From receptive profiles to a metric model of V1

In this work we show how to construct connectivity kernels induced by the receptive profiles of simple cells of the primary visual cortex (V1). These kernels are directly defined by the shape of such profiles: this provides a metric model for the functional architecture of V1, whose global geometry is determined by the reciprocal interactions between local elements. Our construction adapts to any bank of filters chosen to represent a set of receptive profiles, since it does not require any structure on the parameterization of the family. The connectivity kernel that we define carries a geometrical structure consistent with the well-known properties of long-range horizontal connections in V1, and it is compatible with the perceptual rules synthesized by the concept of association field. These characteristics are still present when the kernel is constructed from a bank of filters arising from an unsupervised learning algorithm.Comment: 25 pages, 18 figures. Added acknowledgement

A metric model of the visual cortex

The purpose of this thesis is the development of a model for the geometry of the connectivity of the primary visual cortex (V1), by means of functional analysis tools on metric measure spaces. The metric structure proposed to describe the internal connections of V1 implements a notion of correlation between neurons, based on their feature selectivity: this is expressed through a connectivity kernel that is directly induced by the local feature analysis performed by the cells. Such kernel carries a geometrical structure consistent with the well-known properties of long-range horizontal connections in V1, and it is compatible with the perceptual rules synthesized by the concept of association field. Moreover, its construction can be applied to banks of filters not necessarily obtained through a group representation, and possibly only numerically known. This model is then applied to insert biologically inspired connections in deep learning algorithms, to enhance their ability to perform pattern completion in image classification tasks. The main novelty in our approach lies in its ability to recover global geometric properties of the functional architecture of V1 without imposing any parameterization or invariance, but rather by exploiting the local information naturally encoded in the behavior of single V1 neurons in presence of a visual stimulus

Emergence of Lie Symmetries in Functional Architectures Learned by CNNs

In this paper we study the spontaneous development of symmetries in the early layers of a Convolutional Neural Network (CNN) during learning on natural images. Our architecture is built in such a way to mimic some properties of the early stages of biological visual systems. In particular, it contains a pre-filtering step l(0) defined in analogy with the Lateral Geniculate Nucleus (LGN). Moreover, the first convolutional layer is equipped with lateral connections defined as a propagation driven by a learned connectivity kernel, in analogy with the horizontal connectivity of the primary visual cortex (V1). We first show that the l(0) filter evolves during the training to reach a radially symmetric pattern well approximated by a Laplacian of Gaussian (LoG), which is a well-known model of the receptive profiles of LGN cells. In line with previous works on CNNs, the learned convolutional filters in the first layer can be approximated by Gabor functions, in agreement with well-established models for the receptive profiles of V1 simple cells. Here, we focus on the geometric properties of the learned lateral connectivity kernel of this layer, showing the emergence of orientation selectivity w.r.t. the tuning of the learned filters. We also examine the short-range connectivity and association fields induced by this connectivity kernel, and show qualitative and quantitative comparisons with known group-based models of V1 horizontal connections. These geometric properties arise spontaneously during the training of the CNN architecture, analogously to the emergence of symmetries in visual systems thanks to brain plasticity driven by external stimuli

Il metodo di steepest descent per la regolarizzazione di immagini

Scopo della tesi è la descrizione di un metodo per il calcolo di minimi di funzionali, basato sulla steepest descent. L'idea principale è quella di considerare un flusso nella direzione opposta al gradiente come soluzione di un problema di Cauchy in spazi di Banach, che sotto l'ipotesi di Palais-Smale permette di determinare minimi. Il metodo viene applicato al problema di denoising e segmentazione in elaborazione di immagini: vengono presentati metodi classici basati sull'equazione del calore, il total variation ed il Perona Malik. Nell'ultimo capitolo il grafico di un'immagine viene considerato come varietà, che induce una metrica sul suo dominio, e viene nuovamente utilizzato il metodo di steepest descent per costruire algoritmi che tengano conto delle caratteristiche geometriche dell'immagine