Sparse Dictionary-Based Solution of Dynamic Inverse Problems
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arXiv
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| Format: | Preprint |
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2026
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| author | Mason-Mackay, Aidan Calvetti, Daniela Somersalo, Erkki Aarnio, Antti Kettunen, Mikko Gröhn, Ekaterina Paasonen Olli Kolehmainen, Ville |
| author_facet | Mason-Mackay, Aidan Calvetti, Daniela Somersalo, Erkki Aarnio, Antti Kettunen, Mikko Gröhn, Ekaterina Paasonen Olli Kolehmainen, Ville |
| contents | In ill-posed dynamic inverse problems expected spatial features and temporal correlation between frames can be leveraged to improve the quality of the computed solution, in particular when the available data are limited and the dimensionality of the unknown is large. One way to take advantage of the spatial and temporal traits believed to characterize the solution is to encode them into the entries of a dictionary, and to seek the solution as a sparse linear combination of the dictionary atoms. To promote a vector of coefficients with mostly vanishing entries, we consider a stochastic extension of the dictionary coding problem model with a random hierarchical sparsity promoting prior. We compute the Maximum A Posteriori (MAP) estimate of the coefficient vector using the Iterative Alternating Sequential Algorithm (IAS), which has been demonstrated to efficiently solve inverse problems with minimal need for parameter tuning. The proposed methodology is tested on real-world dynamic Computed Tomography and MRI datasets, where it is compared to the popular Alternating Direction Method of Minimizers (ADMM). The computed examples show the that proposed methodology is competitive with the ADMM for compressed sensing, with a significantly lower sensitivity to hyper-parameter selection. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_18593 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Sparse Dictionary-Based Solution of Dynamic Inverse Problems Mason-Mackay, Aidan Calvetti, Daniela Somersalo, Erkki Aarnio, Antti Kettunen, Mikko Gröhn, Ekaterina Paasonen Olli Kolehmainen, Ville Numerical Analysis 47A52, 65J22, 94A08 In ill-posed dynamic inverse problems expected spatial features and temporal correlation between frames can be leveraged to improve the quality of the computed solution, in particular when the available data are limited and the dimensionality of the unknown is large. One way to take advantage of the spatial and temporal traits believed to characterize the solution is to encode them into the entries of a dictionary, and to seek the solution as a sparse linear combination of the dictionary atoms. To promote a vector of coefficients with mostly vanishing entries, we consider a stochastic extension of the dictionary coding problem model with a random hierarchical sparsity promoting prior. We compute the Maximum A Posteriori (MAP) estimate of the coefficient vector using the Iterative Alternating Sequential Algorithm (IAS), which has been demonstrated to efficiently solve inverse problems with minimal need for parameter tuning. The proposed methodology is tested on real-world dynamic Computed Tomography and MRI datasets, where it is compared to the popular Alternating Direction Method of Minimizers (ADMM). The computed examples show the that proposed methodology is competitive with the ADMM for compressed sensing, with a significantly lower sensitivity to hyper-parameter selection. |
| title | Sparse Dictionary-Based Solution of Dynamic Inverse Problems |
| topic | Numerical Analysis 47A52, 65J22, 94A08 |
| url | https://arxiv.org/abs/2602.18593 |