In this paper, we propose a generalized penalization technique and a convex constraint minimization approach for the $p$-harmonic flow problem following the ideas in [Kang & March, IEEE T. Image Process., 16 (2007), 2251-2261]. We use fast algorithms to solve the subproblems, such as the dual projection methods, primal-dual methods and augmented Lagrangian methods. With a special penalization term, some special algorithms  are presented. Numerical experiments are given to demonstrate the performance of the proposed methods. We successfully show that our algorithms are effective and efficient due to two reasons: the solver for subproblem is fast in essence and there is no need to solve the subproblem accurately (even 2 inner iterations of the subproblem are enough). It is also observed that  better PSNR values are produced using the new algorithms.

This paper proposes a novel multi-scale fluid flow data assimilation approach, which integrates and complements the advantages of a Bayesian sequential assimilation technique, the Weighted Ensemble Kalman filter (WEnKF) [27]. The data assimilation proposed in this work incorporates measurement brought by an efficient multiscale stochastic formulation of the well-known Lucas-Kanade (LK) estimator. This estimator has the great advantage to provide uncertainties associated to the motion measurements at different scales. The proposed assimilation scheme benefits from this multiscale uncertainty information and enables to enforce a physically plausible dynamical consistency of the estimated motion fields along the image sequence. Experimental evaluations are presented on synthetic and real fluid flow sequences.

In this paper, a fast algorithm for Euler's elastica functional is proposed, in which the Euler's elastica functional is reformulated as a constrained minimization problem. Combining the augmented Lagrangian method and operator splitting techniques, the resulting saddle-point problem is solved by a serial of subproblems. To tackle the nonlinear constraints arising in the model, a novel fixed-point-based approach is proposed so that all the subproblems either is a linear problem or has a closed-form solution. We show the good performance of our approach in terms of speed and reliability using numerous numerical examples on synthetic, real-world and medical images for image denoising, image inpainting and image zooming problems.

We present an efficient implementation of volumetric anisotropic image diffusion filters on modern programmable graphics processing units (GPUs), where the mathematics behind volumetric diffusion is effectively reduced to the diffusion in 2D images. We hereby avoid the computational bottleneck of a time consuming eigenvalue decomposition in $\mathbb{R}^3$. Instead, we use a projection of the Hessian matrix along the surface normal onto the tangent plane of the local isodensity surface and solve for the remaining two tangent space eigenvectors. We derive closed formulas to achieve this and prevent the GPU code from branching. We show that our most complex volumetric anisotropic diffusion filters gain a speed up of more than 600 compared to a CPU solution.

The paper is devoted to an approach for image inpainting developed on the basis of neurogeometry of vision and sub-Riemannian geometry. Inpainting is realized by completing damaged isophotes (level lines of brightness) by optimal curves for the left-invariant sub-Riemannian problem on the group of rototranslations (motions) of a plane SE(2). The approach is considered as anthropomorphic inpainting since these curves satisfy the variational principle discovered by neurogeometry of vision.  A parallel algorithm and software to restore monochrome binary or halftone images represented as series of isophotes were developed. The approach and the algorithm for computation of completing arcs are presented in detail.

This article describes the implementation of a simple wavelet-based optical-flow motion estimator dedicated to continuous motions such as fluid flows. The wavelet representation of the unknown velocity field is considered. This scale-space representation, associated to a simple gradient-based optimization algorithm, sets up a well-defined multiresolution framework for the optical flow estimation. Moreover, a very simple closure mechanism, approaching locally the solution by high-order polynomials is provided by truncating the wavelet basis at fine scales. Accuracy and efficiency of the proposed method are evaluated on image sequences of turbulent fluid flows.

We consider the linear and non-linear enhancement of diffusion weighted magnetic resonance images (DW-MRI) to use contextual information in denoising and inferring fiber crossings. We describe the space of DW-MRI images in a moving frame of reference, attached to fiber fragments which allows for convection-diffusion along the fibers. Because of this approach, our method is naturally able to handle crossings in data. We will perform experiments showing the ability of the enhancement to infer information about crossing structures, even in diffusion tensor images (DTI) which are incapable of representing crossings themselves. We will present a novel non-linear enhancement technique which performs better than linear methods in areas around ventricles, thereby eliminating the need for additional preprocessing steps to segment out the ventricles. We pay special attention to the details of implementation of the various numerical schemes.

One of the classical optimization models for image segmentation is the well known Markov Random Fields (MRF) model. This model is a discrete optimization problem, which is shown here to formulate many continuous models used in image segmentation. In spite of the presence of MRF in the literature, the dominant perception has been that the model is not effective for image segmentation. We show here that the reason for the non-effectiveness is due to the lack of access to the optimal solution. Instead of solving optimally, heuristics have been engaged. Those heuristic methods cannot guarantee the quality of the solution nor the running time of the algorithm. Worse still, heuristics do not link directly the input functions and parameters to the output thus obscuring what would be ideal choices of parameters and functions which are to be selected by users in each particular application context.
We describe here how MRF can model and solve efficiently several known continuous models for image segmentation and describe briefly a very efficient polynomial time algorithm, which is provably fastest possible, to solve optimally the MRF problem. The MRF algorithm is enhanced here compared to the algorithm in Hochbaum (2001) by allowing the set of assigned labels to be any discrete set. Other enhancements include dynamic features that permit adjustments to the input parameters and solves optimally for these changes with minimal computation time. Several new theoretical results on the properties of the algorithm are proved here and are demonstrated for images in the context of medical and biological imaging. An interactive implementation tool for MRF is described, and its performance and flexibility in practice are demonstrated via computational experiments.
We conclude that many continuous models common in image segmentation have discrete analogs to various special cases of MRF and as such are solved optimally and efficiently, rather than with the use of continuous techniques, such as PDE methods, which restrict the type of functions used and furthermore, can only guarantee convergence to a local minimum.

In this paper, we explore the use of the diffusion geometry framework for the fusion of geometric and photometric information in local and global shape descriptors. Our construction is based on the definition of a diffusion process on the shape manifold embedded into a high-dimensional space where the embedding coordinates represent the photometric information. Experimental results show that such data fusion is useful in coping with different challenges of shape analysis where pure geometric and pure photometric methods fail.

Partial similarity of shapes is a challenging problem arising in many important applications in computer vision, shape analysis, and graphics, e.g. when one has to deal with partial information and acquisition artifacts. The problem is especially hard when the underlying shapes are non-rigid and are given up to a deformation. Partial matching is usually approached by computing local descriptors on a pair of shapes and then establishing a point-wise non-bijective correspondence between the two, taking into account possibly different parts. In this paper, we introduce an alternative correspondence-less approach to matching fragments to an entire shape undergoing a non-rigid deformation. We use region-wise local descriptors and optimize over the integration domains on which the integral descriptors of the two parts match. The problem is regularized using the Mumford-Shah functional. We show an efficient discretization based on the Ambrosio-Tortorelli approximation generalized to triangular point clouds and meshes, and present experiments demonstrating the success of the proposed method.

Detecting similarity between non-rigid shapes is one of the fundamental problems in computer vision. In order to measure the similarity the shapes must first be aligned. As opposite to rigid alignment that can be parameterized using a small number of unknowns representing rotations, reflections and translations, non-rigid alignment is not easily parameterized. Majority of the methods addressing this problem boil down to a minimization of a certain distortion measure. The complexity of a matching process is exponential by nature, but it can be heuristically reduced to a quadratic or even linear for shapes which are smooth two-manifolds. Here we model the shapes using both local and global structures, employ these to construct a quadratic dissimilarity measure, and provide a hierarchical framework for minimizing it to obtain sparse set of corresponding points. These correspondences may serve as an initialization for dense linear correspondence search.

Image deconvolution problems with a symmetric point-spread function arise in many areas of science and engineering. These problems often are solved by the Richardson-Lucy method, a nonlinear iterative method. We first show a convergence result for the Richardson-Lucy method. The proof sheds light on why the method may converge slowly. Subsequently, we describe an iterative active set method that imposes the same constraints on the computed solution as the Richardson-Lucy method. Computed examples show the latter method to yield better restorations than the Richardson-Lucy method and typically require less computational effort.

Total variation regularization has good performance in noise removal and edge preservation but lacks in texture restoration. Here we present a texture-preserving strategy to restore images contaminated by blur and noise. According to a texture detection strategy,  we apply spatially adaptive fractional order diffusion. A fast algorithm based on the half-quadratic technique is used to minimize the resulting objective function. Numerical results show the effectiveness of our strategy.

In this paper, we present a surface reconstruction via 2D strokes and a vector field on the strokes based on a two-step method. In the first step, from sparse strokes drawn by artists and a given vector field on the strokes, we propose a nonlinear vector interpolation combining total variation (TV) and $H^1$ regularization with a curl-free constraint for obtaining a dense vector field. In the second step, a height map is obtained by integrating the dense vector field in the first step. Jump discontinuities in surface and discontinuities of surface gradients can be well reconstructed without any surface distortion. We also provide a fast and efficient algorithm for solving the proposed functionals. Since vectors on the strokes are interpreted as a projection of surface gradients onto the plane, different types of strokes are easily devised to generate geometrically crucial structures such as ridge, valley, jump, bump, and dip on the surface. The stroke types help users to create a surface which they intuitively imagine from 2D strokes. We compare our results with conventional methods via many examples.