Isaac Weiss

Geometric invariants and object recognition

We discuss the role of the general invariance concept in object recognition, and review the classical and recent literature on projective invariance. Invariants help solve major problems of object recognition. For instance, different images of the same object often differ from each other, because of the different viewpoint from which they were obtained. To match the two images, common methods thus need to find the correct viewpoint, a difficult problem that can involve search in a high dimensional space of all possible points of view and/or finding point correspondences. Geometric invariants are shape descriptors, computed from the geometry of the shape, that remain unchanged under geometric transformations such as changing the viewpoint. Thus they can be matched without search. Deformations of objects are another important class of geometric changes for which invariance is useful.

Projective invariants of shapes

A major goal of computer vision is object recognition, which involves matching of images of an object, obtained from different, unknown points of view. Since there are infinitely many points of view, one is faced with the problem of a search in a multidimensional parameter space. A related problem is the stereo reconstruction of 3-D surfaces from multiple 2-d images. The author proposes to solve these fundamental problems by using geometrical properties of the visible shape that are invariant to a change in the point of view. To obtain such invariants, he starts from classical theories for differential and algebraic invariants not previously used in image understanding. As they stand, these theories are not directly applicable to vision. He suggests extensions and adaptations of these methods to the needs of machine vision. He then studies general projective transformations, which include both perspective and orthographic projections as special cases.<<ETX>>

Noise-resistant invariants of curves

Projective invariants are shape descriptors that are independent of the point of view from which the shape is seen, and, therefore, are of major importance in object recognition. They make it possible to match an image of an object to one stored in a database without the need for searching for the correct viewpoint. An invariant representation of a general curve is obtained. The calculation is local and does not suffer from the occlusion problem of global descriptors. To make the method robust, differentiation techniques that give much more reliable results than previous ones are developed. These differentiation methods are useful in many other applications as well. >

Logo recognition using geometric invariants

The problem of logo recognition is of great interest in the document domain, especially for databases, because of its potential for identifying the source of the document and its generality as a recognition problem. By recognizing the logo, one obtains semantic information about the document, which may be useful in deciding whether or not to analyze the textual components. A multi-level stages approach to logo recognition which uses global invariants to prune the database and local affine invariants to obtain a more refined match is presented. An invariant signature which can be used for matching under a variety of transformations is obtained. The authors provide a method of computing Euclidean invariants and show how to extend them to capture similarity, affine, and projective invariants when necessary. They implement feature detection, feature extraction, and local invariant algorithms and successfully demonstrate the approach on a small database.<<ETX>>

Pedestrian Detection via Periodic Motion Analysis

We describe algorithms for detecting pedestrians in videos acquired by infrared (and color) sensors. Two approaches are proposed based on gait. The first employs computationally efficient periodicity measurements. Unlike other methods, it estimates a periodic motion frequency using two cascading hypothesis testing steps to filter out non-cyclic pixels so that it works well for both radial and lateral walking directions. The extraction of the period is efficient and robust with respect to sensor noise and cluttered background. In order to integrate shape and motion, we convert the cyclic pattern into a binary sequence by Maximal Principal Gait Angle (MPGA) fitting in the second method. It does not require alignment and continuously estimates the period using a Phase-locked Loop. Both methods are evaluated by experimental results that measure performance as a function of size, movement direction, frame rate and sequence length.

Smoothed differentiation filters for images

Computation of the derivatives of an image defined on a lattice structure is of paramount importance in computer vision. The solution implies least square fitting of a continuous function to a neighborhood centered on the site where the value of the derivative is sought. We present a systematic approach to the problem involving orthonormal bases spanning the vector space defined over the neighborhood. Derivatives of any order can be obtained by convolving the image with a priori known filters. We show that if orthonormal polynomial bases are employed the filters have closed form solutions. The same filter is obtained when the fitted polynomial functions have one consecutive degree. Moment preserving properties, sparse structure for some of the filters, and relationship to the Marr-Hildreth and Canny edge detectors are also proven. Expressions for the filters corresponding to fitting polynomials up to degree six and differentiation orders up to five, for the cases of unweighted data and data weighted by the discrete approximation of a Gaussian, are given in the appendices.

Line fitting in a noisy image

The conventional least-squared-distance method of fitting a line to a set of data points is unreliable when the amount of random noise in the input (such as an image) is significant compared with the amount of data correlated to the line itself. Points which are far away from the line (outliers) are usually just noise, but they contribute the most to the distance averaging, skewing the line from its correct position. The author presents a statistical method of separating the data of interest from random noise, using a maximum-likelihood principle. >

Shape reconstruction on a varying mesh

A central class of image understanding problems is concerned with reconstructing a shape from an incomplete data set, such as fitting a surface to (partially) given contours. A new theory for solving such problems is presented. Unlike the current heuristic methods, the method used starts from fundamental principles that should be followed by any reconstruction method, regardless of its mathematical or physical implementation. A mathematical procedure which conforms to these principles is presented. One major advantage of the method is the ability to handle shapes containing both smooth and sharp parts without using thresholds. A sharp variation, such as a corner, requires a high-resolution mesh for adequate representation, while slowly varying sections can be represented with sparser mesh points. Unlike current methods, this procedure fits the surface on a varying mesh. The mesh is constructed automatically to be more dense at parts of the image that have more rapid variation. Analytical examples are given in simple cases, followed by numerical experiments. >

High-order differentiation filters that work

Reliable derivatives of digital images have always been hard to obtain, especially (but not only) at high orders. We analyze the sources of errors in traditional filters, such as derivatives of the Gaussian, that are used for differentiation. We then study a class of filters which is much more suitable for our purpose, namely filters that preserve polynomials up to a given order. We show that the errors in differentiation can be corrected using these filters. We derive a condition for the validity domain of these filters, involving some characteristics of the filter and of the shape. Our experiments show a very good performance for smooth functions. >

Point-to-line mappings as Hough transforms

In 1962 [US Patent 3069654], Hough used a linear point-to-line mapping (PTLM) to detect large sets of collinear points in an image, by mapping the points into concurrent lines and detecting peaks where many lines intersect. In 1972, Duda and Hart [Commun. ACM 15 (1972) 11] pointed out that Houghs method is not practical, because the peaks need not lie in a bounded region. They (and others after them) therefore developed methods of detecting sets of collinear points using nonlinear point-to-curve mappings that map collinear points into concurrent curves whose intersections do lie in a bounded range. In this paper we show that any PTLM that maps collinear points into concurrent lines must be linear, and that no such PTLM can map all the sets of collinear points in an image into peaks that lie in a bounded region; thus Duda and Harts objection applies to any PTLM-based Hough transform.