Donat un conjunt d'objectes, el concepte genèric de mediana està definit com l'objecte amb la suma de distàncies a tot el conjunt, més petita. Sovint, aquest concepte és usat per a obtenir el representant del conjunt. En el reconeixement estructural de patrons, els grafs han estat usats normalment per a representar objectes complexos. En el domini dels grafs, el concepte de mediana és conegut com median graph. Potencialment, té les mateixes aplicacions que el concepte de mediana per poder ser usat com a representant d'un conjunt de grafs. Tot i la seva simple definició i les potencials aplicacions, s'ha demostrat que el seu càlcul és una tasca extremadament complexa. Tots els algorismes existents només han estat capaços de treballar amb conjunts petits de grafs, i per tant, la seva aplicació ha estat limitada en molts casos a usar dades sintètiques sense significat real. Així, tot i el seu potencial, ha restat com un concepte eminentment teòric. L'objectiu principal d'aquesta tesi doctoral és el d'investigar a fons la teoria i l'algorísmica relacionada amb el concepte de medinan graph, amb l'objectiu final d'extendre la seva aplicabilitat i lliurar tot el seu potencial al món de les aplicacions reals. Per això, presentem nous resultats teòrics i també nous algorismes per al seu càlcul. Des d'un punt de vista teòric aquesta tesi fa dues aportacions fonamentals. Per una banda, s'introdueix el nou concepte d'spectral median graph. Per altra banda es mostra que certes de les propietats teòriques del median graph poden ser millorades sota determinades condicions. Més enllà de les aportacioncs teòriques, proposem cinc noves alternatives per al seu càlcul. La primera d'elles és una conseqüència directa del concepte d'spectral median graph. Després, basats en les millores de les propietats teòriques, presentem dues alternatives més per a la seva obtenció. Finalment, s'introdueix una nova tècnica per al càlcul del median basat en el mapeig de grafs en espais de vectors, i es proposen dos nous algorismes més. L'avaluació experimental dels mètodes proposats utilitzant una base de dades semi-artificial (símbols gràfics) i dues amb dades reals (mollècules i pàgines web), mostra que aquests mètodes són molt més eficients que els existents. A més, per primera vegada, hem demostrat que el median graph pot ser un bon representant d'un conjunt d'objectes utilitzant grans quantitats de dades. Hem dut a terme experiments de classificació i clustering que validen aquesta hipòtesi i permeten preveure una pròspera aplicació del median graph a un bon nombre d'algorismes d'aprenentatge. / Given a set of objects, the generic concept of median is defined as the object with the smallest sum of distances to all the objects in the set. It has been often used as a good alternative to obtain a representative of the set. In structural pattern recognition, graphs are normally used to represent structured objects. In the graph domain, the concept analogous to the median is known as the median graph. By extension, it has the same potential applications as the generic median in order to be used as the representative of a set of graphs. Despite its simple definition and potential applications, its computation has been shown as an extremely complex task. All the existing algorithms can only deal with small sets of graphs, and its application has been constrained in most cases to the use of synthetic data with no real meaning. Thus, it has mainly remained in the box of the theoretical concepts. The main objective of this work is to further investigate both the theory and the algorithmic underlying the concept of the median graph with the final objective to extend its applicability and bring all its potential to the world of real applications. To this end, new theory and new algorithms for its computation are reported. From a theoretical point of view, this thesis makes two main contributions. On one hand, the new concept of spectral median graph. On the other hand, we show that some of the existing theoretical properties of the median graph can be improved under some specific conditions. In addition to these theoretical contributions, we propose five new ways to compute the median graph. One of them is a direct consequence of the spectral median graph concept. In addition, we provide two new algorithms based on the new theoretical properties. Finally, we present a novel technique for the median graph computation based on graph embedding into vector spaces. With this technique two more new algorithms are presented. The experimental evaluation of the proposed methods on one semi-artificial and two real-world datasets, representing graphical symbols, molecules and webpages, shows that these methods are much more ecient than the existing ones. In addition, we have been able to proof for the first time that the median graph can be a good representative of a class in large datasets. We have performed some classification and clustering experiments that validate this hypothesis and permit to foresee a successful application of the median graph to a variety of machine learning algorithms.

El reconeixement de patrons és la tasca que pretén distingir objectes entre diferents classes. Quan aquesta tasca es vol solucionar de forma automàtica un pas crucial és el com representar formalment els patrons a l'ordinador. En funció d'aquests formalismes, podem distingir entre el reconeixement estadístic i l'estructural. El primer descriu objectes com un conjunt de mesures col·locats en forma del que s'anomena un vector de característiques. El segon assumeix que hi ha relacions entre parts dels objectes que han de quedar explícitament representades i per tant fa servir estructures relacionals com els grafs per codificar la seva informació inherent. Els espais vectorials són una estructura matemàtica molt flexible que ha permès definir diverses maneres eficients d'analitzar patrons sota la forma de vectors de característiques. De totes maneres, la representació vectorial no és capaç d'expressar explícitament relacions binàries entre parts dels objectes i està restrigida a mesurar sempre, independentment de la complexitat dels patrons, el mateix nombre de característiques per cadascun d'ells. Les representacions en forma de graf presenten la situació contrària. Poden adaptar-se fàcilment a la complexitat inherent dels patrons però introdueixen un problema d'alta complexitat computational, dificultant el disseny d'eines eficients per al procés i l'anàlisis de patrons. Resoldre aquesta paradoxa és el principal objectiu d'aquesta tesi. La situació ideal per resoldre problemes de reconeixement de patrons seria el representar-los fent servir estructures relacionals com els grafs, i a l'hora, poder fer ús del ric repositori d'eines pel processament de dades del reconeixement estadístic. Una solució elegant a aquest problema és la de transformar el domini dels grafs en el domini dels vectors, on podem aplicar qualsevol algorisme de processament de dades. En altres paraules, assignant a cada graf un punt en un espai vectorial, automàticament tenim accés al conjunt d'algorismes del món estadístic per aplicar-los al domini dels grafs. Aquesta metodologia s'anomena graph embedding. En aquesta tesi proposem de fer una associació de grafs a vectors de característiques de forma simple i eficient fixant l'atenció en la informació d'etiquetatge dels grafs. En particular, comptem les freqüències de les etiquetes dels nodes així com de les aretes entre etiquetes determinades. Tot i la seva localitat, aquestes característiques donen una representació prou robusta de les propietats globals dels grafs. Primer tractem el cas de grafs amb etiquetes discretes, on les característiques són sencilles de calcular. El cas continu és abordat com una generalització del cas discret, on enlloc de comptar freqüències d'etiquetes, ho fem de representants d'aquestes. Ens trobem que les representacions vectorials que proposem pateixen d'alta dimensionalitat i correlació entre components, i tractem aquests problems mitjançant algorismes de selecció de característiques. També estudiem com la diversitat de diferents representacions pot ser explotada per tal de millorar el rendiment de classificadors base en el marc d'un sistema de múltiples classificadors. Finalment, amb una extensa evaluació experimental mostrem com la metodologia proposada pot ser calculada de forma eficient i com aquesta pot competir amb altres metodologies per a la comparació de grafs. / Pattern recognition is the task that aims at distinguishing objects among different classes. When such a task wants to be solved in an automatic way a crucial step is how to formally represent such patterns to the computer. Based on the different representational formalisms, we may distinguish between statistical and structural pattern recognition. The former describes objects as a set of measurements arranged in the form of what is called a feature vector. The latter assumes that relations between parts of the underlying objects need to be explicitly represented and thus it uses relational structures such as graphs for encoding their inherent information. Vector spaces are a very flexible mathematical structure that has allowed to come up with several efficient ways for the analysis of patterns under the form of feature vectors. Nevertheless, such a representation cannot explicitly cope with binary relations between parts of the objects and it is restricted to measure the exact same number of features for each pattern under study regardless of their complexity. Graph-based representations present the contrary situation. They can easily adapt to the inherent complexity of the patterns but introduce a problem of high computational complexity, hindering the design of efficient tools to process and analyze patterns. Solving this paradox is the main goal of this thesis. The ideal situation for solving pattern recognition problems would be to represent the patterns using relational structures such as graphs, and to be able to use the wealthy repository of data processing tools from the statistical pattern recognition domain. An elegant solution to this problem is to transform the graph domain into a vector domain where any processing algorithm can be applied. In other words, by mapping each graph to a point in a vector space we automatically get access to the rich set of algorithms from the statistical domain to be applied in the graph domain. Such methodology is called graph embedding. In this thesis we propose to associate feature vectors to graphs in a simple and very efficient way by just putting attention on the labelling information that graphs store. In particular, we count frequencies of node labels and of edges between labels. Although their locality, these features are able to robustly represent structurally global properties of graphs, when considered together in the form of a vector. We initially deal with the case of discrete attributed graphs, where features are easy to compute. The continuous case is tackled as a natural generalization of the discrete one, where rather than counting node and edge labelling instances, we count statistics of some representatives of them. We encounter how the proposed vectorial representations of graphs suffer from high dimensionality and correlation among components and we face these problems by feature selection algorithms. We also explore how the diversity of different embedding representations can be exploited in order to boost the performance of base classifiers in a multiple classifier systems framework. An extensive experimental evaluation finally shows how the methodology we propose can be efficiently computed and compete with other graph matching and embedding methodologies.

Detecção de objetos é um problema clássico em visão computacional, presente em aplicações como vigilância automatizada, análise de imagens médicas e recuperação de informação. Dentre as abordagens existentes na literatura para resolver esse problema, destacam-se métodos baseados em reconhecimento de pontos-chave que podem ser interpretados como diferentes implementações de um mesmo arcabouço. O objetivo desta pesquisa de doutorado é desenvolver e avaliar uma versão generalizada desse arcabouço, na qual reconhecimento de pontos-chave é substituído por reconhecimento de grafos-chave. O potencial da pesquisa reside na riqueza de informação que um grafo pode apresentar antes e depois de ser reconhecido. A dificuldade da pesquisa reside nos problemas que podem ser causados por essa riqueza, como maldição da dimensionalidade e complexidade computacional. Três contribuições serão incluídas na tese: a descrição detalhada de um arcabouço para detecção de objetos baseado em grafos-chave, implementações fiéis que demonstram sua viabilidade e resultados experimentais que demonstram seu desempenho. / Object detection is a classic problem in computer vision, present in applications such as automated surveillance, medical image analysis and information retrieval. Among the existing approaches in the literature to solve this problem, we can highlight methods based on keypoint recognition that can be interpreted as different implementations of a same framework. The objective of this PhD thesis is to develop and evaluate a generalized version of this framework, on which keypoint recognition is replaced by keygraph recognition. The potential of the research resides in the information richness that a graph can present before and after being recognized. The difficulty of the research resides in the problems that can be caused by this richness, such as curse of dimensionality and computational complexity. Three contributions are included in the thesis: the detailed description of a keygraph-based framework for object detection, faithful implementations that demonstrate its feasibility and experimental results that demonstrate its performance.

Detecção de objetos é um problema clássico em visão computacional, presente em aplicações como vigilância automatizada, análise de imagens médicas e recuperação de informação. Dentre as abordagens existentes na literatura para resolver esse problema, destacam-se métodos baseados em reconhecimento de pontos-chave que podem ser interpretados como diferentes implementações de um mesmo arcabouço. O objetivo desta pesquisa de doutorado é desenvolver e avaliar uma versão generalizada desse arcabouço, na qual reconhecimento de pontos-chave é substituído por reconhecimento de grafos-chave. O potencial da pesquisa reside na riqueza de informação que um grafo pode apresentar antes e depois de ser reconhecido. A dificuldade da pesquisa reside nos problemas que podem ser causados por essa riqueza, como maldição da dimensionalidade e complexidade computacional. Três contribuições serão incluídas na tese: a descrição detalhada de um arcabouço para detecção de objetos baseado em grafos-chave, implementações fiéis que demonstram sua viabilidade e resultados experimentais que demonstram seu desempenho. / Object detection is a classic problem in computer vision, present in applications such as automated surveillance, medical image analysis and information retrieval. Among the existing approaches in the literature to solve this problem, we can highlight methods based on keypoint recognition that can be interpreted as different implementations of a same framework. The objective of this PhD thesis is to develop and evaluate a generalized version of this framework, on which keypoint recognition is replaced by keygraph recognition. The potential of the research resides in the information richness that a graph can present before and after being recognized. The difficulty of the research resides in the problems that can be caused by this richness, such as curse of dimensionality and computational complexity. Three contributions are included in the thesis: the detailed description of a keygraph-based framework for object detection, faithful implementations that demonstrate its feasibility and experimental results that demonstrate its performance.

Architectural floor plans are scale-accurate 2D drawings of one level of a building, seen from above, which convey structural and semantic information related to rooms, walls, symbols, textual data, etc. They consist of lines, curves, symbols, and textual markings, showing the relationships between rooms and all physical features, required for the proper construction or renovation of the building. First, this thesis provides a thorough study of state-of-the-art on symbol spotting methods for architectural drawings, an application domain providing the document image analysis and graphic recognition communities with an interesting set of challenges linked to the sheer complexity and density of embedded information, that have yet to be resolved. Second, we propose a hybrid method that capitalizes on strengths of both vector-based and pixel-based symbol spotting techniques. In the description phase, the salient geometric constituents of a symbol are extracted by a variety of vectorization techniques, including a proposed voting-based algorithm for finding partial ellipses. This enables us to better handle local shape irregularities and boundary discontinuities, as well as partial occlusion and overlap. In the matching phase, the spatial relationship between the geometric primitives is encoded via a primitive-aware proximity graph. A statistical approach is then used to rapidly yield a coarse localization of symbols within the plan. Localization is further refined with a pixel-based step implementing a modified cross-correlation function. Experimental results on the public SESYD synthetic dataset and real-world images demonstrate that our approach clearly outperforms other popular symbol spotting approaches. Traditional on-the-fly symbol spotting methods are unable to address the semantic challenge of graphical notation variability, i.e. low intra-class symbol similarity, an issue that is particularly important in architectural floor plan analysis. The presence of occlusion and clutter, characteristic of real-world plans, along with a varying graphical symbol complexity from almost trivial to highly complex, also pose challenges to existing spotting methods. Third, we address all the above issues by leveraging recent advances in deep learning-based neural networks and adapting an object detection framework based on the YOLO (You Only Look Once) architecture. We propose a training strategy based on tiles, avoiding many issues particular to deep learning-based object detection networks related to the relatively small size of symbols compared to entire floor plans, aspect ratios, and data augmentation. Experimental results demonstrate that our method successfully detects architectural symbols with low intra-class similarity and of variable graphical complexity, even in the presence of heavy occlusion and clutter. / Graduate

Die Dissertation beschäftigt sich mit Labeling Problemen. Dieses Forschungsgebiet bildet einen wichtigen Teil der strukturellen Mustererkennung, in der die Struktur des zu erkennenden Objektes explizit berücksichtigt wird. Die entwickelte Theorie wird auf die Aufgabe der Stereorekonstruktion angewendet. / The thesis studies the class of labeling problems. This theory contributes to the new stream in pattern recognition in which structure is explicitly taken into account. The developed theory is applied to practical problem of stereo reconstruction.

Die Dissertation beschäftigt sich mit Labeling Problemen. Dieses Forschungsgebiet bildet einen wichtigen Teil der strukturellen Mustererkennung, in der die Struktur des zu erkennenden Objektes explizit berücksichtigt wird. Die entwickelte Theorie wird auf die Aufgabe der Stereorekonstruktion angewendet. / The thesis studies the class of labeling problems. This theory contributes to the new stream in pattern recognition in which structure is explicitly taken into account. The developed theory is applied to practical problem of stereo reconstruction.

Esta tese de doutorado aborda os tópicos de modelagem e de reconhecimento de objetos estruturados, ou sistemas estruturados de objetos, em imagens. Um objeto ou sistema estruturado é aquele que pode ser descrito através de elementos primitivos que o compõem e pelas relações existentes entre esses elementos. Por exemplo, uma aeronave pode ser descrita pelos seguintes elementos primitivos: asas direita e esquerda, fuselagem e cockpit. O aspecto relacional de um objeto estruturado direciona sua representação computacional e seu reconhecimento em imagens ao paradigma estrutural de reconhecimento de padrões. Contudo, a variabilidade das características dos seus elementos primitivos é melhor representada através do paradigma estatístico de reconhecimento de padrões. Devido à complementaridade dos paradigmas, a conjunção dessas abordagens é um tema de pesquisa de interesse atual. Para conjugar esses dois aspectos, esta tese propôs uma metodologia que combina o conhecimento a priori das relações que caracterizam um objeto estruturado com dados estatísticos coletados de amostras desse objeto, num modelo híbrido denominado grafo estatístico-relacional (GER). Segundo essa representação, foi estudada uma abordagem probabilística para reconhecer um objeto estruturado em imagens. Nesse cenário, o GER modelo é considerado uma variável aleatória, enquanto uma rotulação de uma imagem de entrada é interpretada como uma potencial observação do modelo. A tarefa de reconhecimento foi então formulada como um problema de otimização, que busca maximizar a probabilidade da observação de acordo com o modelo. O método foi aplicado à modelagem de órgãos abdominais em imagens de ressonância magnética não-contrastadas. Esses órgãos apresentam um arranjo espacial consistente em imagens distintas, além de propriedades de aparência e anatômicas variáveis, o que vem ao encontro da proposta da representação por GER e da abordagem probabilística para o reconhecimento dos órgãos em novas imagens. / The purpose of this thesis was to propose a formalism for the problems of modeling and recognition of a structured object, or a system of structured objects, in images. A structured object is one that may be described in terms of its compound primitive elements and their inherent relations. For instance, an aircraft may be described in terms of the following primitives: right and left wings, fuselage, and cockpit. The relational aspect of structured objects leads these problems to solutions in structural pattern recognition, which describes patterns as primitives and relations. Nevertheless, the variability of primitive elements and of their relations is better modeled by traditional statistical pattern recognition methods. Because of the complementary capabilities of these approaches, the fusion of both has recently been pointed out as a trend in computer vision. To consider these sources of information, the methodology presented herein combines relational cues inherent to a structured object with statistical information learned from a set of object samples. A hybrid model of a structured object is represented by means of a statistical relational graph (SRG). The SRG is a prototype attributed relational graph (ARG) in which nodes represent primitive elements and arcs link nodes representing related primitives. Each node or arc is associated with attributes which are parameters of probability distributions that describe random variables representing primitive or relational attributes. Based on this representation, a probabilistic approach was proposed to tackle the problem of recognizing a structured object in an input image. The model SRG is interpreted as a random variable, whereas a labeling of the input image is considered a potential observation of the model. The recognition task was formulated as the optimization of an objective-function that is actually a probability measure to be maximized. The proposed approach was applied to the modeling of abdominal organs in non-contrasted magnetic resonance images. These organs present consistent spatial arrangement in distinct images, as well as varying appearance and anatomical properties, which meet the principle of the SRG representation and the associated probabilistic recognition scenario.

Esta tese de doutorado aborda os tópicos de modelagem e de reconhecimento de objetos estruturados, ou sistemas estruturados de objetos, em imagens. Um objeto ou sistema estruturado é aquele que pode ser descrito através de elementos primitivos que o compõem e pelas relações existentes entre esses elementos. Por exemplo, uma aeronave pode ser descrita pelos seguintes elementos primitivos: asas direita e esquerda, fuselagem e cockpit. O aspecto relacional de um objeto estruturado direciona sua representação computacional e seu reconhecimento em imagens ao paradigma estrutural de reconhecimento de padrões. Contudo, a variabilidade das características dos seus elementos primitivos é melhor representada através do paradigma estatístico de reconhecimento de padrões. Devido à complementaridade dos paradigmas, a conjunção dessas abordagens é um tema de pesquisa de interesse atual. Para conjugar esses dois aspectos, esta tese propôs uma metodologia que combina o conhecimento a priori das relações que caracterizam um objeto estruturado com dados estatísticos coletados de amostras desse objeto, num modelo híbrido denominado grafo estatístico-relacional (GER). Segundo essa representação, foi estudada uma abordagem probabilística para reconhecer um objeto estruturado em imagens. Nesse cenário, o GER modelo é considerado uma variável aleatória, enquanto uma rotulação de uma imagem de entrada é interpretada como uma potencial observação do modelo. A tarefa de reconhecimento foi então formulada como um problema de otimização, que busca maximizar a probabilidade da observação de acordo com o modelo. O método foi aplicado à modelagem de órgãos abdominais em imagens de ressonância magnética não-contrastadas. Esses órgãos apresentam um arranjo espacial consistente em imagens distintas, além de propriedades de aparência e anatômicas variáveis, o que vem ao encontro da proposta da representação por GER e da abordagem probabilística para o reconhecimento dos órgãos em novas imagens. / The purpose of this thesis was to propose a formalism for the problems of modeling and recognition of a structured object, or a system of structured objects, in images. A structured object is one that may be described in terms of its compound primitive elements and their inherent relations. For instance, an aircraft may be described in terms of the following primitives: right and left wings, fuselage, and cockpit. The relational aspect of structured objects leads these problems to solutions in structural pattern recognition, which describes patterns as primitives and relations. Nevertheless, the variability of primitive elements and of their relations is better modeled by traditional statistical pattern recognition methods. Because of the complementary capabilities of these approaches, the fusion of both has recently been pointed out as a trend in computer vision. To consider these sources of information, the methodology presented herein combines relational cues inherent to a structured object with statistical information learned from a set of object samples. A hybrid model of a structured object is represented by means of a statistical relational graph (SRG). The SRG is a prototype attributed relational graph (ARG) in which nodes represent primitive elements and arcs link nodes representing related primitives. Each node or arc is associated with attributes which are parameters of probability distributions that describe random variables representing primitive or relational attributes. Based on this representation, a probabilistic approach was proposed to tackle the problem of recognizing a structured object in an input image. The model SRG is interpreted as a random variable, whereas a labeling of the input image is considered a potential observation of the model. The recognition task was formulated as the optimization of an objective-function that is actually a probability measure to be maximized. The proposed approach was applied to the modeling of abdominal organs in non-contrasted magnetic resonance images. These organs present consistent spatial arrangement in distinct images, as well as varying appearance and anatomical properties, which meet the principle of the SRG representation and the associated probabilistic recognition scenario.