implementation of an adaptive finite element approximation of the mumford shah functional we present and detail a method for the numerical solving of the mumford shah problem based on a finite element method and on adaptive meshes we start with the formulation introduced in 13 detail its numerical implementation and then propose a variant which is proved to converge to the mumford shah problem a few experiments are illustrated

scale space and edge detection using anisotropic diffusion a new definition of scale space is suggested and a class of algorithms used to realize a diffusion process is introduced the diffusion coefficient is chosen to vary spatially in such a way as to encourage intraregion smoothing rather than interregion smoothing it is shown that the no new maxima should be generated at coarse scales property of conventional scale space is preserved as the region boundaries in the approach remain sharp a high quality edge detector which successfully exploits global information is obtained experimental results are shown on a number of images parallel hardware implementations are made feasible because the algorithm involves elementary local operations replicated over the image

solution of the stochastic radiative transfer equation with rayleigh scattering using rvt technique abstract this paper considers the solution of the radiative transfer equation in a semi infinite continuous stochastic medium with rayleigh scattering the available solutions in literature for this stochastic integro differential equation side are represented only by the ensemble averaged solutions here the deterministic solution of the problem is obtained at first then the random variable transformation rvt technique together with an integral transformation to the input stochastic process is implemented to get the complete stochastic solution of the problem this solution is represented by the probability density function of the solution process moreover the nth statistical moment of the solution is derived numerical results are given for exponential and gaussian distributions of the input stochastic process also these results are given for different boundary conditions

solution of the stochastic transport equation of neutral particles with anisotropic scattering using rvt technique much of the literatures are directed toward the development of a mathematical formalism for a rigorous estimation of the ensemble average of the solution process of a stochastic differential equation sde the random variable transformation technique rvt is a powerful technique to get the complete solution for the sde represented by the probability density function of the solution process in this paper the rvt technique together with a simple integral transformation to the input stochastic process are implemented to get the complete solution of the one speed transport equation for neutral particles in a semi infinite stochastic medium with linear anisotropic scattering the extinction function of the medium input stochastic process is assumed to be a continuous random function of position the probability density function and hence the higher order statistical moments of the solution process are presented numerical results are given for different distributions of the input stochastic process

piecewise linear approximation of vector valued images and curves via second order variational model variational models are known to work well for addressing image restoration regularization problems however most of the methods proposed in the literature are defined for scalar inputs and are used on multiband images such as rgb or multispectral imagery by the composition of a simple band wise processing this involves suboptimal results and may introduce artifacts only in a few cases variational models are extended to the case of vector valued inputs however the known implementations are restricted to the first order models while the second order models are never considered thus typical problems of the first order models such as the staircasing effect cannot be overtaken this paper considers a second order functional model to function approximation with free discontinuities given by blake zisserman bz and proposes an efficient minimization algorithm in the case of vector valued inputs in the bz model the hessian of the solution is penalized outside a set of finite length therefore the solution is forced to be piecewise linear moreover the model allows the formation of free discontinuities and free gradient discontinuities the proposed algorithm is applied to difficult color image restoration regularization problems and to piecewise linear approximation of curves in space

a proposed technique of sfem on solving ordinary random differential equation the stochastic finite element sfem is a new method to solve random differential equation using the known deterministic finite element technique fem adapted to stochastic techniques for solving stochastic differential equations in this paper a proposed technique of fem and random variable transformation rvt is used to solve a differential equation with random excitation the technique shows high accuracy when solving a case study compared with the exact solution

a developed solution of the stochastic milne problem using probabilistic transformations abstract this paper considers the solution of the milne problem of radiative transfer with isotropic scattering in a continuous stochastic medium properties of the medium are assumed to be continuous random functions of the spatial dimensions the available solutions in literature for this stochastic integro differential equation side are represented only by the ensemble average of the radiant energy density in this paper a developed algorithm based on the implementation of the random variable transformation technique together with an integral transformation to the stochastic properties is introduced a complete stochastic solution represented by the probability density function p d f of the radiant energy density is obtained using the closed form of the p d f the nth moment of the stochastic solution is evaluated in realization of this work exponential and gaussian statistics for the medium properties are assumed results are physically acceptable and found to be compatible with those in the literature

an approximate probabilistic solution of a random sir type epidemiological model using rvt technique abstract this paper adapts a multi dimensional random variable transformation technique to derive a comprehensive stochastic description of the susceptible infected recovered epidemiological model an approximate solution of a system of nonlinear differential equations that characterizes this model is deterministically obtained and from this approximation we derive the first probability density functions for the solution processes of susceptible infected and recovered percentages these probability density functions are used to find the approximate mean and variance functions as well as the confidence intervals taking a general situation the infection contact rate the recovery rate and the initial conditions are taken to be random variables with arbitrary distributions to test the validity of the theoretical findings associated to the proposed random epidemiological model some numerical results are tabulated and graphically presented through an illustrative example

the mathematics of infectious diseases many models for the spread of infectious diseases in populations have been analyzed mathematically and applied to specific diseases threshold theorems involving the basic reproduction number r 0 the contact number sigma and the replacement number r are reviewed for the classic sir epidemic and endemic models similar results with new expressions for r 0 are obtained for mseir and seir endemic models with either continuous age or age groups values of r 0 and sigma are estimated for various diseases including measles in niger and pertussis in the united states previous models with age structure heterogeneity and spatial structure are surveyed

anisotropic diffusion of multivalued images with applications to color filtering a general framework for anisotropic diffusion of multivalued images is presented we propose an evolution equation where at each point in time the directions and magnitudes of the maximal and minimal rate of change in the vector image are first evaluated these are given by eigenvectors and eigenvalues of the first fundamental form in the given image metric then the image diffuses via a system of coupled differential equations in the direction of minimal change the diffusion strength is controlled by a function that measures the degree of dissimilarity between the eigenvalues we apply the proposed framework to the filtering of color images represented in cie l a b space

fourth order partial differential equations for noise removal a class of fourth order partial differential equations pdes are proposed to optimize the trade off between noise removal and edge preservation the time evolution of these pdes seeks to minimize a cost functional which is an increasing function of the absolute value of the laplacian of the image intensity function since the laplacian of an image at a pixel is zero if the image is planar in its neighborhood these pdes attempt to remove noise and preserve edges by approximating an observed image with a piecewise planar image piecewise planar images look more natural than step images which anisotropic diffusion second order pdes uses to approximate an observed image so the proposed pdes are able to avoid the blocky effects widely seen in images processed by anisotropic diffusion while achieving the degree of noise removal and edge preservation comparable to anisotropic diffusion although both approaches seem to be comparable in removing speckles in the observed images speckles are more visible in images processed by the proposed pdes because piecewise planar images are less likely to mask speckles than step images and anisotropic diffusion tends to generate multiple false edges speckles can be easily removed by simple algorithms such as the one presented in this paper

using fem rvt technique for solving a randomly excited ordinary differential equation with a random operator the technique of stochastic finite element sfem which is the finite element technique fem adapted to stochastic problems can be re described to use random variable transformation technique rvt a new fem rvt technique was successfully used in solving stochastic problems with random excitation m el tawil w el tahhan a hussein a proposed technique of sfem on solving ordinary random differential equation j appl math comput 161 2005 35 47 in this paper the technique is adapted to solve a randomly excited differential equation with a random operator the technique shows high accuracy when solving a case study compared with the exact solution finally a problem with unknown exact solution is solved using this technique

existence theorem for a minimum problem with free discontinuity set we study the variational problem where is an open set in n n 2g l q l 1 q o andh n 1 is the n 1 dimensional hausdorff measure

optimal approximations by piecewise smooth functions and associated variational problems abstract this reprint will introduce and study the most basic properties of three new variational problems which are suggested by applications to computer vision in computer vision a fundamental problem is to appropriately decompose the domain r of a function g x y of two variables this problem starts by describing the physical situation which produces images assume that a three dimensional world is observed by an eye or camera from some point p and that g1 rho represents the intensity of the light in this world approaching the point sub 1 from a direction rho if one has a lens at p focusing this light on a retina or a film in both cases a plane domain r in which we may introduce coordinates x y then let g x y be the strength of the light signal striking r at a point with coordinates x y g x y is essentially the same as sub 1 rho possibly after a simple transformation given by the geometry of the imaging syste the function g x y defined on the plane domain r will be called an image what sort of function is g the light reflected off the surfaces si of various solid objects o sub i visible from p will strike the domain r in various open subsets r sub i when one object o1 is partially in front of another object o2 as seen from p but some of object o2 appears as the background to the sides of o1 then the open sets r1 and r2 will have a common boundary the edge of object o1 in the image defined on r and one usually expects the image g x y to be discontinuous along this boundary jhd

variational segmentation of vector valued images with gradient vector flow in this paper we extend the gradient vector flow field for robust variational segmentation of vector valued images rather than using scalar edge information we define a vectorial edge map derived from a weighted local structure tensor of the image that enables the diffusion of the gradient vectors in accurate directions through the 4d gradient vector flow equation to reduce the contribution of noise in the structure tensor image channels are weighted according to a blind estimator of contrast the method is applied to biological volume delineation in dynamic pet imaging and validated on realistic monte carlo simulations of numerical phantoms as well as on real images

applications of vector fields to image processing we use rotational and curvature properties of vector fields to identify critical features of an image using vector analysis and dif ferential geometry we establish the properties needed and then use these properties in three ways first our results make it theoretically possible to identify extremal edges of an intensity function f x y of two variables by considering the gradient vector field v f there is also enough information in f to find regions of high curvature i e high curvature of the level paths of f for color images we use the vector field v i q in application the image is partitioned into a grid of squares on the boundary of each square v v is sampled and these unit vectors are used as the tangents of a curve the rotation number or topological degree and the average curvature f are computed for each square analysis of these numbers yields infor mation on edges and curvature experimental results from both simu lated and real data are described

approximation of functional depending on jumps by elliptic functional via t convergence we show how it is possible to approximate the mumford shah see 29 image segmentation functional by elliptic functionals defined on sobolev spaces the heuristic idea is to consider functionals h u z with z ranging between 0 and 1 and related to the set k the minimizing zh are near to 1 in a neighborhood of the set k and far from the neighborhood they are very small the neighborhood shrinks as h for a similar approach to the problem compare kulkarni see 25 the approximation of h to takes place in a variational sense the de giorgi f convergence

smoothing for signals with discontinuities using higher order mumford shah models minimizing the mumford shah functional is frequently used for smoothing signals or time series with discontinuities a significant limitation of the standard mumford shah model is that linear trends and in general polynomial trends in the data are not well preserved this can be improved by building on splines of higher order which leads to higher order mumford shah models in this work we study these models in the univariate situation we discuss important differences to the first order mumford shah model and we obtain uniqueness results for their solutions as a main contribution we derive fast minimization algorithms for mumford shah models of arbitrary orders we show that the worst case complexity of all proposed schemes is quadratic in the length of the signal remarkably they thus achieve the worst case complexity of the fastest solver for the piecewise constant mumford shah model which is the simplest model of the class further we obtain stability results for the proposed algorithms we complement these results with a numerical study our reference implementation processes signals with more than 10 000 elements in less than 1 s

random variable transformation for generalized stochastic radiative transfer in finite participating slab media abstract the stochastic radiative transfer problem is studied in a participating planar finite continuously fluctuating medium the problem is considered for specular and diffusly reflecting boundaries with linear anisotropic scattering random variable transformation rvt technique is used to get the complete average for the solution functions that are represented by the probability density function pdf of the solution process in the rvt algorithm a simple integral transformation to the input stochastic process the extinction function of the medium is applied this linear transformation enables us to rewrite the stochastic transport equations in terms of the optical random variable x and the optical random thickness l then the transport equation is solved deterministically to get a closed form for the solution as a function of x and l so the solution is used to obtain the pdf of the solution functions applying the rvt technique among the input random variable l and the output process the solution functions the obtained averages of the solution functions are used to get the complete analytical averages for some interesting physical quantities namely reflectivity and transmissivity at the medium boundaries in terms of the average reflectivity and transmissivity the average of the partial heat fluxes for the generalized problem with internal source of radiation are obtained and represented graphically

subsets of r which support hypergroups with polynomial characters abstract it is shown that if a hypergroup h with h an infinite subset of r has polynomial characters of every degree and if either h is compact or the polynomials are orthogonal with respect to a measure supported on h then those polynomials are essentially the jacobi polynomials related results are obtained for polynomial product formulas