Parameter estimation in sparse representation based face hallucination

•Automatic parameter determination in sparse representation based face hallucination.•Patch size choice is justified from the results of compressive sensing theory.•Regularization parameter is analytically tractable under MAP framework.

Owning to the excellent ability to characterize the sparsity of natural images, ℓ1ℓ1 norm sparse representation is widely applied to face hallucination. However, the determination on two key parameters such as patch size and regularization parameter has not been satisfactorily resolved yet. To this end, we proposed a novel parameter estimation method to identify them in an analytical way. In particular, the optimal patch size is derived from the sufficient condition for reliable sparse signal recovery established in compressive sensing theory. Furthermore, by interpreting ℓ1ℓ1 norm SR as the corresponding maximum a posteriori estimator with Laplace prior constraint, we obtain an explicit expression for regularization parameter in statistics of reconstruction errors and coefficients. Our proposed method can significantly reduce the computational cost of parameter determination while without sacrificing numerical precision and eventual face hallucination performance. Experimental results on degraded images in simulation and real-world scenarios validate its effectiveness.