# Fast and Efficient Union of Sparse Orthonormal Transforms via DCT and Bayesian Optimization +
## 3 Proposed Algorithm
### Parameter Setting via Bayesian Optimization
end while  Observe \(f(\lambda_{s})\)  end while

## 4 Results

### Experimental Environment

We experimented with our methods by using the images shown in Figure 4. For equivalent comparison, we resized the images to \(256\times 256\) pixels and segmented them to \(4\times 4\) or \(8\times 8\) patches. To measure the performances of different algorithms, we focused on image compression, more specifically, reconstruction with a limited number of coefficients, by comparing PSNR (dB) with the number of used coefficients. Sparse-coding-based algorithms depend on the value of \(\lambda\) in their formulations, Equations (3) and (11), because they control the trade-off between the mean square error (MSE) and sparsity. In the compression sense, the optimal value varied from the number of coefficients or bases used in the compression scheme. In this study, we found the value for each target sparsity level via Bayesian optimization with expected implementation and the Matern 5/2 covariance kernel function. For Bayesian optimization, we used the 'bayesopt' library offered by MATLAB. For equivalent comparisons, all experiments were implemented using MATLAB R2021a in Windows 10 Education, equipped with an Intel i7-9700 CPU with 32 GB RAM. All the algorithms used in these experiments are conducted with the same stopping condition. We set the stopping condition to the difference of objective functions between the present and 10 past iterations. Figure 3: The objective function of Bayesian optimization for \(\lambda\) in Figure 2.

### Comparison with Sparse Orthonormal Transform

This subsection compares the SOT and the proposed method with different numbers of dictionaries for energy compaction. Figures 5 and 6 compare the objective qualities in PSNR (dB) for each number of retained bases. As shown in Figures 5 and 6, our method outperformed SOT (constructed by one orthonormal dictionary) in PSNR (dB). We concluded that this result is due to two reasons: (a) Sparse coding algorithms work better based on the input data classified according to their structure than whole unstructured input data, and (b) the dictionary from a small dataset is more adaptive and more representative than that from a large dataset. Furthermore, the difference in reconstruction errors between the proposed method and the SOT in Figure 6 is less than the difference in Figure 5. Small patches have simpler and more dominant directional information than large patches, whereas large patches usually are more complex and have diverse orientations, leading to the difference. In the next section, we verify whether analysis (a) is correct. Figure 5: The object quality comparison: PSNR (dB) versus the number of retained coefficients for 4 \(\times\) 4 patches between SOT and our methods. (**a**) Lena, (**b**) peppers, (**c**) Barbara, (**d**) house. Figure 6: _Cont._

Figure 6: The object quality comparison: PSNR (dB) versus the number of retained coefficients for 8 \(\times\) 8 patches between SOT and our methods. (**a**) [PERSON], (**b**) peppers, (**c**) [PERSON], (**d**) house. Figure 2:0 The object quality comparison: PSNR (dB) versus the number of retained coefficients for 4 \(\times\) 4 patches between classification methods with different numbers of orthogonal dictionaries. (**a**) [PERSON], (**b**) peppers, (**c**) [PERSON], (**d**) house. ### Comparison with an Overcomplete Dictionary

For a comparison of the proposed method with the overcomplete dictionary-based algorithm, we used a UONB with optimization [10] and block coordinate relaxation algorithm, which is faster than the orthogonal matching pursuit. We performed the experiment using different numbers of dictionaries of the proposed method and UONB, from two to five. Figures 9 and 10 show the differences between our method and UONB. In these figures, the proposed method shows a more powerful performance for small patches. In \(8\times 8\) cases, the performance graphs of UONB and the proposed method have crossing points at approximately four retained coefficients.