Abstract
Efficient representations of multivariate functions are critical for the design of state-of-the-art methods of data restoration and image reconstruction. In this work, we consider the representation of spatio-temporal data such as temporal sequences (videos) of 2- and 3-dimensional images, where conventional separable representations are usually very inefficient, due to their limitations in handling the geometry of the data. To address this challenge, we define a class E(A)⊂L2(R4) of functions of 4 variables dominated by hypersurface singularities in the first three coordinates that we apply to model 4-dimensional data corresponding to temporal sequences (videos) of 3-dimensional objects. To provide an efficient representation for this type of data, we introduce a new multiscale directional system of functions based on cylindrical shearlets and prove that this new approach achieves superior approximation properties with respect to conventional multiscale representations. We illustrate the advantages of our approach by applying a discrete implementation of the new representation to a challenging problem from dynamic tomography. Numerical results confirm the potential of our novel approach with respect to conventional multiscale methods.
Original language | English |
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Article number | 115206 |
Journal | Journal of Computational and Applied Mathematics |
Volume | 429 |
Early online date | 11 Mar 2023 |
DOIs | |
Publication status | Published - 30 Sept 2023 |
Bibliographical note
Funding Information:All authors acknowledge the support of the IT for Science group 3 3 of the University of Helsinki for the high performance computing cluster Turso. TAB was partially supported by the Royal Society, United Kingdom through the Newton International Fellowship grant n. NIF\R1\201695 and by the Academy of Finland, Finland through the postdoctoral grant, decision number 330522 . DL acknowledges support of NSF-DMS 1720487 and 172045 . TH acknowledges support of the Emil Aaltonen Foundation junior researcher grant no. 200029 and the Academy of Finland, Finland Project 310822 .
Data availability: No data was used for the research described in the article.
Keywords
- Dynamic tomography
- Multiscale analysis
- Regularization
- Shearlets
- Sparse approximations
- Spatio-temporal data
ASJC Scopus subject areas
- Computational Mathematics
- Applied Mathematics