We present a design method for designing 3D Mth band eigenfilters with 48-hedral symmetry, along with their application to sampling structure conversion of 3D signals defined on the face centered cubic (FCC) or body centered cubic (BCC) lattice to cartesian cubic (CC) lattice. While we present experimental results for FCC and BCC lattices, since these are commonly used 3D lattices, the design formulations we present are valid for other 3D lattices with 48-hedral symmetry as well. The sampling structure conversion of a signal sampled on FCC (or BCC) lattice to the CC lattice is a 3D interpolation problem, with the upsampling operation defined using the FCC (or BCC) lattice. In the design formulation for 3D eigenfilters, we impose the Mth band constraint, the so-called zero direct component (DC) leakage constraint, and the 48-hedral symmetry of the 3D filter impulse response. The Mth band constraint ensures that the original input samples are preserved at the output and the zero DC leakage results in the suppression of the zero (DC) frequency in the aliases due to the upsampling operation, thus improving the quality of the interpolated output. The symmetry results in the reduction of independent parameters in the filter design. Notation: Boldfaced lower-case letters (m) are used to represent vectors and boldfaced upper-case letters (M) are used for matrices. M T denotes the transpose of M and M −T denotes the inverse of transpose of M. The lattice generated by matrix M, denoted as LAT(M), is defined as the set of vectors m such that m = Mn for integer vectors n. Here, M is called the generating matrix for the lattice LAT(M). The generating matrix for a given lattice is not unique [1]. N(M) is the set of integer vectors of the form Mn, n ∈ [0, 1) d where d is the dimension. SPD(M), the symmetric parallelepiped of M, is defined as the set of vectors m such that m = Mn, n ∈ [-1, 1) d . In this paper, unless explicitly mentioned otherwise, we consider the 3D case (d = 3).
AUTHORSQutubuddin Saifee is currently pursuing his Ph. D at the (IIT, Bombay). He has been on the faculty of IIT, Bombay for over fifteen years to date. His areas of interest in research include communications and signal processing with special emphasis on wavelets and multi-resolution signal processing. Details of his professional qualifications and achievements can be found at the
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