Local Features in Natural Images via Singularity Theory

Authors: James Damon, Peter Giblin, Gareth Haslinger



Includes geometric features of objects, shade/shadow features, and apparent contours

Analyzes generic changes under viewer movement

Amply illustrated with computer generated images

Yields approach for edge detection where multiple curves meet

Employs singularity theory on semi-analytic stratified spaces

Combines singularity methods with geometry of surfaces

Builds on earlier results in singularity theory and shows necessity of topological methods

This monograph considers a basic problem in the computer analysis of natural images, which are images of scenes involving multiple objects that are obtained by a camera lens or a viewer’s eye. The goal is to detect geometric features of objects in the image and to separate regions of the objects with distinct visual properties. When the scene is illuminated by a single principal light source, we further include the visual clues resulting from the interaction of the geometric features of objects, the shade/shadow regions on the objects, and the “apparent contours”.  We do so by a mathematical analysis using a repertoire of methods in singularity theory. This is applied for generic light directions of both the “stable configurations” for these interactions, whose features remain unchanged under small viewer movement, and the generic changes which occur under changes of view directions. These may then be used to differentiate between objects and determine their shapes and positions.

Table of contents

Front Matter

Preliminaries

Front Matter
Introduction
Overview

Mathematical Basis for Analysis of Feature-Shade/Shadow-Contours

Front Matter
Apparent Contours for Projections of Smooth Surfaces
Abstract Classification of Singularities Preserving Features
Singularity Equivalence Groups Capturing Interactions
Methods for Classification of Singularities
Methods for Topological Classification of Singularities

The Classification of Interactions Involving Feature-Shade/Shadow-Contours

Front Matter
Stratifications of Generically Illuminated Surfaces with Geometric Features
Realizations of Abstract Mappings Representing Projection Singularities
Statements of the Main Classification Results

Classifications of Interactions of Pairs of Feature-Shade/Shadow-Contours

Front Matter
Stable View Projections and Transitions Involving Shade/Shadow Curves on a Smooth Surface (SC)
Transitions Involving Views of Geometric Features (FC)

Classifications of Multiple Interactions

Front Matter
Transitions Involving Geometric Features and Shade/Shadow Curves (SFC)
Classifications of Stable Multilocal Configurations and Their Generic Transitions

Back Matter