Optimización de una energía mediante cortes de grafos. Segmentación de imágenes

La segmentación de imágenes puede plantearse como un problema de minimización de una energía discreta. Nos enfrentamos así a una doble cuestión: definir una energía cuyo mínimo proporcione la segmentación buscada y, una vez definida la energía, encontrar un mínimo absoluto de la misma. La primera parte de esta tesis aborda el segundo problema, y la segunda parte, en un contexto más aplicado, el primero. Las técnicas de minimización basadas en cortes de grafos permiten obtener el mínimo de una energía discreta en tiempo polinomial mediante algoritmos de tipo min-cut/max-flow. Sin embargo, estas técnicas solo pueden aplicarse a energías que son representabas por grafos. Un importante reto es estudiar qué energías son representabas así como encontrar un grafo que las represente, lo que equivale a encontrar una función gadget con variables adicionales. En la primera parte de este trabajo se estudian propiedades de las funciones gadgets que permiten acotar superiormente el número de variables adicionales. Además se caracterizan las energías con cuatro variables que son representabas, definiendo gadgets con dos variables adicionales. En la segunda parte, más práctica, se aborda el problema de segmentación de imágenes médicas, base en muchas ocasiones para la diagnosis y el seguimiento de terapias. La segmentación multi-atlas es una potente técnica de segmentación automática de imágenes médicas, con tres aspectos importantes a destacar: el tipo de registro entre los atlas y la imagen objetivo, la selección de atlas y el método de fusión de etiquetas. Este último punto puede formularse como un problema de minimización de una energía. A este respecto introducimos dos nuevas energías representables. La primera, de orden dos, se utiliza en la segmentación en hígado y fondo de imágenes abdominales obtenidas mediante tomografía axial computarizada. La segunda, de orden superior, se utiliza en la segmentación en hipocampos y fondo de imágenes cerebrales obtenidas mediante resonancia magnética. ABSTRACT The image segmentation can be described as the problem of minimizing a discrete energy. We face two problems: first, to define an energy whose minimum provides the desired segmentation and, second, once the energy is defined we must find its global minimum. The first part of this thesis addresses the second problem, and the second part, in a more applied context, the first problem. Minimization techniques based on graph cuts find the minimum of a discrete energy in polynomial time via min-cut/max-flow algorithms. Nevertheless, these techniques can only be applied to graph-representable energies. An important challenge is to study which energies are graph-representable and to construct graphs which represent these energies. This is the same as finding a gadget function with additional variables. In the first part there are studied the properties of gadget functions which allow the number of additional variables to be bounded from above. Moreover, the graph-representable energies with four variables are characterised and gadgets with two additional variables are defined for these. The second part addresses the application of these ideas to medical image segmentation. This is often the first step in computer-assisted diagnosis and monitoring therapy. Multiatlas segmentation is a powerful automatic segmentation technique for medical images, with three important aspects that are highlighted here: the registration between the atlas and the target image, the atlas selection, and the label fusion method. We formulate the label fusion method as a minimization problem and we introduce two new graph-representable energies. The first is a second order energy and it is used for the segmentation of the liver in computed tomography (CT) images. The second energy is a higher order energy and it is used for the segmentation of the hippocampus in magnetic resonance images (MRI).

Column Generation Algorithms for Nonlinear Optimization II: Numerical Investigations

García et al. present a class of column generation (CG) algorithms for nonlinear programs. Its main motivation from a theoretical viewpoint is that under some circumstances, finite convergence can be achieved, in much the same way as for the classic simplicial decomposition method; the main practical motivation is that within the class there are certain nonlinear column generation problems that can accelerate the convergence of a solution approach which generates a sequence of feasible points. This algorithm can, for example, accelerate simplicial decomposition schemes by making the subproblems nonlinear. This paper complements the theoretical study on the asymptotic and finite convergence of these methods given in [1] with an experimental study focused on their computational efficiency. Three types of numerical experiments are conducted. The first group of test problems has been designed to study the parameters involved in these methods. The second group has been designed to investigate the role and the computation of the prolongation of the generated columns to the relative boundary. The last one has been designed to carry out a more complete investigation of the difference in computational efficiency between linear and nonlinear column generation approaches. In order to carry out this investigation, we consider two types of test problems: the first one is the nonlinear, capacitated single-commodity network flow problem of which several large-scale instances with varied degrees of nonlinearity and total capacity are constructed and investigated, and the second one is a combined traffic assignment model

On the role of global flow instability analysis in closed loop flow control

Control of linear flow instabilities has been demonstrated to be an effective theoretical flow control methodology, capable of modifying transitional flows on canonical geometries such as the plane channel and the flat-plate boundary layer. Extending the well-developed theoretical flow control techniques to flows over or through complex geometries requires addressing the issue of efficient capturing of the leading members of the global eigenspectrum pertinent to such flows. The present contribution describes state-of-the-art modal global instability analysis methodologies recently developed in our group, based on matrix formation and time-stepping, respectively. The relative performance of these algorithms is assessed on the recovery of BiGlobal and TriGlobal eigenspectra in the spanwise periodic and the cubic lid-driven cavity, respectively; the adjoint eigenspectrum in the latter flow is recovered for the first time. For three-dimensional flows without any homogeneous spatial direction, the time-stepping methodology was found to outperform the matrix-forming approach and permit recovering the leading TriGlobal eigenmodes in an three-dimensional open cavity of aspect ratio L : D : W = 5 : 1 : 1; theoretical flow control of this configuration is underway.