main-content

## Über dieses Buch

This book proposes a semi-discrete version of the theory of Petitot and Citti-Sarti, leading to a left-invariant structure over the group SE(2,N), restricted to a finite number of rotations. This apparently very simple group is in fact quite atypical: it is maximally almost periodic, which leads to much simpler harmonic analysis compared to SE(2). Based upon this semi-discrete model, the authors improve on previous image-reconstruction algorithms and develop a pattern-recognition theory that also leads to very efficient algorithms in practice.

## Inhaltsverzeichnis

### Chapter 1. Introduction

Abstract
This chapter is an introduction to the book. Here we start by presenting the classical Citti-Petitot-Sarti model for the structure of the visual cortex, with particular attention to its connections with control theory, followed by our semi-discrete improvement of the model, which will be the main subject of this monograph. We also present how this structure can be exploited for image reconstruction and then introduce our novel framework for texture analsysis and pattern recognition, via the bispectrum. We end the chapter with a discussion on the organization of the monograph.
Dario Prandi, Jean-Paul Gauthier

### Chapter 2. Preliminaries

Abstract
This chapter introduces the concepts that are the main subject of the rest of this work, along with the essential tools that are needed. After a brief introduction on harmonic analysis in non-commutative groups, we introduce the general setting considered in this work, alongside with some essential facts on its representation theory. Afterwards, we recall some basic notions on almost-periodic functions and and a precise construction that allows to select some relevant subspaces. Finally we present our models for natural images (compactly supported functions of $$L^2({\mathbb R}^2)$$) and textures (properly selected finite-dimensional subspaces of almost-periodic functions in the plane).
Dario Prandi, Jean-Paul Gauthier

### Chapter 3. Lifts

Abstract
Let $${\mathbb {G}}= {\mathbb {K}}\ltimes {\mathbb {H}}$$ be a semi-direct product, as considered in Sect. 2.​2. Here, we are interested in operators $$L:L^2({\mathbb {H}})\rightarrow L^2({\mathbb {G}})$$, which we call lift operators for obvious reasons. Observe that, via the isomorphism $$\sigma ^*:B_2({\mathbb {H}})\rightarrow L^2({\mathbb {H}}^\flat )$$ any lift $$L:L^2({\mathbb {H}}^\flat )\rightarrow L^2({\mathbb {G}}^\flat )$$ induces a lift $$L':B_2({\mathbb {H}})\rightarrow B_2({\mathbb {G}})$$ of Besicovitch almost periodic functions. We are mainly interested in identifying the action of the quasi-regular representation on $$f\in L^2({\mathbb {H}})$$ by analyzing the Fourier transform of the lift $$Lf\in L^2({\mathbb {G}})$$. Thus, the first, and more natural, requirement on the lift operation is to intertwine the quasi-regular representation acting on $$L^2({\mathbb {H}})$$ with the left regular representations on $$L^2({\mathbb {G}})$$. We call these type of lifts left-invariant. We show that, under some mild regularity assumptions on L, left-invariant lifts coincide with wavelet transforms, as defined in Sect. 2.​2.​3. These kind of lifts have been extensively studied in, e.g., [33], and related works. Unfortunately, left-invariant lifts have a huge drawback for our purposes: they never have an invertible non-commutative Fourier transform $$\widehat{Lf}(T^\lambda )$$. The second part of this chapter is then devoted to the generalization of the concept of cyclic lift, introduced in [73] exactly to overcome the above problem. In this general context, we will present a cyclic lift as a combination of an almost-left-invariant lift and a centering operation, as defined in Definition 2.​2. As a consequence, we obtain a precise characterization of the invertibility of $$\widehat{Lf}(T^\lambda )$$ for these lifts.
Dario Prandi, Jean-Paul Gauthier

### Chapter 4. Almost-Periodic Interpolation and Approximation

Abstract
In this chapter, following [41], we present a method to interpolate or approximate a given function $$f:{\mathbb G}\rightarrow {\mathbb C}$$ (or $$F:{\mathbb H}\rightarrow {\mathbb C}$$) by an AP functions in $${{\mathrm{AP}}}_F({\mathbb G})$$, i.e., AP functions whose Fourier transform is supported in a given discrete and finite set $$F\subset \widehat{\mathbb G}$$. (See Sect. 2.​3.​1.) In order to do this, we generalize the well-known decomposition of the 2D Fourier transform on the plane in polar coordinates, via the Fourier-Bessel operator. In the first part of the chapter, exploiting the deep connection between (4.1) and the group of rototranslations SE(2), we generalize the former to $${{\mathrm{AP}}}_F({\mathbb G})$$ functions. In particular, we show how a discrete operator that we call the generalized Fourier-Bessel operator plays a crucial role in this generalization. We then consider the problem of interpolating functions $$\psi :{\mathbb G}\rightarrow {\mathbb C}$$ on $${\mathbb K}$$-invariant finite sets $$\tilde{E}\subset {\mathbb G}$$ via $${{\mathrm{AP}}}_F({\mathbb G})$$ functions. The last part of the chapter is devoted to particularize (and slightly generalize) the above results to the relevant case for image processing, i.e., $${\mathbb G}=SE(2,N)$$. Indeed we present numerical algorithms for the (exact) evaluation, interpolation, and approximation of $${{\mathrm{AP}}}_F(SE(2,N))$$ functions on finite sets of spatial samples $$\tilde{E}\subset {\mathbb R}^2$$, invariant under the action of $${\mathbb Z}_N$$. This is an instance of a very general problem, and can be seen as a generalization of the discrete Fourier Transform and its inverse, that act on regular square grids, i.e., invariant under the the action of SE(2, 4).
Dario Prandi, Jean-Paul Gauthier

### Chapter 5. Pattern Recognition

Abstract
This chapter is the heart of the book and contains the main contributions. Here, we present a framework for pattern recognition on groups, based on Fourier invariants. Our aim is to give an effective procedure for discriminate functions up to the action of the left-regular representation of some group. In the first part of the chapter, we introduce the simplest Fourier-based invariants that we will focus on: the power spectrum and the bispectrum, and we show that the latter are weakly complete with respect to the left-regular representation $$\varLambda$$ of $${\mathbb G}$$, i.e., generic functions $$f, g\in L^2({\mathbb G})$$ have the same bispectrum if and only if $$f=\varLambda (a)g$$ for some $$a\in {\mathbb G}$$. In the second part of the chapter, we focus on the problem of discriminating functions on $$L^2({\mathbb H})$$ under the action of the semi-direct product $${\mathbb G}={\mathbb K}\ltimes {\mathbb H}$$, as given by its quasi-regular representation $$\pi$$. For this aim, we will exploit the lifts presented in Chap. 3 by showing the bispectrum to be weakly complete for regular cyclic lifts but not for left-invariant ones. This yields us to consider stronger invariants, the rotational power spectrum and rotational bispectrum invariants. We then prove the main theorem of the chapter: Theorem 5.5, which states that, up to a centering operator, these invariants are weakly complete on left-invariant lifts of function in $$L^2({\mathbb H})$$. Some stronger version of this theorem are also presented in the case of real functions and when $${\mathbb H}={\mathbb R}^2$$. Finally, we conclude the chapter by presenting the extension of this theory to almost-periodic functions.
Dario Prandi, Jean-Paul Gauthier

### Chapter 6. Image Reconstruction

Abstract
In this chapter we apply non-commutative Fourier analysis in order to build numerically efficient algorithms for heat diffusion on groups and its application to image reconstruction. The first section of the chapter is devoted to recalling definition and some basic properties of hypoelliptic diffusions on Lie groups. These constructions are then extended to the semi-discrete semi-direct products that are the focus of the monograph. In particular, we present an efficient approach for the heat diffusion on lifted functions and on almost-periodic functions. These results are then particularized to the case of SE(2, N), and an associated image reconstruction algorithm is presented.
Dario Prandi, Jean-Paul Gauthier

### Chapter 7. Applications

Abstract
This chapter collects the results of numerical testing in image processing applications of the various concepts explained throughout this work: AP interpolation and approximation, image reconstruction, and pattern recognition. These are mostly taken from the already mentioned papers [9, 11, 12, 15, 41].
Dario Prandi, Jean-Paul Gauthier

### Backmatter

Weitere Informationen