The Community for Technology Leaders
RSS Icon
Issue No.06 - November/December (2010 vol.16)
pp: 1495-1504
Usman Alim , Simon Fraser University,Burnaby, BC, Canada
Torsten Möller , Simon Fraser University, Burnaby, BC, Canada
Laurent Condat , GREYC, A joint CNRS-UCBN-ENSICAEN Research Unit, Cedex,France
We investigate the use of a Fourier-domain derivative error kernel to quantify the error incurred while estimating the gradient of a function from scalar point samples on a regular lattice. We use the error kernel to show that gradient reconstruction quality is significantly enhanced merely by shifting the reconstruction kernel to the centers of the principal lattice directions. Additionally, we exploit the algebraic similarities between the scalar and derivative error kernels to design asymptotically optimal gradient estimation filters that can be factored into an infinite impulse response interpolation prefilter and a finite impulse response directional derivative filter. This leads to a significant performance gain both in terms of accuracy and computational efficiency. The interpolation prefilter provides an accurate scalar approximation and can be re-used to cheaply compute directional derivatives on-the-fly without the need to store gradients. We demonstrate the impact of our filters in the context of volume rendering of scalar data sampled on the Cartesian and Body-Centered Cubic lattices. Our results rival those obtained from other competitive gradient estimation methods while incurring no additional computational or storage overhead.
Derivative, Gradient, Reconstruction, Sampling, Lattice, Body Centered Cubic Lattice, Interpolation, Approximation, Frequency Error Kernel
Usman Alim, Torsten Möller, Laurent Condat, "Gradient Estimation Revitalized", IEEE Transactions on Visualization & Computer Graphics, vol.16, no. 6, pp. 1495-1504, November/December 2010, doi:10.1109/TVCG.2010.160
[1] U. R. Alim and T. Möller, A fast Fourier transform with rectangular output on the BCC and FCC lattices. In Proceedings of the Eighth International Conference on Sampling Theory and Applications (SampTA'09), Marseille, France, May 18–22, 2009.
[2] M. J. Bentum, T. Malzbender, and B. B. Lichtenbelt, Frequency analysis of gradient estimators in volume rendering. IEEE Transactions on Visualization and Computer Graphics, 2 (3): 242–254, Sept. 1996.
[3] T. Blu, P. Thévenaz, and M. Unser, Linear interpolation revitalized. IEEE Transactions on Image Processing, 13 (5): 710–719, May 2004.
[4] T. Blu and M. Unser, Quantitative Fourier analysis of approximation techniques: Part I—Interpolators and projectors. IEEE Transactions on Signal Processing, 47 (10): 2783–2795, October 1999.
[5] R. Brekelmans, L. Driessen, H. Hamers, and D. Den, Hertog. Gradient estimation schemes for noisy functions. Journal of Optimization Theory and Applications, 126 (3): 529–551, 2005.
[6] R. Brekelmans, L. Driessen, H. Hamers, and D. Den, Hertog. Gradient estimation using Lagrange interpolation polynomials. Journal of Optimization Theory and Applications, 136 (3): 341–357, 2008.
[7] L. Condat and T. Möller, Quantitative error analysis for the reconstruction of derivatives. Research Report hal-00462203, GREYC lab., Caen, France, Sept. 2009.
[8] L. Condat and D. Van De Ville, Quasi-interpolating spline models for hexagonally-sampled data. IEEE Transactions on Image Processing, 16 (5): 1195–1206, May 2007.
[9] B. Csébfalvi., An evaluation of prefiltered reconstruction schemes for volume rendering. IEEE Transactions on Visualization and Computer Graphics, 14 (2): 289–301, 2008.
[10] B. Csébfalvi and B. Domonkos, Prefiltered gradient reconstruction for volume rendering. Journal of WSCG, 17(1–3):49–56, 2009.
[11] C. de Boor, K. Höllig, and S. Riemenschneider, Box Splines. Springer Verlag, 1993.
[12] D. E. Dudgeon and R. M. Mersereau, Multidimensional Digital Signal Processing. Prentice-Hall, Inc., Englewood-Cliffs, NJ, 1st edition, 1984.
[13] S. C. Dutta Roy and B. Kumar, Handbook of Statistics, volume 10, chapter Digital Differentiators, pages 159–205. Elsevier Science Publishers B. V., North Holland, 1993.
[14] Y. C. Eldar and M. Unser, Nonideal sampling and interpolation from noisy observations in shift-invariant spaces. IEEE Trans. Image Proc. 54 (7): 2636–2651, July 2006.
[15] A. Entezari, R. Dyer, and T. Möller, Linear and Cubic Box Splines for the Body Centered Cubic Lattice. In Proceedings of the IEEE Conference on Visualization, pages 11–18, Oct. 2004.
[16] A. Entezari, D. Van De Ville, and T. Möller, Practical box splines for reconstruction on the body centered cubic lattice. IEEE Transactions on Visualization and Computer Graphics, 14 (2): 313–328, 2008.
[17] H. Farid and E. Simoncelli, Differentiation of discrete multi-dimensional signals. IEEE Transactions on Image Processing, 13 (4): 496–508, 2004.
[18] B. Finkbeiner, U. R. Alim, D. V. D. Ville, and T. Möller., High-quality volumetric reconstruction on optimal lattices for computed tomography. Computer Graphics Forum (Proceedings of the Eurographics/IEEE-VGTC Symposium on Visualization 2009 (Euro Vis 2009)), 28(3): 1023–1030, 2009.
[19] M. E. Goss, An adjustable gradient filter for volume visualization image enhancement. In Graphics Interface, pages 67–74, 1994.
[20] Z. Hossain, U. R. Alim, and T. Möller, Towards high quality gradient estimation on regular lattices. IEEE Transactions on Visualization and Computer Graphics, 99(PrePrints), 2010.
[21] S. R. Marschner and R. J. Lobb, An evaluation of reconstruction filters for volume rendering. In R. D. Bergeron, and A. E. Kaufman editors, Proceedings of the IEEE Conference on Visualization, pages 100–107. IEEE Computer Society Press, Oct. 1994.
[22] T. Meng, B. Smith, A. Entezari, A. E. Kirkpatrick, D. Weiskopf, L. Kalan-tari, and T. Moller, On visual quality of optimal 3D sampling and reconstruction. In Graphics Interface 2007, pages 265–272, May 2007.
[23] R. Mersereau and T. Speake, The processing of periodically sampled multidimensional signals. IEEE Transactions on Acoustics, Speech, and Signal Processing, ASSP- 31 (1): 188–194, 1983.
[24] T. Möller, R. Machiraju, K. Mueller, and R. Yagel, A comparison of normal estimation schemes. Proceedings of the IEEE Conference on Visualization, pages 19–26, Oct. 1997.
[25] T. Möller, K. Mueller, Y. Kurzion, R. Machiraju, and R. Yagel, Design of accurate and smooth filters for function and derivative reconstruction. Proceedings of the Symposium on Volume Visualization, pages 143–151, Oct 1998.
[26] N. Neophytou and K. Mueller, Space-time points: 4D splatting on efficient grids. In VVS '02: Proceedings of the 2002 IEEE Symposium on Volume Visualization, pages 97–106, Piscataway, NJ, USA, 2002.
[27] D. P. Petersen and D. Middleton, Sampling and reconstruction of wave-number-limited functions in N-dimensional euclidean spaces. Information and Control, 5 (4): 279–323, Dec. 1962.
[28] S. Riachy, S. Bachalany, M. Mboup, and J.-P. Richard, An algebraic method for multi-dimensional derivative estimation. In Proc. of 16th Med. Conf. on Control and Automation, pages 356–361, June 2008.
[29] C. E. Shannon, Communication in the presence of noise. Proceedings of the Institute of Radio Engineers, 37 (1): 10–21, 1949.
[30] G. Strang and G. J. Fix, A Fourier analysis of the finite element variational method. Constructive Aspects of Functional Analysis, pages 796–830, 1971.
[31] T. Theuβl, T. Möller, and E. Gröller, Optimal Regular Volume Sampling. In Proceedings of the IEEE Conference on Visualization 2001, pages 91–98, Oct 2001.
[32] T. Theuβl, H. Helwig, and E. Gröller, Mastering windows: improving reconstruction. In Proc. of the IEEE Symposium on Volume Visualization (VVS), pages 101–108, 2000.
[33] P. Thévenaz, T. Blu, and M. Unser, Interpolation revisited. IEEE Transactions on Medical Imaging, 19 (7): 739–758, July 2000.
[34] M. Unser, A general Hilbert space framework for the discretization of continuous signal processing operators. In Proceedings of the SPIE Conference on Mathematical Imaging: Wavelet Applications in Signal and Image Processing III, volume 2569, pages 51–61, San Diego CA, USA, July 9–14, 1995. Part I.
[35] M. Unser, Sampling-50 years after Shannon. Proceedings of the IEEE, 88 (4): 569–587, 2000.
[36] M. Unser, A. Aldroubi, and M. Eden, B-Spline signal processing: Part I—Theory. IEEE Transactions on Signal Processing, 41 (2): 821–833, February 1993.
[37] E. Young, Vector and tensor analysis. Marcel Dekker Inc, 1992.
13 ms
(Ver 2.0)

Marketing Automation Platform Marketing Automation Tool