Fast Cosine Transform for FCC Lattices

Abstract

Voxel representation and processing is an important issue in a broad spectrum of applications. E.g., 3D imaging in biomedical engineering applications, video game development and volumetric displays are often based on data representation by voxels. By replacing the standard sampling lattice with a face-centered lattice one can obtain the same sampling density with less sampling points and reduce aliasing error, as well. We introduce an analog of the discrete cosine transform for the facecentered lattice relying on multivariate Chebyshev polynomials. A fast algorithm for this transform is deduced based on algebraic signal processing theory and the rich geometry of the special unitary Lie group of degree four.Index Terms-discrete cosine transform (DCT), fast Fourier transform (FFT), FCC lattices, Chebyshev polynomials, volumetric image representation 978-1-5386-5346-3/18/$31.00

The Discrete Cosine Transform on Triangles

“…The Coxeter-Dynkin diagrams can be partitioned into series denoted by A n , B n , C n , and D n (and 5 additional special cases) [8]. In [5,6] A n -type Chebyshev polynomials were used. In this work we rely on multivariate Chebyshev polynomials of B 2 -type.…”

“…These techniques were used in [5] to derive a cosine transform together with its fast algorithm on the face-centered cubic lattice and in [6] on the hexagonal lattice. Both approaches relied on multivariate Chebyshev polynomials.…”

FFT and orthogonal discrete transform on weight lattices of semi-simple Lie groups

“…In several variables it is not known [51] if there are, up to affine-linear coordinate changes, any examples of polynomials with this property except the monomials and multivariate Chebyshev polynomials. The second approach in combination with multivari-ate Chebyshev polynomials was used to derive fast algorithms for undirected hexagonal [43] and FCC lattices [50].…”

“…Fast transforms for regular undirected lattices have been derived for the hexagonal lattice [43] and for the FCC lattice [50]. Both algorithms are special cases of a whole family, based on generalization of the Chebyshev polynomials to multivariate polynomials intimately connected to Lie theory.…”