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Local Features in Natural Images via Singularity Theory

  • Book
  • © 2016

Overview

Authors:
  1. James Damon

    1. Department of Mathematics, University of North Carolina , Chapel Hill, USA

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  2. Peter Giblin

    1. Dept. of Mathematical Sciences, University of Liverpool Dept. of Mathematical Sciences, Liverpool, United Kingdom

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  3. Gareth Haslinger

    1. Ainsdale, United Kingdom

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  • 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

Part of the book series: Lecture Notes in Mathematics (LNM, volume 2165)

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Table of contents (14 chapters)

  1. Front Matter

    Pages i-x
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  2. Preliminaries

    1. Front Matter

      Pages 1-1
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    2. Introduction

      • James Damon, Peter Giblin, Gareth Haslinger
      Pages 3-10

    3. Overview

      • James Damon, Peter Giblin, Gareth Haslinger
      Pages 11-20

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

    1. Front Matter

      Pages 21-21
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    2. Apparent Contours for Projections of Smooth Surfaces

      • James Damon, Peter Giblin, Gareth Haslinger
      Pages 23-33

    3. Abstract Classification of Singularities Preserving Features

      • James Damon, Peter Giblin, Gareth Haslinger
      Pages 35-39

    4. Singularity Equivalence Groups Capturing Interactions

      • James Damon, Peter Giblin, Gareth Haslinger
      Pages 41-71

    5. Methods for Classification of Singularities

      • James Damon, Peter Giblin, Gareth Haslinger
      Pages 73-99

    6. Methods for Topological Classification of Singularities

      • James Damon, Peter Giblin, Gareth Haslinger
      Pages 101-114

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

    1. Front Matter

      Pages 115-115
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    2. Stratifications of Generically Illuminated Surfaces with Geometric Features

      • James Damon, Peter Giblin, Gareth Haslinger
      Pages 117-134

    3. Realizations of Abstract Mappings Representing Projection Singularities

      • James Damon, Peter Giblin, Gareth Haslinger
      Pages 135-155

    4. Statements of the Main Classification Results

      • James Damon, Peter Giblin, Gareth Haslinger
      Pages 157-177

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

    1. Front Matter

      Pages 179-179
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    2. Stable View Projections and Transitions Involving Shade/Shadow Curves on a Smooth Surface (SC)

      • James Damon, Peter Giblin, Gareth Haslinger
      Pages 181-191

    3. Transitions Involving Views of Geometric Features (FC)

      • James Damon, Peter Giblin, Gareth Haslinger
      Pages 193-212

  6. Classifications of Multiple Interactions

    1. Front Matter

      Pages 213-213
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    2. Transitions Involving Geometric Features and Shade/Shadow Curves (SFC)

      • James Damon, Peter Giblin, Gareth Haslinger
      Pages 215-241

    3. Classifications of Stable Multilocal Configurations and Their Generic Transitions

      • James Damon, Peter Giblin, Gareth Haslinger
      Pages 243-252
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Keywords

About this book

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.

Authors and Affiliations

  • Department of Mathematics, University of North Carolina , Chapel Hill, USA

    James Damon

  • Dept. of Mathematical Sciences, University of Liverpool Dept. of Mathematical Sciences, Liverpool, United Kingdom

    Peter Giblin

  • Ainsdale, United Kingdom

    Gareth Haslinger

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