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.
Dario Prandi was born in 1986. He received is PhD in applied mathematics from École Polyechnique, Palaiseau, France, and SISSA, Trieste, Italy, in 2014, and is currently a CNRS researcher at Laboratoire des Signaux et des Systèmes, CentraleSupélec, Gif-sur-Yvette. His research interests include sub-Riemannian geometry, image processing and, neuroscience.
Jean-Paul Gauthier was born in 1952. He received his PhDs in computer science and physics in 1978 and 1982 respectively. He started his career as a CNRS research worker, and from 1988 onward was a professor at various French universities. In 2007 he became a professor at Toulon University, and has been professor emeritus since 2017. His fields of interest include control theory and its applications, image processing, and sub-Riemannian geometry. He was featured in a review by the American Mathematical Society in 2001, and he has been a member of Institut Universitaire de France (IUF) since 1992.
1 Introduction.- 2 Preliminaries.- 3 Lifts.- 4 Almost-periodic interpolation and approximation.- 5 Pattern recognition.- 6 Image reconstruction.- 7 Applications.- 8 Appendix: A Circulant matrices.- 9 Appendix B: Bispectrally admissible sets.
"The book is written in a very accessible fashion and the proposed methods are described in depth. It is suitable for graduate students and for researchers and professionals interested in image reconstruction and pattern recognition." (Krzystof Gdawiec, zbMATH 1415.68007, 2019)