euler s elastica and curvature based inpainting image inpainting is a special image restoration problem for which image prior models play a crucial role euler s elastica was first introduced to computer vision by mumford algebraic geometry and its applications springer verlag new york 1994 pp 491 506 as a curve prior model by functionalizing the elastica energy masnou and morel proceedings of the 5th ieee international conference image processing 3 1998 pp 259 263 proposed an elastica based variational inpainting model the current paper is intended to contribute to the development of its mathematical foundation and the study of its properties and connections to the earlier works of bertalmio sapiro caselles and ballester siggraph 2000 acm press new york 2000 and chan and shen j visual comm image rep 12 2001 pp 436 449 a computational scheme based on numerical pdes is presented which allows the automatic handling of topologically complex inpainting domains

difference approximations for the second order wave equation difference approximations are derived for the second order wave equation in one and two space dimensions without first writing it as a first order system both the dirichlet and the neumann problems are treated for the one dimensional case relations between the boundary error and the interior phase error are derived for a fully second order accurate discretization as well as a scheme that is fourth order accurate in the interior and second order accurate at the boundary general two dimensional domains are considered for the dirichlet problem where the domain is embedded in a cartesian grid and the boundary conditions are approximated by interpolation a stable conservative scheme is derived where the time step is determined only by the interior discretization formula discretization cells cut by the boundary are treated implicitly but the resulting scheme becomes explicit because the implicit dependence only is pointwise numerical examples are provided to verify the stability and accuracy of the proposed method

geometry based demosaicking demosaicking is a particular case of interpolation problems where from a scalar image in which each pixel has either the red the green or the blue component we want to interpolate the full color image state of the art demosaicking algorithms perform interpolation along edges but these edges are estimated locally we propose a level set based geometric method to estimate image edges inspired by the image in painting literature this method has a time complexity of o s where s is the number of pixels in the image and compares favorably with the state of the art algorithms both visually and in most relevant image quality measures

variational exemplar based image colorization in this paper we address the problem of recovering a color image from a grayscale one the input color data comes from a source image considered as a reference image reconstructing the missing color of a grayscale pixel is here viewed as the problem of automatically selecting the best color among a set of color candidates while simultaneously ensuring the local spatial coherency of the reconstructed color information to solve this problem we propose a variational approach where a specific energy is designed to model the color selection and the spatial constraint problems simultaneously the contributions of this paper are twofold first we introduce a variational formulation modeling the color selection problem under spatial constraints and propose a minimization scheme which computes a local minima of the defined nonconvex energy second we combine different patch based features and distances in order to construct a consistent set of possible color candidates this set is used as input data and our energy minimization automatically selects the best color to transfer for each pixel of the grayscale image finally the experiments illustrate the potentiality of our simple methodology and show that our results are very competitive with respect to the state of the art methods

an explicit fourth order orthogonal curvilinear staggered grid fdtd method for maxwell s equations the explicit fourth order staggered finite difference time domain scheme previously proposed for a cartesian grid is extended to maxwell s equations in an orthogonal curvilinear coordinate system and applied to electromagnetic wave problems a simple technique is also presented for generating orthogonal curvilinear grids that conform to the material boundaries and interfaces of the problem numerical experiments are presented to illustrate the efficiency and accuracy of the method

harmonic maps with defects two problems concerning maps with point singularities from a domain c 3 tos2 are solved the first is to determine the minimum energy of when the location and topological degree of the singularities are prescribed in the second problem is the unit ball and g is given on we show that the only cases in whichg x x minimizes the energy isg const org x rx withr a rotation extensions of these problems are also solved e g points are replaced by holes 3 s2 is replaced by n sn 1 or by n pn 1 the latter being appropriate for the theory of liquid crystals

fast image and video colorization using chrominance blending colorization the task of coloring a grayscale image or video involves assigning from the single dimension of intensity or luminance a quantity that varies in three dimensions such as red green and blue channels mapping between intensity and color is therefore not unique and colorization is ambiguous in nature and requires some amount of human interaction or external information a computationally simple yet effective approach of colorization is presented in this paper the method is fast and it can be conveniently used on the fly permitting the user to interactively get the desired results promptly after providing a reduced set of chrominance scribbles based on the concepts of luminance weighted chrominance blending and fast intrinsic distance computations high quality colorization results for still images and video are obtained at a fraction of the complexity and computational cost of previously reported techniques possible extensions of the algorithm introduced here included the capability of changing the colors of an existing color image or video as well as changing the underlying luminance and many other special effects demonstrated here

restoration of color images by vector valued bv functions and variational calculus we analyze a variational problem for the recovery of vector valued functions and compute its numerical solution the data of the problem are a small set of complete samples of the vector valued function and some significant incomplete information where the former are missing the incomplete information is assumed as the result of a distortion with values in a lower dimen sional manifold for the recovery of the function we minimize a functional which is formed by the discrepancy with respect to the data and total variation regularization constraints we show the existence of minimizers in the space of vector valued bounded variation functions for the computa tion of minimizers we provide a stable and efficient method first we approximate the functional by coercive functionals on w 1 2 in terms of convergence then we realize approximations of minimiz ers of the latter functionals by an iterative procedure to solve the pde system of the corresponding euler lagrange equations the numerical implementation comes naturally by finite element dis cretization we apply the algorithm to the restoration of color images from limited color information and gray levels where the colors are missing the numerical experiments show that this scheme is very fast and robust the reconstruction capabilities of the model are shown also from very limited randomly distributed color data several examples are included from the real restoration problem of a mantegna s art frescoes in italy

orientation diffusions diffusions are useful for image processing and computer vision because they provide a convenient way of smoothing noisy data analyzing images at multiple scales and enhancing discontinuities a number of diffusions of image brightness have been defined and studied so far they may be applied to scalar and vector valued quantities that are naturally associated with intervals of either the real line or other flat manifolds some quantities of interest in computer vision and other areas of engineering that deal with images are defined on curved manifolds typical examples are orientation and hue that are defined on the circle generalizing brightness diffusions to orientation is not straightforward especially in the case where a discrete implementation is sought an example of what may go wrong is presented a method is proposed to define diffusions of orientation like quantities first a definition in the continuum is discussed then a discrete orientation diffusion is proposed the behavior of such diffusions is explored both analytically and experimentally it is shown how such orientation diffusions contain a nonlinearity that is reminiscent of edge process and anisotropic diffusion a number of open questions are proposed

the stability of numerical boundary treatments for compact high order finite difference schemes the stability characteristics of various compact fourth and sixth order spatial operators are assessed with the theory of gustafsson kreiss and sundstrom g k s for the semidiscrete initial boundary value problem these results are generalized to the fully discrete case with a recently developed theory of kreiss and wu in all cases favorable comparisons are obtained between g k s theory eigenvalue determination and numerical simulation the conventional definition of stability then is sharpened to include only those spatial discretizations that are asymptotically stable bounded left half plane eigenvalues many of the higher order schemes that are g k s stable are not asymptotically stable a series of compact fourth and sixth order schemes that are both asymptotically and g k s stable for the scalar case are then developed

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

diffusion of general data on non flat manifolds viaharmonic maps theory the direction diffusion case in a number of disciplines directional data provides a fundamental source of information a novel framework for isotropic and anisotropic diffusion of directions is presented in this paper the framework can be applied both to denoise directional data and to obtain multiscale representations of it the basic idea is to apply and extend results from the theory of harmonic maps and in particular harmonic maps in liquid crystals this theory deals with the regularization of vectorial data while satisfying the intrinsic unit norm constraint of directional data we show the corresponding variational and partial differential equations formulations for isotropic diffusion obtained from an l 2 norm and edge preserving diffusion obtained from an l norm in general and an l 1 norm in particular in contrast with previous approaches the framework is valid for directions in any dimensions supports non smooth data and gives both isotropic and anisotropic formulations in addition the framework of harmonic maps here described can be used to diffuse and analyze general image data defined on general non flat manifolds that is functions between two general manifolds we present a number of theoretical results open questions and examples for gradient vectors optical flow and color images

color image enhancement via chromaticity diffusion a novel approach for color image denoising is proposed in this paper the algorithm is based on separating the color data into chromaticity and brightness and then processing each one of these components with partial differential equations or diffusion flows in the proposed algorithm each color pixel is considered as an n dimensional vector the vectors direction a unit vector gives the chromaticity while the magnitude represents the pixel brightness the chromaticity is processed with a system of coupled diffusion equations adapted from the theory of harmonic maps in liquid crystals this theory deals with the regularization of vectorial data while satisfying the intrinsic unit norm constraint of directional data such as chromaticity both isotropic and anisotropic diffusion flows are presented for this n dimensional chromaticity diffusion flow the brightness is processed by a scalar median filter or any of the popular and well established anisotropic diffusion flows for scalar image enhancement we present the underlying theory a number of examples and briefly compare with the current literature

numerical methods for p harmonic flows and applications to image processing we propose in this paper an alternative approach for computing p harmonic maps and flows instead of solving a constrained minimization problem on sn 1 we solve an unconstrained minimization problem on the entire space of functions this is possible using the projection on the sphere of any arbitrary function then we show how this formulation can be used in practice for problems with both isotropic and anisotropic diffusion with applications to image processing using a new finite difference scheme

filling in by joint interpolation of vector fields and gray levels a variational approach for filling in regions of missing data in digital images is introduced the approach is based on joint interpolation of the image gray levels and gradient isophotes directions smoothly extending in an automatic fashion the isophote lines into the holes of missing data this interpolation is computed by solving the variational problem via its gradient descent flow which leads to a set of coupled second order partial differential equations one for the gray levels and one for the gradient orientations the process underlying this approach can be considered as an interpretation of the gestaltist s principle of good continuation no limitations are imposed on the topology of the holes and all regions of missing data can be simultaneously processed even if they are surrounded by completely different structures applications of this technique include the restoration of old photographs and removal of superimposed text like dates subtitles or publicity examples of these applications are given we conclude the paper with a number of theoretical results on the proposed variational approach and its corresponding gradient descent flow

dual norms and image decomposition models following a recent work by y meyer decomposition models into a geometrical component and a textured component have recently been proposed in image processing in such approaches negative sobolev norms have seemed to be useful to modelize oscillating patterns in this paper we compare the properties of various norms that are dual of sobolev or besov norms we then propose a decomposition model which splits an image into three components a first one containing the structure of the image a second one the texture of the image and a third one the noise our decomposition model relies on the use of three different semi norms the total variation for the geometrical component a negative sobolev norm for the texture and a negative besov norm for the noise we illustrate our study with numerical examples

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

a domain decomposition method for the acoustic wave equation with discontinuous coefficients and grid change a domain decomposition technique is proposed for the computation of the acoustic wave equation in which the bulk modulus and density fields are allowed to be discontinuous at the interfaces inside each subdomain the method presented coincides with the second order finite difference schemes traditionally used in geophysical modeling however the possibility of assigning to each subdomain its own space step makes numerical simulations much less expensive another interest of the method lies in the fact that its hybrid variational formulation naturally leads to exact equations for gridpoints on the interfaces transposing babuska brezzi s formalism on mixed and hybrid finite elements provides a suitable functional framework for this domain decomposition formulation and shows that the inf sup condition remains the basic requirement for convergence to occur

summation by parts projections and stability ii we have derived stability results for high order finite difference approximations of mixed hyperbolic parabolic initial boundary value problems ibvp the results are obtained using summation by parts and a new way of representing general linear boundary conditions as an orthogonal projection by slightly rearranging the analytic equations we can prove strict stability for hyperbolic parabolic ibvp furthermore we generalize our technique so as to yield strict stability on curvilinear non smooth domains in two space dimensions finally we show how to incorporate inhomogeneous boundary data while retaining strict stability using the same procedure one can prove strict stability in higher dimensions as well

a modified tv stokes model for image processing we introduce and investigate the modified total variation tv stokes model for two classical image processing tasks i e image restoration and image inpainting the modified tv stokes model is a two step model based on a tv minimization in each step and the use of geometric information of the image in the first step a smoothed and divergence free tangential field of the given image is recovered and in the second step the image is reconstructed from the corresponding normals the existence and the uniqueness of the solution to the minimization problems are established for both steps of the model numerical examples and comparisons are presented to illustrate the effectiveness of the model

convergent cartesian grid methods for maxwell s equations in complex geometries a convergent second order cartesian grid finite difference scheme for the solution of maxwell s equations is presented the scheme employs a staggered grid in space and represents the physical location of the material and metallic boundaries correctly hence eliminating problems caused by staircasing and contrary to the popular yee scheme enforces the correct jump conditions on the field components across material interfaces a detailed analysis of the accuracy of the new embedding scheme is presented confirming its second order global accuracy furthermore the scheme is proven to be a bounded error scheme and thus convergent conditions for fully discrete stability is furthermore established this enables the derivation of bounds for fully discrete stability with cfl restrictions being almost identical to those of the much simpler yee scheme the analysis exposes that the effects of staircasing as well as a lack of properly enforced jump conditions on the field components have significant consequences for the global accuracy it is among other things shown that for cases in which a field component is discontinuous along a grid line as happens at general two and three dimensional material interfaces the yee scheme may exhibit local divergence and loss of global convergence to validate the analysis several one and two dimensional test cases are presented showing an improvement of typically 1 to 2 orders of accuracy at little or no additional computational cost over the yee scheme which in most cases exhibits first order accuracy c 2001 academic press

daily precipitation modeling by discrete autoregressive moving average processes to account for the statistical dependence among the daily precipitation magnitudes and among the daily precipitation occurrence times various models from the discrete autoregressive moving average darma family are constructed for and applied to the daily precipitation data in indiana the structure of the darma family is specified by the marginal distribution of independent identically distributed i i d discrete random variables and by their random linear combination a fundamental property of the darma family is that its correlation structure can be specified independently of its marginal density the darma correlation structure remains invariant under transformation of the marginal density this property is satisfied by the indiana daily precipitation data and it facilitates their modeling by the darma family of processes it is shown that the markov chain a popular model for the daily precipitation sequences is a specified model within darma family the indiana daily precipitation data is transformed into a multistate discrete precipitation sequence by discretizing the daily precipitation amount into a number of magnitude states the stochastic process of the daily precipitation sequences is analyzed in terms of the locally stationary seasons defined within the paper in each season the candidate darma models are identified in terms of the covariance function within the season the parameter estimation of these candidate models is done by fitting their theoretical autocorrelation functions to their empirical counterparts and the state probabilities are obtained through the multistate mean run lengths the theoretical run length distributions of the candidate models are derived for each of the precipitation magnitude states including the dry state the goodness of fit tests of the candidate models and the best model selection are based on the theoretical multistate run length distributions of the daily precipitation data supplement data tables calculations etc is available with entire article on microfiche order from american geophysical union 2000 florida avenue n w washington dc 20009 document w84 001 2 50 payment must accompany order

block pseudospectral methods for maxwell s equations ii two dimensional discontinuous coefficient case block pseudospectral bps methods are examined for maxwell s equations in two dimensional inhomogeneous media for the case of a rectangular strip with a straight line interface blocks may be coupled via fictitious points or a generalization of characteristic outflow conditions the bps methods generalize to curvilinear strips described by a change of variables such strips can conform to interfaces while overlapping with a high order free space grid to form a composite grid method numerical experiments on strips with dielectrics lossy materials perfect conductors and absorbing layers indicate that the two coupling methods are comparable and accurate with just 3 4 points per wavelength full composite examples are included to demonstrate high accuracy and geometric flexibility

directional processing of color images theory and experimental results the processing of color image data using directional information is studied the class of vector directional filters vdf which was introduced by the authors in a previous work is further considered the analogy of vdf to the spherical median is shown and their relation to the spatial median is examined moreover their statistical and deterministic properties are studied which demonstrate their appropriateness in image processing vdf result in optimal estimates of the image vectors in the directional sense this is very important in the case of color images where the vectors direction signifies the chromaticity of a given color issues regarding the practical implementation of vdf are also considered in addition efficient filtering schemes based on vdf are proposed which include adaptive and or double window structures experimental and comparative results in image filtering show very good performance measures when the error is measured in the l a b space l a b is known as a space where equal color differences result in equal distances and therefore it is very close to the human perception of colors moreover an indication of the chromaticity error is obtained by measuring the error on the maxwell triangle the results demonstrate that vdf are very accurate chromaticity estimators

high order finite difference methods for wave propagation in discontinuous media high order finite difference approximations are derived for the second order wave equation with discontinuous coefficients on rectangular geometries the discontinuity is treated by splitting the domain at the discontinuities in a multi block fashion each sub domain is discretized with compact second derivative summation by parts operators and the blocks are patched together to a global domain using the projection method this guarantees a conservative strictly stable and high order accurate scheme the analysis is verified by numerical simulations in one and two spatial dimensions

variational models for image colorization via chromaticity and brightness decomposition colorization refers to an image processing task which recovers color in grayscale images when only small regions with color are given we propose a couple of variational models using chromaticity color components to colorize black and white images we first consider total variation minimizing tv colorization which is an extension from tv inpainting to color using chromaticity model second we further modify our model to weighted harmonic maps for colorization this model adds edge information from the brightness data while it reconstructs smooth color values for each homogeneous region we introduce penalized versions of the variational models we analyze their convergence properties and we present numerical results including extension to texture colorization

image colorization by fusion of color transfers based on dft and variance features abstract color transfer methods usually suffer from spatial color coherency problem in order to address this problem this paper develops a fused color transfer method for image colorization our idea is to design a local student s t test to screen the incoherent colors in the preliminary colorization results obtained by a simple color transfer method with dft and variance features furthermore we propose a variational fusion model to inpaint these incoherent colors and fuse the other useful colors together we also present an efficient algorithm for solving the fusion model numerically and show the convergence of the algorithm finally experimental results are reported to demonstrate the effectiveness of the proposed method and its performance is competitive with those of the other testing methods

time compact difference methods for wave propagation in discontinuous media in b gustafsson and e mossberg siam j sci comput 26 2004 pp 259 271 a fourth order one step method was constructed for the solution of wave propagation problems the method is based on the first order system form of the pde and uses a staggered grid both in space and time the method was also applied with good results to a problem with discontinuous coefficients without using any special procedure across the discontinuity in this paper we will analyze a model problem from acoustics and demonstrate the theoretical foundation for this behavior furthermore we shall present more detailed numerical experiments which confirm the theoretical results

mathematical models for local nontexture inpaintings dedicated to stanley osher on the occasion of his 60th birthday abstract inspired by the recent work of bertalmio et al on digital inpaintings siggraph 2000 we develop general mathematical models for local inpaintings of nontexture images on smooth regions inpaintings are connected to the harmonic and biharmonic extensions and inpainting orders are analyzed for inpaintings involving the recovery of edges we study a variational model that is closely connected to the classical total variation tv denoising model of rudin osher and fatemi phys d 60 1992 pp 259 268 other models are also discussed based on the mumford shah regularity comm pure appl math xlii 1989 pp 577 685 and curvature driven diffusions cdd of chan and shen j visual comm image rep 12 2001 the broad applications of the inpainting models are demonstrated through restoring scratched old photos disocclusion in vision analysis text removal digital zooming and edge based image coding