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The image processing task of the recovery of an image from a noisy or compromised image is an illposed inverse problem. To solve this problem, it is necessary to incorporate prior information about the smoothness, or the structure, of the solution, by incorporating regularization. Here, we consider linear blur operators with an efficiently-found singular value decomposition. Then, regularization is obtained by employing a truncated singular value expansion for image recovery. In this study, we focus on images for which the image blur operator is separable and can be represented by a Kronecker product such that the associated singular value decomposition is expressible in terms of the singular value decompositions of the separable components. The truncation index k can then be identified without forming the full Kronecker product of the two terms. This report investigates the problem of learning an optimal k using two methods. For one method to learn k we assume the knowledge of the true images, yielding a supervised learning algorithm based on the average relative error. The second method uses the method of generalized cross validation and does not require knowledge of the true images. The approach is implemented and demonstrated to be successful for Gaussian, Poisson and salt and pepper noise types across noise levels with signal to noise ratios as low as 10. This research contributes to the field by offering insights into the use of the supervised and unsupervised estimators for the truncation index, and demonstrates that the unsupervised algorithm is not only robust and computationally efficient, but is also comparable to the supervised method.