This work revolves around Gabor filters. It involves theoretical aspects such as the uncertainty principle as well as practical issues such as filter bank design and application of Gabor filters to texture segmentation. This thesis is composed of three main parts. In the first part, we explore the role of the metric of the Gabor feature space, and its usefulness in capturing what we refer to as...

We prove a quantitative stability result for the Heisenberg–Pauli–Weyl inequality. This leads to next, and next-to-next order correction terms in

We introduce higher order (polynomial) extensions of the unique (up to isomorphisms) non trivial central extension of the Heisenberg algebra. Using the boson representation of the latter, we construct the corresponding polynomial analogue of the Weyl C*-algebra and use this result to deduce the explicit form of the composition law of the associated generalization of the 1-dimensional Heisenberg...

The aim of this paper is to show that any continuous ∗-homomorphism of L(C)(with twisted convolution as multiplication) into B(L(R)) is essentially a Weyl transform. From this we deduce a similar characterisation for the group Fourier transform on the Heisenberg group, in terms of convolution.

The parentage between Weyl pairs, generalized Pauli group and unitary group is investigated in detail. We start from an abstract definition of the Heisenberg-Weyl group on the field R and then switch to the discrete Heisenberg-Weyl group or generalized Pauli group on a finite ring Zd. The main characteristics of the latter group, an abstract group of order d noted Pd, are given (conjugacy class...

A careful study of the physical properties of a family of coherent states on the circle, introduced some years ago by deBì evre and González in [DG 92], is carried out. They were obtained from the Weyl–Heisenberg coherent states in L 2 (R) by means of the Weil– Brezin–Zak transformation, they are labeled by the points of the cylinder S 1 × R, and they provide a realization of L 2 (S 1) by entir...