Biomedical Imaging Group, Swiss Federal Institute of Technology Lausanne, CH-1015 Lausanne, Switzerland.
IEEE Trans Image Process. 2003;12(9):1080-90. doi: 10.1109/TIP.2003.812329.
The LeGall 5/3 and Daubechies 9/7 filters have risen to special prominence because they were selected for inclusion in the JPEG2000 standard. 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 formulae 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.
LeGall 5/3 和 Daubechies 9/7 滤波器因其被选入 JPEG2000 标准而备受关注。我们确定了它们的关键数学特征：Riesz 界、逼近阶和正则性（Hölder 和 Sobolev）。我们给出了逼近理论的数量，如 L2 误差的渐近常数和分析空间与合成空间之间的角度，它们刻画了相对于正交投影的性能损失。我们还推导出了新的渐近误差公式，其中显示出的边界常数与小波的第一个非零矩的大小成正比。Daubechies 9/7 脱颖而出，因为它非常接近正交，但与具有四个零点的其他小波相比，这对其渐近性能略有不利。
