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Registration and modeling of shapes are two important problems in computer vision and pattern recognition. Despite enormous progress made over the past decade, these problems are still open. In this paper, we advance the state of the art in both directions. First, we consider an efficient registration method that aims to recover a one-to-one correspondence between shapes and introduce measures of uncertainties driven from the data which explain the local support of the recovered transformations. To this end, a free-form deformation is used to describe the deformation model. The transformation is combined with an objective function defined in the space of implicit functions used to represent shapes. Once the registration parameters have been recovered, we introduce a novel technique for model building and statistical interpretation of the training examples based on a variable bandwidth kernel approach. The support on the kernels varies spatially and is determined according to the uncertainties of the registration process. Such a technique introduces the ability to account for potential registration errors in the model. Handwritten character recognition and knowledge-based object extraction in medical images are examples of applications that demonstrate the potentials of the proposed framework. Index Terms--Shape registration, shape modeling, uncertainty estimates, nonparametric statistics, variable metric kernels.

It is well-known that the linear scale-space theory in computer vision is mainly based on the Gaussian kernel. The purpose of the paper is to propose a scale-space theory based on B-spline kernels. Our aim is twofold. On one hand, we present a general framework and show how B-splines provide a flexible tool to design various scale-space representations: continuous scale-space, dyadic scale-space frame, and compact scale-space representation. In particular, we focus on the design of continuous scale-space and dyadic scale-space frame representation. A general algorithm is presented for fast implementation of continuous scale-space at rational scales. In the dyadic case, efficient frame algorithms are derived using B-spline techniques to analyze the geometry of an image. Moreover, the image can be synthesized from its multiscale local partial derivatives. Also, the relationship between several scale-space approaches is explored. In particular, the evolution of wavelet theory from traditional scale-space filtering can be well understood in terms of B-splines. On the other hand, the behavior of edge models, the properties of completeness, causality, and other properties in such a scale-space representation are examined in the framework of B-splines. It is shown that, besides the good properties inherited from the Gaussian kernel, the B-spline derived scale-space exhibits many advantages for modeling visual mechanism with regard to the efficiency, compactness, orientation feature, and parallel structure. Index Terms - Image modeling, B-spline, wavelet, scale-space, scaling theorem, fingerprint theorem.

The literature on recursive estimation of structure and motion from monocular image sequences comprises a large number of apparently unrelated models and estimation techniques. We propose a framework that allows us to derive and compare all models by following the idea of dynamical system reduction. The "natural" dynamic model, derived from the rigidity constraint and the projection model, is first reduced by explicitly decoupling structure (depth) from motion. Then, implicit decoupling techniques are explored, which consist of imposing that some function of the unknown parameters is held constant. By appropriately choosing such a function. not only can we account for models seen so far in the literature, but we can also derive novel ones. Index Terms - Visual motion estimation, epipolar geometry, motion decoupling, compensation, fixation, parallax, output stabilization, model reduction.

This paper concerns the extraction of rotation invariant texture features and the use of such features in script identification from document images. Rotation invariant texture features are computed based on an extension of the popular multi-channel Gabor filtering technique, and their effectiveness is tested with 300 randomly rotated samples of 15 Brodatz textures. These features are then used in an attempt to solve a practical but hitherto mostly overlooked problem in document image processing - the identification of the script of a machine printed document. Automatic script and language recognition is an essential front-end process for the efficient and correct use of OCR and language translation products in a multilingual environment. Six languages (Chinese, English, Greek, Russian, Persian, and Malayalam) are chosen to demonstrate the potential of such a texture-based approach in script identification. Index Terms - Texture, rotation invariance, Gabor filtering, script and language recognition, document image processing.

The three best-known criteria in two-view motion analysis are based, respectively, on the distances between points and their corresponding epipolar lines, on the gradient-weighted epipolar errors, and on the distances between points and the reprojections of their reconstructed points. The last one has a better statistical interpretation, but is, however, significantly slower than the first two. In this paper, I show that, given a reasonable initial guess of the epipolar geometry, the last two criteria are equivalent when the epipoles are at infinity, and differ from each other only a little even when the epipoles are in the image, as shown experimentally. The first two criteria are equivalent only when the epipoles are at infinity and when the observed object/scene has the same scale in the two images. This suggests that the second criterion is sufficient in practice because of its computational efficiency. Experiments with several thousand computer simulations and four sets of real data confirm the analysis. The result is valid for both calibrated and uncalibrated images. Index Terms - Motion analysis, multiple-view geometry, 3D reconstruction, optimization criteria, algorithmic comparison, structure from motion, uncalibrated images.

Junctions are important features for image analysis and form a critical aspect of image understanding tasks such as object recognition. We present a unified approach to detecting (location of the center of the junction), classifying (by the number of wedges - lines, corners, three-junctions such as T or Y junctions, or four-junctions such as X-junctions), and reconstructing junctions (in terms of radius size, the angles of each wedge and the intensity in each of the wedges) in images. Our main contribution is a modeling of the junction which is complex enough to handle all these issues and yet simple enough to admit an effective dynamic programming solution. Broadly, we use a template deformation framework along with a gradient criterium to detect radial partitions of the template. We use the minimum description length principle to obtain the optimal number of partitions that best describes the junction. Kona [27] is an implementation of this model. We (quantitatively) demonstrate the stability and robustness of the detector by analyzing its behavior in the presence of noise, using synthetic/controlled apparatus. We also present a qualitative study of its behavior on real images. Index Terms - Junctions, corners, feature detection, low-level vision, minimum description length (MDL) principle, energy minimization.

Despite the intuitive interpretation of the skeleton as the "center line of the shape," it is surprisingly hard to get skeletonization algorithms that simultaneously produce all the "good" properties we expect (e.g., well-centered, well-connected, rotation-invariant, efficient, robust, accurately reflecting the shape). In this paper, we introduce the new concept of "veinerization," which produces a graph that contains all the "topological" information needed to derive a wide variety of skeletons. Theoretically, the main contribution is to provide a homogeneous framework for integration of the major concepts described in other related works on digital skeletonization. In practice, the new aspect of this approach is to provide the user with different criteria for selecting the most suitable skeleton for a given application; e.g., the user can select a suitable threshold for obtaining the desirable balance between "having a skeleton without noisy prunes" and "having a skeleton that reflects the initial shape." This algorithm has been tested on numerous kinds of patterns, including pathological ones like fractal sets well-known for the complexity of their shapes. Index Terms - Digital skeleton, veinerization, skeletonization, context adaptation, digital topology, graphs, fractal sets.

A variety of analytic and probabilistic models in connection to Markov random fields (MRFs) have been proposed in the last decade for solving low level vision problems involving discontinuities. This paper presents a systematic study of these models and defines a general discontinuity adaptive (DA) MRF model. By analyzing the Euler equation associated with the energy minimization, it shows that the fundamental difference between different models lies in the behavior of interaction between neighboring points, which is determined by the a priori smoothness constraint encoded into the energy function. An important necessary condition is derived for the interaction to be adaptive to discontinuities to avoid oversmoothing. This forms the basis on which a class of adaptive interaction functions (AIFs) is defined. The DA model is defined in terms of the Euler equation constrained by this class of AIFs. Its solution is [C.sup.1] continuous and allows arbitrarily large but bounded slopes in dealing with dis-continuities. Because of the continuous nature, it is stable to changes in parameters and data, a good property for regularizing ill-posed problems. Experimental results are shown. Index Terms - Discontinuities, energy functions, Euler equation, computer vision, Markov random fields, minimization, regularization.

Gibbs random field (GRF) models and features from cooccurrence matrices are typically considered as separate but useful tools for texture discrimination. In this paper we show an explicit relationship between cooccurrences and a large class of GRF's. This result comes from a new framework based on a set-theoretic concept called the 'aura set' and on measures of this set, 'aura measures.' This framework is also shown to be useful for relating different texture analysis tools: We show how the aura set can be constructed with morphological dilation, how its measure yields cooccurrences, and how it can be applied to characterizing the behavior of the Gibbs model for texture. In particular, we show how the aura measure generalizes, to any number of gray levels and neighborhood order, some properties previously known for just the binary, nearest-neighbor GRF. Finally, we illustrate how these properties can guide one's intuition about the types of GRF patterns which are most likely to form.

An image representation scheme using a set of block pattern models (BPM's) is introduced. These models consist of three categories (constant, oriented, and irregular), with the last one consisting of two subgroups: textured and mixed. They are constructed to represent three basic types of image patterns: shade, edge, and texture. Image representation using these models requires considerably fewer bits than the original pixel-wise description and yet characterizes perceptually significant features more effectively. In particular, the parameterization of the oriented model using oriented basis functions lays a mathematical foundation for designing directional operators that are desirable in many image processing applications. Algorithms for model classification, model parameter estimation, and image reconstruction from model parameters are presented, and these provide the necessary vehicles for applying the proposed representation scheme to various image processing tasks. The applications of the proposed models in image coding, image zooming, and image smoothing are described. The coding system approximates each image block by a BPM and further quantizes the model parameters. Satisfactory coded images have been obtained at bit rates between 0.5 approximates 0.6 bpp (bits per pixel) with a high-rate realization and between 0.3 approximates 0.5 bpp with a low-rate realization. The high-rate realization has a simple structure suitable for real-time implementation. The methods for image zooming and smoothing are similar, where both adapt the processing for each pixel according to the model of its neighborhood. By using directional filters in oriented regions, edges and lines are rendered sharper in a smoother manner than with conventional linear filtering approaches, which leads to significant improvement in perceived image quality.

Although most optical flow techniques presume brightness constancy, it is well-known that this constraint is often violated, producing poor estimates of image motion. This paper describes a generalized formulation of optical flow estimation based on models of brightness variations that are caused by time-dependent physical processes. These include changing surface orientation with respect to a directional illuminant, motion of the illuminant, and physical models of heat transport in infrared images. With these models, we simultaneously estimate the 2D image motion and the relevant physical parameters of the brightness change model. The estimation problem is formulated using total least squares (TLS), with confidence bounds on the parameters. Experiments in four domains, with both synthetic and natural inputs, show how this formulation produces superior estimates of the 2D image motion. Index Terms--Optical flow, physics-based brightness variation, total least squares.

We have recently proposed an approach to recognition that uses regions to determine the pose of objects while allowing for partial occlusion of the regions. Regions introduce an attractive alternative to existing global and local approaches, since, unlike global features, they can handle occlusion and segmentation errors, and unlike local features they are not as sensitive to sensor errors, and they are easier to match. The region-based approach also uses image information directly, without the construction of intermediate representations, such as algebraic descriptions, which may be difficult to reliably compute. In this paper, we further analyze properties of the method for planar objects undergoing projective transformations. In particular, we prove that three visible regions are sufficient to determine the transformation uniquely and that for a large class of objects, two regions are insufficient for this purpose. However, we show that when several regions are available, the pose of the object can generally be recovered even when some or all regions are significantly occluded. Our analysis is based on investigating the flow patterns of points under projective transformations in the presence of fixed points. Index Terms--Object recognition, pose estimation with regions.

Constructing a model for data in [R.sup.2] is a common problem in many scientific fields, including pattern recognition, computer vision, and applied mathematics. Often little is known about the process which generated the data or its statistical properties. For example, in fitting a piecewise linear model, the number of pieces, as well as the knot locations, may be unknown. Hence, the method used to build the statistical model should have few assumptions, yet, still provide a model that is optimal in some sense. Such methods can be designed through the use of genetic algorithms. In this paper, we examine the use of genetic algorithms to fit piecewise linear functions to data in [R.sup.2]. The number of pieces, the location of the knots, and the underlying distribution of the data are assumed to be unknown. We discuss existing methods which attempt to solve this problem and introduce a new method which employs genetic algorithms to optimize the number and location of the pieces. Experimental results are presented which demonstrate the performance of our method and compare it to the performance of several existing methods. We conclude that our method represents a valuable tool for fitting both robust and nonrobust piecewise linear functions. Index Terms--Genetic algorithms, optimization, statistical data analysis, splines.

Determining the rigid transformation relating 2D images to known 3D geometry is a classical problem in photogrammetry and computer vision. Heretofore, the best methods for solving the problem have relied on iterative optimization methods which cannot be proven to converge and/or which do not effectively account for the orthonormal structure of rotation matrices. We show that the pose estimation problem can be formulated as that of minimizing an error metric based on collinearity in object (as opposed to image) space. Using object space collinearity error, we derive an iterative algorithm which directly computes orthogonal rotation matrices and which is globally convergent. Experimentally, we show that the method is computationally efficient, that it is no less accurate than the best currently employed optimization methods, and that it outperforms all tested methods in robustness to outliers. Index Terms--Pose estimation, absolute orientation, optimization, weak-erspective camera models, numerical optimization.

This paper presents a new variational framework for detecting and tracking multiple moving objects in image sequences. Motion detection is performed using a statistical framework for which the observed interframe difference density function is approximated using a mixture model. This model is composed of two components, namely, the static (background) and the mobile (moving objects) one. Both components are zero-mean and obey Laplacian or Gaussian law. This statistical framework is used to provide the motion detection boundaries. Additionally, the original frame is used to provide the moving object boundaries. Then, the detection and the tracking problem are addressed in a common framework that employs a geodesic active contour objective function. This function is minimized using a gradient descent method, where a flow deforms the initial curve towards the minimum of the objective function, under the influence of internal and external image dependent forces. Using the level set formulation scheme, complex curves can be detected and tracked while topological changes for the evolving curves are naturally managed. To reduce the computational cost required by a direct implementation of the level set formulation scheme, a new approach named Hermes is proposed. Hermes exploits aspects from the well-known front propagation algorithms (Narrow Band, Fast Marching) and compares favorably to them. Very promising experimental results are provided using real video sequences. Index Terms--Front propagation, geodesic active contours, level set theory, motion detection, tracking.