# A Flexible and Simple Lossless DWT Filter Bank Using a MAXFLAT FIR Half-Band Filter
## 4 Performance Evaluation  
For performance evaluation, a reversible/irreversible JPEG 2000 (part 1) coding scheme [28; 32] was used based on five grayscale images with resolution of 256 \(\times\) 256 pixels. To demonstrate the effectiveness of the proposed s-filter bank, the 2, 3, 5 and 7-taps s-filter banks were compared with the (reversible) 5/3 and (irreversible) 9/7 filter banks [28; 32]. Figure 4: The magnitude responses of two-taps filters: (**a**) half-band lowpass filter with order of 1 and (**b**) comb filter with order of 7. Then, the s-filter bank structure of Figure 2b was chosen creating relatively narrower high-pass bandwidth than Figure 2a. This is due to the fact that the narrow high-frequency bandwidth (by directly using a highpass filter) has lower entropy than the wide case (by lower-band subtraction loop). In the lossless (reversible) mode, each image was split into 4, 7 and 10 subbands and average entropy (bit-rate) was compared. Further, in the lossy (irreversible) mode, each image was split into the seven subband decompositions and coded at 0.6-0.4 bit per pixel (bpp) for Peak Signal to Noise Ratio (PSNR) comparison. Table 2 shows simulation results compared in the lossless compression mode of the JPEG2000 where integer operations by adding and shifting are performed. It can be seen that the proposed s-filter bank using a 3-tap filter (described in Table 1) provides lower bit rate(entropy) for test images than using the 5/3 filter bank, which is attributed to achieving a higher image compression ratio despite the use of only one 3-tap filter. Further, the PSNR comparison of lossy image compression is also indicated in Table 3. The proposed s-filter banks are experimentally verified to have PSNR performance that is better than or similar to the 9/7 filter banks, although providing a much more computationally efficient solution due to integer operations. Table 4 shows the comparison of computational complexity for two-channel analytic filter banks. It can be found that the proposed s-filter banks exhibit significantly low computational complexity of less than 50% of the (reversible) 5/3 and (irreversible) 9/7 filter banks. Consequently, the comparison results suggest that proposed lossless s-filter banks not only achieve relatively high image compression performance, but also provide low computational complexity without calculation error due to integer operation. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Filter Bank 1 (Reversible)**} & \multirow{2}{*}{**Number of Subbands**} & \multicolumn{5}{c}{**Test Image**} \\ \cline{3-7}  & & **[PERSON]** & **[PERSON]** & **[PERSON]** & **[PERSON]** & **[PERSON]** \\ \hline JPEG2000 & 4 & 4.16 & 4.22 & 3.76 & 3.86 & 4.89 \\
5/3 & 7 & 3.45 & 3.57 & 3.25 & 3.16 & 4.44 \\
10 & 3.30 & 3.44 & 3.14 & 3.00 & 4.33 \\ \hline s-filter bank & 4 & 3.90 & 4.04 & 3.53 & 3.64 & 4.65 \\
3-tap & 7 & 2.95 & 3.19 & 2.77 & 2.77 & 3.95 \\ (K = 1) & 10 & 2.74 & 2.99 & 2.59 & 2.56 & 3.76 \\ \hline \hline \end{tabular}  
* The simulations were performed in the lossless (reversible) JPEG2000 standard.  
\end{table} 
Table 2: Lossless compression bit-rate comparison (bpp).