The local Fourier dictionary: A natural tool for data analysis, in Wavelet Applications in Signal and Image Processing VII (M. A. Unser, A. Aldroubi, and A. F. Laine, eds.), Proc. SPIE 3813, pp.610-624, 1999. Invited paper.


The local Fourier dictionary contains a large number of localized complex exponential functions. Representations of a function using the dictionary elements locally inherit many nice properties of the conventional Fourier representation, such as translation invariance and orientation selectivity. In this paper, after giving an intuitive review of its construction, we describe an algorithm to recover location-dependent shifts of local features in signals for matching and registration, and propose a best local translation basis selected from the local Fourier basis. Then we will report our preliminary results on the statistical analysis of natural scene images using the local Fourier dictionary, whose purpose is to examine the importance of sparsity, statistical independence, and orientation selectivities in representation and modeling of such images.

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