A Performance Analysis of Fast Gabor Transform Methods

complex exponentials. Bastiaans [2] dubbed this family of functions the Gabor elementary functions (GEF). He also Computation of the finite discrete Gabor transform can be accomplished in a variety of ways. Three representative methgeneralized the notion of Gabor’s expansion to include ods (matrix inversion, Zak transform, and relaxation network) non-Gaussian windows. were evaluated in terms of execution speed, accuracy, and The Gabor transform has several features which make stability. The relaxation network was the slowest method tested. it attractive for image representation. First, the nonstationIts strength lies in the fact that it makes no explicit assumptions ary nature of many natural signals [3, 4] mean that local, about the basis functions; in practice it was found that convernot global, frequency information is desirable; the Gabor gence did depend on basis choice. The matrix method requires transform provides this. The GEF also enjoy the property a separable Gabor basis (i.e., one that can be generated by of minimal joint uncertainty. Each GEF is thus maximally taking a Cartesian product of one-dimensional functions), but concentrated in space and spatial frequency; this property is faster than the relaxation network by several orders of magnicarries over to higher dimensional GEF also [5, 6]. It has tude. It proved to be a stable and highly accurate algorithm. The Zak–Gabor algorithm requires that all of the Gabor basis also been shown empirically [7] that the entropy of the functions have exactly the same envelope and gives no freedom Gabor transform of an image is much lower than the enin choosing the modulating function. Its execution, however, tropy of the pixel representation of the image. In fact, is very stable, accurate, and by far the most rapid of the three recent work indicates that Gabor expansions can provide methods tested.  1997 Academic Press better signal compression that can the DCT for very low bit rates [8]. The Gabor expanison is therefore recognized as a potential image compression tool [9–11].