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Keywords = sparse orthonormal transform

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18 pages, 1534 KB  
Article
RIP Sensing Matrices Construction for Sparsifying Dictionaries with Application to MRI Imaging
by Jinn Ho, Wen-Liang Hwang and Andreas Heinecke
Signals 2024, 5(4), 794-811; https://doi.org/10.3390/signals5040044 - 2 Dec 2024
Viewed by 2281
Abstract
Practical applications of compressed sensing often restrict the choice of its two main ingredients. They may (i) prescribe the use of particular redundant dictionaries for certain classes of signals to become sparsely represented or (ii) dictate specific measurement mechanisms which exploit certain physical [...] Read more.
Practical applications of compressed sensing often restrict the choice of its two main ingredients. They may (i) prescribe the use of particular redundant dictionaries for certain classes of signals to become sparsely represented or (ii) dictate specific measurement mechanisms which exploit certain physical principles. On the problem of RIP measurement matrix design in compressed sensing with redundant dictionaries, we give a simple construction to derive sensing matrices whose compositions with a prescribed dictionary have with high probability the RIP in the klog(n/k) regime. Our construction thus provides recovery guarantees usually only attainable for sensing matrices from random ensembles with sparsifying orthonormal bases. Moreover, we use the dictionary factorization idea that our construction rests on in the application of magnetic resonance imaging, in which also the sensing matrix is prescribed by quantum mechanical principles. We propose a recovery algorithm based on transforming the acquired measurements such that the compressed sensing theory for RIP embeddings can be utilized to recover wavelet coefficients of the target image, and show its performance on examples from the fastMRI dataset. Full article
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18 pages, 1666 KB  
Article
Fast and Efficient Union of Sparse Orthonormal Transforms via DCT and Bayesian Optimization
by Gihwan Lee and Yoonsik Choe
Appl. Sci. 2022, 12(5), 2421; https://doi.org/10.3390/app12052421 - 25 Feb 2022
Cited by 4 | Viewed by 2123
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 [...] Read more.
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 (UONB), 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. Full article
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