Tibério S. Caetano

Graphical Models and Point Pattern Matching

This paper describes a novel solution to the rigid point pattern matching problem in Euclidean spaces of any dimension. Although we assume rigid motion, jitter is allowed. We present a noniterative, polynomial time algorithm that is guaranteed to find an optimal solution for the noiseless case. First, we model point pattern matching as a weighted graph matching problem, where weights correspond to Euclidean distances between nodes. We then formulate graph matching as a problem of finding a maximum probability configuration in a graphical model. By using graph rigidity arguments, we prove that a sparse graphical model yields equivalent results to the fully connected model in the noiseless case. This allows us to obtain an algorithm that runs in polynomial time and is provably optimal for exact matching between noiseless point sets. For inexact matching, we can still apply the same algorithm to find approximately optimal solutions. Experimental results obtained by our approach show improvements in accuracy over current methods, particularly when matching patterns of different sizes

Rich-club phenomenon across complex network hierarchies

The “rich-club phenomenon” in complex networks is characterized when nodes of higher degree are more interconnected than nodes with lower degree. The presence of this phenomenon may indicate several interesting high-level network properties, such as tolerance to hub failures. Here, the authors investigate the existence of this phenomenon across the hierarchies of several real-world networks. Their simulations reveal that the presence or absence of this phenomenon in a network does not imply its presence or absence in the network’s successive hierarchies, and that this behavior is even nonmonotonic in some cases.

A probabilistic model for the human skin color

We present a multivariate statistical model to represent the human skin color. There are no limitations regarding whether the person is white or black, once the model is able to learn automatically the ethnicity of the person involved. We propose to model the skin color in the chromatic subspace, which is by default normalized with respect to illumination. First, skin samples from both white and black people are collected. These samples are then used to estimate a parametric statistical model, which consists of a mixture of Gaussian probability density functions (pdfs). Estimation is performed by a learning process based on the expectation-maximization (EM) algorithm. Experiments are carried out and receiver operating characteristics (ROC curves) are obtained to analyse the performance of the estimated model. The results are compared to those of models that use a single Gaussian density.

Graphical models for graph matching: Approximate models and optimal algorithms

Comparing scene, pattern or object models to structures in images or determining the correspondence between two point sets are examples of attributed graph matching. In this paper we show how such problems can be posed as one of inference over hidden Markov random fields. We review some well known inference methods studied over past decades and show how the Junction Tree framework from Graphical Models leads to algorithms that outperform traditional relaxation-based ones. 2004 Elsevier B.V. All rights reserved.

Estimating labels from label proportions

Consider the following problem: given sets of unlabeled observations, each set with known label proportions, predict the labels of another set of observations, also with known label proportions. This problem appears in areas like e-commerce, spam filtering and improper content detection. We present consistent estimators which can reconstruct the correct labels with high probability in a uniform convergence sense. Experiments show that our method works well in practice.

Performance evaluation of single and multiple-Gaussian models for skin color modeling

We present an experimental setup to evaluate the relative performance of single Gaussian models and Gaussian mixture models for skin color modeling. Firstly, a sample set of 1,120,000 skin pixels from a number of ethnic groups is selected and represented in the chromaticity space. Parameter estimation for both the single Gaussian and seven (with 2 to 8 Gaussian components) Gaussian mixture models is performed. For the mixture models, learning is carried out via the expectation-maximisation (EM) algorithm. In order to compare performances achieved by the 8 different models, we apply to each model a test set of 800 images - none from the training set. True skin regions, representing ground truth, are manually selected, and false positive and true positive rates are computed for each value of a specific threshold. Finally, receiver operating characteristics (ROC) curves are plotted for each model, making it possible to analyze and compare their relative performances. Results obtained show that, for medium to high true positive rates, mixture models (with 2 to 8 components) outperform the single Gaussian model. Nevertheless, for low false positive rates, all the models behave similarly.

Graph Rigidity, Cyclic Belief Propagation, and Point Pattern Matching

A recent paper (Caetano et al., 2006) proposed a provably optimal, polynomial time method for performing near-isometric point pattern matching by means of exact probabilistic inference in a chordal graphical model. Its fundamental result is that the chordal graph in question is shown to be globally rigid, implying that exact inference provides the same matching solution as exact inference in a complete graphical model. This implies that the algorithm is optimal when there is no noise in the point patterns. In this paper, we present a new graph which is also globally rigid but has an advantage over the graph, its maximal clique size is smaller, rendering inference significantly more efficient. However, this graph is not chordal and thus standard junction tree algorithms cannot be directly applied. Nevertheless, we show that loopy belief propagation in such a graph converges to the optimal solution. This allows us to retain the optimality guarantee in the noiseless case, while substantially reducing both memory requirements and processing time. Our experimental results show that the accuracy of the proposed solution is indistinguishable when there is noise in the point patterns.

Learning high-order MRF priors of color images

In this paper, we use large neighborhood Markov random fields to learn rich prior models of color images. Our approach extends the monochromatic Fields of Experts model (Roth & Black, 2005a) to color images. In the Fields of Experts model, the curse of dimensionality due to very large clique sizes is circumvented by parameterizing the potential functions according to a product of experts. We introduce simplifications to the original approach by Roth and Black which allow us to cope with the increased clique size (typically 3x3x3 or 5x5x3 pixels) of color images. Experimental results are presented for image denoising which evidence improvements over state-of-the-art monochromatic image priors.

Shape classification through structured learning of matching measures

Many traditional methods for shape classification involve establishing point correspondences between shapes to produce matching scores, which are in turn used as similarity measures for classification. Learning techniques have been applied only in the second stage of this process, after the matching scores have been obtained. In this paper, instead of simply taking for granted the scores obtained by matching and then learning a classifier, we learn the matching scores themselves so as to produce shape similarity scores that minimize the classification loss. The solution is based on a max-margin formulation in the structured prediction setting. Experiments in shape databases reveal that such an integrated learning algorithm substantially improves on existing methods.

An MRF and Gaussian Curvature Based Shape Representation for Shape Matching

Matching and registration of shapes is a key issue in Computer Vision, Pattern Recognition, and Medical Image Analysis. This paper presents a shape representation framework based on Gaussian curvature and Markov random fields (MRFs) for the purpose of shape matching. The method is based on a surface mesh model in R3, which is projected into a two-dimensional space and there modeled as an extended boundary closed Markov random field. The surface is homeomorphic to S2. The MRF encodes in the nodes entropy features of the corresponding similarities based on Gaussian curvature, and in the edges the spatial consistency of the meshes. Correspondence between two surface meshes is then established by performing probabilistic inference on the MRF via Gibbs sampling. The technique combines both geometric, topological, and probabilistic information, which can be used to represent shapes in three dimensional space, and can be generalized to higher dimensional spaces. As a result, the representation can be used for shape matching, registration, and statistical shape analysis.