Level Set Evolution Equation Research Articles

Fractional guidance-based level set evolution for noisy image segmentation with intensity inhomogeneity

Intensity inhomogeneity (IIH) and noise are ubiquitous in images acquired by a variety of imaging modalities, which pose an ongoing challenge for segmentation methods used in many applications. This paper proposes a level set evolution equation based on fractional derivative to address the IIH and noise issues in image segmentation, which integrates the information from the input and guidance images. The proposed model is a linear diffusion equation of level set function with nonlinear edge-stopping and guidance sources, in which the edge-stopping source is derived from the input image and a local statistical model (Bayesian formulation) in the level set framework, whereas the guidance source is derived from the guidance image and the fitting term of the popular Chan-Vese model. The guidance source ensures swift contour movement toward the boundary of the objects and enhances the robustness to noise and contour initialization, while the edge-stopping source is used to stop the evolving contour on the boundary of the objects later in the evolution. To tackle the challenge of obtaining accurate guidance images for many applications, an adaptive variable-order of fractional derivative robust to IIH and noise is introduced to generate the desired guidance image from the input image. The proposed diffusion equation with sources is solved numerically by the series splitting method and finite difference scheme. Experimental results on synthetic and real images show that the proposed model has higher performance in terms of accuracy and efficiency of segmentation, compared to several state-of-the-art level set models.

A sweeping optimization algorithm for the global cosine fitting energy image segmentation model

AbstractImage segmentation plays a pivotal role in image processing. Level set model is a traditional variation image segmentation method. In order to achieve level set evolution equation, the level set energy functionals are minimized with the gradient descent methods and then the partial differential equation (PDE) was solved by finite difference scheme. Slow speed is one of its disadvantages. We propose a sweep optimization algorithm based on global cosine fitting (GCF) energy. Instead of calculating the PDE and the curvature, the proposed sweeping algorithm directly calculates the energy change when a pixel moves from one side of evolving contour to the other. It checks whether the GCF energy is decreased or not. The proposed algorithm has many advantages. For example, independent of initial level set contour positions and parameters, need not consider the Courant Friedrichs Lew condition and the regularization energy term. The proposed algorithm can be easily extended to high dimension image segmentation. The experiments on synthetic images, noise images and real images demonstrate the effectiveness of the proposed sweeping optimization algorithm.

A two-grid method for level-set based topology optimization with GPU-acceleration

In this paper, several accelerating strategies for the numerical methods of topology optimization are proposed. In our implementation, the finite element method is used for discretization, and the novelty of this research lies in two aspects. Two different meshes with variable element sizes are used to discretize the state equation and the level-set evolution equation. On the other hand, GPU-based accelerating schemes are implemented both for finite element assembler and sparse linear solver. Such combination is shown numerically to be effective to reduce the computational costs. Numerical illustrations are presented on several benchmark problems for topology optimization, such as the Cantilever and MBB problems. Compared with the traditional finite element calculations, our GPU-based two-grid schemes run up to 20 times faster without losing numerical accuracy.

A morphological approach to piecewise constant active contour model incorporated with the geodesic edge term

Traditional level set-based image segmentation method has to solve the level set evolution equation which is the Euler–Lagrange equation of the energy functional defined on the image domain. Solving level set evolution equation is very time-consuming, and reinitialization is usually needed. The level set evolution equation can also be solved by mathematical morphology. The morphological implementation is very simple, fast and stable. The piecewise constant active contour model incorporated with the geodesic edge term is a hybrid active contour model which combines two active contour models which are active contour model without edges and geodesic active contour model. In this paper, the mathematical morphology-based level set evolution method is applied to the piecewise constant active contour model incorporated with the geodesic edge term. The curvature morphological operator is also improved. Experimental results show that, compared with the original piecewise constant active contour model incorporated the geodesic edge term, the new mathematical morphology-based model can segment images more accurately and there are significant gains in simplicity, speed and stability. The new mathematical morphology-based model is also compared to morphological piecewise constant active contour model, morphological geodesic active contour model, traditional piecewise constant active contour model and two other active contour models. Results show that the proposed method gets the segmentation result with faster speed and higher accuracy.

An Efficient Multi-Scale Local Binary Fitting-Based Level Set Method for Inhomogeneous Image Segmentation

An efficient level set model based on multiscale local binary fitting (MLBF) is proposed for image segmentation. By introducing multiscale idea into the LBF model, the proposed MLBF model can effectively and efficiently segment images with intensity inhomogeneity. In addition, by adding a reaction diffusion term into the level set evolution (LSE) equation, the regularization of the level set function (LSF) can be achieved, thus completely eliminating the time-consuming reinitialization process. In the implementation phase, in order to greatly improve the efficiency of the numerical solution of the level set segmentation model, we introduce three strategies: The first is the additive operator splitting (AOS) solver which is used for breaking the restrictions on time step; the second is the salient target detection mechanism which is used to achieve full automatic initialization of the LSE process; the third is the sparse filed method (SFM) which is used to restrict the groups of pixels that need to be updated in a small strip region. Under the combined effect of these three strategies, the proposed model achieves very high execution efficiency in the following aspects: contour location accuracy, speed of evolution convergence, robustness against initial contour position, and robustness against noise interference.

Synthetic Aperture Radar Image Segmentation with Reaction Diffusion Level Set Evolution Equation in an Active Contour Model

This paper presents a method for synthetic aperture radar (SAR) image segmentation by draing upon a reaction–diffusion (RD) level set evolution (LSE) equation. The well-known RD theory consists of two main parts: reaction and diffusion terms. We first constructed the reaction term using an energy functional, which integrates the gamma statistical distribution with region–edge information from SAR images that can simultaneously suppress speckle noise and drive the active contour toward the object boundaries. Then, we used partial differential equation-based LSE to solve the proposed energy functional. Finally, a diffusion term was introduced into the LSE to ensure stability of the level set function and regularize the segmented region. The experimental results of both simulated and real SAR images showed that the proposed model has good robustness against a speckle noise as well as higher segmentation efficiency and accuracy than some existing models.

A multiphysics model for radiofrequency activation of soft hydrated tissues

A multi-physics model has been developed to investigate the effects of cellular level mechanisms on the electro-thermo-mechanical response of hydrated soft tissues with radiofrequency (RF) activation. A micromechanical model generates an equation of state (EOS) that provides the additional pressure arising from evaporation of intra- and extracellular water as well as temperature to the continuum level thermo-mechanical model. A level set method is used to capture the interfacial evolution of tissue damage with the level set evolution equation derived from the second law of thermodynamics, which is consistent with Griffith’s fracture evolution criterion. The discretized equations are solved simultaneously using a Krylov subspace based iterative solver (GMRES) with block preconditioning that effectively deflates the spectrum of the system matrix, resulting in exponential convergence of the Arnoldi iterations. Example problems, including experimental validation, illustrate the computational accuracy and efficiency of the technique.

An efficient level set method based on multi-scale image segmentation and hermite differential operator

In this paper, an efficient and robust level set method is presented to segment the images with intensity inhomogeneity. The multi-scale segmentation idea is incorporated into energy functional construction and a new Hermite differential operator is designed to numerically solve the level set evolution equation. Firstly, the circular shape window is used to define local region so as to approximate the image as well as intensity inhomogeneity. Then, multi-scale statistical analysis is performed on intensities of local circular regions centered in each pixel. So, the multi-scale local energy term can be constructed by fitting multi-scale approximation of inhomogeneity-free image in a piecewise constant way. To avoid the time-consuming re-initialization procedure, a new double-well potential function is adopted to construct the penalty energy term. Finally, the multi-scale segmentation is performed by minimizing the total energy functional. Here, a new differential operator based on Hermite polynomial interpolation is proposed to solve the minimization. The experiments and comparisons with three popular local region-based methods on images with different levels of intensity inhomogeneity have demonstrated the efficiency and robustness of the proposed method.

A new active contour remote sensing river image segmentation algorithm inspired from the cross entropy

The CV (Chan–Vese) model is a piecewise constant approximation of the Mumford and Shah model. It assumes that the original image can be segmented into two regions such that each region can be represented as constant grayscale value. In fact, the objective functional of the CV model actually finds a segmentation of the image such that the within-class variance is minimized. This is equivalent to the Otsu image thresholding algorithm which also aims to minimize the within-class variance. Similarly to the Otsu image thresholding algorithm, cross entropy is another widely used image thresholding algorithm and it finds a segmentation such that the cross entropy of the segmented image and the original image is minimized. Inspired from the cross entropy, a new active contour image segmentation algorithm is proposed. The region term in the new objective functional is the integral of the logarithm of the ratio between the grayscale of the original image and the mean value computed from the segmented image weighted by the grayscale of the original image. The new objective functional can be solved by the level set evolution method. A distance regularized term is added to the level set evolution equation so the level set need not be reinitialized periodically. A fast global minimization algorithm of the objective functional is also proposed which incorporates the edge term originated from the geodesic active contour model. Experimental results show that, the algorithm proposed can segment images more accurately than the CV model and the implementation speed of the fast global minimization algorithm is fast.

Efficient methods for implicit geometrical representation of complex material microstructures

SUMMARYWe present two methods for initializing distance functions on adaptively‐refined finite element meshes to represent complex material microstructures from segmented x‐ray tomographic data. Implicit microstructure representation combined with the extended FEM allows modelers to represent complex material microstructures with consistent mesh quality and accuracy. In the first method, a level set evolution equation is formulated and solved by the Galerkin method on an adaptively‐refined mesh. We show that the convergence and stability of this method is optimal for the order of elements used. In the second approach, we initialize the distance field by the fast marching method on a uniform grid, and then project the solution onto the finite element mesh by least‐squares. We show that this latter approach is superior in speed and accuracy. As an example problem, both methods are demonstrated in the initialization of distance fields for two inclusion phases within a Al‐7075 alloy. Copyright © 2014 John Wiley & Sons, Ltd.

An adaptive level set evolution equation for contour extraction

Level set evolution without re-initialization is a novel variational level set method for edge contour extraction, which has many advantages over the traditional level set formulations. The resulting evolution equation is derived from the minimization of an energy functional which is a linear combination of a weighted length, weighted area and deviation penalization energy. However, the zero level set always keeps evolving in only one direction during evolution, depending on the sign of the scale parameter associated with the weighted area. This inconvenient makes the evolution highly sensitive to the contour initializations. In this paper, we propose an adaptive level set evolution equation following this method, wherein the scale parameter associated with the weighted area is modified as an adaptive variable sign function and one of two terms associated with the weighted length is removed to reduce computational cost. The proposed equation avoids completely the intrinsic limitation mentioned above and offers many advantages over the original equation, as illustrated by several examples of contour extraction, such as robustness to noise and detection of objects with discontinuous boundaries.

Robust Contour Tracking by Combining Region and Boundary Information

This paper presents a new object tracking model that systematically combines region and boundary features. Besides traditional region features (intensity/color and texture), we design a new boundary-based object detector for accurate and robust tracking in low-contrast and complex scenes, which usually appear in the commonly used monochrome surveillance systems. In our model, region feature-based energy terms are characterized by probability models, and boundary feature terms include edge and frame difference. With a new weighting term, a novel energy functional is proposed to systematically combine the region and boundary-based components, and it is minimized by a level set evolution equation. For an efficient computational cost, motion information is utilized for new frame level set initialization. Compared with region feature-based models, the experimental results show that the proposed model significantly improves the performance under different circumstances, especially for objects in low-contrast and complex environments.

Road extraction from remote sensing images combining-region growing with level set

A color region growing algorithm based on PCA(Principal Component Analysis) was proposed.Level set method was employed to extract urban roads from high-resolution remote sensing images.Firstly,the region growing algorithm was applied to the preliminary road segmentation,and then the roughly obtained road region was used to construct the initial level set function,which then evolved stably according to a level set evolution equation without re-initialization.Finally,a regularizing local edge algorithm based on level set without re-initialization was presented,and utilized to regularize the local erroneous road curve due to obstacles.Experimental results show that the method is efficient and practical for extracting complete roads from high-resolution remote sensing images,with less manual intervention and stronger anti-noise ability.

Adaptive grid based on geometric conservation law level set method for time dependent PDE

AbstractA new method for mesh generation is formulated based on the level set functions, which are solutions of the standard level set evolution equation with the Cartesian coordinates as initial values (Liao et al. J Comput Phys 159 (2000), 103–122; Osher and Sethian J Comput Phys 79 (1988), 12; Sethian, Level set methods and fast marching methods, Cambridge University Press, 1999; Di et al. J Sci Comput 31 (2007), 75–98). The intersection of the level contours of the evolving functions form a new grid at each time. The velocity vector in the evolution equation is chosen according to the Geometric Conservation Law (GCL) method (Cao et al., SIAM J Sci Comput 24 (2002), 118–142.). This method has precise control over the Jacobian of transformation because of using the GCL method. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009

A level set characteristic Galerkin finite element method for free surface flows

We present a numerical method for free surface flows that couples the incompressible Navier-Stokes equations with the level set method in the finite element framework. The implicit characteristic-Galerkin approximation together with the fractional four-step algorithm is employed to discretize the governing equations. The schemes for solving the level set evolution and reinitialization equations are verified with several benchmark cases, including stationary circle, rotation of a slotted disk and stretching of a circular fluid element. The results are compared with those calculated from the level set finite volume method of Yue et al. , which employed the third-order essentially non-oscillatory (ENO) schemes for advection of the level set function in a generalized curvilinear coordinate system

Motion tracking as spatio-temporal motion boundary detection

The purpose of this study is to investigate tracking of moving objects in a sequence of images by detecting the surface generated by motion boundaries in the space–time domain. Estimation of this spatio-temporal surface is formulated as a Bayesian image partitioning problem. Minimization of the resulting energy functional seeks a solution biased toward smooth closed surfaces which coincide with motion boundaries, have small area, and partition the image into regions of contrasting motion activity. The Euler–Lagrange partial differential equations of minimization are expressed as level set evolution equations to obtain a topology independent and numerically stable algorithm. The formulation does not require estimation of the image motion field or assume a known background. It allows multiple non-simultaneous independent motions to occur and can account for camera motion without prior estimation of this motion. The analysis assumes short-range image motion. With moving cameras, it assumes that this short-range motion varies smoothly everywhere except across motion boundaries.

Level-Set-Based Deformation Methods for Adaptive Grids

A new method for generating adaptive moving grids is formulated based on physical quantities. Level set functions are used to construct the adaptive grids, which are solutions of the standard level set evolution equation with the Cartesian coordinates as initial values. The intersection points of the level sets of the evolving functions form a new grid at each time. The velocity vector in the evolution equation is chosen according to a monitor function and is equal to the node velocity. A uniform grid is then deformed to a moving grid with desired cell volume distribution at each time. The method achieves precise control over the Jacobian determinant of the grid mapping as the traditional deformation method does. The new method is consistent with the level set approach to dynamic moving interface problems.