Variational Weighted ℓ p − ℓ q $\ell _p-\ell _q$ Regularization for Hyperspectral Image Restoration Under Mixed Noise

Abstract

In this paper, we propose to use weighted ${\ell }_2$-norm for approximating the solution of general ${\ell }_p-{\ell }_q$-norm regularization problem for recovering hyperspectral images (HSI) corrupted by a mixture of Gaussian-impulse noise. As a special case of $p,q\in \{1,2\}$, we design an optimization framework to accommodate the combined effect of different noise sources. An initial impulse noise pre-detection phase decouples the raw noisy HSI data into impulse and Gaussian corrupted pixels. Gaussian corrupted pixels are handled by data-fidelity term in ${\ell }_2-\mathrm{norm}$ while impulse corrupted pixels possess more Laplacian like behavior; modeled using ${\ell }_1-\mathrm{norm}$. Solutions of problems involving ${\ell }_1-\mathrm{norm}$ in data fidelity and regularization terms complicate the optimization process but are less sensitive to the outlier pixels. On the other hand, the least square solutions for the data misfit are computationally efficient but generates solutions which are quite sensitive to the outlier pixels; which is the characteristic of impulse corrupted pixels. Therefore, in this paper, we decouple the set of pixels into two distinct parts; handled using two separate data fidelity terms. Total variation (TV) is used on the Casorati matrix representation of the input data to exploit similarity along both spatial and spectral dimensions. The resulting optimization problem is reformulated as iteratively reweighted least square for the general ${\ell }_p-{\ell }_q$-norm problem for $p=\{1,2\}$ for data fidelity terms and $q=1$ for the TV regularization term. Experiments conducted over synthetically corrupted HSI data and images obtained from real HSI sensors confirm the suitability of the proposed weighted norm optimization framework (WNOF) over a wide range of degradation scenarios.