Enhancing joint reconstruction and segmentation with non-convex Bregman iteration

All imaging modalities such as computed tomography (CT), emission tomography and magnetic resonance imaging (MRI) require a reconstruction approach to produce an image. A common image processing task for applications that utilise those modalities is image segmentation, typically performed posterior to the reconstruction. We explore a new approach that combines reconstruction and segmentation in a unified framework. We derive a variational model that consists of a total variation regularised reconstruction from undersampled measurements and a Chan-Vese based segmentation. We extend the variational regularisation scheme to a Bregman iteration framework to improve the reconstruction and therefore the segmentation. We develop a novel alternating minimisation scheme that solves the non-convex optimisation problem with provable convergence guarantees. Our results for synthetic and real data show that both reconstruction and segmentation are improved compared to the classical sequential approach

Efficient Algorithms for Mumford-Shah and Potts Problems

In this work, we consider Mumford-Shah and Potts models and their higher order generalizations. Mumford-Shah and Potts models are among the most well-known variational approaches to edge-preserving smoothing and partitioning of images. Though their formulations are intuitive, their application is not straightforward as it corresponds to solving challenging, particularly non-convex, minimization problems. The main focus of this thesis is the development of new algorithmic approaches to Mumford-Shah and Potts models, which is to this day an active field of research. We start by considering the situation for univariate data. We find that switching to higher order models can overcome known shortcomings of the classical first order models when applied to data with steep slopes. Though the existing approaches to the first order models could be applied in principle, they are slow or become numerically unstable for higher orders. Therefore, we develop a new algorithm for univariate Mumford-Shah and Potts models of any order and show that it solves the models in a stable way in O(n^2). Furthermore, we develop algorithms for the inverse Potts model. The inverse Potts model can be seen as an approach to jointly reconstructing and partitioning images that are only available indirectly on the basis of measured data. Further, we give a convergence analysis for the proposed algorithms. In particular, we prove the convergence to a local minimum of the underlying NP-hard minimization problem. We apply the proposed algorithms to numerical data to illustrate their benefits. Next, we apply the multi-channel Potts prior to the reconstruction problem in multi-spectral computed tomography (CT). To this end, we propose a new superiorization approach, which perturbs the iterates of the conjugate gradient method towards better results with respect to the Potts prior. In numerical experiments, we illustrate the benefits of the proposed approach by comparing it to the existing Potts model approach from the literature as well as to the existing total variation type methods. Hereafter, we consider the second order Mumford-Shah model for edge-preserving smoothing of images which –similarly to the univariate case– improves upon the classical Mumford-Shah model for images with linear color gradients. Based on reformulations in terms of Taylor jets, i.e. specific fields of polynomials, we derive discrete second order Mumford-Shah models for which we develop an efficient algorithm using an ADMM scheme. We illustrate the potential of the proposed method by comparing it with existing methods for the second order Mumford-Shah model. Further, we illustrate its benefits in connection with edge detection. Finally, we consider the affine-linear Potts model for the image partitioning problem. As many images possess linear trends within homogeneous regions, the classical Potts model frequently leads to oversegmentation. The affine-linear Potts model accounts for that problem by allowing for linear trends within segments. We lift the corresponding minimization problem to the jet space and develop again an ADMM approach. In numerical experiments, we show that the proposed algorithm achieves lower energy values as well as faster runtimes than the method of comparison, which is based on the iterative application of the graph cut algorithm (with α-expansion moves)

Variational Multi-Task Models for Image Analysis: Applications to Magnetic Resonance Imaging

This thesis deals with the study and development of several variational multi-task models for solving inverse problems in imaging, with a particular focus on Magnetic Resonance Imaging (MRI). In most image processing problems, one usually deals with the reconstruction task, i.e., the task of reconstructing an image from indirect measurements, and then performs various operations, one after the other (i.e. sequentially), to improve the quality of the reconstruction and to extract useful information. However, recent developments in a variational context, have shown that performing those tasks jointly (i.e. in a multi-task framework) offers great benefits, and this is the perspective that we follow in this thesis. We go beyond traditional sequential approaches and set a new basis for variational multi-task methods for MRI analysis. We demonstrate that by sharing representation between tasks and carefully interconnecting them, one can create synergies across challenging problems and reduce error propagation. More precisely, firstly we propose a multi-task variational model to tackle the problems of image reconstruction and image segmentation using non-convex Bregman iteration. We describe theoretical and numerical details of the problem and its optimisation scheme. Moreover, we show that our multi-task model achieves better results in several examples and MRI applications than existing approaches in the same context. Secondly, we show that our approach can be extended to a multi-task reconstruction and segmentation model for the nonlinear inverse problem of velocity-encoded MRI. In this context, the aim is to estimate not only the magnitude from MRI data, but also the phase and its flow information, whilst simultaneously identify regions of interest through the segmentation task. Finally, we go beyond two-task frameworks and introduce for the first time a variational multi-task model to handle three imaging tasks. To this end, we design a variational multi-task framework addressing reconstruction, super-resolution and registration for improving the quality of MRI reconstruction. We demonstrate that our model is theoretically well-motivated and it outperforms sequential models whilst requiring less computational cost. Furthermore, we show through experimental results the potential of this approach for clinical applications

Reconstruction algorithms for Magnetic Resonance Imaging

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2013.Cataloged from PDF version of thesis.Includes bibliographical references (p. 135-142).This dissertation presents image reconstruction algorithms for Magnetic Resonance Imaging (MRI) that aims to increase the imaging efficiency. Algorithms that reduce imaging time without sacrificing the image quality and mitigate image artifacts are proposed. The goal of increasing the MR efficiency is investigated across multiple imaging techniques: structural imaging with multiple contrasts preparations, Diffusion Spectrum Imaging (DSI), Chemical Shift Imaging (CSI), and Quantitative Susceptibility Mapping (QSM). The main theme connecting the proposed methods is the utilization of prior knowledge on the reconstructed signal. This prior often presents itself in the form of sparsity with respect to either a prespecified or learned signal transformation.by Berkin Bilgic.Ph.D

Trends in Mathematical Imaging and Surface Processing

Motivated both by industrial applications and the challenge of new problems, one observes an increasing interest in the field of image and surface processing over the last years. It has become clear that even though the applications areas differ significantly the methodological overlap is enormous. Even if contributions to the field come from almost any discipline in mathematics, a major role is played by partial differential equations and in particular by geometric and variational modeling and by their numerical counterparts. The aim of the workshop was to gather a group of leading experts coming from mathematics, engineering and computer graphics to cover the main developments

Proceedings of the second "international Traveling Workshop on Interactions between Sparse models and Technology" (iTWIST'14)

The implicit objective of the biennial "international - Traveling Workshop on Interactions between Sparse models and Technology" (iTWIST) is to foster collaboration between international scientific teams by disseminating ideas through both specific oral/poster presentations and free discussions. For its second edition, the iTWIST workshop took place in the medieval and picturesque town of Namur in Belgium, from Wednesday August 27th till Friday August 29th, 2014. The workshop was conveniently located in "The Arsenal" building within walking distance of both hotels and town center. iTWIST'14 has gathered about 70 international participants and has featured 9 invited talks, 10 oral presentations, and 14 posters on the following themes, all related to the theory, application and generalization of the "sparsity paradigm": Sparsity-driven data sensing and processing; Union of low dimensional subspaces; Beyond linear and convex inverse problem; Matrix/manifold/graph sensing/processing; Blind inverse problems and dictionary learning; Sparsity and computational neuroscience; Information theory, geometry and randomness; Complexity/accuracy tradeoffs in numerical methods; Sparsity? What's next?; Sparse machine learning and inference.Comment: 69 pages, 24 extended abstracts, iTWIST'14 website: http://sites.google.com/site/itwist1

Variational Methods for Discrete Tomography

Image reconstruction from tomographic sampled data has contoured as a stand alone research area with application in many practical situations, in domains such as medical imaging, seismology, astronomy, flow analysis, industrial inspection and many more. Already existing algorithms on the market (continuous) fail in being able to model the analysed object. In this thesis, we study discrete tomographic approaches that enable the addition of constraints in order to better fit the description of the analysed object and improve the end result. A particular focus is set on assumptions regarding the signals' sampling methodology, point at which we look towards the recently introduced Compressive Sensing (CS) approach, that has shown to return remarkable results based on how sparse a given signal is. However, research done in the CS field does not accurately relate to real world applications, as objects usually surrounding us are considered to be piece-wise constant (not sparse on their own) and the properties of the sensing matrices from the viewpoint of CS do not re ect real acquisition processes. Motivated by these shortcomings, we study signals that are sparse in a given representation, e.g. the forward-difference operator (total variation) and develop reconstruction diagrams (phase transitions) with the help of linear programming, convex analysis and duality that enable the user to pin-point the type of objects (with regard to their sparsity) which can be reconstructed, given an ensemble of acquisition directions. Moreover, a closer look is given to handling large data volumes, by adding different perturbations (entropic, quadratic) to the already constrained linear program. In empirical assessments, perturbation has lead to an increased reconstruction rate. Needless to say, the topic of this thesis is motivated by industrial applications where the acquisition process is restricted to a maximum of nine cameras, thus returning a severely undersampled inverse problem

Advanced methods for mapping the radiofrequency magnetic fields in MRI

As MRI systems have increased in static magnetic field strength, the radiofrequency (RF) fields that are used for magnetisation excitation and signal reception have become significantly less uniform. This can lead to image artifacts and errors when performing quantitative MRI. A further complication arises if the RF fields vary substantially in time. In the first part of this investigation temporal variations caused by respiration were explored on a 3T scanner. It was found that fractional changes in transmit field amplitude between inhalation and expiration ranged from 1% to 14% in the region of the liver in a small group of normal subjects. This observation motivated the development of a pulse sequence and reconstruction method to allow dynamic observation of the transmit field throughout the respiratory cycle. However, the proposed method was unsuccessful due to the inherently time-consuming nature of transmit field mapping sequences. This prompted the development of a novel data reconstruction method to allow the acceleration of transmit field mapping sequences. The proposed technique posed the RF field reconstruction as a nonlinear least-squares optimisation problem, exploiting the fact that the fields vary smoothly. It was shown that this approach was superior to standard reconstruction approaches. The final component of this thesis presents a unified approach to RF field calibration. The proposed method uses all measured data to estimate both transmit and receive sensitivities, whilst simultaneously insisting that they are smooth functions of space. The resulting maps are robust to both noise and imperfections in regions of low signal