- Minimal Surfaces in Sub-Riemannian Structures and Functional Geometry of the Visual Cortex
- Baspinar, Emre <1988>
Subject
- MAT/05 Analisi matematica
Description
- We develop geometrical models of vision consistent with the characteristics of the visual cortex and study geometric flows in the relevant model geometries. We provide a novel sub-Riemannian model of the primary visual cortex, which models orientation-frequency selective phase shifted cortex cell behavior and the associated horizontal connectivity. We develop an image enhancement algorithm using sub-Riemannian diffusion and Laplace-Beltrami flow in the model framework. We provide two geometric models for multi-scale orientation map and orientation-frequency preference map construction which employ Bargmann transform in high dimensional cortical spaces. We prove the uniqueness of the solution to sub-Riemannian mean curvature flow equation in the Heisenberg group geometry. An iterative diffusion process followed by a maximum selection mechanism was proposed by Citti and Sarti in the sub-Riemannian setting of the roto-translation group. They conjectured that this two-fold procedure is equivalent to a mean curvature flow. However a complete proof was missing, even in the Euclidean setting. We prove in the Euclidean setting that this two fold procedure is equivalent to mean curvature flow.
Date
- 2018-05-14
Type
- Doctoral Thesis
- PeerReviewed
Format
- application/pdf
Identifier
urn:nbn:it:unibo-22824
Baspinar, Emre (2018) Minimal Surfaces in Sub-Riemannian Structures and Functional Geometry of the Visual Cortex, [Dissertation thesis], Alma Mater Studiorum Università di Bologna. Dottorato di ricerca in Matematica