This talk will cover two seemingly disparate problems under the umbrella of an exact rank constraint: one in statistical and radar signal processing focusing on the estimation of structured covariance matrices and the other a key image processing/vision task called robust alignment of images. In the first part, we look at the regularized maximum likelihood (ML) estimation of structured covariance matrices (SCM) that arise in radar space time adaptive processing. The underlying physical model enables knowledge of the rank of the semi-definite component of the SCM which is a key constraint to incorporate. We will show that despite the presence of the challenging non-convex rank constraint, a closed form result that achieves the global optima is indeed possible. We also investigate scenarios where the knowledge of the rank may be imprecise and use expected likelihood approaches to determine the rank and other constraints of interest. Several uniqueness results in determining imprecise constraints are provided in this context. The second part of the talk sets up a key open problem in computer vision and image processing – that of batch aligning images. Technically, we formulate the alignment problem as rank-1 and sparse matrix decomposition over transformed images, and develop an efficient algorithm for solving this challenging nonconvex optimization problem. The algorithm reduces to a sequence of sub-problems, where we analytically establish exact recovery conditions, provide convergence and optimality guarantees. From a practical viewpoint, we demonstrate via results that an exact vs. approximate capture of the rank helps improve results over state of the art image alignment approaches based on robust PCA and other competing methods.