Inpainting in Discrete Sobolev Spaces: Structural Information for Uncertainty Reduction
In this article, we introduce a new mathematical functional whose minimization determines the quality of the solution for the exemplar-based inpainting-by-patch problem. The new functional expression includes finite difference terms in a similar fashion to what happens in the theoretical Sobolev spa...
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| Other Authors: | |
| Format: | Article |
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Multidisciplinary Digital Publishing Institute
2023
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| Online Access: | https://hdl.handle.net/1721.1/152035 |
| _version_ | 1824458096327524352 |
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| author | Seracini, Marco Brown, Stephen R. |
| author2 | Massachusetts Institute of Technology. Department of Earth, Atmospheric, and Planetary Sciences |
| author_facet | Massachusetts Institute of Technology. Department of Earth, Atmospheric, and Planetary Sciences Seracini, Marco Brown, Stephen R. |
| author_sort | Seracini, Marco |
| collection | MIT |
| description | In this article, we introduce a new mathematical functional whose minimization determines the quality of the solution for the exemplar-based inpainting-by-patch problem. The new functional expression includes finite difference terms in a similar fashion to what happens in the theoretical Sobolev spaces: its use reduces the uncertainty in the choice of the most suitable values for each point to inpaint. Moreover, we introduce a probabilistic model by which we prove that the usual principal directions, generally employed for continuous problems, are not enough to achieve consistent reconstructions in the discrete inpainting asset. Finally, we formalize a new priority index and new rules for its dynamic update. The quality of the reconstructions, achieved using a reduced neighborhood size of more than <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>95</mn><mo>%</mo></mrow></semantics></math></inline-formula> with respect to the current state-of-the-art algorithms based on the same inpainting approach, further provides the experimental validation of the method. |
| first_indexed | 2024-09-23T11:37:45Z |
| format | Article |
| id | mit-1721.1/152035 |
| institution | Massachusetts Institute of Technology |
| last_indexed | 2025-02-19T04:20:27Z |
| publishDate | 2023 |
| publisher | Multidisciplinary Digital Publishing Institute |
| record_format | dspace |
| spelling | mit-1721.1/1520352024-11-21T14:56:02Z Inpainting in Discrete Sobolev Spaces: Structural Information for Uncertainty Reduction Seracini, Marco Brown, Stephen R. Massachusetts Institute of Technology. Department of Earth, Atmospheric, and Planetary Sciences In this article, we introduce a new mathematical functional whose minimization determines the quality of the solution for the exemplar-based inpainting-by-patch problem. The new functional expression includes finite difference terms in a similar fashion to what happens in the theoretical Sobolev spaces: its use reduces the uncertainty in the choice of the most suitable values for each point to inpaint. Moreover, we introduce a probabilistic model by which we prove that the usual principal directions, generally employed for continuous problems, are not enough to achieve consistent reconstructions in the discrete inpainting asset. Finally, we formalize a new priority index and new rules for its dynamic update. The quality of the reconstructions, achieved using a reduced neighborhood size of more than <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>95</mn><mo>%</mo></mrow></semantics></math></inline-formula> with respect to the current state-of-the-art algorithms based on the same inpainting approach, further provides the experimental validation of the method. 2023-09-06T19:23:00Z 2023-09-06T19:23:00Z 2023-08-18 2023-08-25T12:37:10Z Article http://purl.org/eprint/type/JournalArticle https://hdl.handle.net/1721.1/152035 Applied Sciences 13 (16): 9405 (2023) PUBLISHER_CC http://dx.doi.org/10.3390/app13169405 Creative Commons Attribution http://creativecommons.org/licenses/by/4.0/ application/pdf Multidisciplinary Digital Publishing Institute Multidisciplinary Digital Publishing Institute |
| spellingShingle | Seracini, Marco Brown, Stephen R. Inpainting in Discrete Sobolev Spaces: Structural Information for Uncertainty Reduction |
| title | Inpainting in Discrete Sobolev Spaces: Structural Information for Uncertainty Reduction |
| title_full | Inpainting in Discrete Sobolev Spaces: Structural Information for Uncertainty Reduction |
| title_fullStr | Inpainting in Discrete Sobolev Spaces: Structural Information for Uncertainty Reduction |
| title_full_unstemmed | Inpainting in Discrete Sobolev Spaces: Structural Information for Uncertainty Reduction |
| title_short | Inpainting in Discrete Sobolev Spaces: Structural Information for Uncertainty Reduction |
| title_sort | inpainting in discrete sobolev spaces structural information for uncertainty reduction |
| url | https://hdl.handle.net/1721.1/152035 |
| work_keys_str_mv | AT seracinimarco inpaintingindiscretesobolevspacesstructuralinformationforuncertaintyreduction AT brownstephenr inpaintingindiscretesobolevspacesstructuralinformationforuncertaintyreduction |