Research Article New Robust Principal Component Analysis for Joint Image Alignment and Recovery via Affine Transformations and Frobenius and L 2,1 Norms Habte Tadesse Likassa Department of Statistics, Ambo University, Ambo, Ethiopia Correspondence should be addressed to Habte Tadesse Likassa; habte.tade@yahoo.com Received 29 January 2020; Revised 6 March 2020; Accepted 7 March 2020; Published 10 April 2020 Academic Editor: Niansheng Tang Copyright © 2020 Habte Tadesse Likassa. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. is paper proposes an effective and robust method for image alignment and recovery on a set of linearly correlated data via Frobenius and L 2,1 norms. e most popular and successful approach is to model the robust PCA problem as a low-rank matrix recovery problem in the presence of sparse corruption. e existing algorithms still lack in dealing with the potential impact of outliers and heavy sparse noises for image alignment and recovery. us, the new algorithm tackles the potential impact of outliers and heavy sparse noises via using novel ideas of affine transformations and Frobenius and L 2,1 norms. To attain this, affine transformations and Frobenius and L 2,1 norms are incorporated in the decomposition process. As such, the new algorithm is more resilient to errors, outliers, and occlusions. To solve the convex optimization involved, an alternating iterative process is also considered to alleviate the complexity. Conducted simulations on the recovery of face images and handwritten digits demonstrate the effectiveness of the new approach compared with the main state-of-the-art works. 1. Introduction Image alignment and recovery [1] has found applications in a variety of areas such as medical imaging, wireless sensor networks, surveillance, batch image denoising, and com- putational imaging. Image recovery can also be used in background extraction, where the low-rank component corresponds to the background and the sparse component captures the foreground. However, this problem faces some severe challenges such as illumination variation, occlusion, outliers, and heavy sparse noises. It is thus important to develop robust image recovery algorithms to tackle the abovementioned adverse effects. A variety of algorithms have been reported for image alignment and recovery problem. For example, Peng et al. [2] considered a robust algorithm for sparse and low-rank de- composition (RASL) to remove the potential impact of outliers and sparse errors incurred by corruption and occlusion, but it still lacks to perform well when the potential impact of outliers and heavy sparse noise is large in a large number of images. To tackle this problem, Likassa et al. [3] addressed a modified RASL via incorporating affine transformation with rank prior information which boosted the performance of the algorithm. Ebadi and Izquierdo [4] proposed an efficient robust principal component analysis by using some approximation for image recovery. However, they do not have the potential to remove the impact of outliers in big data. A robust principal com- ponent analysis (RPCA) algorithm was addressed in [5] based on convex program, which is guaranteed to recover the low- rank matrix despite gross sparse errors; however, the existing RPCA method is known to be extremely fragile to the presence of gross corruptions. To tackle this dilemma, Chen et al. [6] proposed a nonconvex plus quadratic penalized low-rank and sparse decomposition (NQLSD) method to fit the low-rank model and then used a robust fitting function to reduce the influence of corruption and occlusion on image alignment, where it is still questionable due to large time complexity. Song et al. [7] addressed an online robust image alignment approach Hindawi International Journal of Mathematics and Mathematical Sciences Volume 2020, Article ID 8136384, 9 pages https://doi.org/10.1155/2020/8136384