Hybrid Compressed Sensing of images

We consider the problem of recovering a signal/image (x) with a k-sparse representation, from hybrid (complex and real), noiseless linear samples (y) using a mixture of complex-valued sparse and real-valued dense projections within a single matrix. The proposed Hybrid Compressed Sensing (HCS) employs the complex-sparse part of the projection matrix to divide the n-dimensional signal (x) into subsets. In turn, each subset of the signal (coefficients) is mapped onto a complex sample of the measurement vector (y). Under a worst-case scenario of such sparsity-induced mapping, when the number of complex sparse measurements is sufficiently large then this mapping leads to the isolation of a significant fraction of the k non-zero coefficients into different complex measurement samples from y. Using a simple property of complex numbers (namely complex phases) one can identify the isolated non-zeros of x. After reducing the effect of the identified non-zero coefficients from the compressive samples, we utilize the real-valued dense submatrix to form a full rank system of equations to recover the signal values in the remaining indices (that are not recovered by the sparse complex projection part). We show that the proposed hybrid approach can recover a k-sparse signal (with high probability) while requiring only m ≈ 3√n/2k real measurements (where each complex sample is counted as two real measurements). We also derive expressions for the optimal mix of complex-sparse and real-dense rows within an HCS projection matrix. Further, in a practical range of sparsity ratio (k/n) suitable for images, the hybrid approach outperforms even the most complex compressed sensing frameworks (namely basis pursuit with dense Gaussian matrices). The theoretical complexity of HCS is less than the complexity of solving a full-rank system of m linear equations. In practice, the complexity can be lower than this bound.