This paper describes a periodic motion based pedestrian segmentation algorithm for videos acquired from moving platforms. Given a sequence of bounding boxes containing the detected and tracked walking human, the goal is to analyze the low D structure by considering every object sample as a point in the high D manifold space and use the learned structure for segmentation. In this work, we introduce a novel bottom-up learning approach. We represent the human stride as a cascade of models with increasing parameter numbers. These parameters describe the dynamics of pedestrians from coarse to fine. By applying the learned manifold structure, we can predict the location of body parts, especially legs, with high accuracy at every frame. The segmentation in consecutive images is done by EM clustering. With the accuracy for prediction using the twin-pendulum model, EM is more likely to converge to global maximums. Experimental results for real videos are presented The algorithm has demonstrated a reliable performance for videos acquired from moving platforms.

This paper describes an efficient pedestrian detection system for videos acquired from moving platforms. Given a detected and tracked object as a sequence of images within a bounding box, we describe the periodic signature of its motion pattern using a twin-pendulum model. Then a principle gait angle is extracted in every frame providing gait phase information. By estimating the periodicity from the phase data using a digital phase locked loop (dPLL), we quantify the cyclic pattern of the object, which helps us to continuously classify it as a pedestrian. Past approaches have used shape detectors applied to a single image or classifiers based on human body pixel oscillations, but ours is the first to integrate a global cyclic motion model and periodicity analysis. Novel contributions of this paper include: i) development of a compact shape representation of cyclic motion as a signature for a pedestrian, ii) estimation of gait period via a feedback loop module, and iii) implementation of a fast online pedestrian classification system which operates on videos acquired from moving platforms.

In this paper, we present a new method for representing and recognizing objects, based on invariants of the object's regions. We apply the method to articulated objects in low-resolution, noisy range images. Articulated objects such as a back-hoe can have many degrees of freedom, in addition to the unknown variables of viewpoint. Recognizing such an object in an image can involve a search in a high-dimensional space that involves all these unknown variables. Here, we use invariance to reduce this search space to a manageable size. The low resolution of our range images makes it hard to use common features such as edges to find invariants. We have thus developed a new "featureless" method that does not depend on feature detection. Instead of local features, we deal with whole regions of the object. We define a "transform" that converts the image into an invariant representation on a grid, based on invariant descriptors of entire regions centered around the grid points. We use these region-based invariants for indexing and recognition. While the focus here is on articulation, the method can be easily applied to other problems such as the occlusion of fixed objects.

In this work, we treat major problems of object recognition which have received relatively little attention lately. Among them are the loss of depth information in the projection from a 3D object to a single 2D image, and the complexity of finding feature correspondences between images. We use geometric invariants to reduce the complexity of these problems. There are no geometric invariants of a projection from 3D to 2D. However, given certain modeling assumptions about the 3D object, such invariants can be found. The modeling assumptions can be either a particular model or a generic assumption about a class of models. Here, we use such assumptions for single-view recognition. We find algebraic relations between the invariants of a 3D model and those of its 2D image under general projective projection. These relations can be described geometrically as invariant models in a 3D invariant space, illuminated by invariant "light rays," and projected onto an invariant version of the given image. We apply the method to real images.

Articulated objects can have many degrees of freedom. Recognizing such an object in a single image can involve a search in a high-dimensional space that involves all these degrees of freedom, in addition to the usual unknown viewpoint. In this paper we use invariance to reduce this search space to a manageable size. Our method avoids feature detection for improved robustness. We apply the method to range images of objects such as back-hoes.

A method has been developed for overcoming some of the major obstacles of object recognition. It represents both 2D images and 3D models from a database by viewpoint invariant descriptors in the same invariant space. As there is a large number of possible matches between the image and model invariants, there is a crucial need to perform this match efficiently. We answer this need by using an adaptation of spatial sorting based on the octree representation of the model points and image lines in three-dimensional space. We show the advantage of the octree method over the traditional "brute-force" approach and we compare the merits of the two methods. This makes it possible for the recognition task to be performed in a reasonable time for a large number of models.

A general paradigm for recognizing 3D objects is offered, and applied to some geometric primitives (spheres, cylinders, cones, and tori). The assumption is that a curve on the surface was measured with high accuracy (for instance, by a sensory robot). Differential invariants of the curve in one method and differential properties of curves and surfaces in the other are then used to recognize the surface. The motivation is twofold: the output of some devices is not surface range data, but such curves. So, surface invariants, which may be simpler in some cases, cannot always be obtained. Also, a considerable speedup is obtained by using curve data, as opposed to surface data which usually contains a much higher number of points.

In this paper we discuss a new approach to invariant signatures for recognizing curves under viewing distortions and partial occlusion. The approach is intended to overcome the ill-posed problem of finding derivatives, on which local invariants usually depend. The basic idea is to use invariant finite differences, with a scale parameter that determines the size of the differencing interval. The scale parameter is allowed to vary so that a "scale space"-like invariant representation of the curve, with larger difference intervals corresponding to larger coarser scales, can be obtained. In this new representation, each traditional local invariant is replaced by a scale-dependent range of invariants. Thus, instead of invariant signature curves we obtain invariant signature surfaces in a 3D invariant "scale space".

There has been considerable work recently on the problem of reconstruction of 3D points sets from two images, taken by uncalibrated cameras. However, the point correspondence has to be given. In this paper we deal with reconstruction of curves rather than points. While we need the correspondence between curves, this is an easier problem because curves are far fewer and more distinctive than points. We derive a simple and general reconstruction method, based on an invariant coordinate system. We then apply it to non-coplanar conics and to combinations of a 3D conic with points. 3D cubics are also discussed. Unlike previous work we do not need to know the epipolar geometry; we recover it from the images.

Geometric invariants are shape descriptors that remain unchanged under geometric transformations such as projection or changing the viewpoint. A new method of obtaining local projective and affine invariants is developed and implemented for real images. Being local, the Invariants are much less sensitive to occlusion than global invariants. The invariants' computation is based on a canonical method. This consists of defining a canonical coordinate system by the intrinsic properties of the shape, independently of the given coordinate system. Since this canonical system is independent of the original one, it is invariant and all quantities defined in it are invariant. The method was applied without the use of a curve parameter. This was achieved by fitting an implicit polynomial to an arbitrary curve in a vicinity of each curve point. Several configurations are treated: a general curve without any correspondence and curves with known correspondences of one or two feature points or lines. Experimental results for different 2D objects in 3D space are presented.>

The conventional ordinary least squares (OLS) method of fitting a line to a set of data points is very unreliable when the amount of random noise in the input (such as an image) is significant compared with the amount of data that is correlated with the lane itself. In this paper we present an analytic method of separating the data of interest from the outliers. We assume that the overall data (i.e., the line data plus the noise) can be modeled as a mixture of two statistical distributions. Applying a variant of the method of moments (MoM) to the assumed model yields an analytic estimate of the desired line.

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.>

A method is presented for finding semi-local projective and affine invariants. The method consists of defining a canonical coordinate system using intrinsic properties of the shape, independently of the given coordinate system. Since this canonical system is independent of the original one, it is invariant and all quantities defined in it are invariant. The method is applied to find local invariants of a general curve with known correspondences of one or two feature points or lines.>

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.>

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.>

A method of obtaining local projective and affine invariants that is more robust than existing methods is presented. These shape descriptors are useful for object recognition because they eliminate the search for the unknown viewpoint. Being local, these invariants are much less sensitive to occlusion than the global ones used elsewhere. The basic ideas are (i) using an implicit curve representation without a curve parameter, thus increasing robustness; and (ii) using a canonical coordinate system which is defined by the intrinsic properties of the shape, regardless of any given coordinate system, and is thus invariant. Several configurations are treated: a general curve without any correspondence, and curves with known correspondence of feature points or lines.>

The projective correspondence between planar point/line sets is determined using conic invariants. In each set six (4+2) randomly chosen points/lines define two conics to which two absolute projective invariants can be associated. Similar invariant values yield six-tuples matched between the sets. The order of features on a conic is not important and thus the probability of finding a match is maximized. Feature correspondence is recovered with a dynamic programming type analysis of the contingency table derived from the ensemble of matched six-tuples. The algorithm is very sensitive to feature accuracy, raising questions about the straightforward use of conic invariants in recognition of arbitrary objects. The procedures employed, however-affine invariant conic fitting to noisy data,-probabilistic two-stage matching and are of general interest.>

Presents a new and more robust method of obtaining local projective and affine invariants. These shape descriptors are useful for object recognition because they eliminate the search for the unknown viewpoint. Being local, the invariants are much less sensitive to occlusion than the global ones used elsewhere. The basic ideas are: (i) employing an implicit curve representation without a curve parameter, thus increasing robustness; and (ii) using a canonical coordinate system which is defined by the intrinsic properties of the shape, regardless of any given coordinate system, and is thus invariant. Several configurations are treated including a general curve without any correspondence, and curves with known correspondence of feature points or lines.>

A systematic approach to least square approximation of images and of their derivatives is presented. Derivatives of any order can be obtained by convolving the image with a priori known filters. It is shown 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 the relationship to the Marr-Hildreth and Canny edge detectors are proven.>

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.>

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.>

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.>

The conventional least-squares 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 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, based on a maximum-likelihood principle.>