# A Semidiscrete Version of the Citti-Petitot-Sarti Model as a Plausible Model for Anthropomorphic Image Reconstruction and Pattern RecognitionPattern Recognition

A Semidiscrete Version of the Citti-Petitot-Sarti Model as a Plausible Model for Anthropomorphic... [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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varLambda$$\end{document} of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb G}$$\end{document}, i.e., generic functions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f, g\in L^2({\mathbb G})$$\end{document} have the same bispectrum if and only if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f=\varLambda (a)g$$\end{document} for some \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a\in {\mathbb G}$$\end{document}. In the second part of the chapter, we focus on the problem of discriminating functions on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2({\mathbb H})$$\end{document} under the action of the semi-direct product \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb G}={\mathbb K}\ltimes {\mathbb H}$$\end{document}, as given by its quasi-regular representation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi$$\end{document}. 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2({\mathbb H})$$\end{document}. Some stronger version of this theorem are also presented in the case of real functions and when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb H}={\mathbb R}^2$$\end{document}. Finally, we conclude the chapter by presenting the extension of this theory to almost-periodic functions.] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

# A Semidiscrete Version of the Citti-Petitot-Sarti Model as a Plausible Model for Anthropomorphic Image Reconstruction and Pattern RecognitionPattern Recognition

Part of the SpringerBriefs in Mathematics Book Series
Springer Journals — Jun 12, 2018
23 pages

/lp/springer-journals/a-semidiscrete-version-of-the-citti-petitot-sarti-model-as-a-plausible-QKHcTCJnBm
Publisher
Springer International Publishing
ISBN
978-3-319-78481-6
Pages
53 –76
DOI
10.1007/978-3-319-78482-3_5
Publisher site
See Chapter on Publisher Site

### 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varLambda$$\end{document} of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb G}$$\end{document}, i.e., generic functions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f, g\in L^2({\mathbb G})$$\end{document} have the same bispectrum if and only if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f=\varLambda (a)g$$\end{document} for some \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a\in {\mathbb G}$$\end{document}. In the second part of the chapter, we focus on the problem of discriminating functions on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2({\mathbb H})$$\end{document} under the action of the semi-direct product \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb G}={\mathbb K}\ltimes {\mathbb H}$$\end{document}, as given by its quasi-regular representation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi$$\end{document}. 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2({\mathbb H})$$\end{document}. Some stronger version of this theorem are also presented in the case of real functions and when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb H}={\mathbb R}^2$$\end{document}. Finally, we conclude the chapter by presenting the extension of this theory to almost-periodic functions.]

Published: Jun 12, 2018

Keywords: Bispectral invariants; Fourier descriptors; Power spectrum

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