0.10 Wavelet-based signal processing and applications  (Page 12/13)

Energy concentration is one of the most important properties for low bitrate transform coding. Suppose for the sample quantization step size $T$ , we have a second set of basis that generate less significant coefficients. The distribution of the significant map indicators is moreskewed, thus require less bits to code. Also, we need to code less number of significant values, thus it may require less bits. In the mean time, asmaller $M$ reduces the second error term as in  [link] . Overall, it is very likely that the new basis improves the rate-distortionperformance. Wavelets have better energy concentration property than theFourier transform for signals with discontinuities. This is one of the main reasons that wavelet based compression methods usually out perform DCTbased JPEG, especially at low bitrate.

Improved wavelet based compression algorithms

The above prototype algorithm works well [link] , [link] , but can be further improved for its various building blocks [link] . As we can see from [link] , the significant map still has considerable structure, which could be exploited.Modifications and improvements use the following ideas:

• Insignificant coefficients are often clustered together. Especially, they often cluster around the same location across severalscales. Since the distance between nearby coefficients doubles for every scale, the insignificant coefficients often form a tree shape, as we cansee from Figure: Discrete Wavelet Transform of the Houston Skyline, using ${\psi }_{D{8}^{\text{'}}}$ with a Gain of $\sqrt{2}$ for Each Higher Scale . These so called zero-trees can be exploited [link] , [link] to achieve excellent results.
• The choice of basis is very important. Methods have been developed to adaptively choose the basis for the signal [link] , [link] . Although they could be computationally very intensive, substantial improvement canbe realized.
• Special run-length codes could be used to code significant map and values [link] , [link] .

  

• Advanced quantization methods could be used to replace the simple scalar quantizer [link] .
• Method based on statistical analysis like classification, modeling, estimation,and prediction also produces impressive result [link] .
• Instead of using one fixed quantization step size, we can successively refine the quantization by using smaller and smaller stepsizes. These embedded schemes allow both the encoder and the decoder to stop at any bit rate [link] , [link] .
• The wavelet transform could be replaced by an integer-to-integer wavelet transform, no quantization is necessary, and the compression islossless [link] .

Other references are: [link] , [link] , [link] , [link] , [link] , [link] , [link] , [link] , [link] , [link] , [link] , [link] , [link] , [link] .

Why are wavelets so useful?

The basic wavelet in wavelet analysis can be chosen so that it is smooth , where smoothness is measured in a variety of ways [link] . To represent $f\left(t\right)$ with $K$ derivatives, one can choose a wavelet $\psi \left(t\right)$ that is $K$ (or more) times continuously differentiable; the penalty for imposing greater smoothness in this senseis that the supports of the basis functions, the filter lengths and hence the computational complexity all increase. Besides, smooth wavelet basesare also the “best bases” for representing signals with arbitrarily many singularities [link] , a remarkable property.