Efficient algorithms for geodesic shooting in diffeomorphic image registration

Diffeomorphic image registration is a common problem in medical image analysis. Here, one searches for a diffeomorphic deformation that maps one image (the moving or template image) onto another image (the fixed or reference image). We can formulate the search for such a map as a PDE constrained optimization problem. These types of problems are computationally expensive. This gives rise to the need for efficient algorithms. After introducing the PDE constrained optimization problem, we derive the first and second order optimality conditions. We discretize the problem using a pseudo-spectral discretization in space and consider Heun's method and the semi-Lagrangian method for the time integration of the PDEs that appear in the optimality system. To solve this optimization problem, we consider an L-BFGS and an inexact Gauss-Newton-Krylov method. To reduce the cost of solving the linear system that arises in Newton-type methods, we investigate different preconditioners. They exploit the structure of the Hessian, and use algorithms to efficiently compute an approximation to its inverse. Further, we build the preconditioners on a coarse grid to further reduce computational costs. The different methods are evaluated for two-dimensional image data (real and synthetic). We study the spectrum of the different building blocks that appear in the Hessian. It is demonstrated that low rank preconditioners are able to significantly reduce the number of iterations needed to solve the linear system in Newton-type optimizers. We then compare different optimization methods based on their overall performance. This includes the accuracy and time-to-solution. L-BFGS turns out to be the best method, in terms of runtime, if we solve solving for large gradient tolerances. If we are interested in computing accurate solutions with a small gradient norm, an inexact Gauss-Newton-Krylov optimizer with the regularization term as preconditioner performs best

Stochastic Bundle Adjustment for Efficient and Scalable 3D Reconstruction

Current bundle adjustment solvers such as the Levenberg-Marquardt (LM) algorithm are limited by the bottleneck in solving the Reduced Camera System (RCS) whose dimension is proportional to the camera number. When the problem is scaled up, this step is neither efficient in computation nor manageable for a single compute node. In this work, we propose a stochastic bundle adjustment algorithm which seeks to decompose the RCS approximately inside the LM iterations to improve the efficiency and scalability. It first reformulates the quadratic programming problem of an LM iteration based on the clustering of the visibility graph by introducing the equality constraints across clusters. Then, we propose to relax it into a chance constrained problem and solve it through sampled convex program. The relaxation is intended to eliminate the interdependence between clusters embodied by the constraints, so that a large RCS can be decomposed into independent linear sub-problems. Numerical experiments on unordered Internet image sets and sequential SLAM image sets, as well as distributed experiments on large-scale datasets, have demonstrated the high efficiency and scalability of the proposed approach. Codes are released at https://github.com/zlthinker/STBA.Comment: Accepted by ECCV 202

A total variation regularization based super-resolution reconstruction algorithm for digital video

Super-resolution (SR) reconstruction technique is capable of producing a high-resolution image from a sequence of low-resolution images. In this paper, we study an efficient SR algorithm for digital video. To effectively deal with the intractable problems in SR video reconstruction, such as inevitable motion estimation errors, noise, blurring, missing regions, and compression artifacts, the total variation (TV) regularization is employed in the reconstruction model. We use the fixed-point iteration method and preconditioning techniques to efficiently solve the associated nonlinear Euler-Lagrange equations of the corresponding variational problem in SR. The proposed algorithm has been tested in several cases of motion and degradation. It is also compared with the Laplacian regularization-based SR algorithm and other TV-based SR algorithms. Experimental results are presented to illustrate the effectiveness of the proposed algorithm.£.published_or_final_versio

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

An algorithm for constrained one-step inversion of spectral CT data

We develop a primal-dual algorithm that allows for one-step inversion of spectral CT transmission photon counts data to a basis map decomposition. The algorithm allows for image constraints to be enforced on the basis maps during the inversion. The derivation of the algorithm makes use of a local upper bounding quadratic approximation to generate descent steps for non-convex spectral CT data discrepancy terms, combined with a new convex-concave optimization algorithm. Convergence of the algorithm is demonstrated on simulated spectral CT data. Simulations with noise and anthropomorphic phantoms show examples of how to employ the constrained one-step algorithm for spectral CT data.Comment: Submitted to Physics in Medicine and Biolog

Proceedings of the FEniCS Conference 2017

Proceedings of the FEniCS Conference 2017 that took place 12-14 June 2017 at the University of Luxembourg, Luxembourg

International Conference on Continuous Optimization (ICCOPT) 2019 Conference Book

The Sixth International Conference on Continuous Optimization took place on the campus of the Technical University of Berlin, August 3-8, 2019. The ICCOPT is a flagship conference of the Mathematical Optimization Society (MOS), organized every three years. ICCOPT 2019 was hosted by the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Berlin. It included a Summer School and a Conference with a series of plenary and semi-plenary talks, organized and contributed sessions, and poster sessions. This book comprises the full conference program. It contains, in particular, the scientific program in survey style as well as with all details, and information on the social program, the venue, special meetings, and more