Fast Quasi-Newton Algorithms for Penalized Reconstruction in Emission Tomography and Further Improvements via Preconditioning

Computationally efficient collimator-detector response compensation in high energy SPECT using 1D convolutions and rotations

Objective. Modeling of the collimator-detector response (CDR) in single photon emission computed tomography (SPECT) reconstruction enables improved resolution and accuracy, and is thus important for quantitative imaging applications such as dosimetry. The implementation of CDR modeling, however, can become a computational bottleneck when there are substantial components of septal penetration and scatter in the acquired data, since a direct convolution-based approach requires large 2D kernels. This work proposes a 1D convolution and rotation-based CDR model that reduces reconstruction times but maintains consistency with models that employ 2D convolutions. To enable open-source development and use of these models in image reconstruction, we release a SPECTPSFToolbox repository for the PyTomography project on GitHub.Approach. A 1D/rotation-based CDR model was formulated and subsequently fit to Monte Carlo (MC) point source data representative of177Lu,131I, and225Ac imaging. Computation times of (i) the proposed 1D/rotation-based model and (ii) a traditional model that uses 2D convolutions were compared for typical SPECT matrix sizes. Both CDR models were then used in the reconstruction of MC, physical phantom, and patient data; the models were compared by quantifying total counts in hot regions of interest (ROIs) and activity contrast between hot ROIs and background regions.Results. For typical matrix sizes in SPECT reconstruction, application of the 1D/rotation-based model provides a two-fold computational speed-up over the 2D model when running on GPU. Only small differences between the 1D/rotation-based and 2D models (order of 1%) were obtained for count and contrast quantification in select ROIs.Significance. A technique for CDR modeling in SPECT was proposed that (i) significantly speeds up reconstruction times, and (ii) yields nearly identical reconstructions to traditional 2D convolution based CDR techniques. The released toolbox will permit open-source development of similar models for different isotopes and collimators.

Joint motion estimation and penalized image reconstruction algorithm with anatomical priors for gated TOF-PET/CT

The presence of respiratory motion not only degrades the reconstructed image but also limits the utilization of anatomical priors in emission tomography. In this study, we explore the potential application of a joint motion estimation and penalized image reconstruction algorithm using anatomical priors in gated time-of-flight positron emission tomography/computed tomography (PET/CT). The algorithm is able to warp both the activity image and the attenuation map to align them with the measured data with the facilitation of anatomical information contained in the attenuation map. Five patient datasets, three acquired in single-bed position and two acquired in whole-body continuous-bed-motion mode, are included. For each patient, the attenuation map is derived from a breath-hold CT. The Parallel Levels Sets (PLS) is chosen as a representative anatomical prior. In addition to demonstrating the reliability of the estimated motion and the benefits of incorporating anatomical prior, preliminary results also indicate that the algorithm shows the potential to reconstruct an activity image in the space corresponding to that of the attenuation map, which could be applied to address the potential misalignment issue in applications involving multiple PET acquisitions but a single CT.

An Investigation of Stochastic Variance Reduction Algorithms for Relative Difference Penalized 3D PET Image Reconstruction

Penalised PET image reconstruction algorithms are often accelerated during early iterations with the use of subsets. However, these methods may exhibit limit cycle behaviour at later iterations due to variations between subsets. Desirable converged images can be achieved for a subclass of these algorithms via the implementation of a relaxed step size sequence, but the heuristic selection of parameters will impact the quality of the image sequence and algorithm convergence rates. In this work, we demonstrate the adaption and application of a class of stochastic variance reduction gradient algorithms for PET image reconstruction using the relative difference penalty and numerically compare convergence performance to BSREM. The two investigated algorithms are: SAGA and SVRG. These algorithms require the retention in memory of recently computed subset gradients, which are utilised in subsequent updates. We present several numerical studies based on Monte Carlo simulated data and a patient data set for fully 3D PET acquisitions. The impact of the number of subsets, different preconditioners and step size methods on the convergence of regions of interest values within the reconstructed images is explored. We observe that when using constant preconditioning, SAGA and SVRG demonstrate reduced variations in voxel values between subsequent updates and are less reliant on step size hyper-parameter selection than BSREM reconstructions. Furthermore, SAGA and SVRG can converge significantly faster to the penalised maximum likelihood solution than BSREM, particularly in low count data.

Measurement of Small-Slope Free-Form Optical Surfaces with the Modified Phase Retrieval

In this paper, we demonstrate the use of the modified phase retrieval as a method for application in the measurement of small-slope free-form optical surfaces. This technique is a solution for the measurement of small-slope free-form optical surfaces, based on the modified phase retrieval algorithm, whose essence is that only two defocused images are needed to estimate the wave front with an accuracy similar to that of the traditional phase retrieval but with less image capturing and computation time. An experimental arrangement used to measure the small-slope free-form optical surfaces using the modified phase retrieval is described. The results of these experiments demonstrate that the modified phase retrieval method can achieve measurements comparable to those of the standard interferometer.

PET Reconstruction With Non-Negativity Constraint in Projection Space: Optimization Through Hypo-Convergence

Standard positron emission tomography (PET) reconstruction techniques are based on maximum-likelihood (ML) optimization methods, such as the maximum-likelihood expectation-maximization (MLEM) algorithm and its variations. Most methodologies rely on a positivity constraint on the activity distribution image. Although this constraint is meaningful from a physical point of view, it can be a source of bias for low-count/high-background PET, which can compromise accurate quantification. Existing methods that allow for negative values in the estimated image usually utilize a modified log-likelihood, and therefore break the data statistics. In this paper, we propose to incorporate the positivity constraint on the projections only, by approximating the (penalized) log-likelihood function by an adequate sequence of objective functions that are easily maximized without constraint. This sequence is constructed such that there is hypo-convergence (a type of convergence that allows the convergence of the maximizers under some conditions) to the original log-likelihood, hence allowing us to achieve maximization with positivity constraint on the projections using simple settings. A complete proof of convergence under weak assumptions is given. We provide results of experiments on simulated data where we compare our methodology with the alternative direction method of multipliers (ADMM) method, showing that our algorithm converges to a maximizer, which stays in the desired feasibility set, with faster convergence than ADMM. We also show that this approach reduces the bias, as compared with MLEM images, in necrotic tumors-which are characterized by cold regions surrounded by hot structures-while reconstructing similar activity values in hot regions.

Benefits of Using a Spatially-Variant Penalty Strength With Anatomical Priors in PET Reconstruction

In this study, we explore the use of a spatially-variant penalty strength in penalized image reconstruction using anatomical priors to reduce the dependence of lesion contrast on surrounding activity and lesion location. This work builds on a previous method to make the local perturbation response (LPR) approximately spatially invariant. While the dependence of lesion contrast on the local properties introduced by the anatomical penalty is intentional, the method aims to reduce the influence from surroundings lying along the lines of response (LORs) but not in the penalty neighborhood structure. The method is evaluated using simulated data, assuming that the anatomical information is absent or well-aligned with the corresponding activity images. Since the parallel level sets (PLS) penalty is convex and has shown promising results in the literature, it is chosen as the representative anatomical penalty and incorporated into the previously proposed preconditioned algorithm (L-BFGS-B-PC) for achieving good image quality and fast convergence rate. A 2D disc phantom with a feature at the center and a 3D XCAT thorax phantom with lesions inserted in different slices are used to study how surrounding activity and lesion location affect the visual appearance and quantitative consistency. A bias and noise analysis is also performed with the 2D disc phantom. The consistency of the algorithm convergence rate with respect to different data noise and background levels is also investigated using the XCAT phantom. Finally, an example of reconstruction for a patient dataset with inserted pseudo lesions is used as a demonstration in a clinical context. We show that applying the spatially-variant penalization with PLS can reduce the dependence of the lesion contrast on the surrounding activity and lesion location. It does not affect the bias and noise trade-off curves for matched local resolution. Moreover, when using the proposed penalization, significant improvement in algorithm convergence rate and convergence consistency is observed.

A Krasnoselskii-Mann Algorithm With an Improved EM Preconditioner for PET Image Reconstruction

This paper presents a preconditioned Krasnoselskii-Mann (KM) algorithm with an improved EM preconditioner (IEM-PKMA) for higher-order total variation (HOTV) regularized positron emission tomography (PET) image reconstruction. The PET reconstruction problem can be formulated as a three-term convex optimization model consisting of the Kullback-Leibler (KL) fidelity term, a nonsmooth penalty term, and a nonnegative constraint term which is also nonsmooth. We develop an efficient KM algorithm for solving this optimization problem based on a fixed-point characterization of its solution, with a preconditioner and a momentum technique for accelerating convergence. By combining the EM precondtioner, a thresholding, and a good inexpensive estimate of the solution, we propose an improved EM preconditioner that can not only accelerate convergence but also avoid the reconstructed image being "stuck at zero." Numerical results in this paper show that the proposed IEM-PKMA outperforms existing state-of-the-art algorithms including, the optimization transfer descent algorithm and the preconditioned L-BFGS-B algorithm for the differentiable smoothed anisotropic total variation regularized model, the preconditioned alternating projection algorithm, and the alternating direction method of multipliers for the nondifferentiable HOTV regularized model. Encouraging initial experiments using clinical data are presented.

Spatially-Compact MR-Guided Kernel EM for PET Image Reconstruction

Positron emission tomography (PET) is a highly sensitive functional and molecular imaging modality which can measure picomolar concentrations of an injected radionuclide. However, the physical sensitivity of PET is limited, and reducing the injected dose leads to low count data and noisy reconstructed images. A highly effective way of reducing noise is to reparameterise the reconstruction in terms of MR-derived spatial basis functions. Spatial basis functions derived using the kernel method have demonstrated excellent noise reduction properties and maintain shared PET-MR detailed structures. However, as previously shown in the literature, the MR-guided kernel method may lead to excessive smoothing of structures that are only present in the PET data. This work makes two main contributions in order to address this problem: first, we exploit the potential of the MR-guided kernel method to form more spatially-compact basis functions which are able to preserve PET-unique structures, and secondly, we consider reconstruction at the native MR resolution. The former contribution notably improves the recovery of structures which are unique to the PET data. These adaptations of the kernel method were compared to the conventional implementation of the MR-guided kernel method and also to MLEM, in terms of ability to recover PET unique structures for both simulated and real data. The spatially-compact kernel method showed clear visual and quantitative improvements in the reconstruction of the PET unique structures, relative to the conventional kernel method for all sizes of PET unique structures investigated, whilst maintaining to a large extent the impressive noise mitigating and detail preserving properties of the conventional MR-guided kernel method. We therefore conclude that a spatially-compact parameterisation of the MR-guided kernel method, should be the preferred implementation strategy in order to obviate unnecessary losses in PET-unique details.