The generalized Gabor transform

The generalized Gabor transform (for image representation) is discussed. For a given function f(t), t∈R, the generalized Gabor transform finds a set of coefficients a_mr such that f(t)=Σ_m=-∞^∞Σ _r=-∞^∞α_mr g(t-mT)exp(i2πrt/T´). The original Gabor transform proposed by D. Gabor (1946) is the special case of T=T´. The computation of the generalized Gabor transform with biorthogonal functions is discussed. The optimal biorthogonal functions are discussed. A relation between a window function and its optimal biorthogonal function is presented based on the Zak (1967) transform when T/T´ is rational. The finite discrete generalized Gabor transform is also derived. Methods of computation for the biorthogonal function are discussed. The relation between a window function and its optimal biorthogonal function derived for the continuous variable generalized Gabor transform can be extended to the finite discrete case. Efficient algorithms for the optimal biorthogonal function and generalized Gabor transform for the finite discrete case are proposed