Weighted structure tensor total variation for image denoising

Based on the variational framework of the image denoising problem, we introduce a novel image denoising regularizer that combines anisotropic total variation model (ATV) and structure tensor total variation model (STV) in this paper. The model can effectively capture the first-order information of the image and maintain local features during the denoising process by applying the matrix weighting operator proposed in the ATV model to the patch-based Jacobian matrix in the STV model. Denoising experiments on grayscale and RGB color images demonstrate that the suggested model can produce better restoration quality in comparison to other well-known methods based on total-variation-based models and the STV model

Variational Methods and Numerical Algorithms for Geometry Processing

In this work we address the problem of shape partitioning which enables the decomposition of an arbitrary topology object into smaller and more manageable pieces called partitions. Several applications in Computer Aided Design (CAD), Computer Aided Manufactury (CAM) and Finite Element Analysis (FEA) rely on object partitioning that provides a high level insight of the data useful for further processing. In particular, we are interested in 2-manifold partitioning, since the boundaries of tangible physical objects can be mathematically defined by two-dimensional manifolds embedded into three-dimensional Euclidean space. To that aim, a preliminary shape analysis is performed based on shape characterizing scalar/vector functions defined on a closed Riemannian 2-manifold. The detected shape features are used to drive the partitioning process into two directions – a human-based partitioning and a thickness-based partitioning. In particular, we focus on the Shape Diameter Function that recovers volumetric information from the surface thus providing a natural link between the object’s volume and its boundary, we consider the spectral decomposition of suitably-defined affinity matrices which provides multi-dimensional spectral coordinates of the object’s vertices, and we introduce a novel basis of sparse and localized quasi-eigenfunctions of the Laplace-Beltrami operator called Lp Compressed Manifold Modes. The partitioning problem, which can be considered as a particular inverse problem, is formulated as a variational regularization problem whose solution provides the so-called piecewise constant/smooth partitioning function. The functional to be minimized consists of a fidelity term to a given data set and a regularization term which promotes sparsity, such as for example, Lp norm with p ∈ (0, 1) and other parameterized, non-convex penalty functions with positive parameter, which controls the degree of non-convexity. The proposed partitioning variational models, inspired on the well-known Mumford Shah models for recovering piecewise smooth/constant functions, incorporate a non-convex regularizer for minimizing the boundary lengths. The derived non-convex non-smooth optimization problems are solved by efficient numerical algorithms based on Proximal Forward-Backward Splitting and Alternating Directions Method of Multipliers strategies, also employing Convex Non-Convex approaches. Finally, we investigate the application of surface partitioning to patch-based surface quadrangulation. To that aim the 2-manifold is first partitioned into zero-genus patches that capture the object’s arbitrary topology, then for each patch a quad-based minimal surface is created and evolved by a Lagrangian-based PDE evolution model to the original shape to obtain the final semi-regular quad mesh. The evolution is supervised by asymptotically area-uniform tangential redistribution for the quads

Constrained TVp - ℓ2 Model for Image Restoration

none3noThe popular total variation (TV) model for image restoration (Rudin et al. in Phys D 60(1–4):259-268, 1992) can be formulated as a Maximum A Posteriori estimator which uses a half-Laplacian image-independent prior favoring sparse image gradients. We propose a generalization of the TV prior, referred to as TVp, based on a half-generalized Gaussian distribution with shape parameter p. An automatic estimation of p is introduced so that the prior better fits the real images’ gradient distribution; we will show that, in general, the estimated p value does not necessarily require to be close to zero. The restored image is computed by using an alternating directions methods of multipliers procedure. In this context, a novel result in multivariate proximal calculus is presented which allows for the efficient solution of the proposed model. Numerical examples show that the proposed approach is particularly efficient and well suited for images characterized by a wide range of gradient distributions.The popular total variation (TV) model for image restoration (Rudin et al. in Phys D 60(1–4):259-268, 1992) can be formulated as a Maximum A Posteriori estimator which uses a half-Laplacian image-independent prior favoring sparse image gradients. We propose a generalization of the TV prior, referred to as TVp, based on a half-generalized Gaussian distribution with shape parameter p. An automatic estimation of p is introduced so that the prior better fits the real images’ gradient distribution; we will show that, in general, the estimated p value does not necessarily require to be close to zero. The restored image is computed by using an alternating directions methods of multipliers procedure. In this context, a novel result in multivariate proximal calculus is presented which allows for the efficient solution of the proposed model. Numerical examples show that the proposed approach is particularly efficient and well suited for images characterized by a wide range of gradient distributions.noneLanza, Alessandro; Morigi, Serena; Sgallari, FiorellaLanza, Alessandro; Morigi, Serena; Sgallari, Fiorell