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