Image reconstruction from the phase or magnitude of its complex wavelet transform

This paper investigates the reconstruction of an image from the phase or magnitude of its complex wavelet transform (CWT). We view the CWT as an approximation to the analytic representation of some real wavelet coefficients and develop the conditions under which a 1D signal is uniquely specified by its analytic phase or magnitude. Then, we extend the uniqueness conditions to multi-resolution and higher dimensions in order to match the situation of the CWT. In the development of the uniqueness conditions, we also gain some insights about the quality of reconstructed images and the geometrical structure of the CWT phase and magnitude representation. Our results for the CWT may also be applied to other localized phase and magnitude representations.