Geometrical PDEs based on second-order derivatives of gauge coordinates in image processing

Image and Vision Computing

Volumn 27, Issue 8, 2009, Pages 1023-1034

  Kuijper, Arjan ^a  

^a JOHANN RADON INSTITUTE FOR COMPUTATIONAL AND APPLIED MATHEMATICS   (Austria)

Author keywords

Gauge coordinates; Geometric image evolution; Multi resolution processing; Nonlinear scale space

Indexed keywords

EQUATIONS OF MOTION; IMAGE SEGMENTATION; MULTIRESOLUTION ANALYSIS;

DE-NOISING; DIFFUSION EQUATIONS; EDGE PRESERVING; GAUGE COORDINATES; GAUSSIAN; GEOMETRIC IMAGE EVOLUTION; INPAINTING; INPUT IMAGE; MEAN CURVATURE MOTION; MULTI-RESOLUTION PROCESSING; NONLINEAR SCALE SPACE; QUANTITATIVE RESULT; REAL-LIFE IMAGES; SECOND ORDER DERIVATIVES;

GAGES;

EID: 67349194317     PISSN: 02628856     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.imavis.2008.09.003     Document Type: Article

References (43)

1. Koenderink J.J. The structure of images. Biological Cybernetics 50 (1984) 363-370
2. Perona P., and Malik J. Scale space and edge detection using anisotropic diffusion. IEEE Transactions on Pattern Analysis and Machine Intelligence 12 7 (1990) 629-639
3. Rudin L.I., Osher S., and Fatemi E. Nonlinear total variation based noise removal algorithms. Physica D 60 (1992) 259-268
4. Alvarez L., Lions P., and Morel J. Image selective smoothing and edge detection by nonlinear diffusion. SIAM Journal on Numerical Analysis 29 (1992) 845-866
5. Alvarez L., Guichard F., Lions P.L., and Morel J.M. Axioms and fundamental equations of image processing. Archives for Rational Mechanics 123 (1993) 199-257
6. Kimia B., and Siddiqi K. Geometric heat equation and non-linear diffusion of shapes and images. Computer Vision and Image Understanding 64 3 (1996) 305-322
7. Caselles V., Morel J.-M., and Sbert C. An axiomatic approach to image interpolation. IEEE Transactions on Image Processing 7 (1996) 376-386
8. A.P. Witkin, Scale-space filtering, in: Proceedings of the Eighth International Joint Conference on Artificial Intelligence, 1983, pp. 1019-1022.
9. Weickert J.A., Ishikawa S., and Imiya A. Linear scale-space has first been proposed in Japan. Journal of Mathematical Imaging and Vision 10 3 (1999) 237-252
10. Iijima T. Basic theory of pattern normalization (for the case of a typical one dimensional pattern). Bulletin of the Electrotechnical Laboratory 26 (1962) 368-388 in Japanese
11. J.A. Weickert, S. Ishikawa, A. Imiya, On the history of Gaussian scale-space axiomatics, in: Sporring et al, [16], 1997, pp. 45-59.
12. A. Kuijper, Exploring and exploiting the structure of saddle points in Gaussian scale space, Computer Vision and Image Understanding, in press. Available from: .
13. Florack L.M.J. Image Structure. Computational Imaging and Vision Series vol. 10 (1997), Kluwer Academic Publisehers, Dordrecht, The Netherlands
14. In: Haar Romeny B.M.t. (Ed). Geometry-Driven Diffusion in Computer Vision of Computational Imaging and Vision Series vol. 1 (1994), Kluwer Academic Publishers, Dordrecht
15. Lindeberg T. Scale-space theory in computer vision. The Kluwer International Series in Engineering and Computer Science (1994), Kluwer Academic Publishers
16. In: Sporring J., Nielsen M., Florack L.M.J., and Johansen P. (Eds). Gaussian Scale-Space Theory of Computational Imaging and Vision Series. second ed. vol. 8 (1997), Kluwer Academic Publishers, Dordrecht
17. Weickert J.A. Anisotropic Diffusion in Image Processing (1998), Teubner, Stuttgart
18. B.M. t. Haar Romeny, L.M.J. Florack, J.J. Koenderink, M.A. Viergever (Eds.), Scale-space theory in computer vision, in: Proceedings of the First International Conference, Scale-Space'97, Utrecht, The Netherlands, in: Lecture Notes in Computer Science, vol. 1252, Springer-Verlag, Berlin, 1997.
19. In: Nielsen M., Johansen P., Olsen O.F., and Weickert J. (Eds). Scale-space theories in computer vision. Lecture Notes in Computer Science vol. 1682 (1999), Springer-Verlag, Heidelberg, Berlin
20. In: Kerckhove M. (Ed). Scale-space and morphology in computer vision. Lecture Notes in Computer Science vol. 2106 (2001), Springer-verlag, Heidelberg, Berlin
21. In: Griffin L., and Lillholm M. (Eds). Scale space methods in computer vision. Lecture Notes in Computer Science vol. 2695 (2003), Springer-Verlag, Heidelberg, Berlin
22. In: Kimmel R., Sochen N., and Weickert J. (Eds). Scale space and PDE methods in computer vision. Lecture Notes in Computer Science vol. 59 (2005), Springer-Verlag, Heidelberg, Berlin
23. Sgallari F., Murli A., Paragios N., space S., and vision v.m.i.c. Lecture Notes in Computer Science. vol. 4485 (2007), Springer-Verlag, Heidelberg, Berlin
24. Duits R., Florack L.M.J., Graaf J.d., and Haar Romeny B.M.t. On the axioms of scale space theory. Journal of Mathematical Imaging and Vision 20 3 (2004) 267-298
25. Haar Romeny B.M.t. Front-end vision and multi-scale image analysis. Computational Imaging and Vision Series Vol. 27 (2003), Kluwer Academic Publishers, Dordrecht, The Netherlands
26. Aubert G., and Kornprobst P. Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations of Applied Mathematical Sciences. second ed. (2006), Springer-Verlag vol. 147
27. P. Kornprobst, R. Deriche, G. Aubert, Image coupling, restoration and enhancement via PDE's, in: Proceedings of the International Conference on Image Processing, vol. 4, 1997, pp. 458-461.
28. Vazquez J.L. Smoothing and Decay Estimates for Nonlinear Diffusion Equations of Porous Medium Type of Oxford Lecture Series. Mathematics and Its Applications Vol. 33 (2006), Oxford University Press
29. A. Kuijper, p-Laplacian driven image processing, in: 14th International Conference on Image Processing, ICIP 2007, San Antonio, Texas, USA, 16-19 September 2007, vol. V, 2007, pp. 257-260.
30. yy = 0. Arkiv für Matematik 7 (1968) 395-425
31. Oberman A.M. A convergent difference scheme for the infinity laplacian: Construction of absolutely minimizing lipschitz extensions. Mathematics of Computation 74 251 (2005) 1217-1230
32. Barenblatt G.I. On self-similar motions of compressible fuids in porous media. Prikl. Mt. Makh. 16 6 (1952) 679-698 in Russian
33. Keeling S.L., and Stollberger R. Nonlinear anisotropic diffusion filtering for multiscale edge enhancement. Inverse Problems 18 1 (2002) 175-190
34. Chen Y., Levine S., and Rao M. Variable exponent, linear growth functionals in image restoration. SIAM Journal of Applied Mathematics 66 4 (2006) 1383-1406
35. Kim S. Pde-based image restoration: A hybrid model and color image denoising. IEEE Transactions on Image Processing 15 5 (2006) 1163-1170
36. Griffin L. Mean, median and mode filtering of images. Proceedings of the Royal Society Series A 456 (2000) 2995-3004
37. Abramowitz M., and Stegun I.A. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. 9th Edition. (1972), Dover, New York
38. E.W. Weisstein, Mathworld - a wolfram web resource, 2006. Available from: .
39. O.F. Olsen, M. Nielsen, Multi-scale gradient magnitude watershed segmentation, in: ICIAP'97 - 9th International Conference on Image Analysis and Processing, in: Lecture Notes in Computer Science, vol. 1310, 1997, pp. 6-13.
40. Cayley A. On contour and slope lines. Lond. Edin. Dublin Phil. Mag. J. Sci. 18 120 (1859) 264-268
41. Maxwell J.C. On hills and dales. Lond. Edin. Dublin Phil. Mag. and J. of Sci. 40 269 (1870) 421-425
42. Niessen W.J., ter Haar Romeny B.M., Florack L.M.J., and Viergever M.A. A general framework for geometry-driven evolution equations. International Journal of Computer Vision 21 3 (1997) 187-205
43. A. Kuijper, Qualitative and quantiztative behaviour of geometrical pdes in image processing, in: Eighth Asian Conference on Computer Vision, ACCV 2007, November 18-22, Tokyo, Japan, 2007, in: Lecture Notes in Computer Science, vol. 4843, 2007, pp. 230-239.