Michel Berthier

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.

The Color Monogenic Signal: Application to Color Edge Detection and Color Optical Flow

The aim of this paper is to define an extension of the analytic signal for a color image. We generalize the construction of the so-called monogenic signal to mappings with values in the vectorial part of the Clifford algebra ℝ5,0. Solving a Dirac equation in this context leads to a multiscale signal (relatively to the Poisson scale-space) which contains both structure and color information. The color monogenic signal can be used in a wide range of applications. Two examples are detailed: the first one concerns a multiscale geometric segmentation with respect to a given color; the second one is devoted to the extraction of the optical flow from moving objects of a given color.

A k-order fuzzy OR operator for pattern classification with k -order ambiguity rejection

In pattern recognition, the membership of an object to classes is often measured by labels. This article mainly deals with the mathematical foundations of labels combination operators, built on t-norms, that extend previous ambiguity measures of objects by dealing not only with two classes ambiguities but also with k classes, k lying between 1 and the number of classes c. Mathematical properties of this family of combination operators are established and a weighted extension is proposed, allowing to give more or less importance to a given class. A classifier with reject options built on the proposed measure is presented and applied on synthetic data. A critical analysis of the results led to derivate some new operators by aggregating previous measures. A modified classifier is proposed and applied to synthetic data as well as to standard real data.

Spinor Fourier Transform for Image Processing

We propose in this paper to introduce a new spinor Fourier transform for both gray-level and color image processing. Our approach relies on the three following considerations: mathematically speaking, defining a Fourier transform requires to deal with group actions; vectors of the acquisition space can be considered as generalized numbers when embedded in a Clifford algebra; the tangent space of the image surface appears to be a natural parameter of the transform we define by means of so-called spin characters. The resulting spinor Fourier transform may be used to perform frequency filtering that takes into account the Riemannian geometry of the image. We give examples of low-pass filtering interpreted as diffusion process. When applied to color images, the entire color information is involved in a really non marginal process.

Clifford–Fourier Transform and Spinor Representation of Images

We propose in this chapter to introduce a spinor representation for images based on the work of T. Friedrich. This spinor representation generalizes the usual Weierstrass representation of minimal surfaces (i.e., surfaces with constant mean curvature equal to zero) to arbitrary surfaces (immersed in \(\mathbb{R}^3\) ). We investigate applications to image processing focusing on segmentation and Clifford–Fourier analysis. All these applications involve sections of the spinor bundle of image graphs, that is spinor fields, satisfying the so-called Dirac equation.

Blockwise similarity in [0,1] via triangular norms and Sugeno integrals - Application to cluster validity

In many fields, e.g. decision-making, numerical values in [0,1] are available and one is often interested in detecting which are similar. In this paper, we propose an operator which is able to detect whether some values can be gathered by blocks with respect to their similarity or not. It combines the values and a kernel function using triangular norms and Sugeno integrals. This operator allows to estimate this blockwise similarity at different levels. For illustration purpose, we use it to define an index suitable for the cluster validity problem in pattern recognition.

Binary codes K-modes clustering for HSI segmentation

An new pixel unsupervised hyperspectral image (HSI) segmentation method is proposed. It relies on a binary incoding of spectral reflectance curve variations of pixels that allows to consider HSI segmentation as a clustering problem in the feature set of binary strings. Using a generalized Hamming distance, a k-modes algorithm is applied to obtain a cluster partionning of the HSI with no use of any spatial information.

Color Representation and Processes with Clifford Algebra

In the literature, colour information of pixels of an image has been represented by different structures. Recently algebraic entities such as quaternions or Clifford algebras have been used to perform image processing for example. We propose to review several contributions for colour image processing by using the Quaternion algebra and the Clifford algebra. First, we illustrate how this formalism can be used to define colour alterations with algebraic operations. We generalise linear filtering algorithms already defined with quaternions and review a Clifford color edge detector. Clifford algebras appear to be an efficient mathematical tool to investigate the geometry of nD images. It has been shown for instance how to use quaternions for colour edge detection or to define an hypercomplex Fourier transform. The aim of the second part of this chapter is to present an example of applications, namely the Clifford Fourier transform of Clifford algebras to colour image processing.

A Riemannian Fourier Transform via Spin Representations

We introduce a new Riemannian Fourier transform for color image processing. The construction involves spin characters and spin representations of complex Clifford algebras. Examples of applications to low-pass filtering are presented.