cwt-SPmag-nov05 - [Ivan W. Selesnick, Richard G. Baraniuk,...

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©ARTVILLE [ Ivan W. Selesnick, Richard G. Baraniuk, and Nick G. Kingsbury ] The Dual-Tree Complex Wavelet Transform [ A coherent framework for multiscale signal and image processing ] T he dual-tree complex wavelet transform ( C WT) 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 -dimen- sional signals, which is substantially lower than the undecimated DWT. The multidimensional (M-D) dual-tree C WT 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 sig- nal and image processing. We use the complex number symbol C in C WT to 1053-5888/05/$20.00©2005IEEE IEEE SIGNAL PROCESSING MAGAZINE [ 123 ] NOVEMBER 2005
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IEEE SIGNAL PROCESSING MAGAZINE [ 124 ] NOVEMBER 2005 avoid confusion with the often-used acronym CWT for the (dif- ferent) continuous wavelet transform. BACKGROUND This article aims to reach two different audiences. The first is the wavelet community, many members of which are unfamiliar with the utility, convenience, and unique properties of complex wavelets. The second is the broader class of signal processing folk who work with applications where the DWT has proven some- what disappointing, such as those involving complex or modulat- ed signals (radar, speech, and music, for example) or higher- dimensional, geometric data (geophysics and imaging, for exam- ple). In these problems, the complex wavelets can potentially offer significant performance improvements over the DWT. THE WAVELET TRANSFORM AND MULTISCALE ANALYSIS Since its emergence 20 years ago, the wavelet transform has been exploited with great success across the gamut of signal processing applications, in the process, often redefining the state-of-the-art performance [102], [112]. In a nutshell, the DWT replaces the infinitely oscillating sinusoidal basis functions of the Fourier transform with a set of locally oscillating basis functions called wavelets . In the classical setting, the wavelets are stretched and shifted versions of a fundamental, real-valued bandpass wavelet ψ( t ) . When carefully chosen and combined with shifts of a real-valued low-pass scaling function φ( t ) , they form an orthonormal basis expansion for one-dimensional (1-D) real-valued continuous-time signals [27]. That is, any finite- energy analog signal x ( t ) can be decomposed in terms of wavelets and scaling functions via x ( t ) = X n =−∞ c ( n )φ( t n ) + X j = 0 X n =−∞ d ( j , n ) 2 j / 2 2 j t n ). ( 1 ) The scaling coefficients c ( n ) and wavelet coefficients d ( j , n ) are computed via the inner products c ( n ) = Z −∞ x ( t t n ) dt ,( 2 ) d ( j , n ) = 2 j / 2 Z −∞ x ( t )ψ( 2 j t n ) .( 3 ) They provide a time-frequency analysis of the signal by measur- ing its frequency content (controlled by the scale factor j ) at dif-
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cwt-SPmag-nov05 - [Ivan W. Selesnick, Richard G. Baraniuk,...

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