High Order DIP-VBTV: An Image Restoration Model Combining a Deep Image Prior and a High Order Total Variation on Vector Bundles

Abstract We introduce a high order Total Variation (TV) for sections of vector bundles over Riemannian manifolds, and which is determined by the following geometric quadruplet: a connection and a positive definite metric on the vector bundle as well as a connection and a positive definite metric on the tangent bundle of the manifold. Then, we insert the high order TV into the Deep Image Prior (DIP), yielding a variational model for the restoration of grey-level and color images. The proposed model can be viewed as a combination of three classes of efficient variational models for image restoration: high order TV-based models which promotes the reconstruction of both edges and fine structures of the original image, geometric TV-based models which can encode extra properties of natural images, and DIP-based models which take benefit of the generative property of neural networks to reconstruct the original image. Experiments conducted up to the order 3 for image deblurring show that the higher the order, the better the results. Moreover, for a given order, the experiments also show that the proposed model outperforms its Euclidean restriction.