Geometric Variational Models for Inverse Problems in Imaging

This dissertation develops geometric variational models for different inverse problems in imaging that are ill-posed, designing at the same time efficient numerical algorithms to compute their solutions. Variational methods solve inverse problems by the following two steps: formulation of a variational model as a minimization problem, and design of a minimization algorithm to solve it. This dissertation is organized in the same manner. It first formulates minimization problems associated with geometric models for different inverse problems in imaging, and it then designs efficient minimization algorithms to compute their solutions. The minimization problem summarizes both the data available from the measurements and the prior knowledge about the solution in its objective functional; this naturally leads to the combination of a measurement or data term and a prior term. Geometry can play a role in any of these terms, depending on the properties of the data acquisition system or the object being imaged. In this context, each chapter of this dissertation formulates a variational model that includes geometry in a different manner in the objective functional, depending on the inverse problem at hand. In the context of compressed sensing, the first chapter exploits the geometric properties of images to include an alignment term in the sparsity prior of compressed sensing; this additional prior term aligns the normal vectors of the level curves of the image with the reconstructed signal, and it improves the quality of reconstruction. A two-step recovery method is designed for that purpose: first, it estimates the normal vectors to the level curves of the image; second, it reconstructs an image matching the compressed sensing measurements, the geometric alignment of normals, and the sparsity constraint of compressed sensing. The proposed method is extended to non-local operators in graphs for the recovery of textures. The harmonic active contours of Chapter 2 make use of differential geometry to interpret the segmentation of an image as a minimal surface manifold. In this case, geometry is exploited in both the measurement term, by coupling the different image channels in a robust edge detector, and in the prior term, by imposing smoothness in the segmentation. The proposed technique generalizes existing active contours to higher dimensional spaces and non-flat images; in the plane, it improves the segmentation of images with inhomogeneities and weak edges. Shape-from-shading is investigated in Chapter 3 for the reconstruction of a silicon wafer from images of printed circuits taken with a scanning electron microscope. In this case, geometry plays a role in the image acquisition system, that is, in the measurement term of the objective functional. The prior term involves a smoothness constraint on the surface and a shape prior on the expected pattern in the circuit. The proposed reconstruction method also estimates a deformation field between the ideal pattern design and the reconstructed surface, substituting the model of shape variability necessary in shape priors with an elastic deformation field that quantifies deviations in the manufacturing process. Finally, the techniques used for the design of efficient numerical algorithms are explained with an example problem based on the level set method. To this purpose, Chapter 4 develops an efficient algorithm for the level set method when the level set function is constrained to remain a signed distance function. The distance function is preserved by the introduction of an explicit constraint in the minimization problem, the minimization algorithm is efficient by the adequate use of variable-splitting and augmented Lagrangian techniques. These techniques introduce additional variables, constraints, and Lagrange multipliers in the original minimization problem, and they decompose it into sub-optimization problems that are simple and can be efficiently solved. As a result, the proposed algorithm is five to six times faster than the original algorithm for the level set method.