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Mathematical Properties of the JPEG2000 Wavelet Filters

M. Unser, T. Blu

IEEE Transactions on Image Processing, vol. 12, no. 9, pp. 1080-1090, September 2003.



The LeGall 5⁄3 and Daubechies 9⁄7 filters have risen to special prominence because they were selected for inclusion in the JPEG2000 standard. Here, we determine their key mathematical features: Riesz bounds, order of approximation, and regularity (Hölder and Sobolev). We give approximation theoretic quantities such as the asymptotic constant for the L2 error and the angle between the analysis and synthesis spaces which characterizes the loss of performance with respect to an orthogonal projection. We also derive new asymptotic error formulæ that exhibit bound constants that are proportional to the magnitude of the first nonvanishing moment of the wavelet. The Daubechies 9⁄7 stands out because it is very close to orthonormal, but this turns out to be slightly detrimental to its asymptotic performance when compared to other wavelets with four vanishing moments.

Erratum

  • p. 1081, second column, Definition 4—Sobolev Regularity, the text should read "The largest real number sc such that" instead of "The smallest real number sc such that".


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