Towards an Unbiased Comparison of CC, BCC, and FCC Lattices in Terms of Prealiasing
In the literature on optimal regular volume sampling, the Body‐Centered Cubic (BCC) lattice has been proven to be optimal for sampling spherically band‐limited signals above the Nyquist limit. On the other hand, if the sampling frequency is below the Nyquist limit, the Face‐Centered Cubic (FCC) latt...
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| Published in | Computer graphics forum Vol. 33; no. 3; pp. 81 - 90 |
|---|---|
| Main Authors | , , , |
| Format | Journal Article |
| Language | English |
| Published |
Oxford
Blackwell Publishing Ltd
01.06.2014
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| Subjects | |
| Online Access | Get full text |
| ISSN | 0167-7055 1467-8659 1467-8659 |
| DOI | 10.1111/cgf.12364 |
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| Abstract | In the literature on optimal regular volume sampling, the Body‐Centered Cubic (BCC) lattice has been proven to be optimal for sampling spherically band‐limited signals above the Nyquist limit. On the other hand, if the sampling frequency is below the Nyquist limit, the Face‐Centered Cubic (FCC) lattice was demonstrated to be optimal in reducing the prealiasing effect. In this paper, we confirm that the FCC lattice is indeed optimal in this sense in a certain interval of the sampling frequency. By theoretically estimating the prealiasing error in a realistic range of the sampling frequency, we show that in other frequency intervals, the BCC lattice and even the traditional Cartesian Cubic (CC) lattice are expected to minimize the prealiasing. The BCC lattice is superior over the FCC lattice if the sampling frequency is not significantly below the Nyquist limit. Interestingly, if the original signal is drastically undersampled, the CC lattice is expected to provide the lowest prealiasing error. Additionally, we give a comprehensible clarification that the sampling efficiency of the FCC lattice is lower than that of the BCC lattice. Although this is a well‐known fact, the exact percentage has been erroneously reported in the literature. Furthermore, for the sake of an unbiased comparison, we propose to rotate the Marschner‐Lobb test signal such that an undue advantage is not given to either lattice. |
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| AbstractList | In the literature on optimal regular volume sampling, the Body‐Centered Cubic (BCC) lattice has been proven to be optimal for sampling spherically band‐limited signals above the Nyquist limit. On the other hand, if the sampling frequency is below the Nyquist limit, the Face‐Centered Cubic (FCC) lattice was demonstrated to be optimal in reducing the prealiasing effect. In this paper, we confirm that the FCC lattice is indeed optimal in this sense in a certain interval of the sampling frequency. By theoretically estimating the prealiasing error in a realistic range of the sampling frequency, we show that in other frequency intervals, the BCC lattice and even the traditional Cartesian Cubic (CC) lattice are expected to minimize the prealiasing. The BCC lattice is superior over the FCC lattice if the sampling frequency is not significantly below the Nyquist limit. Interestingly, if the original signal is drastically undersampled, the CC lattice is expected to provide the lowest prealiasing error. Additionally, we give a comprehensible clarification that the sampling efficiency of the FCC lattice is lower than that of the BCC lattice. Although this is a well‐known fact, the exact percentage has been erroneously reported in the literature. Furthermore, for the sake of an unbiased comparison, we propose to rotate the Marschner‐Lobb test signal such that an undue advantage is not given to either lattice. In the literature on optimal regular volume sampling, the Body-Centered Cubic (BCC) lattice has been proven to be optimal for sampling spherically band-limited signals above the Nyquist limit. On the other hand, if the sampling frequency is below the Nyquist limit, the Face-Centered Cubic (FCC) lattice was demonstrated to be optimal in reducing the prealiasing effect. In this paper, we confirm that the FCC lattice is indeed optimal in this sense in a certain interval of the sampling frequency. By theoretically estimating the prealiasing error in a realistic range of the sampling frequency, we show that in other frequency intervals, the BCC lattice and even the traditional Cartesian Cubic (CC) lattice are expected to minimize the prealiasing. The BCC lattice is superior over the FCC lattice if the sampling frequency is not significantly below the Nyquist limit. Interestingly, if the original signal is drastically undersampled, the CC lattice is expected to provide the lowest prealiasing error. Additionally, we give a comprehensible clarification that the sampling efficiency of the FCC lattice is lower than that of the BCC lattice. Although this is a well-known fact, the exact percentage has been erroneously reported in the literature. Furthermore, for the sake of an unbiased comparison, we propose to rotate the Marschner-Lobb test signal such that an undue advantage is not given to either lattice. [PUBLICATION ABSTRACT] |
| Author | Gröller, Eduard Rautek, Peter Csébfalvi, Balázs Vad, Viktor |
| Author_xml | – sequence: 1 givenname: Viktor surname: Vad fullname: Vad, Viktor organization: Tampere University of Technology, Finland – sequence: 2 givenname: Balázs surname: Csébfalvi fullname: Csébfalvi, Balázs organization: Budapest University of Technology and Economics, Hungary – sequence: 3 givenname: Peter surname: Rautek fullname: Rautek, Peter organization: King Abdullah University of Science and Technology, Saudi Arabia – sequence: 4 givenname: Eduard surname: Gröller fullname: Gröller, Eduard organization: Vienna University of Technology, Austria |
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| Cites_doi | 10.1109/TVCG.2007.70573 10.1109/TVCG.2006.141 10.1016/S0019-9958(62)90633-2 10.1145/1925059.1925071 10.1109/TVCG.2009.87 10.1145/1980462.1980488 10.1109/TIT.2004.840864 10.1016/j.cag.2010.02.002 10.1109/TVCG.2013.7 10.1109/TVCG.2007.70429 10.1109/TVCG.2010.234 10.1109/TIP.2011.2162421 10.1109/TVCG.2011.230 10.1109/TVCG.2010.37 10.1109/VISUAL.1994.346331 10.1109/VISUAL.2001.964498 10.1109/TVCG.2008.115 10.1109/TVCG.2007.70414 10.1109/VISUAL.2004.65 |
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| Copyright | 2014 The Author(s) Computer Graphics Forum © 2014 The Eurographics Association and John Wiley & Sons Ltd. Published by John Wiley & Sons Ltd. 2014 The Eurographics Association and John Wiley & Sons Ltd. |
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| References | Csébfalvi B.: BCC-splines: Generalization of B-splines for the body-centered cubic lattice. Journal of WSCG 16, 1-3 (2008), 81-88. 2 Meng T., Entezari A., Smith B., Möller T., Weiskopf D., Kirkpatrick A.E.: Visual comparability of 3D regular sampling and reconstruction. IEEE Transactions on Visualization and Computer Graphics 17, 10 (2011), 1420-1432. 2, 5, 6, 8 Miyakawa H.: Sampling theorem of stationary stochastic variables in multidimensional space. J. Inst. Electron. Commun. Eng. Jpn. 42, 2 (1959), 421-427. 2, 4, 8 Qiu F., Xu F., Fan Z., Neophytou N., Kaufman A., Mueller K.: Lattice-based volumetric global illumination. IEEE Transactions on Visualization and Computer Graphics 13, 6 (2007), 1576-1583. 2, 4 Csébfalvi B.: An evaluation of prefiltered B-spline reconstruction for quasi-interpolation on the body-centered cubic lattice. IEEE Transactions on Visualization and Computer Graphics 16, 3 (2010), 499-512. 2 Entezari A., Van De Ville D., Möller T.: Practical box splines for reconstruction on the body centered cubic lattice. IEEE Transactions on Visualization and Computer Graphics 14, 2 (2008), 313-328. 2 Csébfalvi B.: Cosine-weighted B-spline interpolation: A fast and high-quality reconstruction scheme for the body-centered cubic lattice. IEEE Transactions on Visualization and Computer Graphics 19, 9 (2013), 1455-1466. 2, 8 Csébfalvi B.: An evaluation of prefiltered reconstruction schemes for volume rendering. IEEE Transactions on Visualization and Computer Graphics 14, 2 (2008), 289-301. 2 Domonkos B., Csébfalvi B.: Evaluation of the linear box-spline filter from trilinear texture samples: A feasibility study. Journal of WSCG 19, 2 (2011), 77-84. 8 Entezari A., Möller T.: Extensions of the Zwart-Powell box spline for volumetric data reconstruction on the Cartesian lattice. IEEE Transactions on Visualization and Computer Graphics 12, 5 (2006), 1337-1344. 2 Kim M., Entezari A., Peters J.: Box spline reconstruction on the face-centered cubic lattice. IEEE Transactions on Visualization and Computer Graphics 14, 6 (2008), 1523-1530. 2, 5, 6, 8 Mirzargar M., Entezari A.: Quasi interpolation with Voronoi splines. IEEE Transactions on Visualization and Computer Graphics 17, 12 (2011), 1832-1841. 2, 4, 5, 6, 8 Petersen D.P., Middleton D.: Sampling and reconstruction of wave-number-limited functions in n-dimensional Euclidean spaces. Information and Control 5, 4 (1962), 279-323. 2, 3, 4, 8 Conway J.H., Sloane N. J. A., Bannai E.: Sphere-packings, lattices, and groups. Springer-Verlag New York, Inc., 1987. 6 Hales T.C.: Cannonballs and honeycombs. AMS 47, 4 (1998), 440-449. 3 Ye W., Entezari A.: A geometric construction of multivariate sinc functions. IEEE Transactions on Image Processing 21, 6 (2012), 2969-2979. 2, 3, 4, 5, 6, 8 Künsch H.R., Agrell E., Hamprecht F.A.: Optimal lattices for sampling. IEEE Transactions on Information Theory 51, 2 (2005), 634-647. 6, 8 Oppenheim A.V., Schafer R.W.: Discrete-Time Signal Processing. Prentice Hall Inc., Englewood Cliffs, 2nd edition, 1989. 3 Hossain Z., Alim U.R., Möller T.: Towards high quality gradient estimation on regular lattices. IEEE Transactions on Visualization and Computer Graphics 17, 4 (2011), 426-439. 3 Mirzargar M., Entezari A.: Voronoi splines. IEEE Transactions on Visualization and Computer Graphics 58, 9 (2010), 4572-4582. 2, 3, 4, 5, 6, 8 Finkbeiner B., Entezari A., Van De Ville D., Möller T.: Efficient volume rendering on the body centered cubic lattice using box splines. Computers & Graphics 34, 4 (2010), 409-423. 8 2013; 19 2010; 34 1959; 42 2010; 16 2010; 58 2006; 12 2001 2012 2010 1962; 5 2008; 16 2009 1987 2008; 14 2005; 51 2006 1994 2004 2011; 17 2011; 19 2007; 13 2012; 21 1998; 47 1989 e_1_2_9_30_2 e_1_2_9_10_2 e_1_2_9_12_2 e_1_2_9_31_2 Mirzargar M. (e_1_2_9_21_2) 2010; 58 Miyakawa H. (e_1_2_9_24_2) 1959; 42 e_1_2_9_14_2 e_1_2_9_13_2 e_1_2_9_16_2 e_1_2_9_15_2 e_1_2_9_18_2 e_1_2_9_19_2 e_1_2_9_20_2 e_1_2_9_23_2 e_1_2_9_22_2 e_1_2_9_7_2 e_1_2_9_4_2 e_1_2_9_3_2 e_1_2_9_2_2 Conway J.H. (e_1_2_9_5_2) 1987 Csébfalvi B. (e_1_2_9_6_2) 2008; 16 e_1_2_9_9_2 Hales T.C. (e_1_2_9_17_2) 1998; 47 Oppenheim A.V. (e_1_2_9_26_2) 1989 e_1_2_9_8_2 Domonkos B. (e_1_2_9_11_2) 2011; 19 e_1_2_9_25_2 e_1_2_9_27_2 e_1_2_9_29_2 e_1_2_9_28_2 |
| References_xml | – reference: Finkbeiner B., Entezari A., Van De Ville D., Möller T.: Efficient volume rendering on the body centered cubic lattice using box splines. Computers & Graphics 34, 4 (2010), 409-423. 8 – reference: Conway J.H., Sloane N. J. A., Bannai E.: Sphere-packings, lattices, and groups. Springer-Verlag New York, Inc., 1987. 6 – reference: Hossain Z., Alim U.R., Möller T.: Towards high quality gradient estimation on regular lattices. IEEE Transactions on Visualization and Computer Graphics 17, 4 (2011), 426-439. 3 – reference: Künsch H.R., Agrell E., Hamprecht F.A.: Optimal lattices for sampling. IEEE Transactions on Information Theory 51, 2 (2005), 634-647. 6, 8 – reference: Ye W., Entezari A.: A geometric construction of multivariate sinc functions. IEEE Transactions on Image Processing 21, 6 (2012), 2969-2979. 2, 3, 4, 5, 6, 8 – reference: Meng T., Entezari A., Smith B., Möller T., Weiskopf D., Kirkpatrick A.E.: Visual comparability of 3D regular sampling and reconstruction. IEEE Transactions on Visualization and Computer Graphics 17, 10 (2011), 1420-1432. 2, 5, 6, 8 – reference: Csébfalvi B.: BCC-splines: Generalization of B-splines for the body-centered cubic lattice. Journal of WSCG 16, 1-3 (2008), 81-88. 2 – reference: Csébfalvi B.: An evaluation of prefiltered B-spline reconstruction for quasi-interpolation on the body-centered cubic lattice. IEEE Transactions on Visualization and Computer Graphics 16, 3 (2010), 499-512. 2 – reference: Oppenheim A.V., Schafer R.W.: Discrete-Time Signal Processing. Prentice Hall Inc., Englewood Cliffs, 2nd edition, 1989. 3 – reference: Entezari A., Van De Ville D., Möller T.: Practical box splines for reconstruction on the body centered cubic lattice. IEEE Transactions on Visualization and Computer Graphics 14, 2 (2008), 313-328. 2 – reference: Qiu F., Xu F., Fan Z., Neophytou N., Kaufman A., Mueller K.: Lattice-based volumetric global illumination. IEEE Transactions on Visualization and Computer Graphics 13, 6 (2007), 1576-1583. 2, 4 – reference: Miyakawa H.: Sampling theorem of stationary stochastic variables in multidimensional space. J. Inst. Electron. Commun. Eng. Jpn. 42, 2 (1959), 421-427. 2, 4, 8 – reference: Hales T.C.: Cannonballs and honeycombs. AMS 47, 4 (1998), 440-449. 3 – reference: Mirzargar M., Entezari A.: Quasi interpolation with Voronoi splines. IEEE Transactions on Visualization and Computer Graphics 17, 12 (2011), 1832-1841. 2, 4, 5, 6, 8 – reference: Csébfalvi B.: Cosine-weighted B-spline interpolation: A fast and high-quality reconstruction scheme for the body-centered cubic lattice. IEEE Transactions on Visualization and Computer Graphics 19, 9 (2013), 1455-1466. 2, 8 – reference: Entezari A., Möller T.: Extensions of the Zwart-Powell box spline for volumetric data reconstruction on the Cartesian lattice. IEEE Transactions on Visualization and Computer Graphics 12, 5 (2006), 1337-1344. 2 – reference: Petersen D.P., Middleton D.: Sampling and reconstruction of wave-number-limited functions in n-dimensional Euclidean spaces. Information and Control 5, 4 (1962), 279-323. 2, 3, 4, 8 – reference: Domonkos B., Csébfalvi B.: Evaluation of the linear box-spline filter from trilinear texture samples: A feasibility study. Journal of WSCG 19, 2 (2011), 77-84. 8 – reference: Mirzargar M., Entezari A.: Voronoi splines. IEEE Transactions on Visualization and Computer Graphics 58, 9 (2010), 4572-4582. 2, 3, 4, 5, 6, 8 – reference: Csébfalvi B.: An evaluation of prefiltered reconstruction schemes for volume rendering. IEEE Transactions on Visualization and Computer Graphics 14, 2 (2008), 289-301. 2 – reference: Kim M., Entezari A., Peters J.: Box spline reconstruction on the face-centered cubic lattice. 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| Snippet | In the literature on optimal regular volume sampling, the Body‐Centered Cubic (BCC) lattice has been proven to be optimal for sampling spherically band‐limited... In the literature on optimal regular volume sampling, the Body-Centered Cubic (BCC) lattice has been proven to be optimal for sampling spherically band-limited... |
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| SubjectTerms | Analysis Body centered cubic lattice Categories and Subject Descriptors (according to ACM CCS) Computer graphics Cubic lattice Errors Face centered cubic lattice I.3.3 [Computer Graphics]: Picture/Image Generation-Display algorithms I.4.10 [Image Processing and Computer Vision]: Image representation-Volumetric Intervals Lattices Optimization Sampling Studies |
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| Title | Towards an Unbiased Comparison of CC, BCC, and FCC Lattices in Terms of Prealiasing |
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