An ensemble of correlated signals are often encountered in many applications of image processing, such as a set of face images of the same subject. In this paper, we propose a new model, called Joint Sparsity Model with Matrix Completion (JSM-MC), which extracts a common component, an innovation component, and a low-rank component from an ensemble of face images. These components have their respective physical significance in terms of representing different types of information in the original ensemble, hence facilitating an analysis task such as recognition. An algorithm is proposed under the model to solve for the components, based on Block Coordinate Descent and Singular Value Thresholding. Experimental results show that the proposed method has unique advantages over existing methods in dealing with challenging face images with extreme illumination conditions or occlusions.