# Fast and Efficient Union of Sparse Orthonormal Transforms via DCT and Bayesian Optimization +
## 3 Proposed Algorithm
### Parameter Setting via Bayesian Optimization
In contrast to UONB, our method exhibits superior performance with an increasing number of dictionaries. In the \(8\times 8\) case, our method outperformed UONB with a small number of coefficients, and UONB showed better performance when using a large number of coefficients. In contrast, the proposed method generally performed better than UONB in the \(4\times 4\) case. The proposed method also performed better with a small patch size. The UONB performed better in the case of more complex patches with large sparsity levels and a large number of dictionaries. For UONB, a large number of orthonormal dictionaries indicated a wider overcomplete dictionary and involved more redundant and expressive representations. Figure 10 illustrates the results at a high sparsity. Figure 2:1 The object quality comparison: PSNR (dB) versus the number of retained coefficients for \(8\times 8\) patches between classification methods with different numbers of orthogonal dictionaries. (**a**) Lena, (**b**) peppers, (**c**) Barbara, (**d**) house. Figure 2:2 _Cont._

Figure 2:3 The object quality comparison: PSNR (dB) versus the number of retained coefficients for \(4\times 4\) different patch sizes between UONB and our methods with different numbers of orthogonal dictionaries. (**a**) Lena, (**b**) peppers, (**c**) Barbara, (**d**) house. ### Processing Time

To reduce the computational time, we attempted to make the best use of the DCT matrix. Table 1 compares the SOT, UONB, and the proposed method in terms of the number of iterations and the computational time (seconds) until convergence. We performed experiments on UONB and the proposed method for three dictionary sizes: L = 2, 3, and 4. Since the SOT uses only an orthogonal dictionary, it is marked only for L = 1. To design an equivalent experimental setting, we set \(\lambda\) values that were optimal to two levels of sparsity: 3 and 5. The time to search \(\lambda\) was not considered. Table 1 shows that our proposed method performed the best in all cases. SOT works better than UONB in terms of computational time. Although the number of iterations required for convergence for SOT was much larger than that required for UONB, SOT was much faster than UONB, because it did not use greedy algorithms. The number of iterations for our proposed method was several times smaller than that for UONB. The degree of reduction varied by approximately two to four times; however, in all cases, our algorithm required fewer iterations. The differences between the computational times was larger than that between the number of iterations. Since our method required fewer \begin{table}
\begin{tabular}{|c|c|c|c|c|c|c|c|} \hline \multirow{2}{*}{**L**} & \multirow{2}{*}{**\# of Retained Coefficients**} & \multicolumn{2}{c|}{**SOT**} & \multicolumn{2}{c|}{**UONB**} & \multicolumn{2}{c|}{**Proposed**} \\ \cline{3-8}  & & **Iterations** & **Time (s)** & **Iterations** & **Time (s)** & **Iterations** & **Time (s)** \\ \hline \multirow{2}{*}{1} & 3 & 2095 & 1.4553 & - & - & - & - \\ \cline{2-8}  & 5 & 2741 & 1.9480 & - & - & - & - \\ \hline \multirow{2}{*}{2} & 3 & - & - & 622 & 49.7882 & **72** & **0.0498** \\ \cline{2-8}  & 5 & - & - & 536 & 82.9780 & **107** & **0.0650** \\ \hline \multirow{3}{*}{3} & 3 & - & - & 628 & 343.3251 & **183** & **0.1174** \\ \cline{2-8}  & 5 & - & - & 505 & 97.5815 & **157** & **0.0872** \\ \hline \multirow{2}{*}{4} & 3 & - & - & 482 & 57.8052 & **195** & **0.0920** \\ \cline{2-8}  & 5 & - & - & 754 & 101.3284 & **192** & **0.1007** \\ \hline \end{tabular}
\end{table} 
Table 1: The processing time and number of iterations for each method. L indicates the number of orthogonal dictionaries. We compare each algorithm with the optimal \(\lambda\) value for two different numbers of retained coefficients. The bold texts indicate minimum results. Figure 10: The object quality comparison: PSNR (dB) versus the number of retained coefficients for \(8\times 8\) different patch sizes between UONB and our methods with different numbers of orthogonal dictionaries.