Sparsity constraint is a priori knowledge of the signal, which means that in some properly chosen basis only a small percentage of the total number of the signal components is nonzero. Sparsity constraint has been used in signal and image processing for a long time. Recent publications have shown that the Sparsity constraint can be used to achieve super-resolution of optical sparse objects beyond the diffraction limit. In this paper we present the quantum theory which establishes the quantum limits of super-resolution for the sparse objects. The key idea of our paper is to use the discrete prolate spheroidal sequences (DPSS) as the sensing basis. We demonstrate both analytically and numerically that this sensing basis gives superior performance of super-resolution over the Fourier basis conventionally used for sensing of sparse signals. The explanation of this phenomenon is in the fact that the DPSS are the eigenfunctions of the optical imaging system while the Fourier basis are not. We investigate the role of the quantum fluctuations of the light illuminating the object, in the performance of reconstruction algorithm. This analysis allows us to formulate the criteria for stable reconstruction of sparse objects with super-resolution. Our results imply that sparsity of the object is not the only parameter which describes super-resolution achievable via sparsity constraint.
© 2012 Optical Society of America
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