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Multilevel adaptive thresholding and shrinkage technique for denoising using Daubechies complex wavelet transform A Kharea, U S Tiwaryb, W Pedryczc and Moongu Jeon*d a Department of Electronics and Communication, University of Allahabad, Allahabad, India b
Indian Institute of Information Technology, Allahabad, India
c
Department of Electrical and Computer Engineering, University of Alberta, Edmonton, Alberta, Canada
d
School of Information and Communications, Gwangju Institute of Science and Technology, Gwangju, Korea
Abstract: In this paper, we have proposed a multilevel soft thresholding technique for noise removal in Daubechies complex wavelet transform domain. Two useful properties of Daubechies complex wavelet transform, approximate shift invariance and strong edge representation, have been explored. Most of the uncorrelated noise gets removed by shrinking complex wavelet coefficients at the lowest level, while correlated noise gets removed by only a fraction at lower levels, so we used multilevel thresholding and shrinkage on complex wavelet coefficients. The proposed method firstly detects strong edges using imaginary components of complex coefficients and then applies multilevel thresholding and shrinkage on complex wavelet coefficients in the wavelet domain at non-edge points. The proposed threshold depends on the variance of wavelet coefficients, the mean and the median of absolute wavelet coefficients at a particular level. Dependence of these parameters makes this method adaptive in nature. Results obtained for one-dimensional signals and two-dimensional images demonstrate an improved denoising performance over other related methods available in the literature. Keywords: Daubechies complex wavelet, denoising, wavelet shrinkage, multilevel thresholding, edge detection
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INTRODUCTION
The discrete wavelet transform (DWT) provides a fast, local, sparse and decorrelated multiresolution analysis of signals. These properties are very much exploited for denoising.1–3 DWT also has serious disadvantages, such as shift sensitivity4 and no phase information.5 Several researchers have provided solutions for minimizing these disadvantages. Some The MS was accepted for publication on 20 February 2010. * Corresponding author: Moongu Jeon, Department of Information and Communications, Gwangju Institute of Science and Technology, Gwangju, Korea; email:
[email protected]
The Imaging Science Journal Vol 58
of them have suggested stationary,6 cycle spinning,7 shiftable8 and steerable9 wavelet transforms, for avoiding shift sensitivity, while the recent trend for avoiding both disadvantages is to use complex-valued wavelet transform.5,10–13 A little has been explored about complex wavelet transform, although complex wavelet transform existed for a long time. The complex wavelet transform extends the idea of real-valued wavelet transform, using complex valued filters in place of real-valued filters. For making approximately shiftinvariant redundant wavelet transform, Gopinath5 has given the concept of phaselet transform that IMAG MP211 # RPS 2010
DOI: 10.1179/136821910X12750339175826
MULTILEVELF SHRINKAGE TECHNIQUE FOR DENOISING
includes the Kingsbury’s dual tree complex wavelet transform (DTCWT)10 as a special case. Kingsbury’s DTCWT has limited redundancy of 2m : 1 for mdimensional signals and has the property of approximate shift invariance. DTCWT uses short-linear phase filters for perfect reconstruction which have good directional selectivity as Gabor-like filters. Despite these advantages, DTCWT use real filters instead of complex filters. Fernandes et al.11 introduced projection-based complex wavelet transform to overcome some of the disadvantages of DTCWT. It consists of projection onto a complex function space followed by a DWT of the complex projection. In both of above complex transforms, use of real filters makes them seemingly complex wavelet transform14 and they are computationally costly as well. For providing phase information, an approximate shift invariance Daubechies complex wavelet transform2 is used, which is proposed by Lawton12 and Lina and Mayrand.13 The Daubechies complex wavelet transform is the natural extension of concepts of Daubechies real-valued wavelet transform. As Daubechies15 has given real solutions for wavelet equation, Lina and Lawton have explored the complex solutions for the equation and proved that complex solutions do exist leading to complex Daubechies wavelet transform. Complex Daubechies wavelets can be made symmetric. The symmetry property of the filter makes it easy to handle the boundary problems for finite length signal.16 A linear phase filter is required to preclude the non-linear phase distortion and to keep the shape of the signal. However, obtaining linear phase for complex Daubechies wavelet is difficult. Recently, a method to achieve both symmetry and approximate linear phase on complex Daubechies wavelet was proposed.17 In the present paper, we have used symmetric and approximately linear phase complex Daubechies wavelet filter, which is nearly shift-invariant and non-redundant. The concept of multiresolution analysis has also been extended for complex wavelet transform, known as complex multiresolution analysis.18 However, the application of complex multiresolution wavelet analysis in signal processing is almost an unexplored area. Clonda et al.18 and Jalobeanu et al.19 used the multiresolution analysis concepts in complex wavelet domain for hierarchical modelling of wavelet coefficients of an image. We have proved that the multiresolution analysis in real wavelet transform IMAG MP211 # RPS 2010
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domain is very much useful in signal denoising applications, due to its capability of removing different fractions of noise at different levels.3 In the case of signal-dependent noise, multilevel noise removal is essential. This framework has been extended in complex wavelet transform domain in the present work. In contrary to removing only Gaussian additive noise from the signal, as in earlier methods9,14,18,20–24 the proposed method can remove various types of noise better without the need for assuming any specific noise model. Removal of different types of noise requires a function which can estimate the noise-free signal for various types of noise. It has been shown in Section 3.3, that the nature of this function is quadratic. We have proposed the novel soft shrinkage method which thresholds the signal up to a particular calculated value of wavelet coefficients and shrinks the higher value coefficients by different proportions. The parameters for this function are calculated through the observed image statistics. This has made the method highly adaptive and universal in the sense that different types of noise in different proportions can be removed. The computed threshold depends on the standard deviation of wavelet coefficient, the mean and the median of absolute wavelet coefficients at each octave of wavelet decomposition. While most of the methods of denoising smooth the signal at all points, the proposed method preserves the strong edge points during the denoising process. These edge points are detected adaptively using the imaginary component of complex scaling coefficients and applies shrinkage at non-edge points. This is very helpful for preserving the contrast of objects in twodimensional images. The proposed method has been compared with some of the earlier thresholding methods1,3,25 using DWT, BLS-GSM denoising,26 SURE-LET denoising,27 Kingsbury’s DTCWT-based denoising,21 HMT-based denoising,28 as well as Clonda’s Daubechies complex wavelet transformbased denoising,18 and it has been shown that the present method yields far better results. The objective of this paper is two-fold: the first is to show the usability of Daubechies complex wavelet transform for denoising and the second is to propose a new edgesensitive multilevel adaptive denoising method in complex wavelet domain, which does not require any parameter except the complex wavelet coefficients. The paper is organized as follows: Section 2 describes the basic concepts of complex Daubechies The Imaging Science Journal Vol 58
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wavelet transform and its properties. In Section 3, the details of the proposed algorithm, including the method for edge detection in complex wavelet domain, the threshold and shrinkage selection, are given. In Section 4, the results of the present method for denoising are shown and compared to other methods and finally, in Section 5, conclusions are given.
2 MULTIRESOLUTION COMPLEX DAUBECHIES WAVELET TRANSFORM ANALYSIS OF SIGNALS
Construction of complex Daubechies wavelet
The basic equation of multiresolution theory is the scaling equation X wðtÞ~2 an wð2t{nÞ (1) n
where an is an coefficient. The an can be real as well as complex valued and San51. Daubechies’s wavelet bases {yj,k(t)} in one dimension are defined through the above scaling function and multiresolution analysis of L2 ð
