Learning Circulant Sensing Kernels
RICE UNIV HOUSTON TX DEPT OF COMPUTATIONAL AND APPLIED MATHEMATICS
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In signal acquisition, Toeplitz and circulant matrices are widely used as sensing operators. They correspond to discrete convolutions and are easily or even naturally realized in various applications. For compressive sensing, recent work has used random Toeplitz and circulant sensing matrices and proved their efficiency in theory, by computer simulations, as well as through physical optical experiments. Motivated by recent work 8, we propose models to learn a circulant sensing matrixoperator for one and higher dimensional signals. Given the dictionary of the signal s to be sensed, the learned circulant sensing matrixoperator is more effective than a randomly generated circulant sensing matrixoperator, and even slightly so than a non-circulant Gaussian random sensing matrix. In addition, by exploiting the circulant structure, we improve the learning from the patch scale in 8 to the much large image scale. Furthermore, we test learning the circulant sensing matrixoperator and the nonparametric dictionary altogether and obtain even better performance. We demonstrate these results using both synthetic sparse signals and real images.
- Numerical Mathematics