A mathematical framework for deep learning in elastic source imaging. (English) Zbl 1400.35234
Summary: An inverse elastic source problem with sparse measurements is our concern. A generic mathematical framework is proposed which extends a low-dimensional manifold regularization in the conventional source reconstruction algorithms thereby enhancing their performance with sparse data-sets. It is rigorously established that the proposed framework is equivalent to the so-called deep convolutional framelet expansion in machine learning literature for inverse problems. Apposite numerical examples are furnished to substantiate the efficacy of the proposed framework.
MSC:
| 35R30 | Inverse problems for PDEs |
| 94A08 | Image processing (compression, reconstruction, etc.) in information and communication theory |
| 42C40 | Nontrigonometric harmonic analysis involving wavelets and other special systems |
| 35L53 | Initial-boundary value problems for second-order hyperbolic systems |
| 35Q74 | PDEs in connection with mechanics of deformable solids |
| 74G75 | Inverse problems in equilibrium solid mechanics |
| 92C55 | Biomedical imaging and signal processing |
Keywords:
elasticity imaging; inverse source problem; deep learning; convolutional neural network; deep convolutional framelets; time-reversalReferences:
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