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65
Image Inpainting
, 2000
"... Inpainting, the technique of modifying an image in an undetectable form, is as ancient as art itself. The goals and applications of inpainting are numerous, from the restoration of damaged paintings and photographs to the removal/replacement of selected objects. In this paper, we introduce a novel a ..."
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Cited by 531 (25 self)
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Inpainting, the technique of modifying an image in an undetectable form, is as ancient as art itself. The goals and applications of inpainting are numerous, from the restoration of damaged paintings and photographs to the removal/replacement of selected objects. In this paper, we introduce a novel algorithm for digital inpainting of still images that attempts to replicate the basic techniques used by professional restorators. After the user selects the regions to be restored, the algorithm automatically fillsin these regions with information surrounding them. The fillin is done in such a way that isophote lines arriving at the regions ’ boundaries are completed inside. In contrast with previous approaches, the technique here introduced does not require the user to specify where the novel information comes from. This is automatically done (and in a fast way), thereby allowing to simultaneously fillin numerous regions containing completely different structures and surrounding backgrounds. In addition, no limitations are imposed on the topology of the region to be inpainted. Applications of this technique include the restoration of old photographs and damaged film; removal of superimposed text like dates, subtitles, or publicity; and the removal of entire objects from the image like microphones or wires in special effects.
Global NonNegative Solutions of a Nonlinear FourthOrder Parabolic Equation for Quantum Systems
"... The existence of nonnegative weak solutions globally in time of a nonlinear fourthorder parabolic equation in one space dimension is shown. This equation arises in the study of interface fluctuations in spin systems and in quantum semiconductor modeling. The problem is considered on a bounded inte ..."
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Cited by 43 (11 self)
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The existence of nonnegative weak solutions globally in time of a nonlinear fourthorder parabolic equation in one space dimension is shown. This equation arises in the study of interface fluctuations in spin systems and in quantum semiconductor modeling. The problem is considered on a bounded interval subject to initial and Dirichlet and Neumann boundary conditions. Further, the initial datum is only assumed to be nonnegative and to satisfy a weak integrability condition. The main difficulty of the existence proof is to ensure that the solutions stay nonnegative and exist globally in time. The first property is obtained by an exponential transformation of variables. Moreover, entropytype estimates allow for the proof of the second property. Results concerning the uniqueness and longtime behaviour are given and the multidimensional problem is discussed. Finally, numerical experiments underlining the preservation of positivity are presented.
Undercompressive Shocks in Thin Film Flows
, 1999
"... Equations of the type h t + (h 2 \Gamma h 3 ) x = \Gammaffl 3 (h 3 h xxx ) x arise in the context of thin liquid films driven by the competing effects of a thermally induced surface tension gradient and gravity. In this paper, we focus on the interaction between the fourth order regulari ..."
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Cited by 37 (11 self)
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Equations of the type h t + (h 2 \Gamma h 3 ) x = \Gammaffl 3 (h 3 h xxx ) x arise in the context of thin liquid films driven by the competing effects of a thermally induced surface tension gradient and gravity. In this paper, we focus on the interaction between the fourth order regularization and the nonconvex flux. Jump initial data, from a moderately thick film to a thin precurser layer, is shown to give rise to a double wave structure that includes an undercompressive wave. This wave, which approaches an undercompressive shock as ffl ! 0; is an accumulation point for a countable family of compressive waves having the same speed. The family appears through a series of bifurcations parameterized by the shock speed. At each bifurcation, a pair of traveling waves is produced, one being stable for the PDE, the other unstable. The conclusions are based primarily on numerical results for the PDE, and on numerical investigations of the ODE describing traveling waves. Fo...
LongTime Asymptotics for Strong Solutions of the Thin Film Equation
 COMMUNICATIONS IN MATHEMATICAL PHYSICS
, 2002
"... In this paper we investigate the largetime behavior of strong solutions to the onedimensional fourth order degenerate parabolic equation ut =−(uuxxx)x, modeling the evolution of the interface of a spreading droplet. For nonnegative initial values u0(x) ∈ H¹(IR), both compactly supported or of fin ..."
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Cited by 27 (6 self)
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In this paper we investigate the largetime behavior of strong solutions to the onedimensional fourth order degenerate parabolic equation ut =−(uuxxx)x, modeling the evolution of the interface of a spreading droplet. For nonnegative initial values u0(x) ∈ H¹(IR), both compactly supported or of finite second moment, we prove explicit and universal algebraic decay in the L¹norm of the strong solution u(x, t) towards the unique (among source type solutions) strong source type solution of the equation with the same mass. The method we use is based on the study of the time decay of the entropy introduced in [13] for the porous medium equation, and uses analogies between the thin film equation and the porous medium equation.
Finitetime blowup of solutions of some longwave unstable thin film equations
 Indiana Univ. Math. J
, 2000
"... ABSTRACT. We consider the family of longwave unstable lubrication equations ht =−(hhxxx)x − (h m hx)x with m ≥ 3. Given a fixed m ≥ 3, we prove the existence of a weak solution that becomes singular in finite time. Specifically, given compactly supported nonnegative initial data with negative energ ..."
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Cited by 27 (9 self)
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ABSTRACT. We consider the family of longwave unstable lubrication equations ht =−(hhxxx)x − (h m hx)x with m ≥ 3. Given a fixed m ≥ 3, we prove the existence of a weak solution that becomes singular in finite time. Specifically, given compactly supported nonnegative initial data with negative energy, there is a time T ∗ < ∞, determined by m and the H1 norm of the initial data, and a compactly supported nonnegative weak solution such that lim supt→T ∗ ‖h(·,t)‖L ∞ = lim supt→T ∗ ‖h(·,t)‖H1 =∞. We discuss the relevance of these singular solutions to an earlier conjecture [Comm. Pure. Appl. Math. 51 (1998), 625661] on when finitetime singularities are possible for longwave unstable lubrication equations. 1.
Fourth order partial differential equations on general geometries
 UNIVERSITY OF CALIFORNIA LOS ANGELES
, 2005
"... We extend a recently introduced method for numerically solving partial differential equations on implicit surfaces (Bertalmío, Cheng, Osher, and Sapiro 2001) to fourth order PDEs including the CahnHilliard equation and a lubrication model for curved surfaces. By representing a surface in ¡ N as the ..."
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Cited by 26 (4 self)
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We extend a recently introduced method for numerically solving partial differential equations on implicit surfaces (Bertalmío, Cheng, Osher, and Sapiro 2001) to fourth order PDEs including the CahnHilliard equation and a lubrication model for curved surfaces. By representing a surface in ¡ N as the level set of a smooth function, φ ¢ we compute the PDE using only finite differences on a standard Cartesian mesh in ¡ N. The higher order equations introduce a number of challenges that are of small concern when applying this method to first and second order PDEs. Many of these problems, such as timestepping restrictions and large stencil sizes, are shared by standard fourth order equations in Euclidean domains, but others are caused by the extreme degeneracy of the PDEs that result from this method and the general geometry. We approach these difficulties by applying convexity splitting methods, ADI schemes, and iterative solvers. We discuss in detail the differences between computing these fourth order equations and computing the first and second order PDEs considered in earlier work. We explicitly derive schemes for the linear fourth order diffusion, the CahnHilliard equation for phase transition in a binary alloy, and surface tension driven flows on complex geometries. Numerical examples validating our methods are presented for these flows for data on general surfaces.
Nonlinear mobility continuity equations and generalized displacement convexity
, 2009
"... We consider the geometry of the space of Borel measures endowed with a distance that is defined by generalizing the dynamical formulation of the Wasserstein distance to concave, nonlinear mobilities. We investigate the energy landscape of internal, potential, and interaction energies. For the intern ..."
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Cited by 21 (4 self)
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We consider the geometry of the space of Borel measures endowed with a distance that is defined by generalizing the dynamical formulation of the Wasserstein distance to concave, nonlinear mobilities. We investigate the energy landscape of internal, potential, and interaction energies. For the internal energy, we give an explicit sufficient condition for geodesic convexity which generalizes the condition of McCann. We take an eulerian approach that does not require global information on the geodesics. As byproduct, we obtain existence, stability, and contraction results for the semigroup obtained by solving the homogeneous Neumann boundary value problem for a nonlinear diffusion equation in a convex bounded domain. For the potential energy and the interaction energy, we present a nonrigorous argument indicating that they are not displacement semiconvex.
A nonlinear fourthorder parabolic equation and related logarithmic Sobolev inequalities
, 2004
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Instabilities in gravity driven flow of thin fluid films
 SIAM Rev
, 2003
"... Abstract. This paper presents theoretical, computational, and experimental aspects of the instability development in the flow of thin fluid films. The theoretical part involves basic fluid mechanics and presents derivation of the thin film equation using lubrication approximation. A simplified versi ..."
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Cited by 19 (1 self)
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Abstract. This paper presents theoretical, computational, and experimental aspects of the instability development in the flow of thin fluid films. The theoretical part involves basic fluid mechanics and presents derivation of the thin film equation using lubrication approximation. A simplified version of this equation is then analyzed analytically using linear stability analysis, and also numerically. The results are then compared directly to experiments. The experimental part outlines the setup, as well as data acquisition and analysis. This immediate comparison to experiments is very useful for gaining better insight into the interpretation of various theoretical and computational results. Key words. nonlinear partial differential equations, perturbation theory, finite difference methods, fluid dynamics
Positive Entropic Schemes for a Nonlinear Fourthorder Parabolic Equation
, 2001
"... A finitedifference scheme with positivitypreserving and entropydecreasing properties for a nonlinear fourthorder parabolic equation arising in quantum systems and interface fluctuations is derived. Initialboundary value problems, the Cauchy problem and a rescaled equation are discussed. Based o ..."
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Cited by 18 (5 self)
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A finitedifference scheme with positivitypreserving and entropydecreasing properties for a nonlinear fourthorder parabolic equation arising in quantum systems and interface fluctuations is derived. Initialboundary value problems, the Cauchy problem and a rescaled equation are discussed. Based on this scheme we elucidate properties of the longtime asymptotics for this equation.