Some known models in phase separation theory (Hele-Shaw, nonlocal mean curvature motion) and their approximations by means of Cahn-Hilliard and nonlocal Allen-Cahn equations are proposed as a tool to generate planar curve-shortening flows without shrinking. This procedure can be seen as a level set approach to area-preserving geometric flows in the spirit of Sapiro and Tannenbaum [38], with application to shape recovery. We discuss the theoretical validation of this method and its implementation to problems of shape recovery in Computer Vision. The results of some numerical experiments on image processing are presented.

We study some numerical properties of a nonconvex variational problem which arises as the continuous limit of a discrete optimization method designed for the smoothing of images with preservation of discontinuities. The functional that has to be minimized fails to attain a minimum value. Instead, minimizing sequences develop gradient oscillations which allow them to reduce the value of the functional. The oscillations of the gradient exhibit analogies with microstructures in ordered materials. The pattern of the oscillations is analysed numerically by using discrete parametrized measures.

The problem of reconstructing a piecewise constant function from a finite number of its Fourier coefficients perturbed by noise is considered. A reconstruction method, based on the computation of the Padè approximants to the Z-transform of the sequence of the noisy Fourier coefficients is proposed. The method is based on the remark that the distribution of the poles of the Padè approximants shows, asymptotically, clusters in the complex plane which allow the identification of the discontinuities of the function. It turns out that the Z-transform is a multiple-valued function and the location of the clusters corresponds to the branch points of such a function. By using this property of the Padè poles, a very effective reconstruction method can be developed. Some numerical experiments are presented to show the feasibility of the method.

We consider a geometric motion associated with the minimization of a curvature dependent functional, which is related to the Willmore functional. Such a functional arises in connection with the image segmentation problem in computer vision theory. We show by using formal asymptotics that the geometric motion can be approximated by the evolution of the zero level set of the solution of a nonlinear fourth-order equation related to the Cahn-Hilliard and Allen-Cahn equations.

This paper deals with finite-difference approximations of Euler equations arising in the variational formulation of image segmentation problems. We illustrate how they can be defined by the following steps: (a) definition of the minimization problem for the Mumford-Shah functional (MSf), (b) definition of a sequence of functionals Gamma-convergent to the MSf, and (c) definition and numerical solution of the Euler equations associated to the k-th functional of the sequence. We define finite difference approximations of the Euler equations, the related solution algorithms, and we present applications to segmentation problems by using synthetic images. We discuss application results, and we mainly analyze computed discontinuity contours and convergence histories of method executions.

The noisy trigonometric moment problem for a finite linear combination of box functions is considered, and a research program, possibly leading to a superresolving method, is outlined and some initial steps are performed. The method is based on the remark that the poles of the Padè approximant to the Z-transform of the noiseless moments show, asymptotically, a regular pattern in the complex plane. The pattern can be described by a set of arcs, connecting points on the unit circle, and a pole density function defined on the arcs. When a moderate noise affects the moments, more arcs are needed to describe the pole pattern, but the noiseless pattern is slightly deformed, still allowing its identification. When this identification is possible, a very effective noise filter and moment extrapolator should be easily constructed. In this paper only some preliminary steps of the above research program are performed. Specifically, the case of one box function is considered. A method for computing the pole patterns, based on the solution of a singular integral equation of Cauchy type, is developed. The method is general enough to be used also for several box functions. Some numerical results, showing the feasibility of the program, are discussed.

In this paper, the modal analysis model, made up by a linear combination of complex exponential functions, is considered. Padè approximants to the Z-transform of a noisy sample are then considered, and the asymptotic locus of their poles is studied. It turns out that this locus is strongly related to the complex exponentials of the model. By exploiting these properties, powerful methods for estimating the model parameters can be devised, which have both denoising and super-resolution capabilities.

We study some properties of a nonconvex variational problem. The infimum of the functional that has to be minimized fails to be attained. Instead, minimizing sequences develop gradient oscillations which allow them to decrease the value of the functional. We show an existence result for a perturbed nonconvex version of the problem, and we study the qualitative properties of the corresponding minimizer. The pattern of the gradient oscillations for the original non perturbed problem is analyzed numerically.

Variational methods for image segmentation try to recover a piecewise smooth function together with a discontinuity set which represents the boundaries of the segmentation. This paper deals with a variational method that constrains the formation of discontinuities along smooth contours. The functional to be minimized, which involves the computation of the geometrical properties of the boundaries, is approximated by a sequence of functionals which can be discretized in a straightforward way. Computer examples of real images are presented to illustrate the feasibility of the method.

Visual reconstruction problems tend to be mathematically ill-posed. They can be reformulated as well-posed variational problems using regularization theory. A generalization of the standard regularization method to visual reconstruction with discontinuities leads to variational problems which include the discontinuity contours in their unknowns. The minimization of the corresponding functionals is a difficult problem. This paper suggests the use of the ?-convergence theory to approximate the functional to be minimized by elliptic functionals, which are more tractable. A ?-convergence theorem which is of relevance to vision applications is discussed, and the results of computer experiments with both synthetic and real images are shown.

The problem of the computation of stereo disparity is approaehed as a mathematically ill-posed problem by using regularization theory. A controlled continuity constraint which provides a local spatial control over the smoothness of the solution enables the problem to be regularized while preserving the disparity discontinuities. The discontinuities are localized during the regularization process by examining the size of the disparity gradient at the gray value edges. An iterative algorithm for the computation of stereo disparity is obtained, and a computer experiment with synthetic data is shown.

The problem of the computation of stereo disparity is approached as a mathematically ill-posed problem by using regularization theory. A controlled continuity constrant which provides a local spatial control over the smoothness of the solution enables the problem to be regularized while preserving the disparity discontinuities. The discontinuities are localized during the regularization process by examining the size of the disparity gradient at the gray value edges. An iterative algorithm for the computation of stereo disparity is obtained, and a computer experiment with synthetic data is shown.

The problem of computing depth from stereo images is approached as a mathematically ill-posed problem by using regularization theory. A variational principle for the reconstruction of surfaces in the presence of depth discontinuities is presented. Discontinuities are preserved using a controlled-continuity stabilizing functional which provides a local control over the smoothness properties of the solution. A discontinuity stabilizing functional imposes a curvilinear smoothness constraint on discontinuities to enable their reconstruction. The location of depth discontinuities is further restricted by using information from intensity edges. An iterative optimization method for the computation of depth is obtained, and a computer experiment with synthetic data is shown.

The computation of stereo disparity is a mathematical ill-posed problem. However, using regularization theory it may be transformed into a well-posed problem. Standard regularization can be to solve ill-posed problems by using stabilizing functionals that impose global smoothness constraints on acceptable solutions. However, the presence of depth discontinuities causes serious difficulties in standard regularizations, since smoothness assumptions do not hold across discontinuities. This paper presents a regularization approach to stereopsis based on controlled-continuity stabilizing functionals. These functionals provide a spatial control over smoothness, allowing the introduction of discontinuities into the solution.An iterative method for the computation of stereo disparity is derived, and the result of a computer simulation with a synthetic stereo pair of image is shown.

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