Abstract: Graph-based representations are effective tools to capture structural information from visual elements. However, retrieving a query graph from a large database of graphs implies a high computational complexity. Moreover, these representations are very sensitive to noise or small changes. In this work, a novel hierarchical graph representation is designed. Using graph clustering techniques adapted from graph-based social media analysis, we propose to generate a hierarchy able to deal with different levels of abstraction while keeping information about the topology. For the proposed representations, a coarse-to-fine matching method is defined. These approaches are validated using real scenarios such as classification of colour images and handwritten word spotting.

Abstract: Graph representation of graphical documents often suffers from noise such as spurious nodes and edges, and their discontinuity. In general these errors occur during the low-level image processing viz. binarization, skeletonization, vectorization etc. Hierarchical graph representation is a nice and efficient way to solve this kind of problem by hierarchically merging node-node and node-edge depending on the distance. But the creation of hierarchical graph representing the graphical information often uses hard thresholds on the distance to create the hierarchical nodes (next state) of the lower nodes (or states) of a graph. As a result, the representation often loses useful information. This paper introduces plausibilities to the nodes of hierarchical graph as a function of distance and proposes a modified algorithm for matching subgraphs of the hierarchical graphs. The plausibility-annotated nodes help to improve the performance of the matching algorithm on two hierarchical structures. To show the potential of this approach, we conduct an experiment with the SESYD dataset.

Abstract: Graph representations have been widely used in pattern recognition thanks to their powerful representation formalism and rich theoretical background. A number of error-tolerant graph matching algorithms such as graph edit distance have been proposed for computing a distance between two labelled graphs. However, they typically suffer from a high
computational complexity, which makes it difficult to apply
these matching algorithms in a real scenario. In this paper, we propose an efficient graph distance based on the emerging field of geometric deep learning. Our method employs a message passing neural network to capture the graph structure and learns a metric with a siamese network approach. The performance of the proposed graph distance is validated in two application cases, graph classification and graph retrieval of handwritten words, and shows a promising performance when compared with
(approximate) graph edit distance benchmarks.

Abstract: Given a set of graphs, the median graph has been theoretically presented as a useful concept to infer a representative of the set. However, the computation of the median graph is a highly complex task and its practical application has been very limited up to now. In this work we present a new genetic algorithm for the median graph computation. A set of experiments on real data, where none of the existing algorithms for the median graph computation could be applied up to now due to their computational complexity, show that we obtain good approximations of the median graph. Finally, we use the median graph in a real nearest neighbour classification showing that it leaves the box of the only-theoretical concepts and demonstrating, from a practical point of view, that can be a useful tool to represent a set of graphs.

Abstract: From its early stages, the community of Pattern Recognition and Computer Vision has considered the importance of leveraging the structural information when understanding images. Usually, graphs have been proposed as a suitable model to represent this kind of information due to their flexibility and representational power able to codify both, the components, objects, or entities and their pairwise relationship. Even though graphs have been successfully applied to a huge variety of tasks, as a result of their symbolic and relational nature, graphs have always suffered from some limitations compared to statistical approaches. Indeed, some trivial mathematical operations do not have an equivalence in the graph domain. For instance, in the core of many pattern recognition applications, there is a need to compare two objects. This operation, which is trivial when considering feature vectors defined in \(\mathbb{R}^n\), is not properly defined for graphs.

In this thesis, we have investigated the importance of the structural information from two perspectives, the traditional graph-based methods and the new advances on Geometric Deep Learning. On the one hand, we explore the problem of defining a graph representation and how to deal with it on a large scale and noisy scenario. On the other hand, Graph Neural Networks are proposed to first redefine a Graph Edit Distance methodologies as a metric learning problem, and second, to apply them in a real use case scenario for the detection of repetitive patterns which define tables in invoice documents. As experimental framework, we have validated the different methodological contributions in the domain of Document Image Analysis and Recognition.