# Fast and Efficient Union of Sparse Orthonormal Transforms via DCT and Bayesian Optimization +
###### Abstract  
Sparse orthonormal transform is based on orthogonal sparse coding, which is relatively fast and suitable in image compression such as analytic transforms with better performance. However, because of the constraints on its dictionary, it has performance limitations. This paper proposes an extension of a sparse orthonormal transform based on unions of orthonormal dictionaries for image compression. Unlike unions of orthonormal bases (IONB), which implement an overcomplete dictionary with several orthonormal dictionaries, the proposed method allocates patches to an orthonormal dictionary based on their directions. The dictionaries are constructed into a discrete cosine transform and an orthonormal matrix. To determine a trade-off parameter between the reconstruction error and sparsity, which hinders efficient implementation, the proposed method adapts Bayesian optimization. The framework exhibits an improved performance with fast implementation to determine the optimal parameter. It is verified that the proposed method performs similar to an overcomplete dictionary with a faster speed via experiments. sparse coding; orthogonal sparse coding; dictionary learning; image transform; sparse orthonormal transform +
Footnote †: This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license ([[https://creativecommons.org/licenses/py/4.0/](https://creativecommons.org/licenses/py/4.0/)]([https://creativecommons.org/licenses/py/4.0/](https://creativecommons.org/licenses/py/4.0/))). similar to those of analytic transforms because it represents the input signal with a minimal basis. [PERSON] et al. [4; 9] formulated a transform with an orthonormal matrix and an \(L_{0}\) norm constraint on the transform coefficients. The transform is easily invertible by its transpose matrix and satisfies [PERSON]'s theorem. They also theoretically proved that the transform is superior to the existing analytic transform, especially Karhunen-Loeve Transform (KLT). It is important because KLT is a well-known optimal transform in the Gaussian process. They show that the SOT is identical to KLT in the Gaussian process and outperforms it in the non-Gaussian process. This SOT heuristically gives better performance than other image orthonormal transforms. However, the performance of the orthonormal dictionary was adequate but not comparable to that of an overcomplete dictionary. The large and wide dictionary could represent sparser and more redundant data. In addition, the orthogonal sparse coding is much faster than the overcomplete coding, but the computation is still slow because of the high number of iterations. In this paper, we extend the SOT to be applied to an overcomplete dictionarywhile reducing the computational time. We also propose a method to find a parameter \(\lambda\) in Equation (1) for efficient implementation. Since the performance of sparse coding is sensitive to \(\lambda\), the use of an optimal \(\lambda\) is important. Our method uses multiple dictionaries to outperform the SOT. Ref. [10] first proposed a sparse coding methodology with multiple dictionaries. The method named union of orthonormal bases (UONB) implements an overcomplete dictionary with several orthogonal dictionaries, and it can find an overcomplete dictionary faster. In this study, we used multiple dictionaries in a different way from [10], because the problem in this study is different from that in [10]. We based our transform on an orthogonal dictionary, such as the SOT. Instead, we classified the image patches and allocated each group to one orthogonal dictionary. Then, we converted the problem into a multiple orthogonal sparse coding problem. To classify input patches and create an efficient framework, we exploited a discrete cosine transform. Before allocating dictionaries, our method requires the construction of a subdataset. We assumed that a dataset with similar directional patches helps design suitable dictionaries.