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Sobolev gradient preconditioning for image-processing PDEs. (English) Zbl 1168.68596

Summary: The article explores the relationship between Sobolev gradients and \(H^{-1}\) mixed methods for a variety of Partial Differential Equations (PDEs) from image processing. A first-order system least-squares problem is used to introduce the method and compare the Euclidean with the Sobolev gradient. The standard two-term decomposition of an image as \(f = u + v\) with \(u\in H^1 \) and \(v\in L^{2} = H^{0}\) yields a second-order linear PDE, while minimizing other \(L^p\) norms give nonlinear PDEs. Finally, a three-term decomposition \(f = u + v + w\) with \(u\in H^1, v \in H^{-1}, w \in H^{0}\) requires the solution of a fourth-order system with the biharmonic operator.

MSC:

68U10 Computing methodologies for image processing
94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
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[1] Guichard, Tutorial Notes, IEEE International Conference on Image Processing (1995)
[2] Sapiro, Geometric Partial Differential Equations and Image Analysis (2001)
[3] Marr, Vision (1981)
[4] Alvarez, Axioms and fundamental equations for image processing, Archive for Rational Mechanics and Analysis 123 (3) pp 199– (1993) · Zbl 0788.68153
[5] Bredies, Mathematical concepts of multiscale smoothing, Applied and Computational Harmonic Analysis 19 pp 141– (2005) · Zbl 1081.65062
[6] Meyer, Oscillating Patterns in Image Processing and Nonlinear Evolution Equations (2001) · Zbl 0987.35003
[7] Brown, A variational approach to an elastic inverse problem, Inverse Problems 21 pp 1953– (2005) · Zbl 1274.35407
[8] Neuberger, Sobolev Gradients and Differential Equations (1997)
[9] Beurling, Dirichlet spaces, Proceedings of the National Academy of Sciences 45 pp 208– (1959)
[10] Richardson, Steepest descent using smooth gradients, Journal of Applied Mathematics and Computation 112 pp 241– (2000) · Zbl 1023.65053
[11] Neuberger, Steepest descent for general systems of linear differential equations in Hilbert space, Springer Lecture Notes 1032 pp 390– (1983) · Zbl 0534.35002
[12] Richardson, Sobolev gradient preconditioning for PDE applications, Iterative Methods in Scientific Computation IV, IMACS Series in Computational and Applied Mathematics 5 pp 223– (1999)
[13] Richardson, High-order Sobolev preconditioning, Nonlinear Analysis 63 pp e1779– (2005)
[14] Neuberger, Sobolev gradients and the Ginzburg-Landau functional, SIAM Journal on Scientific Computing 20 pp 582– (1998) · Zbl 0920.35059
[15] Cai, First-order system least squares for second-order partial differential equations: Part I, SIAM Journal on Numerical Analysis 31 pp 1785– (1994)
[16] Carey, A note on least-squares methods, Communications in Numerical Methods in Engineering 22 pp 83– (2006) · Zbl 1090.65129
[17] Bertoluzza, Stable discretizations of convection-diffusion problems via computable negative-order inner products, SIAM Journal on Numerical Analysis 38 pp 1034– (2000) · Zbl 0974.65104
[18] Bochev, Analysis of least-squares finite element methods for the Stokes equations, Mathematics of Computation 63 pp 479– (1994) · Zbl 0816.65082
[19] Bramble, A least-squares approach based on a discrete minus one inner product for first order systems, Mathematics of Computation 66 pp 935– (1997) · Zbl 0870.65104
[20] Pehlivanov, Error estimates for least squares mixed finite elements, RAIRO-Mathematical Modelling and Numerical Analysis 28 (5) pp 499– (1994) · Zbl 0820.65065
[21] Koenderink, The structure of images, Biological Cybernetics 50 pp 363– (1984) · Zbl 0537.92011
[22] Perona, Scale-space and edge detection using anisotropic diffusion, IEEE Transactions on Pattern Analysis and Machine Intelligence 12 pp 629– (1990)
[23] Rudin, Nonlinear total variation based noise removal algorithms, Physica D 60 pp 259– (1992) · Zbl 0780.49028
[24] Chan, Aspects of total variation regularized L1 function approximation, SIAM Journal on Applied Mathematics 65 pp 1817– (2005)
[25] Bertalmio, Simultaneous structure and texture image inpainting, IEEE Transactions on Image Processing 12 pp 882– (2003)
[26] Osher, Image decomposition and restoration using total variation minimization and the H-1 norm, Multiscale Modeling and Simulation 1 pp 349– (2003) · Zbl 1051.49026
[27] Shen, Piecewise H-1 + H0 + H1 images and the Mumford-Shah-Sobolev model for segmented image decomposition, Applied Math. Research Exp 4 pp 143– (2005)
[28] Henn, A multigrid method for a fourth-order diffusion equation with application to image processing, SIAM Journal on Scientific Computing 27 pp 831– (2005) · Zbl 1096.65105
[29] Lysaker, Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time, IEEE Transactions on Image Processing 12 pp 1579– (2003) · Zbl 1286.94020
[30] Tschumperle, Vector-valued image regularization with PDE’s: a common framework for different applications, IEEE Transactions on Pattern Analysis and Machine Intelligence 27 pp 1– (2005)
[31] Burger, Lecture Notes in Computer Science, in: VLSM 2005 pp 25– (2005)
[32] Groetsch, Non-stationary iterated Tikhonov-Morozov method and third-order differential equations for the evaluation of unbounded operators, Mathematical Methods in the Applied Sciences 23 pp 1287– (2000) · Zbl 0970.65059
[33] Scherzer, Relations between regularization and diffusion filtering, Journal of Mathematical Imaging and Vision 12 pp 43– (2000) · Zbl 0945.68183
[34] Hofer, Energy-minimizing splines in manifolds, ACM Transactions on Graphics 2004 SIGGRAPH 23 pp 284– (2004)
[35] Renka, Minimal surfaces and Sobolev gradients, SIAM Journal of Scientific Computing 16 pp 1412– (1995) · Zbl 0857.35004
[36] Bourdin, Implementation of an adaptive finite elements approximation of the Mumford-Shah functional, Numerische Mathematik 85 (4) pp 609– (2000) · Zbl 0961.65062
[37] Carey, Computational Grids: Generation, Adaptation, and Solution Strategies (1997) · Zbl 0955.74001
[38] Rumpf M An adaptive finite element method for large scale image processing 1999 223 234
[39] Mumford, Optimal approximation by piecewise smooth functions and an associated variational problem, Communications on Pure and Applied Mathematics 42 pp 577– (1989) · Zbl 0691.49036
[40] Ambrosio, Approximation of functionals depending on jumps by elliptic functionals via {\(\Gamma\)}-convergence, Communications on Pure and Applied Mathematics 43 (8) pp 999– (1990) · Zbl 0722.49020
[41] Erlangga, On a class of preconditioners for solving the Helmholtz equation, Applied Numerical Mathematics 50 pp 409– (2004) · Zbl 1051.65101
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