Multiple graph matching and applications

En aplicaciones de reconocimiento de patrones, los grafos con atributos son en gran medida apropiados. Normalmente, los vértices de los grafos representan partes locales de los objetos i las aristas relaciones entre estas partes locales. No obstante, estas ventajas vienen juntas con un severo inconveniente, la distancia entre dos grafos no puede ser calculada en un tiempo polinómico. Considerando estas características especiales el uso de los prototipos de grafos es necesariamente omnipresente. Las aplicaciones de los prototipos de grafos son extensas, siendo las más habituales clustering, clasificación, reconocimiento de objetos, caracterización de objetos i bases de datos de grafos entre otras. A pesar de la diversidad de aplicaciones de los prototipos de grafos, el objetivo del mismo es equivalente en todas ellas, la representación de un conjunto de grafos. Para construir un prototipo de un grafo todos los elementos del conjunto de enteramiento tienen que ser etiquetados comúnmente. Este etiquetado común consiste en identificar que nodos de que grafos representan el mismo tipo de información en el conjunto de entrenamiento. Una vez este etiquetaje común esta hecho, los atributos locales pueden ser combinados i el prototipo construido. Hasta ahora los algoritmos del estado del arte para calcular este etiquetaje común mancan de efectividad o bases teóricas. En esta tesis, describimos formalmente el problema del etiquetaje global i mostramos una taxonomía de los tipos de algoritmos existentes. Además, proponemos seis nuevos algoritmos para calcular soluciones aproximadas al problema del etiquetaje común. La eficiencia de los algoritmos propuestos es evaluada en diversas bases de datos reales i sintéticas. En la mayoría de experimentos realizados los algoritmos propuestos dan mejores resultados que los existentes en el estado del arte.In pattern recognition, the use of graphs is, to a great extend, appropriate and advantageous. Usually, vertices of the graph represent local parts of an object while edges represent relations between these local parts. However, its advantages come together with a sever drawback, the distance between two graph cannot be optimally computed in polynomial time. Taking into account this special characteristic the use of graph prototypes becomes ubiquitous. The applicability of graphs prototypes is extensive, being the most common applications clustering, classification, object characterization and graph databases to name some. However, the objective of a graph prototype is equivalent to all applications, the representation of a set of graph. To synthesize a prototype all elements of the set must be mutually labeled. This mutual labeling consists in identifying which nodes of which graphs represent the same information in the training set. Once this mutual labeling is done the set can be characterized and combined to create a graph prototype. We call this initial labeling a common labeling. Up to now, all state of the art algorithms to compute a common labeling lack on either performance or theoretical basis. In this thesis, we formally describe the common labeling problem and we give a clear taxonomy of the types of algorithms. Six new algorithms that rely on different techniques are described to compute a suboptimal solution to the common labeling problem. The performance of the proposed algorithms is evaluated using an artificial and several real datasets. In addition, the algorithms have been evaluated on several real applications. These applications include graph databases and group-wise image registration. In most of the tests and applications evaluated the presented algorithms have showed a great improvement in comparison to state of the art applications

Q(sqrt(-3))-Integral Points on a Mordell Curve

We use an extension of quadratic Chabauty to number fields,recently developed by the author with Balakrishnan, Besser and M ̈uller,combined with a sieving technique, to determine the integral points overQ(√−3) on the Mordell curve y2 = x3 − 4

From surfaces to objects : Recognizing objects using surface information and object models.

This thesis describes research on recognizing partially obscured objects using surface information like Marr's 2D sketch ([MAR82]) and surface-based geometrical object models. The goal of the recognition process is to produce a fully instantiated object hypotheses, with either image evidence for each feature or explanations for their absence, in terms of self or external occlusion. The central point of the thesis is that using surface information should be an important part of the image understanding process. This is because surfaces are the features that directly link perception to the objects perceived (for normal "camera-like" sensing) and because surfaces make explicit information needed to understand and cope with some visual problems (e.g. obscured features). Further, because surfaces are both the data and model primitive, detailed recognition can be made both simpler and more complete. Recognition input is a surface image, which represents surface orientation and absolute depth. Segmentation criteria are proposed for forming surface patches with constant curvature character, based on surface shape discontinuities which become labeled segmentation- boundaries. Partially obscured object surfaces are reconstructed using stronger surface based constraints. Surfaces are grouped to form surface clusters, which are 3D identity-independent solids that often correspond to model primitives. These are used here as a context within which to select models and find all object features. True three-dimensional properties of image boundaries, surfaces and surface clusters are directly estimated using the surface data. Models are invoked using a network formulation, where individual nodes represent potential identities for image structures. The links between nodes are defined by generic and structural relationships. They define indirect evidence relationships for an identity. Direct evidence for the identities comes from the data properties. A plausibility computation is defined according to the constraints inherent in the evidence types. When a node acquires sufficient plausibility, the model is invoked for the corresponding image structure.Objects are primarily represented using a surface-based geometrical model. Assemblies are formed from subassemblies and surface primitives, which are defined using surface shape and boundaries. Variable affixments between assemblies allow flexibly connected objects. The initial object reference frame is estimated from model-data surface relationships, using correspondences suggested by invocation. With the reference frame, back-facing, tangential, partially self-obscured, totally self-obscured and fully visible image features are deduced. From these, the oriented model is used for finding evidence for missing visible model features. IT no evidence is found, the program attempts to find evidence to justify the features obscured by an unrelated object. Structured objects are constructed using a hierarchical synthesis process. Fully completed hypotheses are verified using both existence and identity constraints based on surface evidence. Each of these processes is defined by its computational constraints and are demonstrated on two test images. These test scenes are interesting because they contain partially and fully obscured object features, a variety of surface and solid types and flexibly connected objects. All modeled objects were fully identified and analyzed to the level represented in their models and were also acceptably spatially located. Portions of this work have been reported elsewhere ([FIS83], [FIS85a], [FIS85b], [FIS86]) by the author