Report
Compressed sensing for inverse problems II: applications to deconvolution, source recovery, and MRI
| العنوان: | Compressed sensing for inverse problems II: applications to deconvolution, source recovery, and MRI |
|---|---|
| المؤلفون: | Alberti, Giovanni S., Felisi, Alessandro, Santacesaria, Matteo, Trapasso, S. Ivan |
| سنة النشر: | 2025 |
| المجموعة: | Computer Science Mathematics |
| مصطلحات موضوعية: | Mathematics - Functional Analysis, Computer Science - Information Theory, Mathematics - Optimization and Control, 42C40, 94A20, 35R30 |
| الوصف: | This paper extends the sample complexity theory for ill-posed inverse problems developed in a recent work by the authors [`Compressed sensing for inverse problems and the sample complexity of the sparse Radon transform', J. Eur. Math. Soc., to appear], which was originally focused on the sparse Radon transform. We demonstrate that the underlying abstract framework, based on infinite-dimensional compressed sensing and generalized sampling techniques, can effectively handle a variety of practical applications. Specifically, we analyze three case studies: (1) The reconstruction of a sparse signal from a finite number of pointwise blurred samples; (2) The recovery of the (sparse) source term of an elliptic partial differential equation from finite samples of the solution; and (3) A moderately ill-posed variation of the classical sensing problem of recovering a wavelet-sparse signal from finite Fourier samples, motivated by magnetic resonance imaging. For each application, we establish rigorous recovery guarantees by verifying the key theoretical requirements, including quasi-diagonalization and coherence bounds. Our analysis reveals that careful consideration of balancing properties and optimized sampling strategies can lead to improved reconstruction performance. The results provide a unified theoretical foundation for compressed sensing approaches to inverse problems while yielding practical insights for specific applications. Comment: 47 pages |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2501.01929 |
| رقم الانضمام: | edsarx.2501.01929 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |