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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.





Chapter 1. Introduction

In this monograph we consider a basic problem in computer imaging for natural images, which are images of objects obtained by projection of their reflected light rays onto a viewing plane, which might be a camera lens or a viewer’s eye. The goal for natural images is to detect the objects in the image and determine their geometric features such as edges, creases, corners, and “marking curves” separating regions of an object with distinct visual properties.
James Damon, Peter Giblin, Gareth Haslinger

Chapter 2. Overview

In this section, we introduce the general framework for approaching the classification of the local features of images of objects in natural images. Two general considerations are that the objects will either be surfaces with boundary edges (representing physical objects that are “thin surfaces”) or 3-dimensional objects whose boundary surfaces exhibit certain geometric features. We allow the surface features to be generic geometric features including generalized notions of creases, corners, marking curves, as well as edges (see below). We give more precise descriptions of the geometric features later in this chapter and Chap. 4
James Damon, Peter Giblin, Gareth Haslinger

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


Chapter 3. Apparent Contours for Projections of Smooth Surfaces

Our goal as described in the overview is to provide a complete analysis of the views of natural scenes involving geometric features, shade/shadow, and apparent contours resulting from viewer movement. Our approach to this will involve progressively adding more detailed structure to simpler situations. The starting point for this is the case where we have a single object whose boundary is a smooth surface \(M \subset \mathbb{R}^{3}\) without geometric features. Hence, for the remainder of this chapter we always assume M is a compact smooth surface without boundary.
James Damon, Peter Giblin, Gareth Haslinger

Chapter 4. Abstract Classification of Singularities Preserving Features

In Chap. 2, we described the geometric features which objects in our images may have. One part of the classification of view projections involves the views of objects with geometric features in the absence of any additional features resulting from illumination. This reduces to the classification of view projections of objects with only geometric features. In this chapter we shall review several abstract classifications of map germs which have already been obtained for this situation. In addition, we shall indicate how these abstract results may be further extended so that they provide the abstract classifications for analyzing additional features beyond those originally considered.
James Damon, Peter Giblin, Gareth Haslinger

Chapter 5. Singularity Equivalence Groups Capturing Interactions

Before introducing the notion of equivalence we will use, we motivate our approach by briefly considering an earlier approach of Henry-Merle et al. [HM, DHM], and Donati-Stolfi [Dn, DS].
James Damon, Peter Giblin, Gareth Haslinger

Chapter 6. Methods for Classification of Singularities

In this chapter we will recall the methods which were introduced by Thom [Le] and especially Mather [MIII, MVI] to classify germs of mappings under various equivalence groups by reducing to the induced actions of Lie groups on jet spaces. This involves using finite determinacy results and Mather’s geometric lemma for actions of Lie groups. This was considerably strengthened by the much improved order of determinacy results from the stronger method of unipotent groups due to Bruce-Du Plessis-Wall [BDW]. These results will be appropriately adapted to apply to our situation.
James Damon, Peter Giblin, Gareth Haslinger

Chapter 7. Methods for Topological Classification of Singularities

As already mentioned, a key part of our investigation involves the abstract classifications of mappings under \(_{\mathcal{V}}\mathcal{A}\)-equivalence for a special semianalytic stratification \(\mathcal{V}\). Initially the stratification is simple, e.g. modeled by a distinguished smooth curve on a smooth surface, or a boundary curve of a smooth surface with boundary. In such cases, there is a finite classification in low codimension.
James Damon, Peter Giblin, Gareth Haslinger

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


Chapter 8. Stratifications of Generically Illuminated Surfaces with Geometric Features

In Chap. 2 we introduced the models for geometric features and the restrictions on the light source. We investigate here the consequences of the light projection being stable for the interaction of the geometric features and the resulting shade/shadow curves. We carry this out by first using the abstract classification of stable germs at geometric feature points, and determining in Sect. 8.1 their distinct geometric realizations to obtain the classification in Theorem 8.7 of the stable projection map germs including visibility for each geometric configuration (FC). Second, we apply this classification in Sect. 8.2 to the light projection maps from geometric feature points to deduce in Theorem 8.8 the classifications of stratifications resulting from the refinement by shade/shadow curves of the stratifications defined by geometric features. We also apply the results to obtain in Theorem 8.9 the classification of stable view projections on geometric features with shade/shadow curves, but without apparent contours (SF).
James Damon, Peter Giblin, Gareth Haslinger

Chapter 9. Realizations of Abstract Mappings Representing Projection Singularities

In Chaps. 6 and 7 we have constructed abstract normal forms for view projections and in Chap. 8 we have identified the various physical situations to which these normal forms apply.
James Damon, Peter Giblin, Gareth Haslinger

Chapter 10. Statements of the Main Classification Results

We now in a position to describe the main results of this monograph concerning the interaction of geometric features, shade/shadow curves and apparent contours in the case of a collection of fixed objects given a fixed generic light source.
James Damon, Peter Giblin, Gareth Haslinger

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


Chapter 11. Stable View Projections and Transitions Involving Shade/Shadow Curves on a Smooth Surface (SC)

In this chapter we consider the case of a smooth surface illuminated from one direction and viewed from a different direction. The shade curve and the apparent contour will then interact, but since these curves are not arbitrary curves on the surface the possible interactions are not necessarily the same as those which are possible on a surface with a marking curve or boundary. Since we assume the light projection is stable, we need only consider the case where it is a fold map or a cusp map.
James Damon, Peter Giblin, Gareth Haslinger

Chapter 12. Transitions Involving Views of Geometric Features (FC)

In Chap. 8 we gave the classification of stable view projections of type (FC). In this chapter we further give the classification of generic transitions of type (FC). We summarize the classification for the five classes of transitions in Theorem 12.1 and in subsequent sections consider the individual cases.
James Damon, Peter Giblin, Gareth Haslinger

Classifications of Multiple Interactions


Chapter 13. Transitions Involving Geometric Features and Shade/Shadow Curves (SFC)

We have already completed the classifications of the realizations of the local transitions for both (SC) in Chap. 11 and (FC) Chap. 12 For (SF) we only consider stable view projections of stable (SF) stratifications which are regular or strata regular, so there is no contribution from apparent contours in the images; and the classification of these view projections for both the local and multilocal cases was completed in Chap. 8 Thus, to complete the classification of stable views and transitions involves two remaining cases. One is for the transitions for the local interaction of all three geometric features, shade/shadow, and apparent contours (SFC); and the second is for the multilocal transitions. In this chapter, we complete the classification for the case (SFC) and in the next chapter we shall complete the classification for multilocal transitions.
James Damon, Peter Giblin, Gareth Haslinger

Chapter 14. Classifications of Stable Multilocal Configurations and Their Generic Transitions

In the preceding chapters we used the results developed in Part II to determine the generic transitions for the local cases. We complete the analysis by treating the remaining cases which involve multilocal classifications. These are of two types. The first arises from the light projection when a distant cast shadow curve intersects a geometric feature. The case where a cast shadow of types smooth curve, C 1-parabola, or V -point occurs on a smooth sheet have already been treated. The second case involves one surface locally occluding part(s) of one or more other surfaces. This corresponds to the case of multigerms for the view projection. In this chapter we classify both the stable multilocal cases yielding the resulting stratifications for cast-shadows and geometric features which were listed in Corollary 8.​10 of Chap. 8, and the stable multilocal views which involve occlusions and their generic transitions.
James Damon, Peter Giblin, Gareth Haslinger


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