Christophe Saint-Jean

Clifford–Fourier Transform for Color Image Processing

The aim of this paper is to define a Clifford–Fourier transform that is suitable for color image spectral analysis. There have been many attempts to define such a transformation using quaternions or Clifford algebras. We focus here on a geometric approach using group actions. The idea is to generalize the usual definition based on the characters of abelian groups by considering group morphisms from ℝ2 to spinor groups Spin(3) and Spin(4). The transformation we propose is parameterized by a bivector and a quadratic form, the choice of which is related to the application to be treated. A general definition for 4D signal defined on the plane is also given; for particular choices of spinors, it coincides with the definitions of S. Sangwine and T. Bulow.

A Metric Approach to nD Images Edge Detection with Clifford Algebras

The aim of this paper is to perform edge detection in color-infrared images from the point of view of Clifford algebras. The main idea is that such an image can be seen as a section of a Clifford bundle associated to the RGBT-space (Red, Green, Blue, Temperature) of acquisition. Dealing with geometric calculus and covariant derivatives of appropriate sections with respect to well-chosen connections allows to get various color and temperature information needed for the segmentation. We show in particular how to recover the first fundamental form of the image embedded in a LSHT-space (Luminance, Saturation, Hue, Temperature) equipped with a metric tensor. We propose applications to color edge detection with some constraints on colors and to edge detection in color-infrared images with constraints on both colors and temperature. Other applications related to different choices of connections, sections and embedding spaces for nD images may be considered from this general theoretical framework.

Color Fourier-Mellin descriptors for image recognition

We propose new sets of Fourier-Mellin descriptors for color images. They are constructed using the Clifford Fourier transform of Batard et al. (2010) [4] and are an extension of the classical Fourier-Mellin descriptors for grayscale images. These are invariant under direct similarity transformations (translations, rotations, scale) and marginal treatment of colors images is avoided. An implementation of these features is given and the choice of the bivector (a distinguished color plane which parameterizes the Clifford Fourier transform) is discussed. The proposed formalism extends and clarifies the notion of direction of analysis as introduced for the quaternionic Fourier-Mellin moments (Guo and Zhu, 2011). Thus, another set of descriptors invariant under this parameter is defined. Our proposals are tested with the purpose of object recognition on well-known color image databases. Their retrieval rates are favorably compared to standard feature descriptors.

An Expectation-Maximization algorithm for the Wishart mixture model: Application to movement clustering

This article presents an Expectation-Maximization algorithm for the Wishart mixture model in which realizations are matrices. Given a set of matrices, an iterative algorithm for estimating the parameters of such a mixture model is proposed. The obtained estimates can be interpreted in terms of mean matrices and scale factors. By applying the maximum a posteriori rule, we get an algorithm for the clustering of a set of matrices. This mixture model is then modified in order to deal with a set of samples. Unfortunately, the samples may be of different sizes. We propose to tackle this problem by considering the cross-product matrix as a signature for each sample. This set of cross-product matrices may be fitted with the proposed Wishart pseudo-mixture model in which the scale parameters of the distribution are not estimated but fixed. Again, we easily get a clustering algorithm from final parameter estimates. The different estimators are studied empirically through an analysis of their bias and variance and are validated onto an artificial dataset. Finally, we apply the Wishart pseudo-mixture model for analyzing motion-captured movements. Given the successive 3D positions of markers over the time, a cross-product matrix is constructed for each movement and put into the proposed classifier. We observe that the recognition rates are higher with our proposed approach than those with other geometric methods. Limits and constraints of the provided models are finally discussed.

New geometric fourier descriptors for color image recognition

This article relies on two recent developments of well known methods which are a color Fourier transform using geometric algebra [1] and Generalized Fourier descriptors defined from the group M2 of the motion of the plane [2]. In this paper, new generalized color Fourier descriptors (GCFD) are proposed. They depend on the choice of a bivector B acting as an analysis plane in a colorimetric space. The relevance of proposed descriptors is discussed on several color image databases. In particular, the influence of parameter B is studied regarding the type of images. It appears that proposed descriptors are more compact with a lower complexity and better classification rate.

Color Object Recognition Based on a Clifford Fourier Transform

The aim of this chapter is to propose two different approaches for color object recognition, both using the recently defined color Clifford Fourier transform. The first one deals with so-called Generalized Fourier Descriptors, the definition of which relies on plane motion group actions. The proposed color extension leads to more compact descriptors, with lower complexity and better recognition rates, than the already existing descriptors based on the processing of the r, g and b channels separately. The second approach concerns color phase correlation for color images. The idea here is to generalize in the Clifford framework the usual means of measuring correlation from the well-known shift theorem. Both methods necessitate to choose a 2-blade B of ℝ4 which corresponds to an analysis plane in the color space. The relevance of proposed methods for classification purposes is discussed on several color image databases. In particular, the influence of parameter B is studied regarding the type of images.

Alive Fishes Species Characterization from Video Sequences

This article presents a method suitable for the characterization of fishes evolving in a basin. It is based on the analysis of video sequences obtained from a fixed camera. One of the main difficulties of analyzing natural scenes acquired from an aquatic environment is the variability of illumination. This disturbs every phase of the whole process. We propose to make each task more robust. In particular, we propose to use a clustering method allowing to provide species parameters estimates that are less sensitive to outliers.

A New Implementation of k-MLE for Mixture Modeling of Wishart Distributions

We describe an original implementation of k-Maximum Likelihood Estimator (k-MLE)[1], a fast algorithm for learning finite statistical mixtures of exponential families. Our version converges to a local maximum of the complete likelihood while guaranteeing not to have empty clusters. To initialize k-MLE, we propose a careful and greedy strategy inspired by k-means++ which selects automatically cluster centers and their number. The paper gives all details for using k-MLE with mixtures of Wishart (WMMs). Finally, we propose to use the Cauchy-Schwartz divergence as a comparison measure between two WMMs and give a general methodology for building a motion retrieval system.

Hartigan’s Method for k-MLE: Mixture Modeling with Wishart Distributions and Its Application to Motion Retrieval

We describe a novel algorithm called \(k\)-Maximum Likelihood Estimator (\(k\)-MLE) for learning finite statistical mixtures of exponential families relying on Hartigan’s \(k\)-means swap clustering method. To illustrate this versatile Hartigan \(k\)-MLE technique, we consider the exponential family of Wishart distributions and show how to learn their mixtures. First, given a set of symmetric positive definite observation matrices, we provide an iterative algorithm to estimate the parameters of the underlying Wishart distribution which is guaranteed to converge to the MLE. Second, two initialization methods for \(k\)-MLE are proposed and compared. Finally, we propose to use the Cauchy-Schwartz statistical divergence as a dissimilarity measure between two Wishart mixture models and sketch a general methodology for building a motion retrieval system.

Clustering with EM: Complex Models vs. Robust Estimation

Clustering multivariate data that are contaminated by noise is a complex issue, particularly in the framework of mixture model estimation because noisy data can significantly affect the parameters estimates. This paper addresses this problem with respect to likelihood maximization using the Expectation-Maximization algorithm. Two different approaches are compared. The first one consists in defining mixture models that take into account noise. The second one is based of robust estimation of the model parameters in the maximization step of EM. Both have been tested separately, then jointly. Finally, a hybrid model is proposed. Results on artificial data are given and discussed.