Faster PET reconstruction with non-smooth priors by randomization and preconditioning

Uncompressed clinical data from modern positron emission tomography (PET) scanners are very large, exceeding 350 million data points (projection bins). The last decades have seen tremendous advancements in mathematical imaging tools many of which lead to non-smooth (i.e. non-differentiable) optimization problems which are much harder to solve than smooth optimization problems. Most of these tools have not been translated to clinical PET data, as the state-of-the-art algorithms for non-smooth problems do not scale well to large data. In this work, inspired by big data machine learning applications, we use advanced randomized optimization algorithms to solve the PET reconstruction problem for a very large class of non-smooth priors which includes for example total variation, total generalized variation, directional total variation and various different physical constraints. The proposed algorithm randomly uses subsets of the data and only updates the variables associated with these. While this idea often leads to divergent algorithms, we show that the proposed algorithm does indeed converge for any proper subset selection. Numerically, we show on real PET data (FDG and florbetapir) from a Siemens Biograph mMR that about ten projections and backprojections are sufficient to solve the MAP optimisation problem related to many popular non-smooth priors; thus showing that the proposed algorithm is fast enough to bring these models into routine clinical practice

Versatile regularisation toolkit for iterative image reconstruction with proximal splitting algorithms

Ill-posed image recovery requires regularisation to ensure stability. The presented open-source regularisation toolkit consists of state-of-the-art variational algorithms which can be embedded in a plug-and-play fashion into the general framework of proximal splitting methods. The packaged regularisers aim to satisfy various prior expectations of the investigated objects, e.g., their structural characteristics, smooth or non-smooth surface morphology. The flexibility of the toolkit helps with the design of more advanced model-based iterative reconstruction methods for different imaging modalities while operating with simpler building blocks. The toolkit is written for CPU and GPU architectures and wrapped for Python/MATLAB. We demonstrate the functionality of the toolkit in application to Positron Emission Tomography (PET) and X-ray synchrotron computed tomography (CT)

해부학적 유도 PET 재구성: 매끄럽지 않은 사전 함수부터 딥러닝 접근까지

학위논문 (박사) -- 서울대학교 대학원 : 의과대학 의과학과, 2021. 2. 이재성.Advances in simultaneous positron emission tomography/magnetic resonance imaging (PET/MRI) technology have led to an active investigation of the anatomy-guided regularized PET image reconstruction algorithm based on MR images. Among the various priors proposed for anatomy-guided regularized PET image reconstruction, Bowsher’s method based on second-order smoothing priors sometimes suffers from over-smoothing of detailed structures. Therefore, in this study, we propose a Bowsher prior based on the l1 norm and an iteratively reweighting scheme to overcome the limitation of the original Bowsher method. In addition, we have derived a closed solution for iterative image reconstruction based on this non-smooth prior. A comparison study between the original l2 and proposed l1 Bowsher priors were conducted using computer simulation and real human data. In the simulation and real data application, small lesions with abnormal PET uptake were better detected by the proposed l1 Bowsher prior methods than the original Bowsher prior. The original l2 Bowsher leads to a decreased PET intensity in small lesions when there is no clear separation between the lesions and surrounding tissue in the anatomical prior. However, the proposed l1 Bowsher prior methods showed better contrast between the tumors and surrounding tissues owing to the intrinsic edge-preserving property of the prior which is attributed to the sparseness induced by l1 norm, especially in the iterative reweighting scheme. Besides, the proposed methods demonstrated lower bias and less hyper-parameter dependency on PET intensity estimation in the regions with matched anatomical boundaries in PET and MRI. Moreover, based on the formulation of l1 Bowsher prior, the unrolled network containing the conventional maximum-likelihood expectation-maximization (ML-EM) module was also proposed. The convolutional layers successfully learned the distribution of anatomically-guided PET images and the EM module corrected the intermediate outputs by comparing them with sinograms. The proposed unrolled network showed better performance than ordinary U-Net, where the regional uptake is less biased and deviated. Therefore, these methods will help improve the PET image quality based on the anatomical side information.양전자방출단층촬영 / 자기공명영상 (PET/MRI) 동시 획득 기술의 발전으로 MR 영상을 기반으로 한 해부학적 사전 함수로 정규화 된 PET 영상 재구성 알고리즘에 대한 심도있는 평가가 이루어졌다. 해부학 기반으로 정규화 된 PET 이미지 재구성을 위해 제안 된 다양한 사전 중 2차 평활화 사전함수에 기반한 Bowsher의 방법은 때때로 세부 구조의 과도한 평활화로 어려움을 겪는다. 따라서 본 연구에서는 원래 Bowsher 방법의 한계를 극복하기 위해 l1 norm에 기반한 Bowsher 사전 함수와 반복적인 재가중치 기법을 제안한다. 또한, 우리는 이 매끄럽지 않은 사전 함수를 이용한 반복적 이미지 재구성에 대해 닫힌 해를 도출했다. 원래 l2와 제안 된 l1 Bowsher 사전 함수 간의 비교 연구는 컴퓨터 시뮬레이션과 실제 데이터를 사용하여 수행되었다. 시뮬레이션 및 실제 데이터에서 비정상적인 PET 흡수를 가진 작은 병변은 원래 Bowsher 이전보다 제안 된 l1 Bowsher 사전 방법으로 더 잘 감지되었다. 원래의 l2 Bowsher는 해부학적 영상에서 병변과 주변 조직 사이에 명확한 분리가 없을 때 작은 병변에서의 PET 강도를 감소시킨다. 그러나 제안 된 l1 Bowsher 사전 방법은 특히 반복적 재가중치 기법에서 l1 노름에 의해 유도된 희소성에 기인한 특성으로 인해 종양과 주변 조직 사이에 더 나은 대비를 보여주었다. 또한 제안된 방법은 PET과 MRI의 해부학적 경계가 일치하는 영역에서 PET 강도 추정에 대한 편향이 더 낮고 하이퍼 파라미터 종속성이 적음을 보여주었다. 또한, l1Bowsher 사전 함수의 닫힌 해를 기반으로 기존의 ML-EM (maximum-likelihood expectation-maximization) 모듈을 포함하는 펼쳐진 네트워크도 제안되었다. 컨볼루션 레이어는 해부학적으로 유도 재구성된 PET 이미지의 분포를 성공적으로 학습했으며, EM 모듈은 중간 출력들을 사이노그램과 비교하여 결과 이미지가 잘 들어맞게 수정했다. 제안된 펼쳐진 네트워크는 지역의 흡수선량이 덜 편향되고 편차가 적어, 일반 U-Net보다 더 나은 성능을 보여주었다. 따라서 이러한 방법들은 해부학적 정보를 기반으로 PET 이미지 품질을 향상시키는 데 유용할 것이다.Chapter 1. Introduction 1 1.1. Backgrounds 1 1.1.1. Positron Emission Tomography 1 1.1.2. Maximum a Posterior Reconstruction 1 1.1.3. Anatomical Prior 2 1.1.4. Proposed l_1 Bowsher Prior 3 1.1.5. Deep Learning for MR-less Application 4 1.2. Purpose of the Research 4 Chapter 2. Anatomically-guided PET Reconstruction Using Bowsher Prior 6 2.1. Backgrounds 6 2.1.1. PET Data Model 6 2.1.2. Original Bowsher Prior 7 2.2. Methods and Materials 8 2.2.1. Proposed l_1 Bowsher Prior 8 2.2.2. Iterative Reweighting 13 2.2.3. Computer Simulations 15 2.2.4. Human Data 16 2.2.5. Image Analysis 17 2.3. Results 19 2.3.1. Simulation with Brain Phantom 19 2.3.2.Human Data 20 2.4. Discussions 25 Chapter 3. Deep Learning Approach for Anatomically-guided PET Reconstruction 31 3.1. Backgrounds 31 3.2. Methods and Materials 33 3.2.1. Douglas-Rachford Splitting 33 3.2.2. Network Architecture 34 3.2.3. Dataset and Training Details 35 3.2.4. Image Analysis 36 3.3. Results 37 3.4. Discussions 38 Chapter 4. Conclusions 40 Bibliography 41 Abstract in Korean (국문 초록) 52Docto

Stochastic Optimisation Methods Applied to PET Image Reconstruction

Positron Emission Tomography (PET) is a medical imaging technique that is used to pro- vide functional information regarding physiological processes. Statistical PET reconstruc- tion attempts to estimate the distribution of radiotracer in the body but this methodology is generally computationally demanding because of the use of iterative algorithms. These algorithms are often accelerated by the utilisation of data subsets, which may result in con- vergence to a limit set rather than the unique solution. Methods exist to relax the update step sizes of subset algorithms but they introduce additional heuristic parameters that may result in extended reconstruction times. This work investigates novel methods to modify subset algorithms to converge to the unique solution while maintaining the acceleration benefits of subset methods. This work begins with a study of an automatic method for increasing subset sizes, called AutoSubsets. This algorithm measures the divergence between two distinct data subset update directions and, if significant, the subset size is increased for future updates. The algorithm is evaluated using both projection and list mode data. The algorithm’s use of small initial subsets benefits early reconstruction but unfortunately, at later updates, the subsets size increases too early, which impedes convergence rates. The main part of this work investigates the application of stochastic variance reduction optimisation algorithms to PET image reconstruction. These algorithms reduce variance due to the use of subsets by incorporating previously computed subset gradients into the update direction. The algorithms are adapted for the application to PET reconstruction. This study evaluates the reconstruction performance of these algorithms when applied to various 3D non-TOF PET simulated, phantom and patient data sets. The impact of a number of algorithm parameters are explored, which includes: subset selection methodologies, the number of subsets, step size methodologies and preconditioners. The results indicate that these stochastic variance reduction algorithms demonstrate superior performance after only a few epochs when compared to a standard PET reconstruction algorithm

On the convergence and sampling of randomized primal-dual algorithms and their application to parallel MRI reconstruction

Stochastic Primal-Dual Hybrid Gradient (SPDHG) is an algorithm to efficiently solve a wide class of nonsmooth large-scale optimization problems. In this paper we contribute to its theoretical foundations and prove its almost sure convergence for convex but neither necessarily strongly convex nor smooth functionals. We also prove its convergence for any sampling. In addition, we study SPDHG for parallel Magnetic Resonance Imaging reconstruction, where data from different coils are randomly selected at each iteration. We apply SPDHG using a wide range of random sampling methods and compare its performance across a range of settings, including mini-batch size and step size parameters. We show that the sampling can significantly affect the convergence speed of SPDHG and for many cases an optimal sampling can be identified

A Fast Convergent Ordered-Subsets Algorithm with Subiteration-Dependent Preconditioners for PET Image Reconstruction

We investigated the imaging performance of a fast convergent ordered-subsets algorithm with subiteration-dependent preconditioners (SDPs) for positron emission tomography (PET) image reconstruction. In particular, we considered the use of SDP with the block sequential regularized expectation maximization (BSREM) approach with the relative difference prior (RDP) regularizer due to its prior clinical adaptation by vendors. Because the RDP regularization promotes smoothness in the reconstructed image, the directions of the gradients in smooth areas more accurately point toward the objective function's minimizer than those in variable areas. Motivated by this observation, two SDPs have been designed to increase iteration step-sizes in the smooth areas and reduce iteration step-sizes in the variable areas relative to a conventional expectation maximization preconditioner. The momentum technique used for convergence acceleration can be viewed as a special case of SDP. We have proved the global convergence of SDP-BSREM algorithms by assuming certain characteristics of the preconditioner. By means of numerical experiments using both simulated and clinical PET data, we have shown that the SDP-BSREM algorithms substantially improve the convergence rate, as compared to conventional BSREM and a vendor's implementation as Q.Clear. Specifically, SDP-BSREM algorithms converge 35\%-50\% faster in reaching the same objective function value than conventional BSREM and commercial Q.Clear algorithms. Moreover, we showed in phantoms with hot, cold and background regions that the SDP-BSREM algorithms approached the values of a highly converged reference image faster than conventional BSREM and commercial Q.Clear algorithms.Comment: 12 pages, 9 figure

The Practicality of Stochastic Optimization in Imaging Inverse Problems

In this work we investigate the practicality of stochastic gradient descent and recently introduced variants with variance-reduction techniques in imaging inverse problems. Such algorithms have been shown in the machine learning literature to have optimal complexities in theory, and provide great improvement empirically over the deterministic gradient methods. Surprisingly, in some tasks such as image deblurring, many of such methods fail to converge faster than the accelerated deterministic gradient methods, even in terms of epoch counts. We investigate this phenomenon and propose a theory-inspired mechanism for the practitioners to efficiently characterize whether it is beneficial for an inverse problem to be solved by stochastic optimization techniques or not. Using standard tools in numerical linear algebra, we derive conditions on the spectral structure of the inverse problem for being a suitable application of stochastic gradient methods. Particularly, we show that, for an imaging inverse problem, if and only if its Hessain matrix has a fast-decaying eigenspectrum, then the stochastic gradient methods can be more advantageous than deterministic methods for solving such a problem. Our results also provide guidance on choosing appropriately the partition minibatch schemes, showing that a good minibatch scheme typically has relatively low correlation within each of the minibatches. Finally, we propose an accelerated primal-dual SGD algorithm in order to tackle another key bottleneck of stochastic optimization which is the heavy computation of proximal operators. The proposed method has fast convergence rate in practice, and is able to efficiently handle non-smooth regularization terms which are coupled with linear operators

On the convergence and sampling of randomized primal-dual algorithms and their application to parallel MRI reconstruction

Stochastic Primal-Dual Hybrid Gradient (SPDHG) is an algorithm to efficiently solve a wide class of nonsmooth large-scale optimization problems. In this paper we contribute to its theoretical foundations and prove its almost sure convergence for convex but neither necessarily strongly convex nor smooth functionals. We also prove its convergence for any sampling. In addition, we study SPDHG for parallel Magnetic Resonance Imaging reconstruction, where data from different coils are randomly selected at each iteration. We apply SPDHG using a wide range of random sampling methods and compare its performance across a range of settings, including mini-batch size and step size parameters. We show that the sampling can significantly affect the convergence speed of SPDHG and for many cases an optimal sampling can be identified