# A Flexible and Simple Lossless DWT Filter Bank Using a MAXFLAT FIR Half-Band Filter
## 1 Introduction  
The discrete wavelet transform (DWT) filter bank using two-band quadrature mirror filter (QMF) structure provides the advantage of separating the input signal into several frequency subbands in both time and wavelet transform domains [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28]. Recently, the DWT filter bank is vastly utilized in large-scale operational applications requiring a computationally demanding task such as remote-sensing (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 [1; 2; 3; 4; 5; 6; 7; 8; 9]. Currently, Joint Photographic Experts Group (JPEG) 2000, available to lossy and lossless DWT, is most utilized for various image applications. To cite an example, [PERSON] et al. [1] dealt with the denoising effect of the JPEG2000 for the compression of noisy images. [PERSON] et al. [2; 3] reported RS image retrieval and classification in a JPEG2000 compressed image archive. Further, [PERSON] et al. [4] presented a wavelet-based multimodal compression method that jointly compresses a medical image and an electrocardiogram (ECG) signal within a JPEG2000 single codec [1; 2; 3; 4; 5; 6; 7; 8; 9]. This research focus on minimizing the amount of computational complexity for image decompression, but there exists the disadvantage that should use two different wavelet filters (Cohen-Daubechies-Wavelet 9/7) causing a lot of computation in the lossy-mode JPEG2000 [1; 2; 3; 4; 5; 6; 7; 8; 9; 10]. For the design of lossless DWT filter bank, a spectral factorization algorithm through energy partitioning in the z-plane was proposed by [PERSON] et al. [11]. However, since these lossless DWT filters have the real (irrational-number) coefficients, there exist problems of not only exponentially increasing computational complexity in the multiple-channel filter bank, but also causing large calculation errors [2; 3; 4]. For this reason, the DWT filter pair (wavelet 5/3) used in the lossless mode of well-known JPEG2000 has integer coefficients for computational-complexity minimization and computing-error free [5; 10; 28]. Nevertheless,existing two-channel lossless DWT filter banks using such filters provide the disadvantage of less flexibility (due to limited error-free conditions) in designing filters, resulting in lower performance due to relatively low image compression ratio and poor denoising efficiency (as compared to lossy DWT filter banks) [1; 2; 3; 4; 5; 6; 7; 10]. A two-channel DWT filter bank consists of analysis and synthesis banks, as shown in Figure 1 where \(H_{L}(z)\), \(H_{U}(z)\) are the transfer functions of analysis bank filters, and \(G_{L}(z)\), \(G_{U}(z)\) are the synthesis filters. The reconstruction signal, in general, suffers from aliasing, amplitude and phase distortions, due to the fact that these filters are not ideal [15; 16; 17]. As is well known, aliasing frequencies are removed completely by substituting \(G_{L}(z)=H_{L}(-z)\) and \(G_{U}(z)=-H_{L}(-z)\). Then, the overall system function \(T(z)\) of the alias-free filter bank can be written as [18; 19; 20; 21; 22; 23; 24; 25; 26]  
\[T(z)=\frac{1}{2}\{H_{L}(z)H_{U}(-z)-H_{L}(-z)H_{U}(z)\} \tag{1}\]  
Apart from this regularity, the unit-amplitude and linear-phase properties are also very important in applications [17]. A system that is free from aliasing, amplitude and phase distortions is called a lossless DWT filter bank that yields a perfect reconstruction (PR) [18; 19; 20; 21; 22; 23; 24]. The reconstructed signal is therefore just a time-delayed version of the transmitted signal \(x[n]\), i.e., \(\hat{x}[n]=cx[n-n_{d}]\) if \(T(z)=cz^{-n_{d}}\) for some nonzero constant \(c\) and some positive integer \(n_{d}\). In [13; 25], the PR conditions for two-band multi-rate DWT filter banks were given to be alias free and to have a unity frequency response.