Empirical Statistics and Stochastic Models for Visual Signals

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Pathologies IV

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The Topology of Normal Singularities of an Algebraic Surface and a Criterion for Simplicity

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An Elementary Theorem in Geometric Invariant Theory

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On the Computational Architecture of the Neocortex I: The Role of the Thalamo-Cortical Loop

This paper proposes that each area of the cortex carries on its calculations with the active participation of a nucleus in the thalamus with which it is reciprocally and topographically connected. Each cortical area is responsible for maintaining and updating the organism's knowledge of a specific aspect of the world, ranging from low level raw data to high level abstract representations, and involving interpreting stimuli and generating actions. In doing this, it will draw on multiple sources of expertise, learned from experience, creating multiple, often conflicting, hypotheses which are integrated by the action of the thalamic neurons and then sent back to the standard input layer of the cortex. Thus this nucleus plays the role of an 'active blackboard' on which the current best reconstruction of some aspect of the world is always displayed. Evidence for this theory is reviewed and experimental tests are proposed. A sequel to this paper will discuss the cortico-cortical loops and propose quite different computational roles for them.Mathematic

Theta Characteristics of an Algebraic Curve

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2D-Shape Analysis Using Conformal Mapping

The study of 2D shapes and their similarities is a central problem in the field of vision. It arises in particular from the task of classifying and recognizing objects from their observed silhouette. Defining natural distances between 2D shapes creates a metric space of shapes, whose mathematical structure is inherently relevant to the classification task. One intriguing metric space comes from using conformal mappings of 2D shapes into each other, via the theory of Teichmüller spaces. In this space every simple closed curve in the plane (a “shape”) is represented by a ‘fingerprint’ which is a diffeomorphism of the unit circle to itself (a differentiable and invertible, periodic function). More precisely, every shape defines to a unique equivalence class of such diffeomorphisms up to right multiplication by a Möbius map. The fingerprint does not change if the shape is varied by translations and scaling and any such equivalence class comes from some shape. This coset space, equipped with the infinitesimal Weil-Petersson (WP) Riemannian norm is a metric space. In this space, the shortest path between each two shapes is unique, and is given by a geodesic connecting them. Their distance from each other is given by integrating the WP-norm along that geodesic. In this paper we concentrate on solving the “welding” problem of “sewing” together conformally the interior and exterior of the unit circle, glued on the unit circle by a given diffeomorphism, to obtain the unique 2D shape associated with this diffeomorphism. This will allow us to go back and forth between 2D shapes and their representing diffeomorphisms in this “space of shapes”. We then present an efficient method for computing the unique shortest path, the geodesic of shape morphing between each two end-point shapes. The group of diffeomorphisms of S^1 acts as a group of isometries on the space of shapes and we show how this can be used to define shape transformations, like for instance ‘adding a protruding limb’ to any shape.Mathematic

Optimal Approximations by Piecewise Smooth Functions and Associated Variational Problems

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The Irreducibility of the Space of Curves of Given Genus

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Statistics of Natural Images and Models

Large calibrated datasets of `random' natural images have recently become available. These make possible precise and intensive statistical studies of the local nature of images. We report results ranging from the simplest single pixel intensity to joint distribution of 3 Haar wavelet responses. Some of these statistics shed light on old issues such as the near scale-invariance of image statistics and some are entirely new. We fit mathematical models to some of the statistics and explain others in terms of local image featuresMathematic