Hostname: page-component-76fb5796d-zzh7m Total loading time: 0 Render date: 2024-04-25T12:12:05.177Z Has data issue: false hasContentIssue false
  • English
  • Français

3D Anisotropic Diffusion on GPUs by Closed-Form Local Tensor Computations

Published online by Cambridge University Press:  28 May 2015

Arjan Kuijper ,
Andreas Schwarzkopf ,
Thomas Kalbe ,
Chandrajit Bajaj ,
Stefan Roth  and
Michael Goesele
Arjan Kuijper*
Affiliation:
Fraunhofer IGD, 64283 Darmstadt, Germany Department of Computer Science, TU Darmstadt, D-64289 Darmstadt, Germany
Andreas Schwarzkopf
Affiliation:
Department of Computer Science, TU Darmstadt, D-64289 Darmstadt, Germany
Thomas Kalbe
Affiliation:
Department of Computer Science, TU Darmstadt, D-64289 Darmstadt, Germany
Chandrajit Bajaj
Affiliation:
ICES-CVC, University of Texas at Austin, Austin, Texas 78712, USA
Stefan Roth
Affiliation:
Department of Computer Science, TU Darmstadt, D-64289 Darmstadt, Germany
Michael Goesele
Affiliation:
Department of Computer Science, TU Darmstadt, D-64289 Darmstadt, Germany
*
Corresponding author.Email: arjan.kuijper@gris.informatik.tu-darmstadt.de
Get access

Abstract

We present an efficient implementation of volumetric anisotropic image diffusion filters on modern programmable graphics processing units (GPUs), where the mathematics behind volumetric diffusion is effectively reduced to the diffusion in 2D images. We hereby avoid the computational bottleneck of a time consuming eigenvalue decomposition in ℝ3. Instead, we use a projection of the Hessian matrix along the surface normal onto the tangent plane of the local isodensity surface and solve for the remaining two tangent space eigenvectors. We derive closed formulas to achieve this and prevent the GPU code from branching. We show that our most complex volumetric anisotropic diffusion filters gain a speed up of more than 600 compared to a CPU solution.

Keywords

68U10Image processingenhancementanisotropic diffusiontensors3D filtering
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 6 , Issue 1 , February 2013 , pp. 72 - 94
Copyright
Copyright © Global Science Press Limited 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Agelas, L. and Masson, R.. Convergence of thefinite volume MPFA O scheme for heterogeneous anisotropic diffusion problems on general meshes. Comptes Rendus Mathematique, 346(17-18):1007–1012, 2008.Google Scholar
[2] Al-Amoudi, A., Castaño-Diez, D., Devos, D.P., Russell, R.B., Johnson, G.T., and Frangakis, A.S.. The three-dimensional molecular structure of the desmosomal plaque. Proc. Natl. Acad. Sci. USA, 108(16):6480–5, 2011.Google Scholar
[3] Bajaj, C.L., Wu, Q., and Xu, G.. Level set based volumetric anisotropic diffusion for 3D image denoising. ICES TR03-10, UTexas, Austin USA, 2003.Google Scholar
[4] Bajaj, C.L. and Xu, G.. Adaptive surfaces fairing by geometric diffusion. In Symp. CAGD, pages 731–737, 2001.Google Scholar
[5] Bajaj, C.L. and Xu, G.. Anisotropic diffusion of surfaces and functions on surfaces. ACM Trans. Graph., 22(1):4–32, 2003.CrossRefGoogle Scholar
[6] Beyer, J., Langer, C., Fritz, L., Hadwiger, M., Wolfsberger, S., and Bühler, K.. Interactive diffusion-based smoothing and segmentation of volumetric datasets on graphics hardware. Methods Inf. Med., 46(3):270–274, 2007.Google Scholar
[7] Binotto, A., Weber, D., Daniel, C., Stork, A., Pereira, C.E., Kuijper, A., and Fellner, D.. Iterative sle solvers over a cpu-gpu platform. In 12th IEEE International Conference on High Performance Computing and Communications, IEEE HPCC-10 (September 1-3, 2010, Melbourne, Australia), pages 305–313. IEEE, 2010.Google Scholar
[8] Duan, Y., Wang, Y., Tai, X.-C., and Hahn, J.. A fast augmented lagrangian method for euler’s elastica model. In Third International Conference on Scale Space Methods and Variational Methods in Computer Vision (SSVM) (May 29th - June 2nd, 2011, Ein Gedi, Israel), LNCS 6667, pages 144–156, 2011.Google Scholar
[9] Durand, F. and Dorsey, J.. Fast bilateral filtering for the display of high-dynamic-range images. ACM Trans. Graph., 21(3):257–266, 2002.Google Scholar
[10] Frangakis, A.S. and Hegerl, R.. Nonlinear anisotropic diffusion in three-dimensional electron microscopy. In Scale-Space Theories in Computer Vision, Second International Conference, Scale-Space’99, LNCS 1682, pages 386–397, 1999.Google Scholar
[11] Galic, I., Weickert, J., Welk, M., Bruhn, A., Belyaev, A., and Seidel, H.-P.. The structure of images. Journal of Mathematical Imaging and Vision, 31:255–269, 2008.Google Scholar
[12] Golub, G.H. and Van Loan, C.F.. Matrix computations (3rd ed.). Johns Hopkins University Press, Baltimore, MD, USA, 1996.Google Scholar
[13] Grewenig, S., Weickert, J., and Bruhn, A.. Parallel implementations of AOS schemes: A fast way of nonlinear diffusion filtering. In Goesele, M., Roth, S., Kuijper, A., Schiele, B., Schindler, K. (Eds.): Pattern Recognition. LNCS 6376, pages 533–542, 2010.Google Scholar
[14] Hadwiger, M., Sigg, C., Scharsach, H., Bühler, K., and Gross, M.H.. Real-time ray-casting and advanced shading of discrete isosurfaces. Comp. Graph. Forum, 24(3):303–312, 2005.CrossRefGoogle Scholar
[15] Jeong, W., Whitaker, R., and Dobin, M.. Interactive 3D seismic fault detection on the graphics hardware. In Proc. Volume Graphics, pages 111–118, 2006.Google Scholar
[16] Kalbe, T., Koch, T., and Goesele, M.. High-quality rendering of varying isosurfaces with cubic1 trivariate c -continuous splines. In Advances in Visual Computing, 5th International Symposium, ISVC 2009 (1), LNCS 5875, pages 596–607, 2009.Google Scholar
[17] Kalbe, T., Schwarzkopf, A., Goesele, M., Kuijper, A., and Bajaj, C.. Volumetric nonlinear anisotropic diffusion on GPUs. In Third International Conference on Scale Space Methods and Variational Methods in Computer Vision (SSVM) (May 29th - June 2nd, 2011, Ein Gedi, Israel), LNCS 6667, pages 62–73, 2011.Google Scholar
[18] Keller, P.J., Schmidt, A.D., Santella, A., Khairy, K., Bao, Z., Wittbrodt, J., and Stelzer, E.H.K.. Fast, high-contrast imaging of animal development with scanned light sheet-based structured-illumination microscopy. Nat. Methods, 7(8):637–42, 2010.Google Scholar
[19] Kindlmann, G., Whitaker, R., Tasdizen, T., and Möller, T.. Curvature-based transfer functions for direct volume rendering: Methods and applications. In Proc. IEEE Vis., pages 513–520, 2003.Google Scholar
[20] Koenderink, J.J.. The structure of images. Biological Cybernetics, 50:363–370, 1984.Google Scholar
[21] Kuijper, A.. Geometrical PDEs based on second order derivatives of gauge coordinates in image processing. Image and Vision Computing, 27(8):1023–1034, 2009.Google Scholar
[22] Kuijper, A., Florack, L.M.J., and Viergever, M.A.. Scale space hierarchy. Journal of Mathematical Imaging and Vision, 18(2):169–189, April 2003.Google Scholar
[23] Kuijper, A. and Florack, L.M.J.. The hierarchical structure of images. IEEE Transactions on Image Processing, 12(9):1067–1079, 2003.Google Scholar
[24] Lindeberg, T.. Scale-Space Theory in Computer Vision. The Kluwer International Series in Engineering and Computer Science. Kluwer Academic Publishers, 1994.Google Scholar
[25] Lindeberg, T.. Generalized Gaussian scale-space axiomatics comprising linear scale-space, affine scale-space and spatio-temporal scale-space. Journal of Mathematical Imaging and Vision, pages 1–46, 2010.Google Scholar
[26] Lipnikov, K., Shashkov, M., Svyatskiy, D., and Vassilevski, Y.. Monotone finite volume schemes for diffusion equations on unstructured triangular and shape-regular polygonal meshes. Journal of Computational Physics, 227(1):492–512, 2007.CrossRefGoogle Scholar
[27] Morigi, S., Rucci, M., and Sgallari, F.. Nonlocal surface fairing. In Third International Conference on Scale Space Methods and Variational Methods in Computer Vision (SSVM) (May 29th - June 2nd, 2011, Ein Gedi, Israel), LNCS 6667, pages 38–49, 2011.Google Scholar
[28] Paris, S., Kornprobst, P., Tumblin, J., and Durand, F.. A gentle introduction to bilateral filtering and its applications. SIGGRAPH course, 2007.Google Scholar
[29] Perona, P. and Malik, J.. Scale-space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Analysis and Machine Intelligence, 12:629–639, 1990.Google Scholar
[30] Schwarzkopf, A.. Volumetrische anisotrope diffusion auf der GPU. Technische Universität Darmstadt, 2010.Google Scholar
[31] Sigg, C.. Representation and Rendering of Implicit Surfaces. PhD thesis, ETH Zurich, 2006.Google Scholar
[32] Tabik, S., Garzon, E.M., Garcia, I., and Fernandez, J.J.. Implementation of anisotropic nonlinear diffusion for filtering 3D images in structural biology on SMP clusters. In Parallel Computing: Current & Future Issues of High-End Computing, volume 33 of Proc. Int. Conf. ParCo, pages 727-734, 2005.Google Scholar
[33] Tasdizen, T., Whitaker, R., Burchard, P., and Osher, S.. Geometric surface smoothing via anisotropic diffusion of normals. In Proc. VIS ‘02, pages 125-132, 2002.Google Scholar
[34] Tomasi, C. and Manduchi, R.. Bilateral filtering for gray and color images. In Proc. IEEE Int. Conf. on Computer Vision ‘98, pages 839–846, 1998.CrossRefGoogle Scholar
[35] Weickert, J.. Scale-space properties of nonlinear diffusion filtering with a diffusion tensor. Technical report, No. 110, Laboratory of Technomathematics, University of Kaiserslautern, Germany, 1994.Google Scholar
[36] Weickert, J., Zuiderveld, K.J., Ter Haar|Romeny, B.M., and Niessen, W.J.. Parallel implementations of AOS schemes: A fast way of nonlinear diffusion filtering. In Proc. 1997 IEEE International Conference on Image Processing (ICIP-97, Santa Barbara, Oct. 26-29, 1997), Vol. 3, pages 396–399, 1997.Google Scholar
[37] Weickert, J.. A review of nonlinear diffusion filtering. In Scale-Space Theory in Computer Vision, volume 1252 of Lecture Notes in Computer Science, pages 1–28. Springer Berlin/Heidelberg, 1997.Google Scholar
[38] Weickert, J.. Anisotropic Diffusion in Image Processing. B.G. Teubner Stuttgart, 1998.Google Scholar
[39] Weickert, J.. Coherence enhancing diffusion filtering. International Journal of Computer Vision, 31:111–127, 1999.CrossRefGoogle Scholar
[40] Weickert, J.. Partial differential equations in image processing and computer vision. Habilitation thesis, University of Mannheim, 2001. http://www.mia.uni-saarland.de/weickert/Papers/habil.pdf.Google Scholar
[41] Zhang, X.F., Chen, WF., Qian, L., and Ye, H.. Affine invariant non-linear anisotropic diffusion smoothing strategy for vector-valued images. Imaging Science Journal, 58(3): 119–124, 2010.Google Scholar
[42] Zhao, Y.. Lattice Boltzman based PDE solver on the GPU. The Visual Computer, 24(5):323–333, 2008.CrossRefGoogle Scholar