The dual-tree complex wavelet transform (CWT) is a relatively recent enhancement to the discrete wavelet transform (DWT), with important additional properties: It is nearly shift invariant and directionally selective in two and higher dimensions. It achieves this with a redundancy factor of only 2 d for d-dimensional signals, which is substantially lower than the undecimated DWT. The multidimensional (M-D) dual-tree CWT is nonseparable but is based on a computationally efficient, separable filter bank (FB). This tutorial discusses the theory behind the dual-tree transform, shows how complex wavelets with good properties can be designed, and illustrates a range of applications in signal and image processing. We use the complex number symbol C in CWT to

Most wavelet-based image coders fail to model the joint coherent behavior of wavelet coefficients near edges. Wedgelets offer a convenient parameterization for the edges in an image, but they have yet to yield a viable compression algorithm. In this paper, we propose an extension of the zerotree-based Space-Frequency Quantization (SFQ) algorithm by adding a wedgelet symbol to its tree-pruning optimization. This incorporates wedgelets into a ratedistortion compression framework and allows simple, coherent descriptions of the wavelet coefficients near edges. The resulting method yields improved visual quality and increased compression efficiency over the standard SFQ technique. 1.

Very low bit rate image coding is an important problem regarding applications such as storage on low memory devices or streaming data on the internet. The state of the art in image compression is to use 2-D wavelets. The advantages of wavelet bases lie in their multiscale nature and in their ability to sparsely represent functions that are piecewise smooth. Their main problem on the other hand, is that in 2-D wavelets are not able to deal with the natural geometry of images, i.e they cannot sparsely represent objects that are smooth away from regular submanifolds. In this paper we propose an approach based on building a sparse representation of images in a redundant geometrically inspired library of functions, followed by suitable coding techniques. Best N-term nonlinear approximations in general dictionaries is, in most cases, a NP-hard problem and sub-optimal approaches have to be followed. In this work we use a greedy strategy, also known as Matching Pursuit to compute the expansion. Finally the last step in our algorithm is an enhancement layer that encodes the residual image: in our simulation we have used a genuine embedded wavelet codec.

It is by now a well-established fact that the usual two-dimensional tensor product wavelet bases are not optimal for representing images consisting of different regions of smoothly varying grey values, separated by smooth boundaries. The chapter starts with a discussion of this phenomenon from a nonlinear approximation point of view, and then proceeds to describe approaches that have been suggested as a remedy. The methods can be sorted roughly into two groups: Adaptive geometry-based approaches such as wedgelets and related constructions on one hand, and directional frames, such as curvelets or ridgelets, on the other. We discuss wedgelets and curvelets in more details, as representatives of the different branches. These systems are first described in the continuous setting, and their construction is motivated by a discussion of their nonlinear approximation properties. We then present digital implementations of the schemes. For wedgelets and related transforms, we present a new method which results in a sig-nificant speedup, in comparison to preexisting implementations. We also give a short description of the contourlet approach to discrete curvelets. In the last section, we present the results of nonlinear approximation exper-iments, comparing wedgelets, contourlets and wavelets, and comment on the potential of the new techniques for image coding.

Abstract. Wavelet is well-suited for smooth images, but unable to economically represent edges. Wedgelet offers one powerful potential approach for describing edges in an image. it provides nearly-optimal representations of objects in the Horizon model, as measured by the minimax description length. In this paper, we proposed a multi-layered compression method based on Cartoon +Texture model which combined wedgelet and wavelet transforms, and where the coefficients of wedgelet and wavelet were coded with Huffman coding, run-length coding and SPIHT. Experiment results showed that the proposed method is effective and feasible in SAR image compression.