Wavelet Transform (779450), страница 7

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The coding techniqueis known as embedded zerotree wavelet coding [131, 1281.We will not study the embedded zerotree coding here in great detail, butwe will give a rough idea of how such a coder works. Most importantly, thequantization of the waveletcoefficientsis carried out successively, using abitplane technique.

One starts with a coarse quantization and refines it inevery subsequent step. Whenever a tree of zeros (pixels quantized to zerowith respect to a given threshold) is identified, it will be coded as a so-calledzerotree by using a single codeword. Starting with coarse quantization ensuresthat a high numberof zerotrees can beidentified at the beginning of the encoding process. During the refinement process, when the quantization step size issuccessively reduced, the number of zerotrees successively decreases. Overallone gets an embedded bitstream where the most important information (interms of signal energy) is coded first. The refinementprocess can be continueduntil one reaches a desired precision. One of the most interesting features ofthis coding technique is that the bitstream can be truncated at any position,resulting in an almost optimal rate-distortion behavior for any bit rate.2638.1 0.

Wavelet-Based Denoising8.10Wavelet-Based DenoisingThe aim of denoising is to remove the noise w ( n ) from a signal+y(n) = z ( n ) w ( n ) .(8.222)For example, w ( n ) may be a Gaussian white noise process, which is statistically independent of z ( n ) .One tries to remove the noise by applying a nonlinear operation to the wavelet representation of y(n). The same problem hasbeen addressed in Chapter 7.3 in the context of the STFT,where it was solvedvia spectral subtraction. In fact, wavelet-based denoising is closely related tospectral subtraction. The maindifference between both approaches lies in thefact that the wavelets used for denoising are real-valued while the STFT iscomplex.(4(b)Figure 8.24. Thresholding techniques; (a) hard; (b) soft thresholding.The denoising procedure is as follows.

First, the signal y(n) is decomposedusing an octave-band filter bank, thus performinga discrete wavelet transform.Then, the wavelet coefficients are manipulated in order to remove the noisecomponent. Two approaches known as hard and soft thresholding have beenproposed for this purpose [43, 421. They use the following non-linearities:y(n), Y(n) > -E< --Ey(n)y(n),07Iy(n)l(hard)(8.223)I -EY(n) - -E, Y(n) > -E+-E,y(n)Iy(n)lFigure 8.24 illustrates both techniques.--EI -E(soft)(8.224)264TransformChapter 8. WaveletBasically, the idea of thresholding is that ~ ( ncan) be representedvia a fewwavelet coefficients, while the noise has wideband characteristics and spreadsout on all coefficients.

For example, this holds true if x ( n ) is a lowpass signal,while w ( n ) is white noise. The thresholding procedure then sets thesmallwavelet coefficients representing w ( n ) to zero, while the large coefficients dueto z(n) are only slightly affected. Thus, provided the threshold E ischosenappropriately, the signal @(n)reconstructed from the manipulated waveletcoefficientswill contain muchlessnoisethan y(n) does. Inpractice,theproblem is to choose E , because the amount of noiseis usually not knowna priori. If E is too small, the noise will not be efficiently removed.

If it is toolarge, the signal will be distorted..

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