Let denote a set of corrupted images from sensors, and let be the corresponding set of low-frequency subimages computed using the LWT. is the number of LWT layers. For simplicity, we assume square images so that . Stack all columns of each into a single vector of dimension , then use these vectors as columns of a matrix . After normalizing the data, we denote by the element of , Display Formula
(7)The cumulative low-frequency subimage matrix is modeled similarly to Eq. (3), Display Formula
(8)in which denotes the noise-free and integrated low-frequency subimage sequence matrix, and denotes the sparse error matrix from which high-frequency content has been attenuated by the selection of LWT coefficients. The low-frequency LWT coefficients are similar across multiple subimages of the same scene. According to the model, is noise-free and will ideally, therefore, consist of identical columns. Accordingly, will be of low rank as required by the matrix completion procedure. Thus, can be estimated via matrix completion and RPCA by solving Display Formula
(9)where the augmented Lagrange multiplier is Display Formula
(10)In this equation, is an estimated positive weighting parameter representing the proportion of the sparse matrix in the low-rank matrix . The default value for this fraction is . is a positive tuning parameter balancing accuracy and computational effort. is the trace of the product and is the iterated Lagrange multiplier.