Peter Giblin

Motion from the frontier of curved surfaces

The frontier of a curved surface is the envelope of contour generators showing the boundary, at least locally, of the visible region swept out under viewer motion. In general, the outlines of curved surfaces (apparent contours) from different viewpoints are generated by different contour generators on the surface and hence do not provide a constraint on viewer motion. We show that frontier points, however, have projections which correspond to a real point on the surface and can be used to constrain viewer motion by the epipolar constraint. We show how to recover viewer motion from frontier points for both continuous and discrete motion, calibrated and uncalibrated cameras. We present preliminary results of an iterative scheme to recover the epipolar line structure from real image sequences using only the outlines of curved surfaces. A statistical evaluation as also performed to estimate the stability of the solution.<<ETX>>

On the local form and transitions of symmetry sets, medial axes, and shocks

In this paper we explore the local geometry of the medial axis (MA) and shocks (SH), and their structural changes under deformations, by viewing these symmetries as subsets of the symmetry set (SS) and present two results. First, we establish that the local form of the medial axis must generically be one of three cases, which we denote by the A notation explained below (here, it merely serves as a reference to sections of the paper): endpoints (A3), interior points (A12), and junctions (A13). The local form of shocks is then derived from a sub-classification of these points into six types. Second, we address the (classical) instabilities of the MA, i.e., abrupt changes in the representation arising from slight changes in shape, as when a new branch appears with slight protrusion. The identification of these ‘transitions’ is clearly crucial in robust object recognition. We show that for the medial axis only two such instabilities are generically possible: (i) when four branches come together (A14), and (ii) when a new branch grows out of an existing one (A1A3). Similarly, there are six cases of shock instabilities, derived as sub-classifications of the MA instabilities. We give an explicit example of a dent forming in an ellipse where many of the transitions described in the paper can be seen to appear.

Growth, motion and 1-parameter families of symmetry sets

Associated to every plane curve there is the locus of centres of circles bitangent to that curve, the so-called symmetry set of the curve. We can view this set as the spine of our curve, which can be recovered by taking the envelope of circles of varying radii along this spine. Varying the symmetry set in some isotopy while keeping the radius function fixed may be viewed as crudely modelling motion of the original curve viewed as a biological object. Fixing the symmetry set and varying the radius function can be considered to model growth crudely. In this paper we shall describe the generic changes in the curves which take place in the process of growth and motion, and outline the corresponding results for spheres centred on a space curve. We also use the idea of a stratified Morse function to describe the generic changes which occur in one parameter families of (full) bifurcation sets in the plane. Applying this to the bifurcation set of distance squared functions we find all the transitions of a symmetry set (and evolute) which occur in a generic isotopy of a plane curve.

Recovery of an unknown axis of rotation from the profiles of a rotating surface

We consider a surface (semitransparent or opaque) in space, viewed by orthogonal projection to a view plane that is rotating uniformly about an unknown axis (equivalently, a surface rotating about an unknown axis and viewed by orthogonal projection to a fixed view plane). We consider profiles of this surface (also known as apparent contours, occluding contours, and outlines), and we do not track marked points or curves nor assume that a correspondence problem has been solved. We show that, provided the angular speed is known, the location of the axis, and hence the surface, can be recovered from measurements on the profiles over an interval of time. If the angular speed is unknown, then there is a one-parameter family of solutions similar to the bas-relief ambiguity. The results are obtained by use of frontier points on the surface, which can also be viewed as points of epipolar tangency. Results of a numerical experiment showed that the performance was best with larger extents of rotation or when the axis was nearly perpendicular to the view direction.

A formal classification of 3D medial axis points and their local geometry

This paper proposes a novel hypergraph skeletal representation for 3D shape based on a formal derivation of the generic structure of its medial axis. By classifying each skeletal point by its order of contact, we shout that generically the medial axis consists of five types of points which are then organized into sheets, curves, and points: (i) sheets (manifolds with boundary) which are the locus of bitangent spheres with regular tangency/sup 1/ A/sub 1//sup 2/. Two types of curves (ii) the intersection curve of three sheets and the locus of centers of tritangent spheres, A/sub 1//sup 3/, and (iii) the boundary of sheets which are the locus of centers of spheres whose radius equals the larger principle curvature, i.e., higher order contact A/sub 3/ points; and two types of points (iv) centers of quad-tangent spheres, A/sub 1//sup 4/, and, (v) centers of spheres with one regular tangency and one higher order tangency, A/sub 1/A/sub 3/ The geometry of the 3D medial axis thus consists of sheets (A/sub 1//sup 2/) bounded by one type of curve (A/sub 3/) on their free end, which corresponds to ridges on the surface, and attached to two other sheets at another type of curves (A/sub 1//sup 3/), which support a generalized cylinder description. The A/sub 3/ curves can only end in A/sub 1/A/sub 3/ points where they must meet an A/sub 1//sup 3/ curve. The A/sub 1//sup 3/ curves can either meet one A/sub 3/ curve or meet three other A/sub 1//sup 3/ curve at an A/sub 1//sup 4/ point. This formal result leads to a compact representation for 3D shape, referred to as the medial axis hypergraph representation consisting of nodes (A/sub 1//sup 4/ and A/sub 1/A/sub 3/ points), links between pairs of nodes (A/sub 1//sup 3/ and A/sub 3/ curves) and hyperlinks between groups of links (A/sub 1//sup 2/ sheets). The description of the local geometry at nodes by itself is sufficient to capture qualitative aspects of shapes, in analogy to 2D. We derive a pointwise reconstruction formula to reconstruct a surface from this medial axis hypergraph. Thus, the hypergraph completely characterizes 3D shape and lays the theoretical foundation for its use in recognition, morphing, design and manipulation of shapes.

Skeletonization using an extended Euclidean distance transform

Abstract A standard method to perform skeletonization is to use a distance transform. Unfortunately, such an approach has the drawback that only the symmetric axis transform can be computed and not the more practical smoothed local symmetries or the more general symmetry set. Using singularity theory we introduce an extended distance transform which may be used to capture more of the symmetries of a shape. We describe the relationship of this extended distance transform to the skeletal shape descriptors themselves, and other geometric phenomena related to the boundary of the curve. We then show how the extended distance transform can be used to derive skeletal descriptions of an object.

Ridges, crests and sub-parabolic lines of evolving surfaces

The ridge lines on a surface can be defined either via contact of the surface with spheres, or via extrema of principal curvatures along lines of curvature. Certain subsets of ridge lines called crest lines have been singled out by some authors for medical imaging applications. There is a related concept of sub-parabolic line on a surface, also defined via extrema of principal curvatures.In this paper we study in detail the structure of the ridge lines, crest lines and sub-parabolic lines on a generic surface, and on a surface which is evolving in a generic (one-parameter) family. The mathematical details of this study are in Bruce et al. (1994c).

Three-dimensional arm movements at constant equi-affine speed

It has long been acknowledged that planar hand drawing movements conform to a relationship between movement speed and shape, such that movement speed is inversely proportional to the curvature to the power of one-third. Previous literature has detailed potential explanations for the power laws existence as well as systematic deviations from it. However, the case of speed-shape relations for three-dimensional (3D) drawing movements has remained largely unstudied. In this paper we first derive a generalization of the planar power law to 3D movements, which is based on the principle that this power law implies motion at constant equi-affine speed. This generalization results in a 3D power law where speed is inversely related to the one-third power of the curvature multiplied by the one-sixth power of the torsion. Next, we present data from human 3D scribbling movements, and compare the obtained speed-shape relation to that predicted by the 3D power law. Our results indicate that the introduction of the torsion term into the 3D power law accounts for significantly more of the variance in speed-shape relations of the movement data and that the obtained exponents are very close to the predicted values.

Affine Invariant Distances, Envelopes and Symmetry Sets

Affine invariant symmetry sets of planar curves are introduced and studied in this paper. Two different approaches are investigated. The first one is based on affine invariant distances, and defines the symmetry set as the closure of the locus of points on (at least) two affine normals and affine-equidistant from the corresponding points on the curve. The second approach is based on affine bitangent conics. In this case the symmetry set is defined as the closure of the locus of centers of conics with (at least) 3-point contact with the curve at two or more distinct points on the curve. This is equivalent to conic and curve having, at those points, the same affine tangent, or the same Euclidean tangent and curvature. Although the two analogous definitions for the classical Euclidean symmetry set are equivalent, this is not the case for the affine group. We present a number of properties of both affine symmetry sets, showing their similarities with and differences from the Euclidean case. We conclude the paper with a discussion of possible extensions to higher dimensions and other transformation groups, as well as to invariant Voronoi diagrams.

Epipolar fields on surfaces

The view lines associated with a family of profile curves of the projection of a surface onto the retina of a moving camera defines a multi-valued vector field on the surface. The integral curves of this field are called epipolar curves and together with a parametrization of the profiles provide a parametrization of regions of the surface. This parametrization has been used in the systematic reconstruction of surfaces from their profiles. We present a complete local investigation of the epipolar curves, including their behaviour in a neighbourhood of a point where the epipolar parametrization breaks down. These results give a systematic way of detecting the gaps left by reconstruction of a surface from profiles. They also suggest methods for filling in these gaps.