A new implementation of the Discrete Wavelet Transform is presented for applications such as image restoration and enhancement. It employs a dual tree of wavelet filters to obtain the real and imaginary parts of the complex wavelet coefficients. This introduces limited redundancy (4 : 1 for 2-dimensional signals) and allows the transform to provide approximate shift invariance and directionally selective filters (properties lacking in the traditional wavelet transform) while preserving the usual properties of perfect reconstruction and computational efficiency. We show how the dual-tree complex wavelet transform can provide a good basis for multiresolution image denoising and de-blurring. 1 INTRODUCTION Although the Discrete Wavelet Transform (DWT) in its maximally decimated form (Mallat's dyadic filter tree [1]) has established an impressive reputation as a tool for image compression, its use for other signal analysis and reconstruction tasks, such as image restoration and enhancem...

this paper we consider how wavelets may be used for image processing. To date, there has been considerable interest in wavelets for image compression, and they are now commonly used by researchers for this purpose, even though the main international standards still use the discrete cosine transform (dct). However for image processing tasks, other than compression, the take-up of wavelets has been less enthusiastic. Here we analyse possible reasons for this and present some new ways to use wavelets which offer significant advantages. A good review of wavelets and their application to compression may be found in Rioul & Vetterli (1991) and in-depth coverage is given in the book by Vetterli & Kovacevic (1995). An issue of the Proceedings of the IEEE (Kovacevic & Daubechies 1996) has been devoted to wavelets and includes many very readable articles by leading experts. In x 2 of this paper we introduce the basic discrete wavelet filter tree and show how it may be used to decompose multi-dimensional signals. In x 3 we show some typical wavelets and illustrate the similar shapes of those which all satisfy the perfect reconstruction constraints. Unfortunately, as explained in x 4, discrete

. We build compactly supported biorthogonal wavelets and perfect reconstruction filter banks for any lattice in any dimension with any number of primal and dual vanishing moments. The resulting scaling functions are interpolating. Our construction relies on the lifting scheme and inherits all of its advantages: fast transform, in-place calculation, and integerto -integer transforms. We show that two lifting steps suffice: predict and update. The predict step can be built using multivariate polynomial interpolation, while update is a multiple of the adjoint of predict. Submitted to IEEE Transactions on Image Processing Over the last decade several constructions of compactly supported wavelets have originated both from signal processing and mathematical analysis. In signal processing, critically sampled wavelet transforms are known as filter banks or subband transforms [32, 43, 54, 56]. In mathematical analysis, wavelets are defined as translates and dilates of one fixed function and ar...

We discuss the shift invariant properties of a new implementation of the Discrete Wavelet Transform, which employs a dual tree of wavelet filters to obtain the real and imaginary parts of complex wavelet coefficients. This introduces limited redundancy (2 m :1 for m-dimensional signals) and allows the transform to provide approximate shift invariance and directionally selective filters (properties lacking in the traditional wavelet transform) while preserving the usual properties of perfect reconstruction and computational efficiency with good well-balanced frequency responses. 1. INTRODUCTION The Discrete Wavelet Transform (DWT) in its maximally decimated form (Mallat's dyadic filter tree [1]) has established an impressive reputation as a tool for signal compression, but its use for other signal analysis and reconstruction tasks has been hampered by two main disadvantages: ffl Lack of shift invariance, which means that small shifts in the input signal can cause major variations i...

In this paper we study the nonsubsampled contourlet transform. We address the corresponding filter design problem using the McClellan transformation. We show how zeroes can be imposed in the filters so that the iterated structure produces regular basis functions. The proposed design framework yields filters that can be implemented efficiently through a lifting factorization. We apply the constructed transform in image noise removal where the results obtained are comparable to the state-of-the art, being superior in some cases.

Abstract—In spite of the success of the standard wavelet transform (WT) in image processing in recent years, the efficiency of its representation is limited by the spatial isotropy of its basis functions built in the horizontal and vertical directions. One-dimensional (1-D) discontinuities in images (edges and contours) that are very important elements in visual perception, intersect too many wavelet basis functions and lead to a nonsparse representation. To efficiently capture these anisotropic geometrical structures characterized by many more than the horizontal and vertical directions, a more complex multidirectional (M-DIR) and anisotropic transform is required. We present a new lattice-based perfect reconstruction and critically sampled anisotropic M-DIR WT. The transform retains the separable filtering and subsampling and the simplicity of computations and filter design from the standard two-dimensional WT, unlike in the case of some other directional transform constructions (e.g., curvelets, contourlets, or edgelets). The corresponding anisotropic basis unctions (directionlets) have directional vanishing moments along any two directions with rational slopes. Furthermore, we show that this novel transform provides an efficient tool for nonlinear approximation of images, achieving the approximation power ( 1 55), which, while slower than the optimal rate ( 2), is much better than ( 1) achieved with wavelets, but at similar complexity. Index Terms—Directional vanishing moments, directionlets, filter banks, geometry, multidirection, multiresolution, separable filtering, sparse image representation, wavelets. I.

We present two- and three-dimensional nonseparable wavelets. They are obtained from discrete-time bases, by iterating filter banks. We consider three sampling lattices: quincunx, separable by two in two dimensions, and FCO. The design methods are based either on cascade structures or on the McClellan transformation, in the quincunx case. We give a few design examples. In particular, the first example of an orthonormal two-dimensional wavelet basis with symmetries, is constructed. During the past decade, the field of filter banks, or subband coding, has established itself firmly as one of the very successful methods for compressing signals ranging from speech to images to video [1, 2]. At the same time, and from another field -- applied mathematics, the theory of wavelets emerged as a powerful tool for providing time-frequency localized expansions of signals [3]. Recently, it has been shown that the two -- filter banks and wavelets, are closely connected, in that one can use iterated fi...

filter bank (DFB) for an efficient directional decomposition of 2-D signals. Due to the nonseparable nature of the system, extending the DFB to higher dimensions while still retaining its attractive features is a challenging and previously unsolved problem. We propose a new family of filter banks, named NDFB, that can achieve the directional decomposition of arbitrary-dimensional ( 2) signals with a simple and efficient tree-structured construction. In 3-D, the ideal passbands of the proposed NDFB are rectangular-based pyramids radiating out from the origin at different orientations and tiling the entire frequency space. The proposed NDFB achieves perfect reconstruction via an iterated filter bank with a redundancy factor of in-D. The angular resolution of the proposed NDFB can be iteratively refined by invoking more levels of decomposition through a simple expansion rule. By combining the NDFB with a new multiscale pyramid, we propose the surfacelet transform, which can be used to efficiently capture and represent surface-like singularities in multidimensional data. Index Terms—Directional decomposition, directional filter banks (DFBs), filter design, high-dimensional transforms, surfacelets. I.

Abstract—We present a new family of two-dimensional and three-dimensional orthogonal wavelets which uses quincunx sampling. The orthogonal refinement filters have a simple analytical expression in the Fourier domain as a function of the order, which may be noninteger. We can also prove that they yield wavelet bases of P @ PA for any H. The wavelets are fractional in the sense that the approximation error at a given scale decays like @ A; they also essentially behave like fractional derivative operators. To make our construction practical, we propose a fast Fourier transform-based implementation that turns out to be surprisingly fast. In fact, our method is almost as efficient as the standard Mallat algorithm for separable wavelets. Index Terms—McClellan transform, nonseparable filter design, quincunx sampling, wavelet transform. I.

In this paper we provide a new method for analyzing multidimensional filter banks. This method enables us to solve various open problems in multidimensional filter bank characterization and design. The essential element in this new approach is the redefinition of polyphase components. It will be shown that a rich set of mathematical tools, in particular algebraic group theory, will become available for use in the analysis of filter banks. We demonstrate the elegance and power of the tool set by employing it for the characterization of multidimensional filter banks, and by applying it to two open problems. The first problem is concerned with the development of a general method to design multichannel ( 2) multidimensional filter banks using transformations, while the second problem is concerned with the derivation of general restrictions on group delays in linear phase filter banks. The treatment of these problems is only an illustration of the power of the tool set of algebraic group th...