IEEE图像去噪精选论文

-
+ 2Nd +3)+ = 1;
N=l
Nd=O, N 2 2
0lNdlN-2
(6)
n = l,2,...,K
calculation of filtersp'M.') and
y ( M . M ) (M.1) = g(M.1)
itN. from: I'
(7)
(8)
- 1 : g ( n )= (-l)"+lh(n);
ABSTRACT
The paper addresses the problem of image denoising using adaptive wavelet transfoniis. Two scale-adaptive versions o f the wavelet transform are implemented and experimentally tested: the classical and the lifting one. In both o f them exemplary test images are contaminated with noise, decomposed into several multiresolution levels, modified via soft thresholding (only detail subimages) and inversely synthesized. On each level the LL image from the previous stage is decomposed into LL, LH, HL and HH subimages using different wavelet or predicdupdate lifting filters (L-lowpass, H-highpass). At each level as the best filters are chosen such ones, for which the normalized energy o f the detail HH subimage is the lowest. Image denoising is realized by soft thresholding o f all detail HH subimages only. Several test images are processed in the paper. Explicit equations for the design of lifting filters are presented.
updated on the base of d(n) and a smoothed signal approximation c(n) is obtained. Filters P(.) and U(.) can be chosen arbitrary. They can be even non-linear (e.g. median) and data-adaptive. Since the signal is reconstructed exactly in reversed order (see fig. I), the transform is always invertable. When the update filter is followed by the predict filter, the ”update-first” lifting scheme is get. Situation wHen both filters are linear has been discussed in detail in [3+4]. Some fruitful graphical illustrations and constraints for filters P(.) and U(.) have been derived and presented there. In particular, relations which should fulfil the filters P ( . ) and U(.) have been given. They assure exact decomposition of the signal x(n) into its low-pass and high-pass components (in figure 1 c(n) and d(n), respectively). In the lifting implementation of the wavelet transform standard high-pass and low-pass decomposition filters g(n) and h(n), respectively, are replaced with the predict and update filters P( .) and U( .) (figure 2). The design problem o f adaptive and non-linear ,,lifted wavelet transforms has been already discussed in literature [3+7]. It has been suggested there that in case of linear filters P(.) and U(.) the length of prediction one can change according to signal characteristics. For instance, the prediction filter should become shorter before and after abrupt change of data. With the help of the presented procedure one can design several pairs of predict & update filters and switch them according to the nature of decomposed data.
1. INTRODUCTION
Second-generation wavelet transforms have been introduced recently [l]. They are obtained using the socalled lifting construction. Lifting implementations of wavelet transform offers several advantages in comparison to the traditional filter bank approach, e.g. faster, in-place calculations and possibility o f designing adaptive, nonlinear, irregular-sampled and integer-to-integer wavelet transforms. It has been shown that any wavelet transfonn can be factored into lifting steps [2]. Block diagrams of the direct and inverse lifting schemes are presented in figure 1. The analyzed signal x(n) is first split into even and odd samples: x,(n) and odd xo(n). Then the odd samples are predicted from the even ones using any function, in particular a linear one. Predicted values are subtracted from x,(n). The resultant signal d(n) represents high-frequency content of x(n) - the so-called detail signal. Finally, the even samples xe(n) are
i f ( M = N = 1)
P
g(3)= I;
W ( N . K ) . Q ( K . N ) ,*(N.I) =g(N.l)
Let us denote the high-pass and low-pass analysis filters of the wavelet transform as g ( n ) and h(n), and the dual synthesis filters as g ( n ) and z ( t 7 ) . After the choice of lifting filters P(.) and U(.) the corresponding wavelet ones can be easily found from the tree-like signal flow diagrams presented in figure 2 or from the following equations:
n=1,2, ...,2 M - 1 :
n = l,2,...,M
g(n)=O; g ( 2 + 2 M d ) = I ;
g ( 2 n - 1) = - p ( n ) ;
h ( K )= C ( K ' N,)u ( ~ " ) ;h(2Md
n=1,2, ...,2 M - 1 :
h(n)=(-l)"g(n); h"(2+2Md)=1;
IMAGE DENOISING USING SCALE-ADAPTIVE LIFTING SCHEMES
Jacwenku.baidu.comk Stepien”, Tomasz ZieZinski*’, Roman Rumian”
’ ) Department of Electronics, *) Department of lnstrumentation and Measurement University of Mining and Metallurgy, al. Mickiewicza 30, 30-059 Krakbw, Poland tel: +(4812) 6172700, fax: +(4812) 6338565, e-mail: stepien@uci.agh.edu.pl
2. DESIGN OF LIFTING FILTERS
In this section we will extend results presented in [2, 61 and propose the explicit solution for the update filter U(.) as well as equations for the corresponding wavelet transform filters. Different choice o f parameters M, h f d , N and Nd leads to different pairs of lifting filters.
2.1 ,,Predict first” version of the forward lifting
Msamples long predict filterp‘M.I):
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0-7803-6297-7/00/$10.0002000 IEEE
2.3 Corresponding wavelet transform filters