This paper deals with the computation of sectional curvature for the manifolds of N landmarks (or feature points) in D dimensions, endowed with the Riemannian metric induced by the group action of diffeomorphisms. The inverse of the metric tensor for these manifolds (i.e., the cometric), when written in coordinates, is such that each of its elements depends on at most $2D$ of the $ND$ coordinates. This makes the matrices of partial derivatives of the cometric very sparse in nature, thus suggesting solving the highly nontrivial problem of developing a formula that expresses sectional curvature in terms of the cometric and its first and second partial derivatives (we call this Mario's formula). We apply such a formula to the manifolds of landmarks, and in particular we fully explore the case of geodesics on which only two points have nonzero momenta and compute the sectional curvatures of 2-planes spanned by the tangents to such geodesics. The latter example gives insight into the geometry of the full manifolds of landmarks.

The statistics of range images from natural environments is a largely unexplored field of research. It closely relates to the statistical modeling of the scene geometry in natural environments, and the modeling of optical natural images. We have used a 3D laser range-finder to collect range images from mixed forest scenes. The images are here analyzed with respect to different statistics.

Large calibrated datasets of 'random' natural images have recently become available. These make possible precise and intensive statistical studies of the local nature of images. We report results ranging from the simplest single pixel intensity to joint distribution of 3 Haar wavelet responses. Some of these statistics shed light on old issues such as the near scale-invariance of image statistics and some are entirely new. We fit mathematical models to some of the statistics and explain others in terms of local image features.

Recently there have been increasing interests in using nonlinear PDEs for applications in computer vision and image processing. In this paper, we propose a general statistical framework for designing a new class of PDEs. For a given application, a Markov random field model p(I) is learned according to the minimax entropy principle so that p(I) should characterize the ensemble of images in our application. P(I) is a Gibbs distribution whose energy terms can be divided into two categories. Subsequently the partial differential equations given by gradient descent on the Gibbs potential are essentially reaction-diffusion equations, where the energy terms in one category produce anisotropic diffusion while the inverted energy terms in the second category produce reaction associated with pattern formation. We call this new class of PDEs the Gibbs Reaction And Diffusion Equations-GRADE and we demonstrate experiments where GRADE are used for texture pattern formation, denoising, image enhancement, and clutter removal.

This paper presents a novel theory for learning generic prior models from a set of observed natural images based on a minimax entropy theory that the authors studied in modeling textures. We start by studying the statistics of natural images including the scale invariant properties, then generic prior models were learnt to duplicate the observed statistics. The learned Gibbs distributions confirm and improve the forms of existing prior models. More interestingly inverted potentials are found to be necessary, and such potentials form patterns and enhance preferred image features. The learned model is compared with existing prior models in experiments of image restoration.

This article proposes a general theory and methodology, called the minimax entropy principle, for building statistical models for images (or signals) in a variety of applications. This principle consists of two parts. The first is the maximum entropy principle for feature binding (or fusion): for a given set of observed feature statistics, a distribution can be built to bind these feature statistics together by maximizing the entropy over all distributions that reproduce them. The second part is the minimum entropy principle for feature selection: among all plausible sets of feature statistics, we choose the set whose maximum entropy distribution has the minimum entropy. Computational and inferential issues in both parts are addressed; in particular, a feature pursuit procedure is proposed for approximately selecting the optimal set of features. The minimax entropy principle is then corrected by considering the sample variation in the observed feature statistics, and an information criterion for feature pursuit is derived. The minimax entropy principle is applied to texture modeling, where a novel Markov random field (MRF) model, called FRAME (filter, random field, and minimax entropy), is derived, and encouraging results are obtained in experiments on a variety of texture images. The relationship between our theory and the mechanisms of neural computation is also discussed.

This article addresses two important themes in early visual computation: it presents a novel theory for learning the universal statistics of natural images, and, it proposes a general framework of designing reaction-diffusion equations for image processing. We studied the statistics of natural images including the scale invariant properties, then generic prior models were learned to duplicate the observed statistics, based on minimax entropy theory. The resulting Gibbs distributions have potentials of the form U(I; /spl Lambda/, S)=/spl Sigma//sub /spl alpha/=1//sup k//spl Sigma//sub x,y//spl lambda//sup (/spl alpha/)/((F/sup (/spl alpha/)/*I)(x,y)) with S={F/sup (1)/, F/sup (2)/,...,F/sup (K)/} being a set of filters and /spl Lambda/={/spl lambda//sup (1)/(),/spl lambda//sup (2)/(),...,/spl lambda//sup (K)/()} the potential functions. The learned Gibbs distributions confirm and improve the form of existing prior models such as line-process, but, in contrast to all previous models, inverted potentials were found to be necessary. We find that the partial differential equations given by gradient descent on U(I; /spl Lambda/, S) are essentially reaction-diffusion equations, where the usual energy terms produce anisotropic diffusion, while the inverted energy terms produce reaction associated with pattern formation, enhancing preferred image features. We illustrate how these models can be used for texture pattern rendering, denoising, image enhancement, and clutter removal by careful choice of both prior and data models of this type, incorporating the appropriate features.

In this paper, a minimax entropy principle is studied, based on which a novel theory, called FRAME (Filters, Random fields And Minimax Entropy) is proposed for texture modeling. FRAME combines attractive aspects of two important themes in texture modeling: multi-channel filtering and Markov random field (MRF) modeling. It incorporates the responses of a set of well selected filters into the distribution over a random field and hence has a much stronger descriptive ability than the traditional MRF models. Furthermore, it interprets and clarifies many previous concepts and methods for texture analysis and synthesis from a unified point of view. Algorithms are proposed for probability inference, stochastic simulation and filter selection. Experiments on a variety of textures are described to illustrate our theory and to show the performance of our algorithms. These experiments demonstrate that many textures previously considered as different categories can be modeled and synthesized in a common framework.

A half-occluded region in a stereo pair is a set of pixels in one image representing points in space visible to that camera or eye only, and not to the other. These occur typically as parts of the background immediately to the left and right sides of nearby occluding objects, and are present in most natural scenes. Previous approaches to stereo either ignored these unmatchable points or attempted to weed them out in a second pass. An algorithm that incorporates them from the start as a strong clue to depth discontinuities is presented. The authors first derive a measure for goodness of fit and a prior based on a simplified model of objects in space, which leads to an energy functional depending both on the depth as measured from a central cyclopean eye and on the regions of points occluded from the left and right eye perspectives. They minimize this using dynamic programming along epipolar lines followed by annealing in both dimensions. Experiments indicate that this method is very effective even in difficult scenes.>

A model is described for image segmentation that tries to capture the low-level depth reconstruction exhibited in early human vision, giving an important role to edge terminations. The problem is to find a decomposition of the domain D of an image that has a minimum of disrupted edges-junctions of edges, crack tips, corners, and cusps-by creating suitable continuations for the disrupted edges behind occluding regions. The result is a decomposition of D into overlapping regions R/sub 1/ union . . . union R/sub n/ ordered by occlusion, which is called the 2.1-D sketch. Expressed as a minimization problem, the model gives rise to a family of optimal contours, called nonlinear splines, that minimize length and the square of curvature. These are essential in the construction of the 2.1-D sketch of an image, as the continuations of disrupted edges. An algorithm is described that constructs the 2.1-D sketch of an image, and gives results for several example images. The algorithm yields the same interpretations of optical illusions as the human visual system.>