Sobolev gradient preconditioning for image-processing PDEs

Communications in Numerical Methods in Engineering

Volumn 24, Issue 6, 2008, Pages 493-504

  Richardson Jr , W B ^a  

^a UNIVERSITY OF TEXAS AT SAN ANTONIO   (United States)

Author keywords

Forward and inverse scale spaces; FOSLS; Image processing; Mixed methods; Negative Sobolev norm; Sobolev gradients

Indexed keywords

PARTIAL DIFFERENTIAL EQUATIONS;

SOBOLEV GRADIENTS;

GRADIENT METHODS;

EID: 48949117329     PISSN: 10698299     EISSN: 10990887     Source Type: Journal    
DOI: 10.1002/cnm.951     Document Type: Conference Paper

References (41)

1. Guichard F, Morel JM. Introduction to partial differential equations in image processing. Tutorial Notes, IEEE International Conference on Image Processing. IEEE: New York, 1995.
2. Sapiro G. Geometric Partial Differential Equations and Image Analysis. Cambridge University Press: New York, 2001.
3. Marr D. Vision. Freeman: New York, 1981.
4. Alvarez L, Guichard F, Lions PL, Morel JM. Axioms and fundamental equations for image processing. Archive for Rational Mechanics and Analysis 1993; 123(3):199-257.
5. Bredies K, Lorenz DA, Maass P. Mathematical concepts of multiscale smoothing. Applied and Computational Harmonic Analysis 2005; 19:141-161.
6. Meyer Y. Oscillating Patterns in Image Processing and Nonlinear Evolution Equations. American Mathematical Society: Providence, RI, 2001.
7. Brown BM, Jais M, Knowles IW. A variational approach to an elastic inverse problem. Inverse Problems 2005; 21:1953-1973.
8. Neuberger JW. Sobolev Gradients and Differential Equations. Springer: Berlin, 1997.
9. Beurling A, Deny J. Dirichlet spaces. Proceedings of the National Academy of Sciences 1959; 45:208-215.
10. Richardson WB. Steepest descent using smooth gradients. Journal of Applied Mathematics and Computation 2000; 112:241-254.
11. Neuberger JW. Steepest descent for general systems of linear differential equations in Hilbert space. Springer Lecture Notes 1983; 1032:390-406.
12. Richardson WB. Sobolev gradient preconditioning for PDE applications. Iterative Methods in Scientific Computation IV, IMACS Series in Computational and Applied Mathematics 1999; 5:223-234.
13. Richardson Jr WB. High-order Sobolev preconditioning. Nonlinear Analysis 2005; 63:e1779-e1787.
14. Neuberger JW, Renka RJ. Sobolev gradients and the Ginzburg-Landau functional. SIAM Journal on Scientific Computing 1998; 20:582-590.
15. Cai Z, Lazarov R, Manteuffel TA, McCormick SF. First-order system least squares for second-order partial differential equations: Part I. SIAM Journal on Numerical Analysis 1994; 31:1785-1799.
16. Carey GF, Richardson WB. A note on least-squares methods. Communications in Numerical Methods in Engineering 2006; 22:83-92.
17. Bertoluzza S, Canuto C, Tabacco A. Stable discretizations of convection-diffusion problems via computable negative-order inner products. SIAM Journal on Numerical Analysis 2000; 38:1034-1055.
18. Bochev PB, Gunzburger MD. Analysis of least-squares finite element methods for the Stokes equations. Mathematics of Computation 1994; 63:479-506.
19. Bramble JH, Lazarov RD, Pasciak JE. A least-squares approach based on a discrete minus one inner product for first order systems. Mathematics of Computation 1997; 66:935-955.
20. Pehlivanov AI, Carey GF. Error estimates for least squares mixed finite elements. RAIRO-Mathematical Modelling and Numerical Analysis 1994; 28(5):499-516.
21. Koenderink JJ. The structure of images. Biological Cybernetics 1984; 50:363-370.
22. Perona A, Malik J. Scale-space and edge detection using anisotropic diffusion. IEEE Transactions on Pattern Analysis and Machine Intelligence 1990; 12:629-639.
23. Rudin LI, Osher SJ, Fatemi E. Nonlinear total variation based noise removal algorithms. Physica D 1992; 60:259-268.
24. 1 function approximation. SIAM Journal on Applied Mathematics 2005; 65:1817-1837.
25. Bertalmio M, Vese L, Sapiro G, Osher S. Simultaneous structure and texture image inpainting. IEEE Transactions on Image Processing 2003; 12:882-889.
26. -l norm. Multiscale Modeling and Simulation 2003; 1:349-370.
27. 1 images and the Mumford-Shah-Sobolev model for segmented image decomposition. Applied Math. Research Exp. (AMRE) 2005; 4:143-167.
28. Henn S, A multigrid method for a fourth-order diffusion equation with application to image processing. SIAM Journal on Scientific Computing 2005; 27:831-849.
29. Lysaker M, Lundervold A, Tai X. Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time. IEEE Transactions on Image Processing 2003; 12:1579-1590.
30. Tschumperle D, Deriche R. Vector-valued image regularization with PDE's: a common framework for different applications. IEEE Transactions on Pattern Analysis and Machine Intelligence 2005; 27:1-12.
31. Burger M, Osher S, Xu J, Gilboa G. Nonlinear inverse scale space methods for image restoration. VLSM 2005. Lecture Notes in Computer Science, vol. 3752. Springer-Verlag, 2005; 25-36.
32. Groetsch CW, Scherzer O. Non-stationary iterated Tikhonov-Morozov method and third-order differential equations for the evaluation of unbounded operators. Mathematical Methods in the Applied Sciences 2000; 23:1287-1300.
33. Scherzer O, Weickert J. Relations between regularization and diffusion filtering. Journal of Mathematical Imaging and Vision 2000; 12:43-63.
34. Hofer M, Pottmann H. Energy-minimizing splines in manifolds. ACM Transactions on Graphics 2004 SIGGRAPH 2004; 23:284-293.
35. Renka RJ, Neuberger JW. Minimal surfaces and Sobolev gradients. SIAM Journal of Scientific Computing 1995; 16:1412-1427.
36. Bourdin B, Chambolle A. Implementation of an adaptive finite elements approximation of the Mumford-Shah functional. Numerische Mathematik 2000; 85(4):609-646.
37. Carey GF. Computational Grids: Generation, Adaptation, and Solution Strategies. Taylor & Francis: Washington DC, 1997.
38. Rumpf M. An adaptive finite element method for large scale image processing. Scale-Space '99. Lecture Notes in Computer Science, vol. 1682. 1999; 223-234.
39. Mumford D, Shah J. Optimal approximation by piecewise smooth functions and an associated variational problem. Communications on Pure and Applied Mathematics 1989; 42:577-585.
40. Ambrosio L, Tortorelli VM. Approximation of functionals depending on jumps by elliptic functionals via T-convergence. Communications on Pure and Applied Mathematics 1990; 43(8):999-1036.
41. Erlangga YA, Vuik C, Oosterlee CW. On a class of preconditioners for solving the Helmholtz equation. Applied Numerical Mathematics 2004; 50:409-425.