A primal-dual active-set method for non-negativity constrained total variation deblurring problems

Master'sMASTER OF SCIENC

Preconditioning for Allen-Cahn variational inequalities with non-local constraints

The solution of Allen-Cahn variational inequalities with mass constraints is of interest in many applications. This problem can be solved both in its scalar and vector-valued form as a PDE-constrained optimization problem by means of a primal-dual active set method. At the heart of this method lies the solution of linear systems in saddle point form. In this paper we propose the use of Krylov-subspace solvers and suitable preconditioners for the saddle point systems. Numerical results illustrate the competitiveness of this approach

A primal-dual active-set method for non-negativity constrained total variation deblurring problems

Abstract—This paper studies image deblurring problems using a total variation-based model, with a non-negativity constraint. The addition of the non-negativity constraint improves the quality of the solutions, but makes the solution process a difficult one. The contribution of our work is a fast and robust numerical algorithm to solve the non-negatively constrained problem. To overcome the nondifferentiability of the total variation norm, we formulate the constrained deblurring problem as a primal-dual program which is a variant of the formulation proposed by Chan, Golub, and Mulet for unconstrained problems. Here, dual refers to a combination of the Lagrangian and Fenchel duals. To solve the constrained primal-dual program, we use a semi-smooth Newton’s method. We exploit the relationship between the semi-smooth Newton’s method and the primal-dual active set method to achieve considerable simplification of the computations. The main advantages of our proposed scheme are: no parameters need significant adjustment, a standard inverse preconditioner works very well, quadratic rate of local convergence (theoretical and numerical), numerical evidence of global convergence, and high accuracy of solving the optimality system. The scheme shows robustness of performance over a wide range of parameters. A comprehensive set of numerical comparisons are provided against other methods to solve the same problem which show the speed and accuracy advantages of our scheme. 1 Index Terms—Image deblurring, non-negativity, primal-dual active-set, semismooth Newton’s method, total variation

Long Range Automated Persistent Surveillance

This dissertation addresses long range automated persistent surveillance with focus on three topics: sensor planning, size preserving tracking, and high magnification imaging. field of view should be reserved so that camera handoff can be executed successfully before the object of interest becomes unidentifiable or untraceable. We design a sensor planning algorithm that not only maximizes coverage but also ensures uniform and sufficient overlapped camera’s field of view for an optimal handoff success rate. This algorithm works for environments with multiple dynamic targets using different types of cameras. Significantly improved handoff success rates are illustrated via experiments using floor plans of various scales. Size preserving tracking automatically adjusts the camera’s zoom for a consistent view of the object of interest. Target scale estimation is carried out based on the paraperspective projection model which compensates for the center offset and considers system latency and tracking errors. A computationally efficient foreground segmentation strategy, 3D affine shapes, is proposed. The 3D affine shapes feature direct and real-time implementation and improved flexibility in accommodating the target’s 3D motion, including off-plane rotations. The effectiveness of the scale estimation and foreground segmentation algorithms is validated via both offline and real-time tracking of pedestrians at various resolution levels. Face image quality assessment and enhancement compensate for the performance degradations in face recognition rates caused by high system magnifications and long observation distances. A class of adaptive sharpness measures is proposed to evaluate and predict this degradation. A wavelet based enhancement algorithm with automated frame selection is developed and proves efficient by a considerably elevated face recognition rate for severely blurred long range face images

Primal-Dual Active-Set Methods for Convex Quadratic Optimization with Applications

Primal-dual active-set (PDAS) methods are developed for solving quadratic optimization problems (QPs). Such problems arise in their own right in optimal control and statistics–two applications of interest considered in this dissertation–and as subproblems when solving nonlinear optimization problems. PDAS methods are promising as they possess the same favorable properties as other active-set methods, such as their ability to be warm-started and to obtain highly accurate solutions by explicitly identifying sets of constraints that are active at an optimal solution. However, unlike traditional active-set methods, PDAS methods have convergence guarantees despite making rapid changes in active-set estimates, making them well suited for solving large-scale problems.Two PDAS variants are proposed for efficiently solving generally-constrained convex QPs. Both variants ensure global convergence of the iterates by enforcing montonicity in a measure of progress. Besides identifying an estimate set estimate, a novel uncertain set is introduced into the framework in order to house indices of variables that have been identified as being susceptible to cycling. The introduction of the uncertainty set guarantees convergence of the algorithm, and with techniques proposed to keep the set from expanding quickly, the practical performance of the algorithm is shown to be very efficient. Another PDAS variant is proposed for solving certain convex QPs that commonly arise when discretizing optimal control problems. The proposed framework allows inexactness in the subproblem solutions, which can significantly reduce computational cost in large-scale settings. By controlling the level inexactness either by exploiting knowledge of an upper bound of a matrix inverse or by dynamic estimation of such a value, the method achieves convergence guarantees and is shown to outperform a method that employs exact solutions computed by direct factorization techniques.Finally, the application of PDAS techniques for applications in statistics, variants are proposed for solving isotonic regression (IR) and trend filtering (TR) problems. It is shown that PDAS can solve an IR problem with n data points with only O(n) arithmetic operations. Moreover, the method is shown to outperform the state-of-the-art method for solving IR problems, especially when warm-starting is considered. Enhancements to themethod are proposed for solving general TF problems, and numerical results are presented to show that PDAS methods are viable for a broad class of such problems

Biological image analysis

In biological research images are extensively used to monitor growth, dynamics and changes in biological specimen, such as cells or plants. Many of these images are used solely for observation or are manually annotated by an expert. In this dissertation we discuss several methods to automate the annotating and analysis of bio-images. Two large clusters of methods have been investigated and developed. A first set of methods focuses on the automatic delineation of relevant objects in bio-images, such as individual cells in microscopic images. Since these methods should be useful for many different applications, e.g. to detect and delineate different objects (cells, plants, leafs, ...) in different types of images (different types of microscopes, regular colour photographs, ...), the methods should be easy to adjust. Therefore we developed a methodology relying on probability theory, where all required parameters can easily be estimated by a biologist, without requiring any knowledge on the techniques used in the actual software. A second cluster of investigated techniques focuses on the analysis of shapes. By defining new features that describe shapes, we are able to automatically classify shapes, retrieve similar shapes from a database and even analyse how an object deforms through time