This paper provides a review of shape analysis methods. Shape analysis methods play an important role in systems for object recognition, matching, registration, and analysis. Researchin shape analysis has been motivated, in part, by studies of human visual form perception systems.

Solid objects in the real world do not pass through each other when they collide. Enforcing this property of "solidness" is important in many interactive graphics applications; for example, solidness makes virtual reality more believable, and solidness is essential for the correctness of vehicle simulators. These applications use a collision-detection algorithm to enforce the solidness of objects. Unfortunately, previous collision-detection algorithms do not adequately address the needs of interactive applications. To work in these applications, a collision-detection algorithm must run at real-time rates, even when many objects can collide, and it must tolerate objects whose motion is specified "on the fly" by a user. This dissertation describes a new collision-detection algorithm that meets these criteria through approximation and graceful degradation, elements of time-critical computing. The algorithm is not only fast but also interruptible, allowing an application to trade accuracy ...

We describe a flexible object recognition and modeling system (FORMS) which represents and recognizes animate objects from their silhouettes. This consists of a model for generating the shapes of animate objects which gives a formalism for solving the inverse problem of object recognition. We model all objects at three levels of complexity: (i) the primitives, (ii) the mid-grained shapes, which are deformations of the primitives, and (iii) objects constructed by using a grammar to join mid-grained shapes together. The deformations of the primitives can be characterized by principal component analysis or modal analysis. When doing recognition the representations of these objects are obtained in a bottom-up manner from their silhouettes by a novel method for skeleton extraction and part segmentation based on deformable circles. These representations are then matched to a database of prototypical objects to obtain a set of candidate interpretations. These interpretations are verified in a...

Robust and time-efficient skeletonization of a (planar) shape, which is connectivity preserving and based on Euclidean metrics, can be achieved by first regularizing the Voronoi diagram (VD) of a shape's boundary points, i.e., by removal of noise-sensitive parts of the tessellation and then by establishing a hierarchic organization of skeleton constituents. Each component of the VD is attributed with a measure of prominence which exhibits the expected invariance under geometric transformations and noise. The second processing step, a hierarchic clustering of skeleton branches, leads to a multiresolution representation of the skeleton, termed skeleton pyramid.

A model of object shape by nets of medial and boundary primitives is justified as richly capturing multiple aspects of shape and yet requiring representation space and image analysis work proportional to the number of primitives. Metrics are described that compute an object representation's prior probability of local geometry by reflecting variabilities in the net's node and link parameter values and that compute a likelihood function measuring the degree of match of an image to that object representation. A paradigm for image analysis of deforming such a model to optimize a posterior probability is described, and this paradigm is shown to be usable as a uniform approach for object definition, object-based registration between images of the same or different imaging modalities, and measurement of shape variation of an abnormal anatomical object compared with a normal. Examples of applications of these methods in radiotherapy, surgery, and psychiatry are given.

The extraction of the centerlines of tubular objects in two and three-dimensional images is a part of many clinical image analysis tasks. One common approach to tubular object centerline extraction is based on intensity ridge traversal. In this paper, we evaluate the effects of initialization, noise, and singularities on intensity ridge traversal and present multiscale heuristics and optimal-scale measures that minimize these effects. Monte Carlo experiments using simulated and clinical data are used to quantify how these "dynamic-scale" enhancements address clinical needs regarding speed, accuracy, and automation. In particular, we show that dynamic-scale ridge traversal is insensitive to its initial parameter settings, operates with little additional computational overhead, tracks centerlines with subvoxel accuracy, passes branch points, and handles significant image noise. We also illustrate the capabilities of the method for medical applications involving a variety of tubular structures in clinical data from different organs, patients, and imaging modalities.

Symmetry is treated as a continuous feature and a Continuous Measure of Distance from Symmetry in shapes is defined. The Symmetry Distance (SD) of a shape is defined to be the minimum mean squared distance required to move points of the original shape in order to obtain a symmetrical shape. This general definition of a symmetry measure enables a comparison of the "amount" of symmetry of different shapes and the "amount" of different symmetries of a single shape. This measure is applicable to any type of symmetry in any dimension. The Symmetry Distance gives rise to a method of reconstructing symmetry of occluded shapes. We extend the method to deal with symmetries of noisy and fuzzy data. Finally, we consider grayscale images as 3D shapes, and use the Symmetry Distance to find the orientation of symmetric objects from their images, and to find locally symmetric regions in images.

A primary goal of statistical shape analysis is to describe the variability of a population of geometric objects. A standard technique for computing such descriptions is principal component analysis. However, principal component analysis is limited in that it only works for data lying in a Euclidean vector space. While this is certainly sufficient for geometric models that are parameterized by a set of landmarks or a dense collection of boundary points, it does not handle more complex representations of shape. We have been developing representations of geometry based on the medial axis description or m-rep. While the medial representation provides a rich language for variability in terms of bending, twisting, and widening, the medial parameters are not elements of a Euclidean vector space. They are in fact elements of a nonlinear Riemannian symmetric space. In this paper, we develop the method of principal geodesic analysis, a generalization of principal component analysis to the manifold setting. We demonstrate its use in describing the variability of medially-defined anatomical objects. Results of applying this framework on a population of hippocampi in a schizophrenia study are presented.

This paper is concerned with the simulation of the Partial Differential Equation (PDE) driven evolution of a closed surface by means of an implicit representation. In most applications, the natural choice for the implicit representation is the signed distance function to the closed surface. Osher and Sethian propose to evolve the distance function with a Hamilton-Jacobi equation. Unfortunately the solution to this equation is not a distance function. As a consequence, the practical application of the level set method is plagued with such questions as when do we have to "reinitialize" the distance function? How do we "reinitialize" the distance function? Etc... which reveal a disagreement between the theory and its implementation. This paper proposes an alternative to the use of Hamilton-Jacobi equations which eliminates this contradiction: in our method the implicit representation always remains a distance function by construction, and the implementation does not differ from the theory...

We propose a model of the spatial visual processes underlying the identification and representation of the shape of primitive spatial regions. We propose that a region's boundaries are sensed at multiple scales by boundariness detectors that give graded responses, that stimulated boundariness detectors of similar scale, s, connect to one another across a distance that is proportional to their scale, and that they connect via cores, where a core encodes the middles and widths of the region and hence is a trace in (x,y,s), i.e., 3D scale space. 3 INTRODUCTION One of the more impressive feats that the human visual system performs is the identification of individual objects from the continuous distribution of light that falls on the retina. To accomplish this task, the observer uses information from the image to identify regions of interest on the basis of spatial changes in luminance, color, texture, motion, etc. He also interprets information from the image on the basis of prior experi...