×
Zhang, Jianping; Chen, Ke; Yu, Bo

A 3D multi-grid algorithm for the Chan-Vese model of variational image segmentation. (English) Zbl 1242.94006

Int. J. Comput. Math. 89, No. 2, 160-189 (2012).
Summary: Variational segmentation models provide effective tools for image processing applications. Although existing models are continually refined to increase their capabilities, solution of such models is often a slow process, since fast methods are not immediately applicable to nonlinear problems. This paper presents an efficient multi-grid algorithm for solving the Chan-Vese model in three dimensions, generalizing our previous work on the topic in two dimensions, but this direct generalized method is of low performance or unfeasible. So here we first present two general smoothers for a nonlinear multi-grid method and then give our three new adaptive smoothers which can optimally choose a parameter of the smoothers automatically; also we analyse them using a local Fourier analysis and our theorem to inform how to obtain an optimal parameter and the best smoother selection. Finally, various advantages of our recommended algorithm are illustrated, using both synthetic and real images.

Cited in 5 Documents

MSC:

94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
65K10 Numerical optimization and variational techniques

Keywords:

image processing; variational problems; optimization; Euler-Lagrange equation; 3D image segmentation; multi-grid algorithm; adaptive smoothers; local Fourier analysis
Cite Review PDF
Full Text: DOI

References:

[1] DOI: 10.1002/cpa.3160430805 · Zbl 0722.49020 · doi:10.1002/cpa.3160430805
[2] Aubert G., Mathematical Problems in Image Processing (2002) · Zbl 1109.35002
[3] Badshah N., Commun. Comput. Phys 4 pp 294– (2008)
[4] DOI: 10.1109/TIP.2009.2014260 · Zbl 1371.94041 · doi:10.1109/TIP.2009.2014260
[5] Badshah N., Commun. Comput. Phys 7 pp 759– (2010)
[6] DOI: 10.1090/S0025-5718-1977-0431719-X · doi:10.1090/S0025-5718-1977-0431719-X
[7] Brandt A., Multigrid Solvers for Non-elliptic and Singular-perturbation Steady-state Problems (1981)
[8] DOI: 10.1137/0731087 · Zbl 0817.65126 · doi:10.1137/0731087
[9] DOI: 10.1007/s10851-007-0002-0 · Zbl 1523.94005 · doi:10.1007/s10851-007-0002-0
[10] Brown, E. S., Chan, T. F. and Bresson, X. ”Completely convex formulation of the Chan-Vese image segmentation model”. Tech. Rep. CAM 10-44, Department of Mathematics, University of California, Los Angeles, USA, 2010 · Zbl 1254.68271
[11] DOI: 10.1109/34.588023 · doi:10.1109/34.588023
[12] Chan T. F., Image Processing and Analysis – Variational PDE, Wavelet, and Stochastic Methods (2005) · Zbl 1095.68127
[13] DOI: 10.1109/83.902291 · Zbl 1039.68779 · doi:10.1109/83.902291
[14] DOI: 10.1007/978-3-642-55987-7_4 · doi:10.1007/978-3-642-55987-7_4
[15] Chang Q. S., Numer. Math. Theor. Method Appl 2 pp 353– (2009)
[16] DOI: 10.1017/CBO9780511543258 · Zbl 1079.65057 · doi:10.1017/CBO9780511543258
[17] DOI: 10.1007/s10915-007-9145-9 · Zbl 1129.65046 · doi:10.1007/s10915-007-9145-9
[18] Cobzas, D., Birkbeck, N., Schmidt, M., Jagersand, M. and Murtha, A. 3D variational brain tumor segmentation using a high dimensional feature set. IEEE 11th International Conference on Computer Vision, ICCV 2007, Rio de Janeiro, Brazil. pp.1–8. New York: IEEE.
[19] DOI: 10.1109/83.951533 · doi:10.1109/83.951533
[20] DOI: 10.1137/080725891 · Zbl 1177.65088 · doi:10.1137/080725891
[21] DOI: 10.1007/s10915-009-9331-z · Zbl 1203.65044 · doi:10.1007/s10915-009-9331-z
[22] DOI: 10.1109/TPAMI.1984.4767475 · doi:10.1109/TPAMI.1984.4767475
[23] DOI: 10.1007/BF00133570 · doi:10.1007/BF00133570
[24] DOI: 10.1109/TIP.2005.854442 · doi:10.1109/TIP.2005.854442
[25] DOI: 10.1007/0-387-21810-6_4 · doi:10.1007/0-387-21810-6_4
[26] Kimmel, R. 2003. Geometric Segmentation of 3D Structures. Proceedings of the 2003 International Conference on Image Processing (ICIP 2003). September14–172003, Barcelona, Catalonia, Spain. Vol. 2, pp.639–642.
[27] DOI: 10.1016/0893-9659(91)90161-N · Zbl 0718.65066 · doi:10.1016/0893-9659(91)90161-N
[28] DOI: 10.1098/rspb.1980.0020 · doi:10.1098/rspb.1980.0020
[29] DOI: 10.1002/cpa.3160420503 · Zbl 0691.49036 · doi:10.1002/cpa.3160420503
[30] Osher S., Level Set Methods and Dynamic Implicit Surfaces (2003) · Zbl 1026.76001 · doi:10.1007/b98879
[31] DOI: 10.1016/0021-9991(88)90002-2 · Zbl 0659.65132 · doi:10.1016/0021-9991(88)90002-2
[32] DOI: 10.1016/0167-2789(92)90242-F · Zbl 0780.49028 · doi:10.1016/0167-2789(92)90242-F
[33] Scherzer O., Variational Methods in Imaging (2009) · Zbl 1177.68245
[34] DOI: 10.1007/11558484_63 · doi:10.1007/11558484_63
[35] DOI: 10.1007/BF00281235 · Zbl 0113.32303 · doi:10.1007/BF00281235
[36] Tao, W. B. and Tai, X.C. ”Multiple piecewise constant active contours for image segmentation using graph cuts optimization”. Tech. Rep. CAM 09-13, Department of Mathematics, University of California, Los Angeles, USA, 2009
[37] Trottenberg U., Multigrid (2001)
[38] DOI: 10.1023/A:1020874308076 · Zbl 1012.68782 · doi:10.1023/A:1020874308076
[39] DOI: 10.1109/83.661190 · doi:10.1109/83.661190
[40] Wienands R., Practical Fourier Analysis for Multigrid Method (2005) · Zbl 1062.65133
[41] DOI: 10.1109/83.661186 · Zbl 0973.94003 · doi:10.1109/83.661186
[42] Yuan, J., Bae, E., Tai, X.C. and Boykov, Y. 2010. ”A study on continuous max-flow and min-cut approaches”. Tech. Rep. CAM 10-61, Department of Mathematics, University of California, Los Angeles, USA
[43] DOI: 10.1109/34.537343 · doi:10.1109/34.537343
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.