Learning optimal smooth invariant subspaces for data approximation

In this article, we consider the problem of approximating a finite set of data (usually huge in applications) by invariant subspaces generated through a small set of smooth functions. The invariance is either by translations under a full-rank lattice or through the action of crystallographic groups. Smoothness is ensured by stipulating that the generators belong to a Paley-Wiener space, that is selected in an optimal way based on the characteristics of the given data. To complete our investigation, we analyze the fundamental role played by the lattice in the process of approximation.

Reconstructing group wavelet transform from feature maps with a reproducing kernel iteration

In this paper we consider the problem of reconstructing an image that is downsampled in the space of its $SE(2)$ wavelet transform, which is motivated by classical models of simple cells receptive fields and feature preference maps in primary visual cortex. We prove that, whenever the problem is solvable, the reconstruction can be obtained by an elementary project and replace iterative scheme based on the reproducing kernel arising from the group structure, and show numerical results on real images.

Optimal translational-rotational invariant dictionaries for images

We provide the construction of a set of square matrices whose translates and rotates provide a Parseval frame that is optimal for approximating a given dataset of images. Our approach is based on abstract harmonic analysis techniques. Optimality is considered with respect to the quadratic error of approximation of the images in the dataset with their projection onto a linear subspace that is invariant under translations and rotations. In addition, we provide an elementary and fully self-contained proof of optimality, and the numerical results from datasets of natural images.

Geometry and dimensionality reduction of feature spaces in primary visual cortex

Some geometric properties of the wavelet analysis performed by visual neurons are discussed and compared with experimental data. In particular, several relationships between the cortical morphologies and the parametric dependencies of extracted features are formalized and considered from a harmonic analysis point of view.

Cortical spatio-temporal dimensionality reduction for visual grouping

The visual systems of many mammals, including humans, is able to integrate the geometric information of visual stimuli and to perform cognitive tasks already at the first stages of the cortical processing. This is thought to be the result of a combination of mechanisms, which include feature extraction at single cell level and geometric processing by means of cells connectivity. We present a geometric model of such connectivities in the space of detected features associated to spatio-temporal visual stimuli, and show how they can be used to obtain low-level object segmentation. The main idea is that of defining a spectral clustering procedure with anisotropic affinities over datasets consisting of embeddings of the visual stimuli into higher dimensional spaces. Neural plausibility of the proposed arguments will be discussed.