Structured Sparsity: Discrete and Convex approaches

Compressive sensing (CS) exploits sparsity to recover sparse or compressible signals from dimensionality reducing, non-adaptive sensing mechanisms. Sparsity is also used to enhance interpretability in machine learning and statistics applications: While the ambient dimension is vast in modern data analysis problems, the relevant information therein typically resides in a much lower dimensional space. However, many solutions proposed nowadays do not leverage the true underlying structure. Recent results in CS extend the simple sparsity idea to more sophisticated {\em structured} sparsity models, which describe the interdependency between the nonzero components of a signal, allowing to increase the interpretability of the results and lead to better recovery performance. In order to better understand the impact of structured sparsity, in this chapter we analyze the connections between the discrete models and their convex relaxations, highlighting their relative advantages. We start with the general group sparse model and then elaborate on two important special cases: the dispersive and the hierarchical models. For each, we present the models in their discrete nature, discuss how to solve the ensuing discrete problems and then describe convex relaxations. We also consider more general structures as defined by set functions and present their convex proxies. Further, we discuss efficient optimization solutions for structured sparsity problems and illustrate structured sparsity in action via three applications.Comment: 30 pages, 18 figure

Shape Calculus for Shape Energies in Image Processing

Many image processing problems are naturally expressed as energy minimization or shape optimization problems, in which the free variable is a shape, such as a curve in 2d or a surface in 3d. Examples are image segmentation, multiview stereo reconstruction, geometric interpolation from data point clouds. To obtain the solution of such a problem, one usually resorts to an iterative approach, a gradient descent algorithm, which updates a candidate shape gradually deforming it into the optimal shape. Computing the gradient descent updates requires the knowledge of the first variation of the shape energy, or rather the first shape derivative. In addition to the first shape derivative, one can also utilize the second shape derivative and develop a Newton-type method with faster convergence. Unfortunately, the knowledge of shape derivatives for shape energies in image processing is patchy. The second shape derivatives are known for only two of the energies in the image processing literature and many results for the first shape derivative are limiting, in the sense that they are either for curves on planes, or developed for a specific representation of the shape or for a very specific functional form in the shape energy. In this work, these limitations are overcome and the first and second shape derivatives are computed for large classes of shape energies that are representative of the energies found in image processing. Many of the formulas we obtain are new and some generalize previous existing results. These results are valid for general surfaces in any number of dimensions. This work is intended to serve as a cookbook for researchers who deal with shape energies for various applications in image processing and need to develop algorithms to compute the shapes minimizing these energies

Parametric Level Set Methods for Inverse Problems

In this paper, a parametric level set method for reconstruction of obstacles in general inverse problems is considered. General evolution equations for the reconstruction of unknown obstacles are derived in terms of the underlying level set parameters. We show that using the appropriate form of parameterizing the level set function results a significantly lower dimensional problem, which bypasses many difficulties with traditional level set methods, such as regularization, re-initialization and use of signed distance function. Moreover, we show that from a computational point of view, low order representation of the problem paves the path for easier use of Newton and quasi-Newton methods. Specifically for the purposes of this paper, we parameterize the level set function in terms of adaptive compactly supported radial basis functions, which used in the proposed manner provides flexibility in presenting a larger class of shapes with fewer terms. Also they provide a "narrow-banding" advantage which can further reduce the number of active unknowns at each step of the evolution. The performance of the proposed approach is examined in three examples of inverse problems, i.e., electrical resistance tomography, X-ray computed tomography and diffuse optical tomography