Evolution equations on Gabor transforms and their applications

Abstract

We introduce a systematic approach to the design, implementation and analysis of left-invariant evolution schemes acting on Gabor transform, primarily for applications in signal and image analysis. Within this approach we relate operators on signals to operators on Gabor transforms. In order to obtain a translation and modulation invariant operator on the space of signals, the corresponding operator on the reproducing kernel space of Gabor transforms must be left invariant, i.e. it should commute with the left regular action of the reduced Heisenberg group Hr. By using the left-invariant vector fields on Hr in the generators of our evolution equations on Gabor transforms, we naturally employ the essential group structure on the domain of a Gabor transform. Here we distinguish between two tasks. Firstly, we consider non-linear adaptive left-invariant convection (reassignment) to sharpen Gabor transforms, while maintaining the original signal. Secondly, we consider signal enhancement via left-invariant diffusion on the corresponding Gabor transform. We provide numerical experiments and analytical evidence for our methods and we consider an explicit medical imaging application.

Analysis of Vessel Connectivities in Retinal Images by Cortically Inspired Spectral Clustering

“…A geometric model of the association fields based on the functional organization of V1 has been proposed in [13]. This geometric approach is part of the research line proposed by [37,45,49,56,63,80] and applications to image processing can be found in [6,23,24]. In this work, a novel mathematical model based on this geometry has been applied to the analysis of retinal images to overcome the above mentioned connectivity problems in vessel tracking.…”

From receptive profiles to a metric model of V1

“…In [1], such a lifting was complemented with the definition of a sub-Riemannian structure in a 5-dimensional space obtained by introducing the features of curvature and intensity, in order to overcome limitations given by crossing and bifurcations in a blood vessel tracking task. In [29], the approach of evolution equations was extended to the range of Gabor transforms, again with applications to the processing of images.…”

Uniqueness of Viscosity Mean Curvature Flow Solution in Two Sub-Riemannian Structures

“…Proof. We first show that the supremum on the whole space is equal to the supremum under the condition that d β (ξ, η) ≤ r. Indeed using (23) and (24) we observe sup ξ,η∈G, 0≤t≤T…”

“…which together with (24) ensures that the maximum is not achieved at t = 0. Now consider the second case where d β (ξ, η) Lip(u 0 ) > Mǫ α 4 .…”