or example, if you are looking for words in long strings of text, you could use a table accessed by indices that are functions of individual words. The table contains the strings where the word appears and the location of the word in the strings. It would be easy then to locate a word by retrieving all of its appearances from the table. This kind of approach was originally proposed for geometric object recognition, making use of indices based on local geometric features that remained invariant to the object transformation. The features were local to handle partial occlusion, and their indexing function was invariant to the relevant transformation, because unlike words in text, geometric features have both location and orientation. For over a decade now, indexing-based approaches have been gaining ground as the method of choice for building working recognition systems that can Geometric Hashing: An Overview HAIM J. WOLFSON Tel Aviv

Abstract-This paper introduces an analytical framework for studying some properties of model acquisition and recognition techniques based on indexing. The goal is to demonstrate that several problems previously associated with the approach can be attributed to the low dimensionality of invariants used. These include limited index selectivity, excessive accumulation of votes in the look-up table buckets, and excessive sensitivity to quantization parameters. Theoretical results demonstrate that using high-dimensional, highly descriptive global invariants produces better results in terms of accuracy, false positive suppression, and computation time. A practical example of high-dimensional global invariants is introduced and used to implement a 2-D shape acquisitionhecognition system. The acquisitiodrecognition system is based on a two-step table look-up mechanism. First, local curve descriptors are obtained by correlating image contour information at short range. Then, seven-dimensional global invariants are computed by correlating triplets of local curve descriptors at longer range. This experimental system is meant to illustrate the behavior of a high-dimensional indexing scheme. Indeed, its performance shows good agreement with the analytical model with respect to database size, fault tolerance, and recognition speed. Model acquisition time is linear to cubic in the number of object features. Object recognition time is constant to linear in the number of models in the database and linear to cubic in the number of features in the image. The system has been tested extensively, with more than 250 arbitrary shapes in the database. Unsupervised shape and subpart acquisition is demonstrated. I.

Abstract. This article introduces a novel representation for three-dimensional (3D) objects in terms of local affine-invariant descriptors of their images and the spatial relationships between the corresponding surface patches. Geometric constraints associated with different views of the same patches under affine projection are combined with a normalized representation of their appearance to guide matching and reconstruction, allowing the acquisition of true 3D affine and Euclidean models from multiple unregistered images, as well as their recognition in photographs taken from arbitrary viewpoints. The proposed approach does not require a separate segmentation stage, and it is applicable to highly cluttered scenes. Modeling and recognition results are presented.

Affine transformations of the plane have been used in a number of model-based recognition systems, in order to approximate the effects of perspective projection. The mathematics underlying these methods is for exact data, where there is no positional uncertainty in the measurement of feature points. In practice, various heuristics are used to adapt the methods to real data with uncertainty. In this paper, we provide a precise analysis of affine point matching under uncertainty. We obtain an expression for the range of affine-invariant values that are consistent with a given set of four points, where each data point lies in a disk of radius e. This analysis reveals that the range of affine-invarint values depends on the actual x-y-positions of the data points. That is, when there is uncertainty in the data then the representation is no longer invariant with respect to the Cartesian coordinate system. This is problematic for the geometric hashing method, because it means that the precomputed lookup table used by that method is not correct when there is positional uncertainty in the sensor data. We analyze the effect that this has on the probability that the geometric hashing method will find false positive matches of a model to an image, and contrast this with a similar analysis of the alignment method.

By combining techniques from geometric hashing and structural indexing, we have developed a new representation for recognition of free-form objects from three dimensional data. The representation comprises descriptive spin-images associated with each oriented point on the surface of an object. Constructed using single point bases, spin-images are data level shape descriptions that are used for efficient matching of oriented points. During recognition, scene spin-images are indexed into a stack of model spin-images to establish point correspondences between a model object and scene data. Given oriented point correspondences, a rigid transformation that maps the model into the scene is calculated and then refined and verified using a modified iterative closest point algorithm. Indexing of oriented points bridges the gap between recognition by global properties and feature based recognition without resorting to error-prone segmentation or feature extraction. It requires no knowledge of th...

Many recent object recognition systems use a small number of pairings of data and model features to compute the 3D transformation from a model coordinate frame into the sensor coordinate system. In the case of perfect image data, these systems seem to work well. With uncertain image data, however, the performance of such methods is less well understood. In this paper, we examine the effects of two-dimensional sensor uncertainty on the computation of three-dimensional model transformations. We use this analysis to bound the uncertainty in the transformation parameters, as well as the uncertainty associated with applying the transformation to map other model features into the image. We also examine the effects of the transformation uncertainty on the effectiveness of recognition methods.

Complex geometric features such as oriented points, lines or 3D frames are increasingly used in image processing and computer vision. However, processing these geometric features is far more difficult than processing points, and a number of paradoxes can arise. We establish in this article the basic mathematical framework required to avoid them and analyze more specifically three basic problems: (1) what is a random distribution of features, (2) how to define a distance between features, (3) and what is the "mean feature" of a number of feature measurements ? We insist on the importance of an invariance hypothesis for these definitions relative to a group of transformations that models the different possible data acquisitions. We develop general methods to solve these three problems and illustrate them with 3D frame features under rigid transformations. The first problem has a direct application in the computation of the prior probability of a false match in classical model-based object recognition algorithms. We also present experimental results of the two other problems for the statistical analysis of anatomical features automatically extracted from 24 three dimensional images of a single patient's head. These experiments successfully confirm the importance of the rigorous requirements presented in this article.

This paper is concerned with accurate and efficient indexing of fingerprint images. We present a model-based approach, which efficiently retrieves correct hypotheses using novel features of triangles formed by the triplets of minutiae as the basic representation unit. The triangle features that we use are its angles, handedness, type, direction, and maximum side. Geometric constraints based on other characteristics of minutiae are used to eliminate false correspondences. Experimental results on live-scan fingerprint images of varying quality and NIST special database 4 (NIST-4) show that our indexing approach efficiently narrows down the number of candidate hypotheses in the presence of translation, rotation, scale, shear, occlusion, and clutter. We also perform scientific experiments to compare the performance of our approach with another prominent indexing approach and show that the performance of our approach is better for both the live scan database and the ink based database NIST-4.

Invariant representations are frequently used in computer vision algorithms to eliminate the effect of an unknown transformation of the data. These representations, however, depend on the order in which the features are considered in the computations. We introduce the class of projective/permutation p -invariants which are insensitive to the labeling of the feature set. A general method to compute the p -invariant of a point set (or of its dual) in the n-dimensional projective space is given. The one-to-one mapping between n 3 points and the components of their p -invariant representation makes it possible to design correspondence algorithms with superior tolerance to positional errors. An algorithm for coplanar points in projective correspondence is described as an application, and its performance is investigated. The use of p -invariants as an indexing tool in object recognition systems may also be of interest.

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