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Fast algorithm for lapped nonorthogonal transform: application to the image Gabor transform

A fast algorithm to solve the problem of the expansion of a one or two-dimensional finite and discrete signal into a lapped non-orthogonal time-modulated set of functions is described, thus providing a solution to the particular problem of the Gabor transform of images. Starting with current methods, such as Bastiaan's auxiliary functions and the ZAK transform, we describe a new algorithm resulting in a significant decrease in CPU time for image Gabor coefficients with only a slight approximation. The complexity of this algorithm is equivalent to the block Fourier transform plus 2 to 4 operations for each pixel. The global complexity is thus in O(M) compared with the most rapid current method in O(M logM). This algorithm is presented in two formalisms: the transform and the filtering formalisms in order to show the interdependence of the two approaches. Finally, theoretical results are demonstrated by an implementation study of the new fast algorithm.