In this survey we consider geometric techniques which have been used to measure the similarity or distance between shapes, as well as to approximate shapes, or interpolate between shapes. Shape is a modality which plays a key role in many disciplines, ranging from computer vision to molecular biology. We focus on algorithmic techniques based on computational geometry that have been developed for shape matching, simplification, and morphing. 1 Introduction The matching and analysis of geometric patterns and shapes is of importance in various application areas, in particular in computer vision and pattern recognition, but also in other disciplines concerned with the form of objects such as cartography, molecular biology, and computer animation. The general situation is that we are given two objects A, B and want to know how much they resemble each other. Usually one of the objects may undergo certain transformations like translations, rotations or scalings in order to be matched with th...

Abstract. We examine the problem of searching a database of threedimensional objects (given in VRML) for objects similar to a given object. We introduce an algorithm which is both iterative andinteractive. Rather than base the search solely on geometric feature similarity, we propose letting the user in uence future search results by marking some of the results of the current search as `relevant ' or `irrelevant', thus indicating personal preferences. A novel approach, based on SVM, is used for the adaptation of the distance measure consistently with these markings, which brings the `relevant ' objects closer and pushes the `irrelevant' objects farther. We show that in practice very few iterations are needed for the system to converge well on what the user \had in mind". 1

Abstract not found

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.

Given two planar sets A and B, we examine the problem of determining the smallest " such that there is a Euclidean motion (rotation and translation) of A that brings each member of A within distance " of some member of B. We establish upper bounds on the combinatorial complexity of this subproblem in model-based computer vision, when the sets A and B contain points, line segments, or (filled-in) polygons. We also show how to use our methods to substantially improve on existing algorithms for finding the minimum Hausdorff distance under Euclidean motion. 1 Author's address: Department of Computer Science, Cornell University, Ithaca, NY 14853. This work was supported by the Advanced Research Projects Agency of the Department of Defense under ONR Contract N00014-92-J-1989, and by ONR Contract N00014-92-J-1839, NSF Contract IRI-9006137, and AFOSR Contract AFOSR-91-0328. 2 Author's address: Department of Computer Science, Johns Hopkins University, Baltimore, MD 21218. This work was suppo...

Figure 1: Symmetry detection on a sculpted model. From left to right: Original model, detected partial and approximate symmetries, color-coded deviations from perfect symmetry as a fraction of the bounding box diagonal. “Symmetry is a complexity-reducing concept [...]; seek it everywhere.” Alan J. Perlis Many natural and man-made objects exhibit significant symmetries or contain repeated substructures. This paper presents a new algorithm that processes geometric models and efficiently discovers and extracts a compact representation of their Euclidean symmetries. These symmetries can be partial, approximate, or both. The method is based on matching simple local shape signatures in pairs and using these matches to accumulate evidence for symmetries in an appropriate transformation space. A clustering stage extracts potential significant symmetries of the object, followed by a verification step. Based on a statistical sampling analysis, we provide theoretical guarantees on the success rate of our algorithm. The extracted symmetry graph representation captures important highlevel information about the structure of a geometric model which in turn enables a large set of further processing operations, including shape compression, segmentation, consistent editing, symmetrization, indexing for retrieval, etc.

Symmetry detection is important in the area of computer vision. A 3-D symmetry detection algorithm is presented in this correspondence. The symmetry detection problem is converted to the correlation of the Gaussian image. Once the Gaussian image of the object has been obtained, the algorithm is independent of the input format. The algorithm can handle different kinds of images or objects. Simulated and real images have been tested in a variety of formats, and the results show that the symmetry can be determined using the Gaussian image.

We consider the problem of measuring the similarity or distance between two finite sets of points in a metric space, and computing the measure. This problem has applications in, e.g., computational geometry, philosophy of science, updating or changing theories, and machine learning. We review some of the distance functions proposed in the literature, among them the minimum distance link measure, the surjection measure, and the fair surjection measure, and supply polynomial time algorithms for the computation of these measures. Furthermore, we introduce the minimum link measure, a new distance function which is more appealing than the other distance functions mentioned. We also present a polynomial time algorithm for computing this new measure. We further address the issue of defining a metric on point sets. We present the metric infimum method that constructs a metric from any distance functions on point sets. In particular, the metric infimum of the minimum link measure is a quite int...

We show that the dynamic Voronoi diagram of k sets of points in the plane, where each set consists of n points moving rigidly, has complexity O(n 2 k 2 s (k)) for some fixed s, where s (n) is the maximum length of a (n; s) Davenport-Schinzel sequence. This improves the result of Aonuma et. al., who show an upper bound of O(n 3 k 4 log k) for the complexity of such Voronoi diagrams. We then apply this result to the problem of finding the minimum Hausdorff distance between two point sets in the plane under Euclidean motion. We show that this distance can be computed in time O((m + n) 6 log(mn)), where the two sets contain m and n points respectively. This work was supported in part by NSF grant IRI-9057928 and matching funds from General Electric and Kodak, and in part by AFOSR under contract AFOSR-91-0328. The second author was also supported by the Eshkol grant 04601-90 from The Israeli Ministry of Science and Technology. 1. Introduction Determining the degree to ...

Measuring the similarity or distance between two sets of points in a metric space is an important problem in machine learning and has also applications in other disciplines e.g. in computational geometry, philosophy of science, methods for updating or changing theories, . . . . Recently Eiter and Mannila have proposed a new measure which is computable in polynomial time. However, it is not a distance function in the mathematical sense because it does not satisfy the triangle inequality.