Efficient preconditioning of laplacian matrices for computer graphics

ACM Transactions on Graphics

Volumn 32, Issue 4, 2013, Pages

  Krishnan, Dilip ^a   Fattal, Raanan ^b   Szeliski, Richard ^c  

^a NEW YORK UNIVERSITY   (United States)

^b HEBREW UNIVERSITY OF JERUSALEM   (Israel)

^c MICROSOFT RESEARCH   (United States)

Author keywords

Computational photography; Laplacians; Matrix preconditioning; Mesh processing; Multigrid

Indexed keywords

COMPUTATIONAL PHOTOGRAPHY; COMPUTER GRAPHICS APPLICATIONS; DISCRETE POISSON EQUATION; LAPLACIANS; MESH PROCESSING; MULTI-GRID; STATE-OF-THE-ART METHODS; TWO-DIMENSIONAL IMAGES;

COLOR PHOTOGRAPHY; IMAGE MATCHING; IMAGE PROCESSING; LAPLACE TRANSFORMS; NUMBER THEORY;

COMPUTER GRAPHICS;

EID: 84880816518     PISSN: 07300301     EISSN: 15577368     Source Type: Journal    
DOI: 10.1145/2461912.2461992     Document Type: Article

References (52)

1. AKSOYLU, B., KHODAKOVSKY, A., AND SCHRÖDER, P. 2005. Multilevel solvers for unstructured surface meshes. SIAM Journal on Scientific Computing 26, 4, 1146-1165.
2. ALCOUFFE, R., BRANDT, A., DENDY, JR, J., AND PAINTER, J. 1981. The multi-grid method for the diffusion equation with strongly discontinuous coefficients. SIAM Journal on Scientific and Statistical Computing 2, 4, 430-454.
3. ARBENZ, P., HETMANUIK, U., LEHOUCQ, R., AND TUMINARO, R. 2003. A comparison of eigensolvers for large-scale 3D modal analysis using amg-preconditioned iterative methods. Int. Journal for Numerical Methods in Engg. 1.
4. BAKER, S., SCHARSTEIN, D., LEWIS, J. P., ROTH, S., BLACK, M. J., AND SZELISKI, R. 2011. A database and evaluation methodology for optical flow. Int. J. Comput. Vision 92 (March), 1-31.
5. BHAT, P., ZITNICK, C. L., COHEN, M., AND CURLESS, B. 2010. Gradientshop: A gradient-domain optimization framework for image and video filtering. ACM Trans. Graph. 29 (April), 10:1- 10:14.
6. BOLZ, J., FARMER, I., GRINSPUN, E., AND SCHRÖODER, P. 2003. Sparse matrix solvers on the GPU: conjugate gradients and multigrid. In ACM SIGGRAPH 2003 Papers, ACM, New York, NY, USA, SIGGRAPH '03, 917-924.
7. BOMAN, E., G., AND HENDRICKSON, B. 2001. On spanning tree preconditioners. Sandia National Labs.
8. BOMAN, E., G., AND HENDRICKSON, B. 2003. Support theory for preconditioning. SIAM J. Matrix Anal. Appl., 3, 694-717.
9. BOUWMEESTER, H., DOUGHERTY, A., AND KNYAZEV, A. V. 2012. Nonsymmetric multigrid preconditioning for conjugate gradient methods. arXiv preprint arXiv:1212.6680.
10. BRANDT, A., BRANNICK, J. J., KAHL, K., AND LIVSHITS, I. 2011. Bootstrap AMG. SIAM J. Scientific Computing 33, 2, 612-632.
11. BRANDT, A. 1973. Multi-level adaptive technique (MLAT) for fast numerical solution to boundary value problems. In Proc. Conf. on Numerical Methods in Fluid Mechanics, vol. 18 of Lecture Notes in Physics. Springer Berlin /Heidelberg, 82-89.
12. BRANDT, A. 1986. Algebraic multigrid theory: The symmetric case. Applied Mathematics and Computation 19, 1U 4, 23-56.
13. BRANDT, A. 2001. Multiscale scientific computation: Review 2001. In Multiscale and Multiresolution Methods, Springer Verlag, 1-96.
14. CRANE, K., WEISCHEDEL, C., AND WARDETZKY, M. 2012. Geodesics in heat. ACM Transactions on Graphics (Proc. SIGGRAPH) 31, 4 (July).
15. DAVIS, T. A. 2006. Direct Methods for Sparse Linear Systems. SIAM.
16. FARBMAN, Z., FATTAL, R., LISCHINSKI, D., AND SZELISKI, R. 2008. Edge-preserving decompositions for multi-scale tone and detail manipulation. ACM Transactions on Graphics (Proc. SIGGRAPH) 27, 3 (August).
17. FARBMAN, Z., FATTAL, R., AND LISCHINSKI, D. 2011. Convolution pyramids. ACM Trans. Graph. 30 (December), 175:1- 175:8.
18. FATTAL, R., LISCHINSKI, D., AND WERMAN, M. 2002. Gradient domain high dynamic range compression. ACM Transactions on Graphics, 249-256.
19. FATTAL, R. 2009. Edge-avoiding wavelets and their applications. In ACM SIGGRAPH papers, ACM, New York, NY, USA, 22:1- 22:10.
20. FLEISHMAN, S., DRORI, I., AND COHEN-OR, D. 2003. Bilateral mesh denoising. ACM Transactions on Graphics (Proc. SIGGRAPH) 22, 3.
21. GOLUB, G., AND VAN LOAN, C. F. 1996. Matrix computation, third edition. The John Hopkins University Press, Baltimore and London.
22. KAZHDAN, M., AND HOPPE, H. 2008. Streaming multigrid for gradient-domain operations on large images. ACM Trans. Graph. 27, 3, 21:1-21:10.
23. KELNER, J. A., ORECCHIA, L., SIDFORD, A., AND ZHU, Z. A. 2013. A simple, combinatorial algorithm for solving SDD systems in nearly-linear time. arXiv preprint arXiv:1301.6628.
24. KINCAID, D., AND CHENEY, W. 1991. Numerical analysis: mathematics of scientific computing. Brooks/Cole Publishing Co., Pacific Grove, CA, USA.
25. KOUTIS, I., MILLER, G. L., AND PENG, R. 2010. Approaching optimality for solving SDD linear systems. Proceedings of FOCS 2010.
26. KOUTIS, I., MILLER, G. L., AND TOLLIVER, D. 2011. Combinatorial preconditioners and multilevel solvers for problems in computer vision and image processing. Computer Vision and Image Understanding, 1638-1646.
27. KOUTIS, I., MILLER, G. L., AND PENG, R. 2011. A nearlym log n time solver for SDD linear systems. In Foundations of Computer Science (FOCS), 2011 IEEE 52nd Annual Symposium on, IEEE, 590-598.
28. KRISHNAN, D., AND SZELISKI, R. 2011. Multigrid and multilevel preconditioners for computational photography. ACM Transactions on Graphics (Proc. SIGGRAPH Asia), 5.
29. KUSHNIR, D., GALUN, M., AND BRANDT, A. 2010. Efficient multilevel eigensolvers with applications to data analysis tasks. Pattern Analysis and Machine Intelligence, IEEE Transactions on 32, 8, 1377-1391.
30. LEVIN, A., LISCHINSKI, D., AND WEISS, Y. 2004. Colorization using optimization. ACM Transactions on Graphics, 689-694.
31. LEVIN, A., RAV-ACHA, A., AND LISCHINSKI, D. 2007. Spectral matting. Proceedings of CVPR.
32. LIPTON, R. J., ROSE, D. J., AND TARJAN, R. E. 1979. Generalized nested dissection. SIAM J. Numer. Anal. 16, 346-358.
33. LISCHINSKI, D., FARBMAN, Z., UYTTENDAELE, M., AND SZELISKI, R. 2006. Interactive local adjustment of tonal values. In ACM SIGGRAPH Papers, ACM, New York, NY, USA, 646-653.
34. LIU, R., AND ZHANG, H. 2007. Mesh segmentation via spectral embedding and contour analysis. Eurographics 26, 3.
35. LIVNE, O., AND BRANDT, A. 2011. Lean algebraic multigrid (LAMG): Fast graph laplacian solver. arXiv:1108.0123v1.
36. MOULTON, J. D., DENDY, JR., J. E., AND HYMAN, J. M. 1998. The black box multigrid numerical homogenization algorithm. J. Comput. Phys. 142 (May), 80-108.
37. PÉREZ, P., GANGNET, M., AND BLAKE, A. 2003. Poisson image editing. ACM Transactions on Graphics (Proc. SIGGRAPH) 22, 3, 313-318.
38. SAAD, Y. 2003. Iterative Methods for Sparse Linear Systems, second ed. Society for Industrial and Applied Mathematics.
39. SHI, L., YU, Y., BELL, N., AND FENG, W.-W. 2006. A fast multigrid algorithm for mesh deformation. In ACM SIGGRAPH 2006 Papers, ACM, New York, NY, USA, SIGGRAPH '06, 1108-1117.
40. SPIELMAN, D. A., AND TENG, S.-H. 2006. Nearly-linear time algorithms for preconditioning and solving symmetric, diagonally dominant linear systems. CoRR abs/cs/0607105.
41. SPIELMAN, D., A. 2010. Algorithms, graph theory and linear equations in Laplacian matrices. Proceedings of the International Congress of Mathematicians (ICM).
42. SUN, J., JIA, J., TANG, C.-K., AND SHUM, H.-Y. 2004. Poisson matting. In ACM SIGGRAPH Papers, ACM, New York, NY, USA, 315-321.
43. SZELISKI, R. 1990. Fast surface interpolation using hierarchical basis functions. Pattern Analysis and Machine Intelligence, IEEE Transactions on 12, 6 (jun), 513-528.
44. SZELISKI, R. 2006. Locally adapted hierarchical preconditioning. Proceedings of ACM SIGGRAPH.
45. TROTTENBERG, U., OOSTERLEE, C., AND SCHULLER, A. 2001. Multigrid. Academic Press.
46. VAIDYA, P. 1990. Solving linear equations with symmetric diagonally dominant matrices by constructing good preconditioners. Tech. rep., Department of Computer Science, University of Illinois at Urbana-Champaign, Urbana, IL.
47. VAN ĚK, P., MANDEL, J., AND BREZINA, M. 1996. Algebraic multigrid by smoothed aggregation for second and fourth order elliptic problems. Computing. 56, 3: 179-196.
48. VAN ĚK, P. 1995. Fast multigrid solver. Applications of Mathematics. 40, 1. 1-20.
49. XU, K., LI, Y., JU, T., HU, S.-M., AND LIU, T.-Q. 2009. Efficient affinity-based edit propagation using K-D tree. In ACM SIGGRAPH Asia papers, ACM, New York, NY, USA, 118:1- 118:6.
50. YSERENTANT, H. 1986. On the multi-level splitting of finite element spaces. Numerische Mathematik 49, 379-412. 10.1007/BF01389538.
51. ZEEUW, P. D. 1990. Matrix-dependent prolongations and restrictions in a blackbox multigrid solver. Journal of Computational and Applied Mathematics 33, 1, 1-27.
52. ZHU, Y., SIFAKIS, E., TERAN, J., AND BRANDT, A. 2010. An efficient multigrid method for the simulation of high-resolution elastic solids. ACM Trans. Graph. 29 (April), 16:1-16:18.