Luc Brun

Graph-Based Representations in Pattern Recognition

Hypergraphs.- Generalization of Two Hypergraphs. Algorithm of Calculation of the Greatest Sub-Hypergraph Common to Two Hypergraphs Annotated by Semantic Information.- Recognition and Detection.- Recognition of Polymorphic Patterns in Parameterized Graphs for 3D Building Reconstruction.- A Graph-Based Representation to Detect Linear Features.- Edge Detection as Finding the Minimum Cost Path in a Graph.- Matching.- Subgraph Transformations for the Inexact Matching of Attributed Relational Graphs.- Efficient Graph Matching for Video Indexing.- Isomorphism between Strong Fuzzy Relational Graphs Based on k-Formulae.- Segmentation.- A Graph Structure for Grey Value and Texture Segmentation.- Discrete Maps: a Framework for Region Segmentation Algorithms.- Image Sequence Segmentation by a Single Evolutionary Graph Pyramid.- Implementation Problems.- Dual Graph Contraction with LEDA.- Implementing Image Analysis with a Graph-Based Parallel Computing Model.- Representation.- The Frontier-Region Graph.- Optimization Techniques on Pixel Neighborhood Graphs for Image Processing.

Introduction to combinatorial pyramids

A pyramid is a stack of image representations with decreasing resolution. Many image processing algorithms run on this hierarchical structure in O(log(n)) parallel processing steps where n is the diameter of the input image. Graph pyramids are made of a stack of successively reduced graphs embedded in the plane. Such pyramids overcome the main limitations of their regular ancestors. The graphs used in the pyramid may be region adjacency graphs or dual graphs. This paper reviews the different hierarchical data structures and introduces a new representation named combinatorial pyramid.

Contraction kernels and combinatorial maps

Graph pyramids are made of a stack of successively reduced graphs embedded in the plane. Such pyramids overcome the main limitations of their regular ancestors. The graphs used in the pyramid may be region adjacency graphs, dual graphs or combinatorial maps. Compared to usual graph data structures, combinatorial maps offer an explicit encoding of the orientation of edges around vertices. Each combinatorial map in the pyramid is generated from the one below by a set of edges to be contracted. This contraction process is controlled by kernels that can be combined in many ways. This paper shows that kernels producing a slow reduction rate can be combined to speed up reduction. Conversely, kernels decompose into smaller kernels that generate a more gradual reduction. We also propose one sequential and one parallel algorithm to compute the contracted combinatorial maps.

Construction of combinatorial pyramids

Irregular pyramids are made of a stack of successively reduced graphs embedded in the plane. Each vertex of a reduced graph corresponds to a connected set of vertices in the level below. One connected set of vertices reduced into a single vertex at the above level is called the reduction window of this vertex. In the same way, a connected set of vertices in the base level graph reduced to a single vertex at a given level is called the receptive field of this vertex. The graphs used in the pyramid may be region adjacency graphs, dual graphs or combinatorial maps. This last type of pyramids are called Combinatorial Pyramids. Compared to usual graph data structures, combinatorial maps encode one graph and its dual within a same formalism and offer an explicit encoding of the orientation of edges around vertices. This paper describes the construction scheme of a Combinatorial Pyramid. We also provide a constructive definition of the notions of reduction windows and receptive fields within the Combinatorial Pyramid framework.

Irregular Pyramids with Combinatorial Maps

This paper presents a new formalism for irregular pyramids based on combinatorial maps. Such pyramid consists of a stack of successively reduced graph. Each smaller graph is deduced from the preceding one by a set of edges which have to be contracted or removed. In order to perform parallel contractions or removals, the set of edges to be contracted or removed has to verify some properties. Such a set of edges is called a Decimation Parameter. A combinatorial map encodes a planar graph thanks to two permutations encoding the edges and their orientation around the vertices. Combining the useful properties of both combinatorial maps and irregular pyramids offers a potential alternative for representing structures at multiple levels of abstraction.

Receptive fields within the combinatorial pyramid framework

A hierarchical structure is a stack of successively reduced image representations. Each basic element of a hierarchical structure is the father of a set of elements in the level below. The transitive closure of this father-child relationship associates to each element of the hierarchy a set of basic elements in the base level image representation. Such a set, called a receptive field, defines the embedding of one element on the original image. Combinatorial pyramids are defined as a stack of successively reduced combinatorial maps, each combinatorial map being defined by two permutations acting on a set of half edges named darts. The basic element of a combinatorial pyramid is thus the dart. This paper defines the receptive field of each dart within a combinatorial pyramid and study the main properties of these sets.

Recognition of Human Actions using Edit Distance on Aclet Strings

In this paper we propose a novel method for human action recognition based on string edit distance. A two layer representation is introduced in order to exploit the temporal sequence of the events: a first representation layer is obtained by using a feature vector obtained from depth images. Then, each action is represented as a sequence of symbols, where each symbol corresponding to an elementary action (aclet) is obtained according to a dictionary previously defined during the learning phase. The similarity between two actions is finally computed in terms of string edit distance, which allows the system to deal with actions showing different length as well as different temporal scales. The experimentation has been carried out on two widely adopted datasets, namely the MIVIA and the MHAD datasets, and the obtained results, compared with state of the art approaches, confirm the effectiveness of the proposed method.

A Graph Kernel Incorporating Molecule's Stereisomerism Information

The prediction of molecules properties through Quantitative Structure Activity (resp. Property) Relationships are two active research fields named QSAR and QSPR. Within these frameworks Graph kernels allow to combine a natural encoding of a molecule by a graph with classical statistical tools such as SVM or kernel ridge regression. Unfortunately some molecules encoded by a same graph and differing only by the three dimensional orientations of their atoms in space have different properties. Such molecules are called stereoisomers. These latter properties can not be predicted by usual graph methods which do not encode stereoisomerism. In this paper we propose a new graph encoding of molecules taking explicitly into account stereoisomerism and propose a new kernel between these structures in order to predict properties related to stereoisomerism.

Approximate Graph Edit Distance by Several Local Searches in Parallel

Solving or approximating the linear sum assignment problem (LSAP) is an important step of several constructive and local search strategies developed to approximate the graph edit distance (GED) of two attributed graphs, or more generally the solution to quadratic assignment problems. Constructive strategies find a first estimation of the GED by solving an LSAP. This estimation is then refined by a local search strategy. While these search strategies depend strongly on the initial assignment, several solutions to the linear problem usually exist. They are not taken into account to get better estimations. All the estimations of the GED based on an LSAP select randomly one solution. This paper explores the insights provided by the use of several solutions to an LSAP, refined in parallel by a local search strategy based on the relaxation of the search space, and conditional gradient descent. Other generators of initial assignments are also considered, approximate solutions to an LSAP and random assignments. Experimental evaluations on several datasets show that the proposed estimation is comparable to more global search strategies in a reduced computational time.

Receptive Fields within the Combinatorial Pyramid Framework

A hierarchical structure is a stack of successively reduced image representations. Each basic element of a hierarchical structure is the father of a set of elements in the level below. The transitive closure of this father-child relationship associates to each element of the hierarchy a set of basic elements in the base level image representation. Such a set, called a receptive field, defines the embedding of one element of the hierarchy on the original image. Using the father-child relationship, global properties of a receptive field may be computed in O(log(m)) parallel processing steps where m is the diameter of the receptive field. Combinatorial pyramids are defined as a stack of successively reduced combinatorial maps, each combinatorial map being defined by two permutations acting on a set of half edges named darts. The basic element of a combinatorial pyramid is thus the dart. This paper defines the receptive field of each dart within a combinatorial pyramid and study the main properties of these sets.