Data-driven Morozov regularization of inverse problems
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866909353649373184 |
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| author | Haltmeier, Markus Kowar, Richard Tiefenthaler, Markus |
| author_facet | Haltmeier, Markus Kowar, Richard Tiefenthaler, Markus |
| contents | The solution of inverse problems is crucial in various fields such as medicine, biology, and engineering, where one seeks to find a solution from noisy observations. These problems often exhibit non-uniqueness and ill-posedness, resulting in instability under noise with standard methods. To address this, regularization techniques have been developed to balance data fitting and prior information. Recently, data-driven variational regularization methods have emerged, mainly analyzed within the framework of Tikhonov regularization, termed Network Tikhonov (NETT). This paper introduces Morozov regularization combined with a learned regularizer, termed DD-Morozov regularization. Our approach employs neural networks to define non-convex regularizers tailored to training data, enabling a convergence analysis in the non-convex context with noise-dependent regularizers. We also propose a refined training strategy that improves adaptation to ill-posed problems compared to NETT's original strategy, which primarily focuses on addressing non-uniqueness. We present numerical results for attenuation correction in photoacoustic tomography, comparing DD-Morozov regularization with NETT using the same trained regularizer, both with and without an additional total variation regularizer. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_14290 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Data-driven Morozov regularization of inverse problems Haltmeier, Markus Kowar, Richard Tiefenthaler, Markus Numerical Analysis The solution of inverse problems is crucial in various fields such as medicine, biology, and engineering, where one seeks to find a solution from noisy observations. These problems often exhibit non-uniqueness and ill-posedness, resulting in instability under noise with standard methods. To address this, regularization techniques have been developed to balance data fitting and prior information. Recently, data-driven variational regularization methods have emerged, mainly analyzed within the framework of Tikhonov regularization, termed Network Tikhonov (NETT). This paper introduces Morozov regularization combined with a learned regularizer, termed DD-Morozov regularization. Our approach employs neural networks to define non-convex regularizers tailored to training data, enabling a convergence analysis in the non-convex context with noise-dependent regularizers. We also propose a refined training strategy that improves adaptation to ill-posed problems compared to NETT's original strategy, which primarily focuses on addressing non-uniqueness. We present numerical results for attenuation correction in photoacoustic tomography, comparing DD-Morozov regularization with NETT using the same trained regularizer, both with and without an additional total variation regularizer. |
| title | Data-driven Morozov regularization of inverse problems |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2310.14290 |