### TABLE 12. The same measurements of the best basis search algorithm for the best wave level transform as in table 11 using the Vaidyanathan24 instead of the Daubechies8 filter.

### Table 2: Modelling results for real spectral data sets

"... In PAGE 7: ...Daubechies, Symmlet, Coiflet, ect.). The best basis in term of error in L2, according to a given threshold, is chosen at compression time. The results of Table2 were achieved for the spectral range 430-830nm sampled every 5nm. The spectral data sets were constructed by a decimation of the original data sets from our goniometer.... ..."

### TABLE 11. The number of times each decomposition level was chosen by the best basis search algorithm of the best wave level transform. The left number corresponds to the 60 transforms with a block length of 2048 and the right number corresponds to the 480 transforms with 256 samples.

### Table 2: Percent error classification. Numbers in brackets represent standard deviations. Wavelet1 uses the first 15 wavelet coefficients of the signal while Wavelet2 stands for a discrimination based on L2 distance of the distributions of the two classes. QDB1 means that first the quadratic discrim- ination best basis is found (see text for details) and then the first 12 or 15 coefficients of this basis are used. In Quadratic2, the discriminating coefficients based on the L2 distance of the distributions of the two classes are found from the same best basis.. In all cases the discrimination is done on each dimension separately.

"... In PAGE 8: ... We have achieved far better results on this frequency band, and thus provide more results to demonstrate the effect of different mother wavelets, different discrimination methods and different number of features. Table2 provides results of the experiments with the wide-band data. The top panel indicates that vast dimensionality reduction is essential for improved performance as the discriminating information is confined in a much smaller dimensional space.... ..."

### Table 1: Wavelet packets decomposition. Coifman and Wickerhauser introduced the best ba- sis algorithm [2], which results in a complete basis called the best basis which minimizes certain informa- tion cost function. Algorithm locally performs com- parison between a parent and the children nodes and helps us to keep the parent node or to \sink quot; deeper toward the children nodes. The entropy of a nonne- gative sequence p, with P pi = 1, was chosen as a popular discriminant measure: H(p) = ? X

"... In PAGE 2: ... Decomposition can be presented as a binary tree structure where nodes of the tree represent subspaces with di erent time-frequency lo- calization characteristics. In Table1 , wavelet packets are represented in a rectangular form. First line de- notes the original signal (a vector in vector space lt;8), while the following lines are calculated by using low (summing - s) and high (di erencing - d) lters.... ..."

### Table 2: Entropies attained by the conventional WPD, sub-optimal SIWPD (d lt;L) and optimal SIWPD (d=L) for acoustic pressure waveforms. The average entropy and the variance are lower when using the SIWPD, and they further decrease when d is larger. Table 1

1997

"... In PAGE 11: ... The number of real multiplications required by these algorithms for expanding a given waveform in its best basis are respectively 2560, 5120, 9216 and 31744. Table2 lists the attained entropies by the best-basis algorithms for an arbitrary subset of ten waveforms. Clearly, the average entropy is lower when using the SIWPD.... ..."

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### Table 1. PSNR(dB) versus bit rate results for DWT and WPT coded images.

"... In PAGE 7: ... The Lagrangian cost function in Equation (1) is used for the best basis selection. The PSNR results are tabulated in Table1 . Some of the WPT encoding results ob- tained by SFQ [9] coder are included for comparison.... ..."

### Table 2. Energy distributions in the 4 sub- bands for BMC-generated DFDs after 1-level wavelet decomposition

in Abstract In

"... In PAGE 2: ... Therefore, the wavelet packet transform is more suitable for DFD compression. Actually, based on the en- ergy distributions in Table2 , we can predict whether the wavelet packet transform DFD coding can outperform the pyramid wavelet transform. For example, for the Mobile sequence, the LL subband of DFD exhibits a relatively high percentage of energy so that the wavelet packet transform should not perform outstanding and maybe even lose its dominance in comparison with the wavelet pyramid.... In PAGE 3: ... bitrate=48kbps, gop_size=10 dwt (BMC) bba_norml (BMC) bba_norml (OBMC) Figure 3. The Coastguard sequence at 48 kBps As mentioned in the previous section, the energy dis- tribution in the subbands is a good indicator for the dom- inance of the pyramidal wavelet decomposition versus al- ternative tree structures: the Coastguard video sequence is explicitly well suited for a BB decomposition because en- ergy found in the 4 subbands of a 1-level decomposition is more or less evenly distributed ( Table2 ). We see that the su- periority of the best basis selection (using the Norm a2a4a3 func- tion [35] as cost measure) holds for (nearly) the entire range even if it is not clearly emphasized in every point.... ..."

### Table 1. \Coi et 6 quot; coe cients for orthogonal wavelet packets. n

1997

"... In PAGE 5: ... The resulting evolution may be called the \reference ow eld evolution quot;. Vorticity elds were computed at 300 equally-spaced future times and then their theoretical dimension was estimated in the wavelet packet best basis de ned by the \Coi et 6 quot; lters listed in Table1 . In Figure 6 may be seen the evolution of theoretical dimension for this reference evolution.... ..."

Cited by 2

### Table 1. \Coi et 6 quot; coe cients for orthogonal wavelet packets. n

1997

"... In PAGE 5: ... The resulting evolution may be called the \reference ow eld evolution quot;. Vorticity elds were computed at 300 equally-spaced future times and then their theoretical dimension was estimated in the wavelet packet best basis de ned by the \Coi et 6 quot; lters listed in Table1 . In Figure 6 may be seen the evolution of theoretical dimension for this reference evolution.... ..."

Cited by 2