Discrete-Continuous ADMM for Transductive Inference in Higher-Order MRFs

This paper introduces a novel algorithm for transductive inference in higher-order MRFs, where the unary energies are parameterized by a variable classifier. The considered task is posed as a joint optimization problem in the continuous classifier parameters and the discrete label variables. In contrast to prior approaches such as convex relaxations, we propose an advantageous decoupling of the objective function into discrete and continuous subproblems and a novel, efficient optimization method related to ADMM. This approach preserves integrality of the discrete label variables and guarantees global convergence to a critical point. We demonstrate the advantages of our approach in several experiments including video object segmentation on the DAVIS data set and interactive image segmentation

Isometry Invariant Shape Priors for Variational Image Segmentation

Variational methods play a fundamental role in mathematical image analysis as a bridge between models and algorithms. A major challenge is to formulate a given model as a feasible optimization problem. There has been a huge leap in that respect concerning local data models in the framework of convex relaxation. But non-local concepts such as the shape of a sought-after object are still difficult to implement. In this thesis we study mathematical representations for shapes and develop shape prior functionals for object segmentation based thereon. A particular focus is set on the isometry invariance of the functionals and the compatibility with existing convex functionals for image labelling. Optimal transport is used as a central modelling and computational tool to compute registrations between different shapes as a basis for a shape similarity measure. This point of view leads to a link between the two somewhat dual representations of a shape by the region it occupies and its outline, allowing to combine their respective strengths. Naively the computational complexity implied by the derived functionals is unfeasible. Therefore suitable hierarchical optimization methods are developed

Segmentation and quantification of spinal cord gray matter–white matter structures in magnetic resonance images

This thesis focuses on finding ways to differentiate the gray matter (GM) and white matter (WM) in magnetic resonance (MR) images of the human spinal cord (SC). The aim of this project is to quantify tissue loss in these compartments to study their implications on the progression of multiple sclerosis (MS). To this end, we propose segmentation algorithms that we evaluated on MR images of healthy volunteers. Segmentation of GM and WM in MR images can be done manually by human experts, but manual segmentation is tedious and prone to intra- and inter-rater variability. Therefore, a deterministic automation of this task is necessary. On axial 2D images acquired with a recently proposed MR sequence, called AMIRA, we experiment with various automatic segmentation algorithms. We first use variational model-based segmentation approaches combined with appearance models and later directly apply supervised deep learning to train segmentation networks. Evaluation of the proposed methods shows accurate and precise results, which are on par with manual segmentations. We test the developed deep learning approach on images of conventional MR sequences in the context of a GM segmentation challenge, resulting in superior performance compared to the other competing methods. To further assess the quality of the AMIRA sequence, we apply an already published GM segmentation algorithm to our data, yielding higher accuracy than the same algorithm achieves on images of conventional MR sequences. On a different topic, but related to segmentation, we develop a high-order slice interpolation method to address the large slice distances of images acquired with the AMIRA protocol at different vertebral levels, enabling us to resample our data to intermediate slice positions. From the methodical point of view, this work provides an introduction to computer vision, a mathematically focused perspective on variational segmentation approaches and supervised deep learning, as well as a brief overview of the underlying project's anatomical and medical background

Joint Symmetry Detection and Shape Matching for Non-Rigid Point Cloud

Despite the success of deep functional maps in non-rigid 3D shape matching, there exists no learning framework that models both self-symmetry and shape matching simultaneously. This is despite the fact that errors due to symmetry mismatch are a major challenge in non-rigid shape matching. In this paper, we propose a novel framework that simultaneously learns both self symmetry as well as a pairwise map between a pair of shapes. Our key idea is to couple a self symmetry map and a pairwise map through a regularization term that provides a joint constraint on both of them, thereby, leading to more accurate maps. We validate our method on several benchmarks where it outperforms many competitive baselines on both tasks.Comment: Under Review. arXiv admin note: substantial text overlap with arXiv:2110.0299

Variational methods and its applications to computer vision

Many computer vision applications such as image segmentation can be formulated in a ''variational'' way as energy minimization problems. Unfortunately, the computational task of minimizing these energies is usually difficult as it generally involves non convex functions in a space with thousands of dimensions and often the associated combinatorial problems are NP-hard to solve. Furthermore, they are ill-posed inverse problems and therefore are extremely sensitive to perturbations (e.g. noise). For this reason in order to compute a physically reliable approximation from given noisy data, it is necessary to incorporate into the mathematical model appropriate regularizations that require complex computations. The main aim of this work is to describe variational segmentation methods that are particularly effective for curvilinear structures. Due to their complex geometry, classical regularization techniques cannot be adopted because they lead to the loss of most of low contrasted details. In contrast, the proposed method not only better preserves curvilinear structures, but also reconnects some parts that may have been disconnected by noise. Moreover, it can be easily extensible to graphs and successfully applied to different types of data such as medical imagery (i.e. vessels, hearth coronaries etc), material samples (i.e. concrete) and satellite signals (i.e. streets, rivers etc.). In particular, we will show results and performances about an implementation targeting new generation of High Performance Computing (HPC) architectures where different types of coprocessors cooperate. The involved dataset consists of approximately 200 images of cracks, captured in three different tunnels by a robotic machine designed for the European ROBO-SPECT project.Open Acces

New Convex Relaxations and Global Optimality in Variational Imaging

Variational methods constitute the basic building blocks for solving many image analysis tasks, be it segmentation, depth estimation, optical flow, object detection etc. Many of these problems can be expressed in the framework of Markov Random Fields (MRF) or as continuous labelling problems. Finding the Maximum A-Posteriori (MAP) solutions of suitably constructed MRFs or the optimizers of the labelling problems give solutions to the aforementioned tasks. In either case, the associated optimization problem amounts to solving structured energy minimization problems. In this thesis we study novel extensions applicable to Markov Random Fields and continuous labelling problems through which we are able to incorporate statistical global constraints. To this end, we devise tractable relaxations of the resulting energy minimization problem and efficient algorithms to tackle them. Second, we propose a general mechanism to find partial optimal solutions to the problem of finding a MAP-solution of an MRF, utilizing only standard relxations