A novel scheme for the detection of object boundaries is presented. The technique is based on active contours evolving in time according to intrinsic geometric measures of the image. The evolving contours naturally split and merge, allowing the simultaneous detection of several objects and both interior and exterior boundaries. The proposed approach is based on the relation between active contours and the computation of geodesics or minimal distance curves. The minimal distance curve lays in a Riemannian space whose metric is defined by the image content. This geodesic approach for object segmentation allows to connect classical "snakes" based on energy minimization and geometric active contours based on the theory of curve evolution. Previous models of geometric active contours are improved, allowing stable boundary detection when their gradients suffer from large variations, including gaps. Formal results concerning existence, uniqueness, stability, and correctness of the evolution are presented as well. The scheme was implemented using an efficient algorithm for curve evolution. Experimental results of applying the scheme to real images including objects with holes and medical data imagery demonstrate its power. The results may be extended to 3D object segmentation as well.

. We present a new method for solving total variation (TV) minimization problems in image restoration. The main idea is to remove some of the singularity caused by the non-differentiability of the quantity jruj in the definition of the TV-norm before we apply a linearization technique such as Newton's method. This is accomplished by introducing an additional variable for the flux quantity appearing in the gradient of the objective function. Our method can be viewed as a primal-dual method as proposed by Conn and Overton [8] and Andersen [3] for the minimization of a sum of Euclidean norms. Experimental results show that the new method has much improved global convergence behaviour than the primal Newton's method. 1. Introduction. During some phases of the manipulation of an image some random noise and blurring is usually introduced. The presence of this noise and blurring makes difficult and inaccurate the latter phases of the image processing. The algorithms for noise removal and debl...

. The study of geometric flows for smoothing, multiscale representation, and analysis of two- and three-dimensional objects has received much attention in the past few years. In this paper, we first survey the geometric smoothing of curves and surfaces via geometric heat-type flows, which are invariant under the groups of Euclidean and affine motions. Second, using the general theory of differential invariants, we determine the general formula for a geometric hypersurface evolution which is invariant under a prescribed symmetry group. As an application, we present the simplest affine invariant flow for (convex) surfaces in three-dimensional space, which, like the affine-invariant curve shortening flow, will be of fundamental importance in the processing of three-dimensional images. Key words. invariant surface evolutions, partial differential equations, geometric smoothing, symmetry groups AMS subject classifications. 35K22, 53A15, 53A55, 53A20, 35B99 PII. S0036139994266311 1. Intro...

A novel approach for the computation of optical flow based on an L type minimization is presented. It is shown that the approach has inherent advantages since it does not smooth the flow-velocity across the edges and hence preserves edge information. A numerical approach based on computation of evolving curves is proposed for computing the optical flow field. Computations are carried out on a number of synthetic and real image sequences in order to illustrate the theory as well as the numerical approach.

In spite of its lack of theoretical justification, nonlinear diffusion filtering has become a powerful image enhancement tool in recent years. The goal of the present paper is to provide a mathematical foundation for continuous nonlinear diffusion filtering as a scale-space transformation which is flexible enough to simplify images without loosing the capability of enhancing edges. By studying the Lyapunov functionals, it is shown that nonlinear diffusion reduces L p norms and central moments and increases the entropy of images. The proposed anisotropic class utilizes a diffusion tensor which may be adapted to the image structure. It permits existence, uniqueness and regularity results, the solution depends continuously on the initial image, and it satisfies an extremum principle. All considerations include linear and certain nonlinear isotropic models and apply to m- dimensional vector-valued images. The results are juxtaposed to linear and morphological scale-spaces. . Keywords....

Abstract—For shapes represented as closed planar contours, we introduce a class of functionals which are invariant with respect to the Euclidean group and which are obtained by performing integral operations. While such integral invariants enjoy some of the desirable properties of their differential counterparts, such as locality of computation (which allows matching under occlusions) and uniqueness of representation (asymptotically), they do not exhibit the noise sensitivity associated with differential quantities and, therefore, do not require presmoothing of the input shape. Our formulation allows the analysis of shapes at multiple scales. Based on integral invariants, we define a notion of distance between shapes. The proposed distance measure can be computed efficiently and allows warping the shape boundaries onto each other; its computation results in optimal point correspondence as an intermediate step. Numerical results on shape matching demonstrate that this framework can match shapes despite the deformation of subparts, missing parts and noise. As a quantitative analysis, we report matching scores for shape retrieval from a database. Index Terms—Integral invariants, shape, shape matching, shape distance, shape retrieval. 1

We accept this thesis as conforming to the required standard

In the past several years, there has been much research devoted to the study of evolutions of plane curves where the velocity of the evolving curve is given by the Euclidean curvature vector. This evolution appears in a number of different

— Article to be published in Computer Aided Geometric Design — We describe a method for multiresolution deformation of closed planar curves that keeps the enclosed area constant. We use a wavelet based multiresolution representation of the curves that are represented by a finite number of control points at each level of resolution. A deformation can then be applied to the curve by modifying one or more control points at any level of resolution. This process is generally known as multiresolution editing to which we add the constraint of constant area. The key contribution of this paper is the efficient computation of the area in the wavelet decomposition form: the area is expressed through all levels of resolution as a bilinear form of the coarse and detail coefficients, and recursive formulas are developed to compute the matrix of this bilinear form. A similar result is also given for the bending energy of the curve. The area constraint is maintained through an optimization process. These contributions allow a real time multiresolution deformation with area constraint of complex curves.

This paper draws on results from Computational Geometry, Differential Geometry, and recent theories on shape representation by the Medial Axis Transform (MAT) in order to introduce a novel technique of multi-scale skeletonization. To this end first the continuous medial axis transform is approximated by the Voronoi tessellation (VT) of the boundary points of a shape. Mapping features of the object boundary evolving under the heat equation onto the VT establishes the so-called skeleton-space, a shape description which combines contour characteristics with region information furnished by the tessellation. The skeleton-space is capable of describing complex shapes characterized by significantly jagged boundaries. In addition, tracking the evolution of characteristic loci of the MAT such as skeleton nodes over varying scale permits to assess the most significant skeleton constituents and to automatically determine feasible scale and pruning parameters. Submitted for publication in the IEEE...