and locally dense triangular mesh, respectively, second column: MLEM reconstructions for each triangular mesh interpolated to pixel mesh, third column: MLEM reconstructions in the original triangular meshes, fourth column: OSL-TV reconstructions for each triangular mesh interpolated to pixel mesh and fifth column: OSL-TV reconstructions in the original triangular mesh.
reconstructions have a leakage and, as stated previously, values of the distribution are lower on average compared with the other reconstructions.The effect of the TV can be seen clearly between the dense and locally dense reconstructions.Further, the RMS, SSIM and PSNR values of the ROI areas of the reconstructions   2. Interestingly, the RMS error is the lowest and SSIM is the highest in the pixel reconstruction.These could be due to the interpolation of the triangular meshes into the pixel mesh, and the difference in smoothness of the background especially in the dense and locally dense meshes.The PSNR values are more evenly distributed over all the reconstructions compared with the RMS and SSIM values.

Conclusions
In this work we demonstrated the use of a locally dense multi-resolution triangular mesh in PET in two simulated 2D cases.Particularly of interest was the locally dense triangular mesh that was formed based on anatomical information.In both cases, the locally dense mesh reconstructions were compared to two generic, the sparse and dense, triangular mesh reconstructions and a pixel reconstruction.The results show that the reconstructions in the locally dense triangular mesh is a potential alternative to the pixel reconstruction as it retains the anatomical shapes better than the pixel reconstruction and has the advantages of both of the generic triangular meshes; computational speed of the sparse triangular mesh and ROI accuracy of the dense triangular mesh.Additionally, we demonstrated that a TV prior is applicable in both cases and also produces generally better quantitative results when compared to the regular MLEM reconstruction.The TV prior could have been applied to the pixel reconstruction as well, but the point in this work was to test that the TV can be specifically applied to the triangular meshes, especially the locally dense triangular mesh.Additionally, we would like to remark that the error metric comparison between the ground truth image and the triangular mesh reconstructions were not optimal in this case as the ground truth image was only available in the pixel mesh and the triangular mesh reconstructions had to be interpolated into the pixel mesh.A more equal comparison would have been to form the ground truth image in the triangular mesh as well and do the comparisons in the same mesh.For a clinical framework, the triangular meshes could be formed in different ways.As the triangular meshes are flexible in geometry, any size or shape for the FOV and for the denser ROIs could be applied.One alternative is that a physician would determine some, or all, of these.For example, the physician could use a CT or MR image to draw the ROIs that should have denser mesh, while all the other voxels outside of these regions would use sparse mesh.Additionally, the physician could also select the size and shape of FOV itself, or, alternatively, the FOV could be formed automatically either as the typical square shape or with a shape that covers only the body.The exact size and shape of FOV, however, would need to be adjusted to accommodate the physician, but the benefit of triangular mesh is that both the size and shape can be selected freely.This allows patient-, or physician-, specific adjustment of the final image/volume itself.
An alternative method for clinical framework would be to automatically select both the ROIs and the size and shape of the FOV.Furthermore, in this case there can be adjustments made beforehand such that the FOV should always follow the outlines of the body or that the FOV should always be square-shaped.An automatic mesh generation could be implemented by first segmenting the CT or MR image.After segmenting the image into specific body regions/organs, the examination could determine the regions where the denser mesh should be applied.In our test case, this would be the heart.We remark that the denser mesh can, and should, be made slightly larger than the segmented regions.This allows to account for possible errors in the CT or MR image, such as the effects of noise, small misalignments and different voxel sizes.Furthermore, since the spatial resolution of PET is lower than that of CT or MRI, a larger mesh is better able to contain the spatial blurring present in PET without introducing distracting variations along the edges of the object.The exact amount of enlargement is a topic of future research though.
In the future, the work presented in this paper could be applied to 3D and expanded to cover also dynamic, 4D, cases.Additionally, even more sparse meshing for the background could be applied, further cutting down the computation time.Furthermore, a more clinically viable test case(s) would be needed.In 3D, the computational advantages of the locally dense mesh would most likely be highlighted more prominently and could be especially useful when applied to whole body imaging.
First data set was based on the NEMA image quality phantom (National Electrical Manufacturers Association (NEMA), 2008) and used a simulated preclinical Siemens Inveon PET scanner(Constantinescu and Mukherjee 2009, Lee et al 2013).The second data set was also simulated in GATE using the XCAT human phantom(Segars et al 2010) with the simulated Siemens biograph mCT scanner(Jakoby et al 2011).For reference, the actual radioactivity distribution input into the GATE simulation is shown in figure 1.The meshes were created in COMSOL multiphysics software and a total of three different types of meshes were created: a dense mesh, a sparse mesh and a locally dense mesh.The system matrix A for the triangular meshes was formed by a C/C++ based closed source MEX code for MATLAB.In the code, the line integral calculations in piecewise linear basis functions are combined with a ray tracing algorithm implemented in a triangle mesh.The ray tracing algorithm is an adaptation from the method used in the open source ValoMC MATLAB-package(Leino et al 2019).For the pixel reconstructions, the system matrix was obtained from OMEGA (open-source MATLAB emission and transmission tomography software) (Wettenhovi et al 2021) using the improved Siddon's algorithm(Siddon 1985, Jacobs et al 1998).

Figure 1 .
Figure 1.The radioactivity distribution input into the GATE simulation in the XCAT case.Units are in Bq.

Figure 2 .
Figure 2. Reconstructions for simulated NEMA data in the whole image domain.Same color scale is used in all figures.In the first row first column: ground truth image and second column: MLEM reconstruction in pixel mesh.In the first column from second to last row: sparse, dense and locally dense triangular mesh, respectively, second column: MLEM reconstructions for each triangular mesh interpolated to pixel mesh, third column: MLEM reconstructions in the original triangular meshes, fourth column: OSL-TV reconstructions for each triangular mesh interpolated to pixel mesh and fifth column: OSL-TV reconstructions in the original triangular mesh.

Figure 3 .
Figure 3. ROI areas for simulated NEMA data.Same color scale is used in all figures.(a) Full ground truth image, with the chosen ROI indicated with a white box and close up of the ROI indicated by the white arrows.(b) First column: ROI for the MLEM pixel reconstruction and ROIs for MLEM reconstructions for sparse, dense and locally dense meshes interpolated to pixel mesh, respectively.Second column: ROIs for MLEM reconstructions in the original triangular meshes.Third column: ROIs for OSL-TV reconstructions for sparse, dense and locally dense meshes interpolated to pixel mesh, respectively.Fourth column: ROIs for OSL-TV reconstructions in the original triangular meshes.

Figure 4 .
Figure 4. Reconstructions for GATE simulated heart data in the whole image domain.Same color scale is used in all figures.In the first row first column: ground truth image and second column: MLEM reconstruction in pixel mesh.In the first column from second to last row: sparse, dense and locally dense triangular mesh, respectively, second column: MLEM reconstructions for each triangular mesh interpolated to pixel mesh, third column: MLEM reconstructions in the original triangular meshes, fourth column: OSL-TV reconstructions for each triangular mesh interpolated to pixel mesh and fifth column: OSL-TV reconstructions in the original triangular mesh.

Figure 5 .
Figure 5. ROI areas for GATE simulated heart data.Same color scale is used in all figures.(a) Full ground truth image, with the chosen ROI indicated with a white box and close up of the ROI indicated by the white arrows.(b) First column: ROI for the MLEM pixel reconstruction and ROIs for MLEM reconstructions for sparse, dense and locally dense meshes interpolated to pixel mesh, respectively.Second column: ROIs for MLEM reconstructions in the original triangular meshes.Third column: ROIs for OSL-TV reconstructions for sparse, dense and locally dense meshes interpolated to pixel mesh, respectively.Fourth column: ROIs for OSL-TV reconstructions in the original triangular meshes.

1 2
Phys.Med.Biol.69(2024)105023P Lesonen et alwhere f is the estimated image, m the measurements, N is the number of LORs and [

Table 2 .
RMS, SSIM and PSNR values for each reconstruction presented in figure4and in 5, compared to the ground truth image and respective ROI.Each reconstruction had 23 iterations and the total computation times for each reconstruction is presented in the last column.The best value for the non-regularized reconstruction is shown in bold.