Abstract. The choice of a proper regularization measure plays an important role in the field of image processing. One classical approach treats color images as two- dimensional surfaces embedded in a five-dimensional spatial-chromatic space. In this case, a natural regularization term arises as the image surface area. Choosing the chromatic coordinates as dominating over the spatial ones, we can think of the image spatial coordinates could as a parameterization of the image surface manifold in a three-dimensional color space. Minimizing the area of the image manifold leads to the Beltrami flow or mean curvature flow of the image surface in the three-dimensional color space, while minimizing the elastica of the image surface yields an additional interesting regularization. Recently, we proposed a color elastica model, which minimizes both the surface area and the elastica of the image manifold. In this paper, we propose to modify the color elastica and introduce two new models for color image regularization. The revised measures are motivated by the relations between the color elastica model, Euler’s elastica model, and the total variation model for gray level images. Compared to our previous color elastica model, the new models are direct extensions of Euler’s elastica model to color images. The proposed models are nonlinear and challenging to minimize. To overcome this difficulty, two operator-splitting methods are suggested. Specifically, nonlinearities are decoupled by the introduction of new vector- and matrix-valued variables. Then, the minimization problems are converted to initial value problems which are time-discretized by operator splitting. Each subproblem, after splitting, either has a closed-form solution or can be solved efficiently. The effectiveness and advantages of the proposed models are demonstrated by comprehensive experiments. The benefits of incorporating the elastica of the image surface as regularization terms compared to common alternatives are empirically validated.

Models related to the Euler's elastica energy have proven to be useful for many applications including image processing. Extending elastica models to color images and multichannel data is a challenging task, as stable and consistent numerical solvers for these geometric models often involve high order derivatives. Like the single channel Euler's elastica model and the total variation models, geometric measures that involve high order derivatives could help when considering image formation models that minimize elastic properties. In the past, the Polyakov action from high energy physics has been successfully applied to color image processing. Here, we introduce an addition to the Polyakov action for color images that minimizes the color manifold curvature. The color image curvature is computed by applying the Laplace--Beltrami operator to the color image channels. When reduced to gray-scale images, while selecting appropriate scaling between space and color, the proposed model minimizes Euler's elastica operating on the image level sets. Finding a minimizer for the proposed nonlinear geometric model is a challenge we address in this paper. Specifically, we present an operator-splitting method to minimize the proposed functional. The nonlinearity is decoupled by introducing three vector-valued and matrix-valued variables. The problem is then converted into solving for the steady state of an associated initial-value problem. The initial-value problem is time split into three fractional steps, such that each subproblem has a closed form solution, or can be solved by fast algorithms. The efficiency and robustness of the proposed method are demonstrated by systematic numerical experiments.

Many objects in real world have convex shapes. It is a difficult task to have representations for convex shapes with good and fast numerical solutions. This paper proposes a method to incorporate convex shape prior for multi-object segmentation using level set method. The relationship between the convexity of the segmented objects and the signed distance function corresponding to their union is analyzed theoretically. This result is combined with Gaussian mixture method for the multiple objects segmentation with convexity shape prior. Alternating direction method of multiplier (ADMM) is adopted to solve the proposed model. Special boundary conditions are also imposed to obtain efficient algorithms for 4th order partial differential equations in one step of ADMM algorithm. In addition, our method only needs one level set function regardless of the number of objects. So the increase in the number of objects does not result in the increase of model and algorithm complexity. Various numerical experiments are illustrated to show the performance and advantages of the proposed method.

Euler's elastica model has a wide range of applications in image processing and computer vision. However, the nonconvexity, the nonsmoothness, and the nonlinearity of the associated energy functional make its minimization a challenging task, further complicated by the presence of high order derivatives in the model. In this article we propose a new operator-splitting algorithm to minimize the Euler elastica functional. This algorithm is obtained by applying an operator-splitting based time discretization scheme to an initial value problem (dynamical flow) associated with the optimality system (a system of multivalued equations). The subproblems associated with the three fractional steps of the splitting scheme have either closed form solutions or can be handled by fast dedicated solvers. Compared with earlier approaches relying on ADMM (Alternating Direction Method of Multipliers), the new method has, essentially, only the time discretization step as free parameter to choose, resulting in a very robust and stable algorithm. The simplicity of the subproblems and its modularity make this algorithm quite efficient. Applications to the numerical solution of smoothing test problems demonstrate the efficiency and robustness of the proposed methodology.

We propose a new methodology for carrying out eikonal based traveltime tomography arising from important applications such as seismic imaging and medical imaging. The new method formulates the traveltime tomography problem as a variational problem for a certain cost functional explicitly with respect to both traveltime and sound speed. Furthermore, the cost functional is penalized to enforce the nonlinear equality constraint associated with the underlying eikonal equation, biharmonically regularized with respect to traveltime, and harmonically regularized with respect to sound speed. To overcome the difficulty associated with the inherent nonlinearity of the eikonal equation, the Euler--Lagrange equation of the penalized-regularized variational problem is reformulated into an equivalent, mixed optimality system. This mixed system is associated with an initial value problem which is solved by an operator-splitting based solution method, and the splitting approach effectively reduces the optimality system into three nonlinear subproblems and three linear subproblems. Moreover, the nonlinear subproblems can be solved pointwise, while the linear subproblems can be reduced to linear second-order elliptic problems. Numerical experiments show that the new method can carry out traveltime tomography successfully and recover sound speeds efficiently.

Variational methods are commonly used to solve noise removal problems. In this paper, we present an augmented Lagrangian-based approach that uses a discrete form of the $L^1$-norm of the mean curvature of the graph of the image as a regularizer, discretization being achieved via a finite element method. When a particular alternating direction method of multipliers is applied to the solution of the resulting saddle-point problem, this solution reduces to an iterative sequential solution of four subproblems. These subproblems are solved using Newton’s method, the conjugate gradient method, and a partial solution variant of the cyclic reduction method. The approach considered here differs from existing augmented Lagrangian approaches for the solution of the same problem; indeed, the augmented Lagrangian functional we use here contains three Lagrange multipliers “only,” and the associated augmentation terms are all quadratic. In addition to the description of the solution algorithm, this paper contains the results of numerical experiments demonstrating the performance of the novel method discussed here.

A block-iterative finite element procedure is presented for two-dimensional fluid dynamics computations on multiply-connected domains based on the vorticity - stream function formulation of the incompressible Navier-Stokes equations. The difficulty associated with the convection term in the vorticity transport equation is addressed by using a streamline-upwind/Petrov-Galerkin scheme.