# Fast and Efficient Union of Sparse Orthonormal Transforms via DCT and Bayesian Optimization +
## 3 Proposed Algorithm
### Parameter Setting via Bayesian Optimization
(**a**) Lena, (**b**) peppers, (**c**) Barbara, (**d**) house. iterations and shorter computation time for each iteration than those of UONB, our method was, on average, hundreds of times faster than UONB. One interesting point is the comparison between SOT and the proposed method. Although the proposed method attempted to find more dictionaries and coefficients than the SOT did, the number of iterations could be reduced by factorizing a dictionary into the DCT matrix and an orthonormal matrix, which prevented an increase in time. ### Effects on a Bayesian Optimization

The Bayesian optimization method was compared with an exhaustive method. We conducted experiments on different image patch sizes and then compared each method by varying the number of iterations. Table 2 shows the PSNR(dB) for each experiment and the corresponding \(\lambda\). We conducted all experiments with \(\lambda\) in the range of 0 to 1. We also performed the exhaustive methods for two different strides: 1/100 and 1/1000. The optimal \(\lambda\) values from the two methods were either similar or different, but the PSNR(dB)s were generally similar. This result indicated that the proposed method with Bayesian optimization performed similar to the exhaustive method but with a much lower number of iterations. ## 5 Conclusions

In this paper, we proposed a novel sparse-coding-based image transform framework as an extension of the SOT for efficient implementation. Overcomplete-dictionary-based methods perform well in terms of sparse representation but require a long time and many resources because of their iterative or greedy optimizations. Moreover, they are suitable for image compression as compared to analytic transforms. Orthogonal sparse coding is similar to analytic transforms such as DCT and KLT. Since the dictionary is square and orthonormal, the transform is invertible and conserves the energy of the data. Thus, orthogonal sparse coding-based transforms for image compression have been proposed over the past few decades. One of these transforms is the SOT. It has been theoretically proven that the SOT outperforms KLT [4]. We extended the SOT based on the unions of several orthonormal \begin{table}
\begin{tabular}{|c|c|c|c|c|c|c|} \hline  & & \multicolumn{2}{c|}{**Methods**} & \multicolumn{2}{c|}{**Bayesian Method**} & \multicolumn{2}{c|}{**Exhaustive Method**} \\ \cline{3-6}  & & **Iterations** & **10** & **30** & **100** & **1000** \\ \hline \multirow{4}{*}{Barbara} & \multirow{2}{*}{4 \(\times\) 4} & PSNR(dB) & 31.7748 & 31.8543 & 31.8395 & **31.8545** \\ \cline{2-6}  & & \(\lambda\) & 0.0396964 & 0.040544 & 0.0200 & 0.0140 \\ \cline{2-6}  & \multirow{2}{*}{8 \(\times\) 8} & PSNR(dB) & 27.01254 & **27.1991** & 25.1582 & 27.1911 \\ \cline{2-6}  & & \(\lambda\) & 0.13714 & 0.063292 & 0.0070 & 0.1330 \\ \hline \multirow{4}{*}{House} & \multirow{2}{*}{4 \(\times\) 4} & PSNR(dB) & 35.5931 & 35.6488 & 35.6092 & **35.6709** \\ \cline{2-6}  & & \(\lambda\) & 0.041138 & 0.026825 & 0.0200 & 0.0150 \\ \cline{2-6}  & \multirow{2}{*}{8 \(\times\) 8} & PSNR(dB) & 30.3563 & **30.3577** & 30.3283 & 30.3559 \\ \cline{2-6}  & & \(\lambda\) & 0.319391 & 0.099313 & 0.1600 & 0.0980 \\ \hline \multirow{4}{*}{Lena} & \multirow{2}{*}{4 \(\times\) 4} & PSNR(dB) & 32.9539 & 33.0129 & 32.9817 & **33.0218** \\ \cline{2-6}  & & \(\lambda\) & 0.051326 & 0.034557 & 0.0400 & 0.0320 \\ \cline{2-6}  & \multirow{2}{*}{8 \(\times\) 8} & PSNR(dB) & 27.7297 & 27.8073 & 27.7918 & **27.8325** \\ \cline{2-6}  & & \(\lambda\) & 0.16529 & 0.069484 & 0.1400 & 0.0940 \\ \hline \multirow{4}{*}{Peppers} & \multirow{2}{*}{4 \(\times\) 4} & PSNR(dB) & 33.1043 & 33.1161 & 33.0836 & **33.1197** \\ \cline{2-6}  & & \(\lambda\) & 0.046513 & 0.06335 & 0.0500 & 0.0350 \\ \cline{1-1} \cline{2-6}  & \multirow{2}{*}{8 \(\times\) 8} & PSNR(dB) & 27.3874 & 27.4054 & 27.3951 & **27.4300** \\ \cline{1-1} \cline{2-6}  & & \(\lambda\) & 0.2068 & 0.21706 & 0.1900 & 0.2050 \\ \hline \end{tabular}
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