René Alquézar

Function-described graphs for modelling objects represented by sets of attributed graphs

We present in this article the model function-described graph (FDG), which is a type of compact representation of a set of attributed graphs (AGs) that borrow from random graphs the capability of probabilistic modelling of structural and attribute information. We define the FDGs, their features and two distance measures between AGs (unclassified patterns) and FDGs (models or classes) and we also explain an efficient matching algorithm. Two applications of FDGs are presented: in the former, FDGs are used for modelling and matching 3D-objects described by multiple views, whereas in the latter, they are used for representing and recognising human faces, described also by several views.

Graph-based representations and techniques for image processing and image analysis☆

In this paper we will discuss the use of some graph-based representations and techniques for image processing and analysis. Instead of making an extensive review of the graph techniques in this field, we will explain how we are using these techniques in an active vision system for an autonomous mobile robot developed in the Institut de Robotica i Informatica Industrial within the project “Active Vision System with Automatic Learning Capacity for Industrial Applications (CICYT TAP98-0473)”. Specifically we will discuss the use of graph-based representations and techniques for image segmentation, image perceptual grouping and object recognition. We first present a generalisation of a graph partitioning greedy algorithm for colour image segmentation. Next we describe a novel fusion of colour-based segmentation and depth from stereo that yields a graph representing every object in the scene. Finally we describe a new representation of a set of attributed graphs (AGs), denominated Function Described Graphs (FDGs), a distance measure for matching AGs with FDGs and some applications for robot vision.

SECOND-ORDER RANDOM GRAPHS FOR MODELING SETS OF ATTRIBUTED GRAPHS AND THEIR APPLICATION TO OBJECT LEARNING AND RECOGNITION

The aim of this article is to present a random graph representation, that is based on second-order relations between graph elements, for modeling sets of attributed graphs (AGs). We refer to these models as Second-Order Random Graphs (SORGs). The basic feature of SORGs is that they include both marginal probability functions of graph elements and second-order joint probability functions. This allows a more precise description of both the structural and semantic information contents in a set of AGs and, consequently, an expected improvement in graph matching and object recognition. The article presents a probabilistic formulation of SORGs that includes as particular cases the two previously proposed approaches based on random graphs, namely the First-Order Random Graphs (FORGs) and the Function-Described Graphs (FDGs). We then propose a distance measure derived from the probability of instantiating a SORG into an AG and an incremental procedure to synthesize SORGs from sequences of AGs. Finally, SORGs are shown to improve the performance of FORGs, FDGs and direct AG-to-AG matching in three experimental recognition tasks: one in which AGs are randomly generated and the other two in which AGs represent multiple views of 3D objects (either synthetic or real) that have been extracted from color images. In the last case, object learning is achieved through the synthesis of SORG models.

A new graph matching method for point-set correspondence using the EM algorithm and Softassign

Finding correspondences between two point-sets is a common step in many vision applications (e.g., image matching or shape retrieval). We present a graph matching method to solve the point-set correspondence problem, which is posed as one of mixture modelling. Our mixture model encompasses a model of structural coherence and a model of affine-invariant geometrical errors. Instead of absolute positions, the geometrical positions are represented as relative positions of the points with respect to each other. We derive the Expectation-Maximization algorithm for our mixture model. In this way, the graph matching problem is approximated, in a principled way, as a succession of assignment problems which are solved using Softassign. Unlike other approaches, we use a true continuous underlying correspondence variable. We develop effective mechanisms to detect outliers. This is a useful technique for improving results in the presence of clutter. We evaluate the ability of our method to locate proper matches as well as to recognize object categories in a series of registration and recognition experiments. Our method compares favourably to other graph matching methods as well as to point-set registration methods and outlier rejectors.

SYNTHESIS OF FUNCTION-DESCRIBED GRAPHS AND CLUSTERING OF ATTRIBUTED GRAPHS

Function-Described Graphs (FDGs) have been introduced by the authors as a representation of an ensemble of Attributed Graphs (AGs) for structural pattern recognition alternative to first-order random graphs. Both optimal and approximate algorithms for error-tolerant graph matching, which use a distance measure between AGs and FDGs, have been reported elsewhere. In this paper, both the supervised and the unsupervised synthesis of FDGs from a set of graphs is addressed. First, two procedures are described to synthesize an FDG from a set of commonly labeled AGs or FDGs, respectively. Then, the unsupervised synthesis of FDGs is studied in he context of clustering a set of AGs and obtaining an FDG model for each cluster. Two algorithms based on incremental and hierarchical clustering, respectively, are proposed, which are parameterized by a graph matching method. Some experimental results both on synthetic data and a real 3D-object recognition application show that the proposed algorithms are effective for clustering a set of AGs and synthesizing the FDGs that describe the classes. Moreover, the synthesized FDGs are shown to be useful for pattern recognition thanks to the distance measure and matching algorithm previously reported.

Smooth point-set registration using neighboring constraints

We present an approach for Maximum Likelihood estimation of correspondence and alignment parameters that benefits from the representational skills of graphs. We pose the problem as one of mixture modeling within the framework of the Expectation-Maximization algorithm. Our mixture model encompasses a Gaussian density to model the point-position errors and a Bernoulli density to model the structural errors. The Gaussian density components are parameterized by the alignment parameters which constrain their means to move according to a similarity transformation model. The Bernoulli density components are parameterized by the continuous correspondence indicators which are updated within an annealing procedure using Softassign. Outlier rejection is modeled as a gradual assignment to the null node. We highlight the analogies of our method to some existing methods. We investigate the benefits of using structural and geometrical information by presenting results of the full rigid version of our method together with its pure geometrical and its pure structural versions. We compare our method to other point-set registration methods as well as to other graph matching methods which incorporate geometric information. We also present a non-rigid version of our method and compare to state-of-the-art non-rigid registration methods. Results show that our method gets either the best performance or similar performance than state-of-the-art methods.

An algebraic framework to represent finite state machines in single-layer recurrent neural networks

In this paper we present an algebraic framework to represent finite state machines (FSMs) in single-layer recurrent neural networks (SLRNNs), which unifies and generalizes some of the previous proposals. This framework is based on the formulation of both the state transition function and the output function of an FSM as a linear system of equations, and it permits an analytical explanation of the representational capabilities of first-order and higher-order SLRNNs. The framework can be used to insert symbolic knowledge in RNNs prior to learning from examples and to keep this knowledge while training the network. This approach is valid for a wide range of activation functions, whenever some stability conditions are met. The framework has already been used in practice in a hybrid method for grammatical inference reported elsewhere (Sanfeliu and Alquzar 1994).

A new incremental method for function approximation using feed-forward neural networks

A sequential method for approximating vectors in Hilbert spaces, called sequential approximation with optimal coefficients and interacting frequencies (SAOCIF), is presented. SAOCIF combines two key ideas. The first one is the optimization of the coefficients. The second one is the flexibility to choose the frequencies. The approximations defined by SAOCIF maintain orthogonal-like properties. The theoretical results obtained prove that, under reasonable conditions, the residue of the approximation obtained with SAOCIF (in the limit) is the best one that can be obtained with any subset of the given set of vectors. In the particular case of L/sup 2/, it can be applied to approximations by algebraic polynomials, Fourier series, wavelets and feed-forward neural networks, among others. Also, a particular algorithm with feed-forward neural networks is presented. The method combines the locality of sequential approximations, where only one frequency is found at every step, with the globality of non-sequential ones, where every frequency interacts with the others. Experimental results show a very satisfactory performance.

A probabilistic integrated object recognition and tracking framework

This paper describes a probabilistic integrated object recognition and tracking framework called PIORT, together with two specific methods derived from it, which are evaluated experimentally in several test video sequences. The first step in the proposed framework is a static recognition module that provides class probabilities for each pixel of the image from a set of local features. These probabilities are updated dynamically and supplied to a tracking decision module capable of handling full and partial occlusions. The two specific methods presented use RGB color features and differ in the classifier implemented: one is a Bayesian method based on maximum likelihood and the other one is based on a neural network. The experimental results obtained have shown that, on one hand, the neural net based approach performs similarly and sometimes better than the Bayesian approach when they are integrated within the tracking framework. And on the other hand, our PIORT methods have achieved better results when compared to other published tracking methods in video sequences taken with a moving camera and including full and partial occlusions of the tracked object.

Clustering of attributed graphs and unsupervised synthesis of function-described graphs

Function-described graphs (FDGs) have been introduced by the authors as a representation of an ensemble of attributed graphs (AGs) for structural pattern recognition as an alternative to first-order random graphs. The unsupervised synthesis of FDGs is studied in the context of clustering a set of AGs and obtaining an FDG model for each cluster. Two algorithms based on incremental and hierarchical clustering, respectively, are proposed, which are parameterized by a graph matching method. Results on 3D object recognition show that these algorithms are effective for clustering a set of AGs and synthesising the FDGs that describe the classes.