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24
Fillingin by joint interpolation of vector fields and gray levels
 IEEE Trans. Image Processing
, 2001
"... Abstract—A variational approach for fillingin regions of missing data in digital images is introduced in this paper. The approach is based on joint interpolation of the image graylevels and gradient/isophotes directions, smoothly extending in an automatic fashion the isophote lines into the holes ..."
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Cited by 124 (21 self)
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Abstract—A variational approach for fillingin regions of missing data in digital images is introduced in this paper. The approach is based on joint interpolation of the image graylevels 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 graylevels 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. Index Terms—Fillingin, Gestalt principles, image gradients, image graylevels, interpolation, partial differential equations, variational approach. I.
Minimizing total variation flow
 Differential and Integral Equations
, 2001
"... (Submitted by: Jerry Goldstein) Abstract. We prove existence and uniqueness of weak solutions for the minimizing total variation flow with initial data in L 1. We prove that the length of the level sets of the solution, i.e., the boundaries of the level sets, decreases with time, as one would expect ..."
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Cited by 44 (6 self)
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(Submitted by: Jerry Goldstein) Abstract. We prove existence and uniqueness of weak solutions for the minimizing total variation flow with initial data in L 1. We prove that the length of the level sets of the solution, i.e., the boundaries of the level sets, decreases with time, as one would expect, and the solution converges to the spatial average of the initial datum as t →∞. We also prove that local maxima strictly decrease with time; in particular, flat zones immediately decrease their level. We display some numerical experiments illustrating these facts. 1. Introduction. Let Ω be a bounded set in R N with Lipschitzcontinuous boundary ∂Ω. We are interested in the problem ∂u Du = div(
Variable exponent, linear growth functionals in image processing
 SIAM Journal on Applied Mathematics
, 2004
"... Abstract. We study a functional with variable exponent, 1 ≤ p(x) ≤ 2, which provides a model for image denoising, enhancement, and restoration. The diffusion resulting from the proposed model is a combination of Total Variation based regularization and Gaussian smoothing. The existence, uniqueness, ..."
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Cited by 28 (1 self)
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Abstract. We study a functional with variable exponent, 1 ≤ p(x) ≤ 2, which provides a model for image denoising, enhancement, and restoration. The diffusion resulting from the proposed model is a combination of Total Variation based regularization and Gaussian smoothing. The existence, uniqueness, and longtime behavior of the proposed model are established. Experimental results illustrate the effectiveness of the model in image restoration.
Analysis of total variation flow and its finite element approximations
 M2AN MATH. MODEL. NUMER. ANAL
, 2002
"... We study the gradient flow for the total variation functional, which arises in image processing and geometric applications. We propose a variational inequality weak formulation for the gradient flow, and establish wellposedness of the problem by the energy method. The main idea of our approach is t ..."
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Cited by 15 (2 self)
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We study the gradient flow for the total variation functional, which arises in image processing and geometric applications. We propose a variational inequality weak formulation for the gradient flow, and establish wellposedness of the problem by the energy method. The main idea of our approach is to exploit the relationship between the regularized gradient flow (characterized by a small positive parameter ε, see (1.7)) and the minimal surface flow [19] and the prescribed mean curvature flow [15]. Since our approach is constructive and variational, finite element methods can be naturally applied to approximate weak solutions of the limiting gradient flow problem. We propose a fully discrete finite element method and establish convergence to the regularized gradient flow problem as h, k → 0, and to the total variation gradient flow problem as h, k, ε → 0 in general cases. Provided that the regularized gradient flow problem possesses strong solutions, which is proved possible if the datum functions are regular enough, we establish practical a priori error estimates for the fully discrete finite element solution, in particular, by focusing on the dependence of the error bounds on the regularization parameter ε. Optimal order error bounds are derived for the numerical solution under the mesh relation k = O(h²). In particular, it is shown that all error bounds depend on 1/ε only in some lower polynomial order for small ε.
Some Qualitative Properties for the Total Variational Flow
"... We prove the existence of a nite extinction time for the solutions of the Dirichlet problem for the total variational ow. For the Neumann problem, we prove that the solutions reach the average of its initial datum in nite time. The asymptotic prole of the solutions of the Dirichlet problem is also s ..."
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Cited by 13 (0 self)
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We prove the existence of a nite extinction time for the solutions of the Dirichlet problem for the total variational ow. For the Neumann problem, we prove that the solutions reach the average of its initial datum in nite time. The asymptotic prole of the solutions of the Dirichlet problem is also studied. It is shown that the proles are non zero solutions of an eigenvalue type problem which seems to be unexplored in the previous literature. The propagation of the support is analyzed in the radial case showing a behaviour enterely dierent to the case of the problem associated to the pLaplacian operator. Finally, the study of the radially symmetric case allows us to point out other qualitative properties which are peculiar of this special class of quasilinear equations. Key words: Total variation ow, nonlinear parabolic equations, asymptotic behaviour, eigenvalue type problem, propagation of the support. AMS (MOS) subject classication: 35K65, 35K55. 1 Introduction Let be a ...
Shock Capturing, Level Sets and PDE Based Methods in Computer Vision and Image Processing: A Review of Osher's Contributions
 J. Comput. Phys
, 2001
"... In this paper we review the algorithm development and applications in high resolution shock capturing methods, level set methods and PDE based methods in computer vision and image processing. The emphasis is on Stanley Osher's contribution in these areas and the impact of his work. We will star ..."
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Cited by 13 (0 self)
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In this paper we review the algorithm development and applications in high resolution shock capturing methods, level set methods and PDE based methods in computer vision and image processing. The emphasis is on Stanley Osher's contribution in these areas and the impact of his work. We will start with shock capturing methods and will review the EngquistOsher scheme, TVD schemes, entropy conditions, ENO and WENO schemes and numerical schemes for HamiltonJacobi type equations. Among level set methods we will review level set calculus, numerical techniques, fluids and materials, variational approach, high codimension motion, geometric optics, and the computation of discontinuous solutions to HamiltonJacobi equations. Among computer vision and image processing we will review the total variation model for image denoising, images on implicit surfaces, and the level set method in image processing and computer vision.
H 1 solutions of a class of fourth order nonlinear equations for image processing
 Discrete and Continuous Dynamical Systems, 10(1 and 2), January and
, 2004
"... Abstract. Recently fourth order equations of the form ut = −∇·((G(Jσu))∇∆u) have been proposed for noise reduction and simplification of two dimensional images. The operator G is a nonlinear functional involving the gradient or Hessian of its argument, with decay in the far field. The operator Jσ is ..."
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Cited by 12 (4 self)
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Abstract. Recently fourth order equations of the form ut = −∇·((G(Jσu))∇∆u) have been proposed for noise reduction and simplification of two dimensional images. The operator G is a nonlinear functional involving the gradient or Hessian of its argument, with decay in the far field. The operator Jσ is a standard mollifier. Using ODE methods on Sobolev spaces, we prove existence and uniqueness of solutions of this problem for H1 initial data. 1. Introduction. Image
Disocclusion By Joint Interpolation Of Vector Fields And Gray Levels
 SIAM Journal Multiscale Modelling and Simulation
, 2003
"... In this paper we study a variational approach for fillingin regions of missing data in 2D and 3D digital images. Applications of this technique include the restoration of old photographs and removal of superimposed text like dates, subtitles, or publicity, or the zooming of images. The approach pre ..."
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Cited by 11 (0 self)
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In this paper we study a variational approach for fillingin regions of missing data in 2D and 3D digital images. Applications of this technique include the restoration of old photographs and removal of superimposed text like dates, subtitles, or publicity, or the zooming of images. The approach presented here, initially introduced in [12], is based on a joint interpolation of the image graylevels and gradient/isophotes directions, smoothly extending the isophote lines into the holes of missing data. The process underlying this approach can be considered as an interpretation of the Gestaltist's principle of good continuation. We study the existence of minimizers of our functional and its approximation by smoother functionals. Then we present the numerical algorithm used to minimize it and display some numerical experiments. Key words. Disocclusion, Elastica, BV functions, Interpolation, Variational approach, # convergence AMS subject classifications. 68U10, 35A15, 65D05, 49J99, 47H06, 1.
On the Use of Dual Norms in Bounded Variation Type Regularization
 Weickert (Eds.), Geometric Properties of Incomplete Data, in: Computational Imaging and Vision
, 2004
"... Recently Y. Meyer gave a characterization of the minimizer of the RudinOsherFatemi functional in terms of the Gnorm. In this work we generalize this result to regularization models with higher order derivatives of bounded variation. This requires us to define generalized Gnorms. We present some ..."
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Cited by 8 (1 self)
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Recently Y. Meyer gave a characterization of the minimizer of the RudinOsherFatemi functional in terms of the Gnorm. In this work we generalize this result to regularization models with higher order derivatives of bounded variation. This requires us to define generalized Gnorms. We present some numerical experiments to support the theoretical considerations.
A Parabolic Quasilinear Problem for Linear Growth Functionals
"... We prove existence and uniqueness of solutions for the Dirichlet problem for quasilinear parabolic equations in divergent form for which the energy functional has linear growth. A tipical example of energy functional we consider is the one given by the nonparametric area integrand f(x; ) = p 1 + kk ..."
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Cited by 7 (2 self)
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We prove existence and uniqueness of solutions for the Dirichlet problem for quasilinear parabolic equations in divergent form for which the energy functional has linear growth. A tipical example of energy functional we consider is the one given by the nonparametric area integrand f(x; ) = p 1 + kk 2 , which corresponds with the timedependent minimal surface equation. We also study the asymptotic behavoiur of the solutions.