A variational formulation for physical noised image segmentation

Image segmentation is a hot topic in image science. In this paper we present a new variational segmentation model based on the theory of Mumford-Shah model. The aim of our model is to divide noised image, according to a certain criterion, into homogeneous and smooth regions that should correspond to structural units in the scene or objects of interest. The proposed region-based model uses total variation as a regularization term, and different fidelity term can be used for image segmentation in the cases of physical noise, such as Gaussian, Poisson and multiplicative speckle noise. Our model consists of five weighted terms, two of them are responsible for image denoising based on fidelity term and total variation term, the others assure that the three conditions of adherence to the data, smoothing, and discontinuity detection are met at once. We also develop a primal-dual hybrid gradient algorithm for our model. Numerical results on various synthetic and real images are provided to compare our method with others,these results show that our proposed model and algorithms are effective.     

Two-Stage Image Segmentation Scheme Based on Inexact Alternating Direction Method

Image segmentation is a fundamental problem in both image processing and computer vision with numerous applications. In this paper, we propose a two-stage image segmentation scheme based on inexact alternating direction method. Specifically, we first solve the convex variant of the Mumford-Shah model to get the smooth solution, and the segmentation is then obtained by applying the K-means clustering method to the solution. Some numerical comparisons are arranged to show the effectiveness of our proposed schemes by segmenting many kinds of images such as artificial images, natural images, and brain MRI images.     

基于直方图初始化的有效的多区域Mumford-Shah图像分割方法（英文）

The Mumford-Shah energy functional is a successful image segmentation model. It is a non-convex variational problem and lacks of good initialization techniques so far. In this paper, motivated by the fact that image histogram is a combination of several Gaussian distributions, and their centers can be considered as approximations of cluster centers, we introduce a histogram-based initialization method to compute the cluster centers. With this technique, we then devise an effective multi-region Mumford-Shah image segmentation method, and adopt the recent proximal alternating minimization method to solve the minimization problem. Experiments indicate that our histogram initialization method is more robust than existing methods,and our segmentation method is very effective for both gray and color images.     

Segmentation of Stochastic Images using Stochastic Extensions of the Ambrosio-Tortorelli and the Random Walker Model

We discuss methods based on stochastic PDEs for the segmentation of images with uncertain gray values resulting from measurement errors and noise. Our approach yields a reliable precision estimate for the segmentation result, and it allows us to quantify the robustness of edges in noisy images and under gray value uncertainty. The ansatz space for such images identifies gray values with random variables. For their discretization we utilize generalized polynomial chaos expansions and the generalized spectral decomposition method. This leads to the stochastic generalization of the Ambrosio-Tortorelli approximation of the Mumford-Shah functional. Moreover, we present the extension of the random walker segmentation for our stochastic images, which is based on an identification of the graph weights with random variables. We demonstrate the performance of the methods on a data set obtained from a digital camera as well as real medical ultrasound data. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

IMAGE SEGMENTATION BY PIECEWISE CONSTANT MUMFORD-SHAH MODEL WITHOUT ESTIMATING THE CONSTANTS

In this work, we try to use the so-called Piecewise Constant Level Set Method （PCLSM） for the Mumford-Shah segmentation model. For image segmentation, the Mumford-Shah model needs to find the regions and the constant values inside the regions for the segmen- tation. In order to use PCLSM for this purpose, we need to solve a minimization problem using the level set function and the constant values as minimization variables. In this work, we test on a model such that we only need to minimize with respect to the level set function, i.e., we do not need to minimize with respect to the constant values. Gradient descent method and Newton method are used to solve the Euler-Lagrange equation for the minimization problem. Numerical experiments are given to show the efficiency and advantages of the new model and algorithms.     

The Inverse Commutant Lifting Problem. II: Hellinger Functional-Model Spaces

In this second part of the paper (Ball and Kheifets in Integral Equ Oper Theory 70(1):17–62, 2011) we further develop the ideas in Kheifets (Operator theory and interpolation, vol OT 115. Birkhäuser, Basel, pp 213–233, 2000; Interpolation theory, systems theory and related topics, vol OT 134. Birkhäuser, Basel, pp 287–317, 2002) to obtain a more concrete function-theoretic form of Theorem 8.4 of the first part in terms of Hellinger model space (Theorems 3.3, 4.3 below). This leads to generalizations of classical results of Arov and to characterizations of the coefficient matrix-measures of the lifting problem in terms of the density properties of the corresponding model spaces. In Sect. 5 we apply our results to the classical Nehari problem.     

3D Multiphase Piecewise Constant Level Set Method Based on Graph Cut Minimization

Segmentation of three-dimensional (3D) complicated structures is of great importance for many real applications. In this work we combine graph cut minimization method with a variant of the level set idea for 3D segmentation based on the Mumford-Shah model. Compared with the traditional approach for solving the Euler-Lagrange equation we do not need to solve any partial differential equations. Instead, the minimum cut on a special designed graph need to be computed. The method is tested on data with complicated structures. It is rather stable with respect to initial value and the algorithm is nearly parameter free. Experiments show that it can solve large problems much faster than traditional approaches.     

A New Proof for the Convergence of an Individual Based Model to the Trait Substitution Sequence

We consider a continuous time stochastic individual based model for a population structured only by an inherited vector trait and with logistic interactions. We consider its limit in a context from adaptive dynamics: the population is large, the mutations are rare and the process is viewed in the timescale of mutations. Using averaging techniques due to Kurtz (in Lecture Notes in Control and Inform. Sci., vol. 177, pp. 186–209, 1992), we give a new proof of the convergence of the individual based model to the trait substitution sequence of Metz et al. (in Trends in Ecology and Evolution 7(6), 198–202, 1992), first worked out by Dieckman and Law (in Journal of Mathematical Biology 34(5–6), 579–612, 1996) and rigorously proved by Champagnat (in Theoretical Population Biology 69, 297–321, 2006): rigging the model such that “invasion implies substitution”, we obtain in the limit a process that jumps from one population equilibrium to another when mutations occur and invade the population.     

Non-local approximation of the Mumford-Shah functional

The Mumford-Shah functional, introduced to study image segmentation problems, is approximated in the sense of -convergence by a sequence of non-local integral functionals. Received June 6, 1996 / Accepted July 11, 1996     

A primal-dual active-set algorithm for bilaterally constrained total variation deblurring and piecewise constant Mumford-Shah segmentation problems

In this paper, we propose a fast primal-dual algorithm for solving bilaterally constrained total variation minimization problems which subsume the bilaterally constrained total variation image deblurring model and the two-phase piecewise constant Mumford-Shah image segmentation model. The presence of the bilateral constraints makes the optimality conditions of the primal-dual problem semi-smooth which can be solved by a semi-smooth Newton’s method superlinearly. But the linear system to solve at each iteration is very large and difficult to precondition. Using a primal-dual active-set strategy, we reduce the linear system to a much smaller and better structured one so that it can be solved efficiently by conjugate gradient with an approximate inverse preconditioner. Locally superlinear convergence results are derived for the proposed algorithm. Numerical experiments are also provided for both deblurring and segmentation problems. In particular, for the deblurring problem, we show that the addition of the bilateral constraints to the total variation model improves the quality of the solutions.     

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.     

Diffusion and growth in an evolving network

We study a simple model of a population of agents whose interaction network co-evolves with knowledge diffusion and accumulation. Diffusion takes place along the current network and, reciprocally, network formation depends on the knowledge profile. Diffusion makes neighboring agents tend to display similar knowledge levels. On the other hand, similarity in knowledge favors network formation. The cumulative nonlinear effects induced by this interplay produce sharp transitions, equilibrium co-existence, and hysteresis, which sheds some light on why multiplicity of outcomes and segmentation in performance may persist resiliently over time in knowledge-based processes.     

A finitely additive white noise approach to nonlinear filtering

An approach to nonlinear filtering theory is developed in which finitely additive white noise replaces the Wiener process in the observation process model. The important case when the signal is a Markov process independent of the noise is investigated in detail. The theory turns out to be simpler than the current theory based on the stochastic calculus. Stochastic partial differential equations are replaced by partial differential equations in which the observation (in the finitely additive set up) occurs as a parameter. Theorems on existence and uniqueness of solutions are obtained. The white noise approach has the advantage that it provides a robust solution to the filtering problem. Furthermore, the robust theory based on the Ito calculus can be recovered from the results of this paper.     

On a problem of Yau regarding a higher dimensional generalization of the Cohn–Vossen inequality

We show that a problem asked by Yau (Open problems in geometry. Chern–a great geometer of the twentieth century, pp. 275–319, 1992) cannot be true in general. The counterexamples are constructed based on the recent work of Wu and Zheng (Examples of positively curved complete Kähler manifolds. Geometry and Analysis, vol. 17, pp. 517–542, 2010).     

Group Extensions and Hall Polynomials

In this paper, we give two proofs of a formula containing the numbers of automorphisms of an Abelian group, of its subgroups, and of its quotient groups. The first proof is based on the use of the theory of Hall polynomials, while the second one uses extension theory for Abelian groups.__________Translated from Matematicheskie Zametki, vol. 78, no. 2, 2005, pp. 180–185.Original Russian Text Copyright © 2005 by G. V. Voskresenskaya.     

A class of quasi-linear elliptic systems arising in image segmentation

We study a quasi-linear system arising in image segmentation in order to determine the minimum of the Mumford-Shah functional. We prove the existence of a solution for a larger class of systems including the previous one. The recursive method used in the proof produces as well an efficient algorithm that can be easily implemented in connection for example with finite element techniques. Results of some numerical experiments are also discussed. Received June 30, 1997     

Image segmentation by level set evolution with region consistency constraint

Image segmentation is a key and fundamental problem in image processing, computer graphics, and computer vision. Level set based method for image segmentation is used widely for its topology flexibility and proper mathematical formulation. However, poor performance of existing level set models on noisy images and weak boundary limit its application in image segmentation. In this paper, we present a region consistency constraint term to measure the regional consistency on both sides of the boundary, this term defines the boundary of the image within a range, and hence increases the stability of the level set model. The term can make existing level set models significantly improve the efficiency of the algorithms on segmenting images with noise and weak boundary. Furthermore, this constraint term can make edge-based level set model overcome the defect of sensitivity to the initial contour. The experimental results show that our algorithm is efficient for image segmentation and outperform the existing state-of-art methods regarding images with noise and weak boundary.     

Simultaneous reconstruction and segmentation for tomography data

We present a Mumford-Shah like approach for the inversion of CT and SPECT-data (Single Photon Emission Computerized Tomography). With this approach we aim at the simultaneous reconstruction and segmentation of activity and density distribution from given tomography data. We assume the functions to be piecewise constant with respect to a set of contours. Shape sensitivity analysis is used to find a descent direction for the cost functional which leads to an update formula for the contour in a level set framework. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Implementation of an adaptive finite-element approximation of the Mumford-Shah functional

Summary. We present and detail a method for the numerical solving of the Mumford-Shah problem, based on a finite element method and on adaptive meshes. We start with the formulation introduced in [13], detail its numerical implementation and then propose a variant which is proved to converge to the Mumford-Shah problem. A few experiments are illustrated. Received October 8, 1998 / Published online: April 20, 2000