This Article 
   
 Share 
   
 Bibliographic References 
   
 Add to: 
 
Digg
Furl
Spurl
Blink
Simpy
Google
Del.icio.us
Y!MyWeb
 
 Search 
   
Toward High-Quality Gradient Estimation on Regular Lattices
April 2011 (vol. 17 no. 4)
pp. 426-439
ASCII Text
x 
 
Zahid Hossain, Usman R. Alim, Torsten Möller, "Toward High-Quality Gradient Estimation on Regular Lattices," IEEE Transactions on Visualization and Computer Graphics, vol. 17, no. 4, pp. 426-439, April, 2011.
 
 
BibTex
x
 
@article{ 10.1109/TVCG.2010.37,
author = {Zahid Hossain and Usman R. Alim and Torsten Möller},
title = {Toward High-Quality Gradient Estimation on Regular Lattices},
journal ={IEEE Transactions on Visualization and Computer Graphics},
volume = {17},
number = {4},
issn = {1077-2626},
year = {2011},
pages = {426-439},
doi = {http://doi.ieeecomputersociety.org/10.1109/TVCG.2010.37},
publisher = {IEEE Computer Society},
address = {Los Alamitos, CA, USA},
}
 
 
RefWorks Procite/RefMan/Endnote
x 
 
TY - JOUR
JO - IEEE Transactions on Visualization and Computer Graphics
TI - Toward High-Quality Gradient Estimation on Regular Lattices
IS - 4
SN - 1077-2626
SP426
EP439
EPD - 426-439
A1 - Zahid Hossain,
A1 - Usman R. Alim,
A1 - Torsten Möller,
PY - 2011
KW - Approximation theory
KW - Taylor series expansion
KW - normal reconstruction
KW - orthogonal projection
KW - body-centered cubic lattice
KW - box splines.
VL - 17
JA - IEEE Transactions on Visualization and Computer Graphics
ER -
 
Zahid Hossain, Simon Fraser University, Burnaby
Usman R. Alim, Simon Fraser University, Burnaby
Torsten Möller, Simon Fraser University, Burnaby
In this paper, we present two methods for accurate gradient estimation from scalar field data sampled on regular lattices. The first method is based on the multidimensional Taylor series expansion of the convolution sum and allows us to specify design criteria such as compactness and approximation power. The second method is based on a Hilbert space framework and provides a minimum error solution in the form of an orthogonal projection operating between two approximation spaces. Both methods lead to discrete filters, which can be combined with continuous reconstruction kernels to yield highly accurate estimators as compared to the current state of the art. We demonstrate the advantages of our methods in the context of volume rendering of data sampled on Cartesian and Body-Centered Cubic lattices. Our results show significant qualitative and quantitative improvements for both synthetic and real data, while incurring a moderate preprocessing and storage overhead.

[1] U.R. Alim and T. Möller, "A Fast Fourier Transform with Rectangular Output on the BCC and FCC Lattices," Proc. Eighth Int'l Conf. Sampling Theory and Applications (SampTA '09), May 2009.
[2] T. Blu and M. Unser, "Quantitative Fourier Analysis of Approximation Techniques: Part I—Interpolators and Projectors," IEEE Trans. Signal Processing, vol. 47, no. 10, pp. 2783-2795, Oct. 1999.
[3] B. Csebfalvi, "An Evaluation of Prefiltered Reconstruction Schemes for Volume Rendering," IEEE Trans. Visualization and Computer Graphics, vol. 14, no. 2, pp. 289-301, Mar./Apr. 2008.
[4] B. Csébfalvi and B. Domonkos, "Pass-Band Optimal Reconstruction on the Body-Centered Cubic Lattice," Proc. Conf. Vision, Modeling, and Visualization 2008, p. 71, Oct. 2008.
[5] B. Csébfalvi and B. Domonkos, "Prefiltered Gradient Reconstruction for Volume Rendering," J. WSCG, vol. 17, nos. 1-3, pp. 49-56, 2009.
[6] C. de Boor, K. Höllig, and S. Riemenschneider, Box Splines. Springer Verlag, 1993.
[7] D. den Hertog, R. Brekelmans, L. Driessen, and H. Hamers, "Gradient Estimation Schemes for Noisy Functions," technical report, 2003.
[8] D.E. Dudgeon and R.M. Mersereau, Multidimensional Digital Signal Processing, first ed. Prentice-Hall, Inc., 1984.
[9] S.C. Dutta Roy and B. Kumar, "Digital Differentiators," Handbook of Statistics, vol. 10, pp. 159-205, Elsevier Science Publishers B.V., 1993.
[10] A. Entezari, R. Dyer, and T. Möller, "Linear and Cubic Box Splines for the Body Centered Cubic Lattice," Proc. IEEE Conf. Visualization, pp. 11-18, Oct. 2004.
[11] A. Entezari, M. Mirzargar, and L. Kalantari, "Quasi Interpolation on the Body Centered Cubic Lattice," Computer Graphics Forum, vol. 28, pp. 1015-1022, 2009.
[12] A. Entezari, D. Van De Ville, and T. Möller, "Practical Box Splines for Reconstruction on the Body Centered Cubic Lattice," IEEE Trans. Visualization and Computer Graphics, vol. 14, no. 2, pp. 313-328, Mar./Apr. 2008.
[13] H. Farid and E. Simoncelli, "Differentiation of Discrete Multi-Dimensional Signals," IEEE Trans. Image Processing, vol. 13, no. 4, pp. 496-508, Apr. 2004.
[14] 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, vol. 28, no. 3, pp. 1023-1030, 2009.
[15] H. Hamers, R. Brekelmans, L. Driessen, and D. den Hertog, "Gradient Estimation Using Lagrange Interpolation Polynomials," technical report, 2003.
[16] G. Kindlmann and J.W. Durkin, "Semi-Automatic Generation of Transfer Functions for Direct Volume Rendering," Proc. IEEE Symp. Volume Visualization, pp. 79-86, 1998.
[17] G. Kindlmann, X. Tricoche, and C.-F. Westin, "Anisotropy Creases Delineate White Matter Structure in Diffusion Tensor MRI," Proc. Ninth Int'l Conf. Medical Image Computing and Computer-Assisted Intervention (MICCAI '06), pp. 126-133, Oct. 2006.
[18] G. Kindlmann, R. Whitaker, T. Tasdizen, and T. Möller, "Curvature-Based Transfer Functions for Direct Volume Rendering: Methods and Applications," Proc. IEEE Conf. Visualization, pp. 513-520, Oct. 2003.
[19] S.R. Marschner and R.J. Lobb, "An Evaluation of Reconstruction Filters for Volume Rendering," Proc. IEEE Conf. Visualization, pp. 100-107, Oct. 1994.
[20] T. Meng, B. Smith, A. Entezari, A.E. Kirkpatrick, D. Weiskopf, L. Kalantari, and T. Möller, "On Visual Quality of Optimal 3D Sampling and Reconstruction," Proc. Conf. Graphics Interface, pp. 265-272, May 2007.
[21] T. Möller, R. Machiraju, K. Mueller, and R. Yagel, "A Comparison of Normal Estimation Schemes," Proc. IEEE Conf. Visualization, pp. 19-26, Oct. 1997.
[22] T. Möller, R. Machiraju, K. Mueller, and R. Yagel, "Evaluation and Design of Filters Using a Taylor Series Expansion," IEEE Trans. Visualization and Computer Graphics, vol. 3, no. 2, pp. 184-199, Apr.-June 1997.
[23] T. Möller, K. Mueller, Y. Kurzion, R. Machiraju, and R. Yagel, "Design of Accurate and Smooth Filters for Function and Derivative Reconstruction," Proc. Symp. Volume Visualization, pp. 143-151, Oct. 1998.
[24] N. Neophytou and K. Mueller, "Space-Time Points: 4D Splatting on Efficient Grids," Proc. 2002 IEEE Symp. Volume Visualization and Graphics (VVS '02), pp. 97-106, 2002.
[25] G. Strang and G.J. Fix, "A Fourier Analysis of the Finite Element Variational Method," Constructive Aspects of Functional Analysis, pp. 796-830, 1971.
[26] G. Strang and G.J. Fix, An Analysis of the Finite Element Method. Prentice Hall, 1973.
[27] H. Sun, N. Kang, J. Zhang, and E.S. Carlson, "A Fourth-Order Compact Difference Scheme on Face Centered Cubic Grids with Multigrid Method for Solving 2D Convection Diffusion Equation," Math. and Computers in Simulation, vol. 63, no. 6, pp. 651-661, 2003.
[28] T. Theußl, T. Möller, and E. Gröller, "Optimal Regular Volume Sampling," Proc. IEEE Conf. Visualization 2001, pp. 91-98, Oct. 2001.
[29] M. Unser, "A General Hilbert Space Framework for the Discretization of Continuous Signal Processing Operators," Proc. SPIE Conf. Math. Imaging: Wavelet Applications in Signal and Image Processing III, pp. 51-61, Part I, July 1995.
[30] M. Unser, "Sampling-50 Years after Shannon," Proc. IEEE, vol. 88, no. 4, pp. 569-587, 2000.
[31] M. Unser, A. Aldroubi, and M. Eden, "B-Spline Signal Processing: Part I—Theory," IEEE Trans. Signal Processing, vol. 41, no. 2, pp. 821-833, Feb. 1993.

Index Terms:
Approximation theory, Taylor series expansion, normal reconstruction, orthogonal projection, body-centered cubic lattice, box splines.
Citation:
Zahid Hossain, Usman R. Alim, Torsten Möller, "Toward High-Quality Gradient Estimation on Regular Lattices," IEEE Transactions on Visualization and Computer Graphics, vol. 17, no. 4, pp. 426-439, April 2011, doi:10.1109/TVCG.2010.37
Usage of this product signifies your acceptance of the Terms of Use.