The eikonal equation and variants of it are of significant interest for problems in computer vision and image processing. It is the basis for continuous versions of mathematical morphology, stereo, shape-from-shading and for recent dynamic theories of shape. Its numerical simulation can be delicate, owing to the formation of singularities in the evolving front and is typically based on level set methods. However, there are more classical approaches rooted in Hamiltonian physics which have yet to be widely used by the computer vision community. In this paper we review the Hamiltonian formulation, which offers specific advantages when it comes to the detection of singularities or shocks. We specialize to the case of Blum's grass fire flow and measure the average outward ux of the vector field that underlies the Hamiltonian system. This measure has very different limiting behaviors depending upon whether the region over which it is computed shrinks to a singular point or a non-singular one. Hence, it is an effective way to distinguish between these two cases. We combine the ux measurement with a homotopy preserving thinning process applied in a discrete lattice. This leads to a robust and accurate algorithm for computing skeletons in 2D as well as 3D, which has low computational complexity. We illustrate the approach with several computational examples.

This paper proposes a statistical framework for computing medial axis of 2D shapes. In this paper the computation of medial axis is posed as a statistical inference problem not as a mathematical transform. This method contributes to three important aspects in computing the medial axis. I) Prior knowledge is adopted for axes and junctions so that the axes around junctions are regularized. II) Multiple interpretations of axes are possible, each being assigned a probability. III) A novel stochastic jump-diusion process is proposed for estimating both axes and junctions in a Markov random eld. We argue that the stochastic algorithm for computing medial axis is compatible with existing algorithms for image segmentation, such as region growing[33], snake[8] and region competition[29]. Thus it provides a new direction for computing medial axis from real textured images. Experiments are demonstrated on both synthetic and real shapes. This algorithm has been successfully applied to shape lear...

This work considers the common problem of completing partially visible artifacts within a 3D scene. Human vision abilities to complete such artifacts are well studied within the realms of perceptual psychology. However, the psychological explanations for completion have received only limited application in the domain of 3D computer vision. Here, we examine prior work in this area of computer vision with reference to psychological accounts of completion and identify remaining challenges for future work.

In this paper, we propose a new variational framework for computing continuous curve skeletons from discrete objects that are suitable for structural shape representation. We have derived a new energy function, which is proportional to some medialness function, such that the minimum cost path between any two medial voxels in the shape is a curve skeleton. We have employed two different medialness functions; the Euclidean distance field and a variant of the magnitude of the gradient vector flow (GVF), resulting in two different energy functions. The first energy controls the identification of the shape topological nodes from which curve skeletons start, while the second one controls the extraction of curve skeletons. The accuracy and robustness of the proposed framework are validated both quantitatively and qualitatively against competing techniques as well as several 3D shapes of different complexity. 1.

Noise presents a major difficulty in implementing various methods of shape analysis currently in use. A way to deal with this problemistopresmoothshapes. However,thisisproblematiconseveral counts. It makes extensions to 3D shapes difficult. The shape may lack sufficiently many pixels in its narrow regions for computing highorder smoothing operators. How constructs such as shape skeletons are affected by smoothing is not at all clear. The objective of this paper is to demonstrate a new approach to shape analysis which does not require presmoothing of the shape. The basic tool is the �gray skeleton � which is the shape skeleton whose points are associated with signi�cance numbers. A pruning method is developed for extracting a "noise-free " skeleton from the gray skeleton. The problem of segmenting shapes is addressed by formulating a segmetation functional in terms of gray skeletons. Fast algorithms for computing and pruning gray skeletons, and for �nding an approximate minimum of the segmentation functional make the approach practical to implement.

We introduce a novel variational method for the extraction of objects with either bilateral or rotational symmetry in the presence of perspective distortion. Information on the symmetry axis of the object and the distorting transformation is obtained as a by-product of the segmentation process. The key idea is the use of a flip or a rotation of the image to segment as if it has been another view of the object. We call this generated image the symmetrical counterpart image. We show that the symmetrical counterpart image and the source image are related by planar projective homography. This homography is determined by the unknown planar projective transformation that distorts the object symmetry. The proposed segmentation method uses a level-set based curve evolution technique. The extraction of the object boundaries is based on the symmetry constraint and the image data. The symmetrical counterpart of the evolving level-set function provides a dynamic shape prior. It supports the segmentation by resolving possible ambiguities due to noise, clutter, occlusions and assimilation with the background. The homography that aligns the symmetrical counterpart to the source level-set is recovered via a registration process carried out concurrently with the segmentation. Promising segmentation results of various images of approximately symmetrical objects are shown.

We present a new skeletal representation along with a matching framework to address the deformable shape recognition problem. The disconnectedness arises as a result of excessive regularization that we use to describe a shape at an attainably coarse scale. Our motivation is to rely on the stable properties of the shape instead of inaccurately measured secondary details. The new representation does not suffer from the common instability problems of traditional connected skeletons, and the matching process gives quite successful results on a diverse database of 2D shapes. An important difference of our approach from the conventional use of the skeleton is that we replace the local coordinate frame with a global Euclidean frame supported by additional mechanisms to handle articulations and local boundary deformations. As a result, we can produce descriptions that are sensitive to any combination of changes in scale, position, orientation and articulation, as well as invariant ones.

Representing a 3D shape by a set of one-dimensional curves that are locally symmetric with respect to its boundary (i.e., curve skeletons) is of importance in several machine intelligence tasks. This paper presents a fast, automatic, and robust variational framework for computing continuous, sub-voxel accurate curve skeletons from volumetric objects. A reference point inside the object is considered a point source that transmits two wave fronts of different energies. The first front (β-front) converts the object into a graph, from which the object salient topological nodes are determined. Curve skeletons are tracked from those nodes along the cost field constructed by the second front (α-front) until the point source is reached. The accuracy and robustness of the proposed work are validated against competing techniques as well as a database of 3D objects. Unlike other state-of-the-art techniques, the proposed framework is highly robust because it avoids locating and classifying skeletal junction nodes, employs a new energy that does not form medial surfaces, and finally extracts curve skeletons that correspond to the most prominent parts of the shape, and are hence less sensitive to noise.

• Compute the local symmetries only at the locations where it can be accurately determined. • Select a shape dependent scale σ * in which its representation is most stable. Disconnected Skeleton • Construct the distance surface ø as the solution of the following linear diffusion equation at a special scale σ*: (1) • Related to the edge strength function v used in

A new method for determining skeletons of 3D shapes is described. It is a combination of the approach based on the ”grass-fire ” technique and Zhu’s approach based on first finding portions of the shape where its width is approximately constant. The method specifically does not require presmoothing of the shape and is robust in the presence of noise. In an appendix, a method based on variational calculus is formulated for determining pruned, smoothed shape skeletons by minimizing a functional. 1