# A Flexible and Simple Lossless DWT Filter Bank Using a MAXFLAT FIR Half-Band Filter
## 5 Conclusions  
The DWT filter bank is vastly utilized in large-scale operational applications requiring a computationally demanding task such as RS image retrieval, classification in a DWT compressed image archive, analyzing quantization noise for medical imaging, image encryption and image deblurring with convolutional neural network. However, existing two-channel filter banks provide the disadvantages of less flexibility (due to limited condition and structure) in designing lowpass and highpass analysis filters, causing undesired reconstruction errors and high computational complexity. In this paper, to address such issues, we proposed a novel two-band lossless s-filter bank with a computationally efficient and error-free structure, which consists of one single (analysis) filter and its mirrored subtraction loop. A unique lossless condition was shown to allow a closed-form half-band polynomial with zero odd-order coefficients, resulting in an explicit formula for directly obtaining integer-coefficient filters with maximally flat (MAXFLAT) half-band frequency response. The simulation results showed that the proposed s-filter banks have better performances than the lossless 5/3 and lossy 9/7 filter banks (of the JPEG2000), although providing low computational complexity of less than 50%. It can be concluded that this new approach allows more flexible and affordable PR DWT bank systems to be built, and it helps solve limited filter-design problems due to obtaining desired lossless filter banks.  
Conceptualization, D.C. and J.J.; methodology, J.J.; software, D.C.; validation, W.C. and Y.K.; formal analysis, D.C.; investigation, D.C.; resources, D.C. and J.J.; data curation, D.C. and Y.K; writing--original draft preparation, D.C.; writing--review and editing, D.C. and J.J.; visualization, W.C.; supervision, D.C.; project administration, D.C. and J.J.; funding acquisition, J.J. All authors have read and agreed to the published version of the manuscript.  
This work was supported by the Korea Institute of Energy Technology Evaluation and Planning (no. 2022400000020) and Korea Evaluation Institute of Industrial Technology (no. 20012884), which are funded by the Ministry of Trade, Industry and Energy of the Republic of Korea.  
Not applicable.  
Data Availability Statement: Data are contained within the article.  
The authors declare no conflict of interest.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Filter Bank**} & \multirow{2}{*}{**Band Split**} & \multicolumn{5}{c}{**Computation**} \\ \cline{3-6}  & & & **Multiplication** & **Addition** & **Shift** & **Total** \\ \hline
9/7 & Low & 5 & 5 & 0 & 18 \\ (irreversible) & High & 4 & 4 & 0 & 18 \\ \hline
5/3 & Low & 0 & 3 & 2 & \\ (reversible) & High & 0 & 2 & 1 & 8 \\ \hline  & 3-tap (\(K=1\)) & Low & 0 & 1 & 0 & 4 \\  & (\(K=2\)) & High & 0 & 3 & 2 & 6 \\ \cline{2-6} s-filter bank & 7-tap (\(K=3\)) & Low & 0 & 1 & 0 & 9 \\ \cline{2-6}  & (\(K=4\)) & High & 0 & 5 & 5 & 11 \\ \cline{2-6}  & 13-tap (\(K=5\)) & Low & 0 & 1 & 0 & 13 \\ \cline{2-6}  & (\(K=5\)) & High & 0 & 6 & 6 & 13 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Computational complexity comparison of two-channel analysis filter banks.