Towards an Unbiased Comparison of CC, BCC, and FCC Lattices in Terms of Prealiasing

• Published in: Computer graphics forum Vol. 33; no. 3; pp. 81 - 90
• Main Authors: Vad, Viktor, Csébfalvi, Balázs, Rautek, Peter, Gröller, Eduard
• Format: Journal Article
• Language: English
• Published: Oxford Blackwell Publishing Ltd 01.06.2014
• Subjects: 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
• ISSN: 0167-7055; 1467-8659; 1467-8659
• DOI: 10.1111/cgf.12364

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.