Generalized wavelet transforms and the cortex transform

This paper presents a generalization of the continuous wavelet transform to higher dimensions. This generalization includes previous higher dimensional generalizations as special cases, without the loss of the elegance of the one-dimensional equations. One noteworthy special case is a transform that is closely related to the cortex transform developed by Watson (1987). This special case firmly relates the cortex transform to wavelet theory. This paper emphasizes the utility of the theory by using it to derive an implementation of the cortex transform. The generalized wavelet analysis introduced in this paper encompasses three important special cases of wavelet analysis: one-dimensional wavelet analysis, rectilinear wavelet analysis, and circular wavelet analysis. The preliminary results presented demonstrate that the wavelet-based cortex transform adequately represents an image and reconstructs it from its representation, provided some care is taken in the implementation