## 2 Preliminary
### Handcrafted Features: Bi-Orthogonal Discrete Wavelet Transform  
For Multi-Resolution Analysis (MRA) [43] of an image, Bi-orthogonal wavelets are preferable over orthogonal ones, due to their flexible nature of transformation - they can be invertible without being necessarily orthogonal. The Bi-orthogonal wavelets have inbuilt properties of symmetry, linear-phase, compact-support, and de-noising. The symmetry and linear phase ensure distortion-less wavelet coefficients; compact support is a guarantee of capturing minute details of a given pattern; the potential of de-noising can improve the recognition ability [44].  
Bi-orthogonal wavelets can be used to develop symmetrical wavelet functions [45], which are useful in the effective edge-representation of images. Since edge representation is a significant aspect in the recognition of handwritten characters, the BDWT becomes the natural choice for handcrafted feature extraction in the proposed scheme.  
The symmetry property needs exact reconstruction of the Bi-orthogonal filter-banks design [46]. The design carries analysis and synthesis filter banks. The scaling and transformation functions associated with the analysis filter bank are given by Equations (1) and (2):  
\[\varnothing(\mathrm{x})=\sqrt{2}\sum_{\mathrm{k}}\mathrm{L}(\mathrm{k})\ \varnothing(2\mathrm{x}-\mathrm{k}) \tag{1}\]  
\[\varphi(\mathrm{x})=\sqrt{2}\sum_{\mathrm{k}}\mathrm{H}(\mathrm{k})\ \varnothing(2\mathrm{x}-\mathrm{k}) \tag{2}\]  
Here, \(\mathrm{L}(\mathrm{k})\) and \(\mathrm{H}(\mathrm{k})\) are responses of filters associated with scaling and transformation functions. For the synthesis-filter bank, the scaling and transformation functions are given by (3) and (4):  
\[\widetilde{\varnothing}(\mathrm{x})=\sqrt{2}\sum_{\mathrm{k}}\overline{ \mathrm{L}}(\mathrm{k})\ \widetilde{\varnothing}(2\mathrm{x}-\mathrm{k}) \tag{3}\]  
\[\widetilde{\varphi}(\mathrm{x})=\sqrt{2}\sum_{\mathrm{k}}\overline{\mathrm{H} }(\mathrm{k})\widetilde{\varnothing}(2\mathrm{x}-\mathrm{k}) \tag{4}\]  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Network** & **Depth** & **Input Size** & **Feature-Vector Size** \\ \hline VGG19-Net & 19 & 224 \(\times\) 224 \(\times\) 3 & 4096 \\ Inception V3-Net & 48 & 299 \(\times\) 299 \(\times\) 3 & 2048 \\ ResNet-50 & 50 & 224 \(\times\) 224 \(\times\) 3 & 2048 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Summary of pre-trained network architecture.  
Here, \(\overline{\mathrm{L}}\)(k) and \(\overline{\mathrm{H}}\)(k) are responses of filters associated with scaling and transformation functions of the synthesis-filter bank. The (3) and (4) are dual of (1) and (2) [47]. For exact reconstruction, the filter responses must satisfy the conditions given in (5), (6) and (7) [48]:  
\[\mathrm{L}(\mathrm{k})\ \overline{\mathrm{L}}(\mathrm{k})=\mathrm{H}( \mathrm{k})\ \overline{\mathrm{H}}(\mathrm{k})=1 \tag{5}\] \[\mathrm{L}(\mathrm{k})\ \ \overline{\mathrm{H}}(\mathrm{k})= \mathrm{H}(\mathrm{k})\ \overline{\mathrm{L}}(\mathrm{k})=0\] (6) \[\overline{\mathrm{L}}(\mathrm{k})\ \mathrm{L}(\mathrm{k})+\ \overline{\mathrm{H}}(\mathrm{k})\ \mathrm{H}(\mathrm{k})=1 \tag{7}\]  
Referring to Equations (1)-(7), all the equations are linear in nature, ensuring a simple design and less complexity.