On a Variational Model for Selective Image Segmentation of Features with Infinite Perimeter

Variational models provide reliable formulation for segmentation of features and their boundaries in an image, following the seminal work of Mumford-Shah (1989, Commun. Pure Appl. Math.) on dividing a general surface into piecewise smooth sub-surfaces. A central idea of models based on this work is to minimize the length of feature’s boundaries (i.e., H1 Hausdorff measure). However there exist problems with irregular and oscillatory object boundaries, where minimizing such a length is not appropriate, as noted by Barchiesi et al. (2010, SIAM J. Multiscale Model. Simu.) who proposed to miminize L2 Lebesgue measure of the γ-neighborhood of the boundaries. This paper presents a dual level set selective segmentation model based on Barchiesi et al. (2010) to automatically select a local feature instead of all global features. Our model uses two level set functions: a global level set which segments all boundaries, and the local level set which evolves and finds the boundary of the object closest to the geometric constraints. Using real life images with oscillatory boundaries, we show qualitative results demonstrating the effectiveness of the proposed method.     

A variational model and its numerical solution for local,selective and automatic segmentation

Variational region-based segmentation models can serve as effective tools for identifying all features and their boundaries in an image. To adapt such models to identify a local feature defined by geometric constraints, re-initializing iterations towards the feature offers a solution in some simple cases but does not in general lead to a reliable solution. This paper presents a dual level set model that is capable of automatically capturing a local feature of some interested region in three dimensions. An additive operator spitting method is developed for accelerating the solution process. Numerical tests show that the proposed model is robust in locally segmenting complex image structures.     

Homotopy method for a mean curvature-based denoising model

Variational image denoising models based on regularization of gradients have been extensively studied. The total variation model by Rudin, Osher, and Fatemi (1992) [38] can preserve edges well but for images without edges (jumps), the solution to this model has the undesirable staircasing effect. To overcome this, mean curvature-based energy minimization models offer one approach for restoring both smooth (no edges) and nonsmooth (with edges) images. As such models lead to fourth order (instead of the usual second order) nonlinear partial differential equations, development of fast solvers is a challenging task. Previously stabilized fixed point methods and their associated multigrid methods were developed but the underlying operators must be regularized by a relatively large parameter. In this paper, we first present a fixed point curvature method for solving such equations and then propose a homotopy approach for varying the regularized parameter so that the Newton type method becomes applicable in a predictor-corrector framework. Numerical experiments show that both of our methods are able to maintain all important information in the image, and at the same time to filter out noise.     

An automatic regularization parameter selection algorithm in the total variation model for image deblurring

Image restoration is an inverse problem that has been widely studied in recent years. The total variation based model by Rudin-Osher-Fatemi (1992) is one of the most effective and well known due to its ability to preserve sharp features in restoration. This paper addresses an important and yet outstanding issue for this model in selection of an optimal regularization parameter, for the case of image deblurring. We propose to compute the optimal regularization parameter along with the restored image in the same variational setting, by considering a Karush Kuhn Tucker (KKT) system. Through establishing analytically the monotonicity result, we can compute this parameter by an iterative algorithm for the KKT system. Such an approach corresponds to solving an equation using discrepancy principle, rather than using discrepancy principle only as a stopping criterion. Numerical experiments show that the algorithm is efficient and effective for image deblurring problems and yet is competitive.     

On a New Diffeomorphic Multi-Modality Image Registration Model and Its Convergent Gauss-Newton Solver

In this work, we propose a new variational model for multi-modal image registration and present an efficient numerical implementation. The model minimizes a new functional based on using reformulated normalized gradients of the images as the fidelity term and higher-order derivatives as the regularizer. A key feature of the model is its ability of guaranteeing a diffeomorphic transformation which is achieved by a control term motivated by the quasi-conformal map and Beltrami coefficient. The existence of the solution of this model is established. To solve the model numerically, we design a Gauss-Newton method to solve the resulting discrete optimization problem and prove its convergence; a multilevel technique is employed to speed up the initialization and avoid likely local minima of the underlying functional. Finally, numerical experiments demonstrate that this new model can deliver good performances for multi-modal image registration and simultaneously generate an accurate diffeomorphic transformation.     

Image denoising using the Gaussian curvature of the image surface

A number of high‐order variational models for image denoising have been proposed within the last few years. The main motivation behind these models is to fix problems such as the staircase effect and the loss of image contrast that the classical Rudin–Osher–Fatemi model [Leonid I. Rudin, Stanley Osher and Emad Fatemi, Nonlinear total variation based noise removal algorithms, Physica D 60 (1992), pp. 259–268] and others also based on the gradient of the image do have. In this work, we propose a new variational model for image denoising based on the Gaussian curvature of the image surface of a given image. We analytically study the proposed model to show why it preserves image contrast, recovers sharp edges, does not transform piecewise smooth functions into piecewise constant functions and is also able to preserve corners. In addition, we also provide two fast solvers for its numerical realization. Numerical experiments are shown to illustrate the good performance of the algorithms and test results. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1066–1089, 2016