J Math Imaging Vis (2007) 28: 125–133 DOI 10.1007/s10851-007-0013-x Computing the Discrete Fourier Transform on a Hexagonal Lattice Andrew Vince · Xiqiang Zheng Published online: 20 June 2007 © Springer Science+Business Media, LLC 2007 Abstract The computation of the Discrete Fourier Trans- form for a general lattice in R d can be reduced to the computation of the standard 1-dimensional Discrete Fourier Transform. We provide a mathematically rigorous but sim- ple treatment of this procedure and apply it to the DFT on the hexagonal lattice. Keywords Discrete Fourier transform · Hexagonal lattice 1 Introduction Traditional image processing algorithms are usually carried out on rectangular arrays, but there is a growing research literature on image processing using other sampling grids [1, 4, 10, 12, 13]. Of particular interest is sampling on a hexagonal grid. Hexagonal grids provide for higher pack- ing density, give a more accurate approximation of circu- lar regions, and exhibit symmetric neighbor adjacency (i.e. the distance from the center of any hexagon to the center of any adjacent hexagon is the same). An extensive list of references can be found in [11]. One of the fundamental tools in image processing is the discrete Fourier transform (DFT) [14]. It is the intention of this paper to present a The authors would like to thank the PYXIS INNOVATION Inc for financial support, and Professors D. Wilson and G. Ritter for stimulating conversations on the Fourier transform. A. Vince ( ) · X. Zheng Department of Mathematics, Little Hall, PO Box 118105, University of Florida, Gainesville, FL 32611-8105, USA e-mail: vince@math.ufl.edu X. Zheng e-mail: xzheng@math.ufl.edu straightforward approach to the DFT for a general lattice L in R d , an approach perhaps more direct than previous treat- ments. The spatial domain of the DFT in this context is a set of coset representatives of the quotient of L by a sub- lattice of L. The computation of the DFT, even in this gen- eral case, can efficiently and easily be reduced to the com- putation of the standard 1-dimensional DFT. This result is then applied to hexagonal arrays, in particular to multires- olution arrays that allow for fast “zooming in” to view fine image detail or “zooming out” to view global image fea- tures. Previous approaches to the DFT on a hexagonal grid in- clude [3, 7], in which the DFT is converted to a square grid, and [15] in which the GBT (generalized balanced ternary) system for indexing a hexagonal grid is used. Our approach to the DFT uses the Smith normal form of a square inte- ger matrix. As pointed out by one of the referees, the use of this normalized form appeared previously in Problem 20, Chap. 2, of [1] and in [6]. This paper contributes a mathe- matically rigorous yet simple approach before applying it to the hexagonal lattice. This introductory section contains general background on the DFT. Section 2 concerns the spatial and frequency do- mains of the DFT for a general lattice. The result on reduc- ing the computation of the lattice DFT to the standard DFT appears in Sect. 3. Section 4 applies the results to the hexag- onal grid. In dimension 1, grid cells are unit intervals centered say, at the points 0, ±1, ±2,... . An image can be thought of as a complex valued function defined on a finite subset, say {0, 1, 2,...,N − 1}, of these points. Let C [N ] denote the vec- tor space of all such functions. The discrete Fourier trans- form is the linear transformation F : C [N ] → C [N ] defined by