Complex Daubechies wavelets: properties and statistical image modelling

This article presents the construction and various properties of complex Daubechies wavelets with a special emphasis on symmetric solutions. Such solutions exhibit interesting relationships between the real and imaginary components of the complex scaling function and the complex wavelet. We present those properties in the context of image processing. Within the framework of statistical modelling, we focus on the redundant description of real images given by the complex multiresolution representation. A hierarchical Markovian Graphical model is then explored. We present an Expectation Maximization algorithm for optimizing the model with observational complex wavelet data. This model is then applied to image estimation and texture classification. In both applications, we demonstrate the benefit brought by the Markovian hypothesis and the performance of the real images's complex multiscale representation. (C) 2003 Elsevier B.V. All rights reserved.