Geometry of Cells Sensible to Curvature and Their Receptive Profiles
Vasiliki Liontou
TL;DR
This work addresses how curvature features can be integrated into the functional architecture of the visual cortex by extending the standard orientation‑position framework to a 4D prolongation endowed with an Engel structure, thereby encoding position, orientation, signed curvature, and scale. It establishes a curvature feature space as the oriented prolongation $\mathbb{S}M$, with horizontality conditions yielding a rank‑2 Engel distribution that is bracket‑generating, and shows that curvature can be naturally paired with scale through the SIM$(2)$ symmetry. By identifying an open submanifold where the Engel structure is left‑invariant under $SIM(2)$, the authors derive left‑invariant generators and connect them to curvature‑sensitive receptive profiles via an uncertainty‑principle framework, producing a family of curvature‑oriented receptive fields from a mother filter. The proposed model thus links cortical modular structure to a geometric control framework, offering a principled way to derive curvature receptive profiles and motivating further study of the global (infinite‑dimensional) generator structure and its implications for visual processing.
Abstract
We propose a model of the functional architecture of curvature sensible cells in the visual cortex that associates curvature with scale. The feature space of orientation and position is naturally enhanced via its oriented prolongation, yielding a 4-dimensional manifold endowed with a canonical Engel structure. This structure encodes position, orientation, signed curvature, and scale. We associate an open submanifold of the prolongation with the quasi-regular representation of the similitude group SIM (2), and find left-invariant generators for the Engel structure. Finally, we use the generators of the Engel structure to characterize curvature-sensitive receptive profiles .