Image Analysis Using a Dual-Tree M -Band
Caroline Chaux, Laurent C. Duval, Jean‐Christophe Pesquet · 2005
o We propose a 2D generalization to the M -band case of the dual-tree decomposition structure (initially proposed by N. Kingsbury and further investigated by I. Selesnick) based on a Hilbert pair of wavelets. We particularly address (i) the construction of the dual basis and (ii) the resulting directional analysis. We also revisit the necessary pre-processing stage in the M -band case. While several reconstructions are possible because of the redundancy of the representation, we propose a new optimal signal reconstruction technique, which minimizes potential estimation errors. The effectiveness of the proposed M - band decomposition is demonstrated via denoising comparisons on several image types (natural, texture, seismics), with various M -band wavelets and thresholding strategies. Signicant im- provements in terms of both overall noise reduction and direction preservation are observed. The classical discrete wavelet transform (DWT) provides a means of implementing a multiscale analysis, based on a critically sampled lter bank with perfect reconstruction. It has been shown to be very effective both theoretically and practically (3) in the processing of certain classes of signals, for instance piecewise smooth signals, having a nite number of discontinuities. But, while decimated transforms yield good compression performance, other data processing applications (analysis, denoising, detection) often require more sophisticated schemes than DWT. One rst drawback usually limiting the practical perfor- mance of DWT algorithms is their shift-variance with respect to the value of the transformed coefcients at a given scale. It often results in shift-variant edge artifacts at the vicinity of jumps, which are not desirable in real-world applications, signal delays being rarely known. A second drawback arises in dimensions greater than one: tensor products of standard wavelets usually possess poor directional properties. The later problem is sensitive in feature detection or denoising applications. A vast majority of the proposed solutions relies on adding some redundancy to the transform. Redundancy based on shift-invariant wavelet trans- forms (see (4), (5) and references therein) suppresses shift