Large-scale deformations of a tubular object, or generalized cylinder, are often defined by a target shape for its center curve, typically using a parametric target curve. This task is non-trivial for free-form deformations or direct manipulation methods because it is hard to manually control the centerline by adjusting control points. Most skeleton-based methods are no better, again due to the small number of manually adjusted control points. In this paper, we propose a method to deform a generalized cylinder based on its skeleton composed of a centerline and orthogonal cross sections. Although we are not the first to use such a skeleton, we propose a novel skeletonization method that tries to minimize the number of intersections between neighboring cross sections by means of a relative curvature condition to detect intersections. The mesh deformation is first defined geometrically by deforming the centerline and mapping the cross sections. Rotation minimizing frames are used during mapping to control twisting. Secondly, given displacements on the cross sections, the deformation is decomposed into finely subdivided regions. We limit distortion at these vertices by minimizing an elastic thin shell bending energy, in linear time. Our method can handle complicated generalized cylinders such as the human colon.

This is the second part of an investigation into the visual clues in illuminated scenes, in terms of the interactions between apparent contours, shade and cast shadow curves, boundary edges, and markings on smooth surfaces, pairs of surfaces meeting in a crease, and triples of surfaces meeting in a corner. We consider both “stable” and “codimension 1” cases, the latter meaning that we list events which occur in a generic “fly-past” of the scene. We assume that there is a single principal source of light. The first part of this work is [Internat. J. Comput. Vision, 82 (2009), pp. 25–47]; in this second part we give details of the cases which involve creases and corners, and the “multilocal” cases where two surfaces separated in space interact via occlusion or cast shadows. We also give some details of the mathematical background to our work; the full mathematical treatment will appear in [Damon, Giblin, and Haslinger, Characterizing Stable Local Features of Illuminated Surfaces and Their Generic Transitions from Viewer Movement, in preparation].

The seven papers in this special section focus on shape analysis and its application in image understanding.

Ridges are characteristic curves of a surface that mark salient intrinsic features of its shape and are therefore valuable for shape matching, surface quality control, visualization and various other applications. Ridges are loci of points on a surface where either of the principal curvatures attain a critical value in its respective principal direction. These curves have complex behavior near umbilics on a surface, and may also pass through certain turning points causing added complexity for ridge computation. We present a new algorithm for numerically tracing ridges on B-Spline surfaces that also accurately captures ridge behavior at umbilics and ridge turning points. The algorithm traverses ridge segments by detecting ridge points while advancing and sliding in principal directions on a surface in a novel manner, thereby computing connected curves of ridge points. The output of the algorithm is a set of curve segments, some or all of which, may be selected for other applications such as those mentioned above. The results of our technique are validated by comparison with results from previous research and with a brute-force domain sampling technique.

We consider the average outward flux through a Jordan curve of the gradient vector field of the Euclidean distance function to the boundary of a 2D shape. Using an alternate form of the divergence theorem, we show that in the limit as the area of the region enclosed by such a curve shrinks to zero, this measure has very different behaviors at medial points than at non-medial ones, providing a theoretical justification for its use in the Hamilton-Jacobi skeletonization algorithm of Siddiqi et al. (2002). We then specialize to the case of shrinking circular neighborhoods and show that the average outward flux measure also reveals the object angle at skeletal points. Hence, formulae for obtaining the boundary curves, their curvatures, and other geometric quantities of interest, can be written in terms of the average outward flux limit values at skeletal points. Thus this measure can be viewed as a Euclidean invariant for shape description: it can be used to both detect the skeleton from the Euclidean distance function, as well as to explicitly reconstruct the boundary from it. We illustrate our results with several numerical simulations.