Single Level Set Research Articles

A single level set function approach for multiple material-phases applied to full-waveform inversion in the time domain

This paper presents a multiple material-phase level-set approach for acoustic full-waveform inversion in the time domain. By using a single level set (LS) function, several level values are used to define virtual boundaries between material phases with different (and known) wave propagation velocities. The aim of the proposed approach is to provide a suitable framework to identify multiple/nested inclusions or a finite number of almost homogeneous sedimentary layers with sharp interfaces between them. The use of a single LS function provides a significant reduction in the number of variables to be identified, when compared with the usual multi-material phase approaches defined by multiple functions, especially for problems with a high number of degrees of freedom. Numerical experiments show satisfactory results in identifying simultaneously different interfaces. Cases with and without inverse crime are evaluated, showing that the approach is reasonably robust in dealing with such a condition.

CDM: A coupled deformable model for image segmentation with speckle noise and severe intensity inhomogeneity

Speckle noise and intensity inhomogeneity are always challenging issues in the area of image segmentation, especially when both difficulties appear simultaneously. Consequently, the majority of existing deformable models yield inadequate results in this situation. This paper aims to address these issues by combining a bias correction term and a despeckling term into a single variational level set architecture, which not only corrects the severe intensity inhomogeneity but also performs image despeckling and segmentation simultaneously. Specifically, edge and region synergetic terms are defined to correct the severe intensity inhomogeneity during image segmentation that eradicates their shortcomings as well as retains their advantages. Further, a despeckling term is constructed, which effectively suppresses the noise while enhancing image details. Since the lack of thorough investigation on despeckling of images degraded by Rayleigh noise distribution, which usually appear in ultrasound (US) images, so we included it in the coupled deformable model (CDM). Moreover, region-based terms eradicate complex re-initialization and numerical instability during the level set evolution, as a result, no distance regularization term is needed. Additionally, Schauder’s fixed point theorem is used to demonstrate the well-posedness of the present system. Extensive numerical experiments on real and synthetic images with speckle noise and severe inhomogeneous intensity demonstrate that the CDM exhibits superior results compared to classical and some recent deformable models in terms of accuracy, robustness, and various quality measures.

Level set-based topological design of multiphase micro-architectured materials using alternating active-phase method

Micro-architectured materials with periodically configured microstructures are widely used in various engineering fields due to their superior mechanical properties. Compared with conventional topology design with one single material, topological design with multiple materials can provide a better solution as the expanded design freedom. This paper presents a level set-based multi-material topological design method for micro-architectured materials involving three phases or more. In this method, a difference-set-based multi-material level set model is employed to represent the topology of each phase, in which N + 1 phases can be precisely represented by the sequential difference-set of N level set functions without any overlaps and redundant regions. By using an alternating active-phase algorithm, the multi-material topology design problem with N + 1 phases is splitted into N(N + 1)/2 binary-phase topological design sub-problems, where each sub-problem involves fewer design variables and volume constraints. The effective properties of multiphase micro-architectured materials are evaluated by a numerical homogenization approach, and their topological evolutions of two active phases in each sub-problem can be readily achieved by updating a single level set function, which greatly facilitate the extension to topological design problems with more phases. Both numerical examples and additive-manufactured specimens demonstrate the effectiveness of the proposed method for designing multiphase micro-architectured materials.

Estimation of non-symmetric and unbounded region of attraction using shifted shape function and R-composition

Sum-of-squares programming is widely used for region of attraction (ROA) estimations of asymptotically stable equilibrium points of nonlinear polynomial systems. However, existing methods yield conservative results, especially for non-symmetric and unbounded regions. In this study, a cost-effective approach for ROA estimation is proposed based on the Lyapunov theory and shape functions. In contrast to existing methods, the proposed method iteratively places the center of a shifted shape function (SSF) close to the boundary of the acquired invariant subset. The set of obtained SSFs yields robust ROA subsets, and R-composition is employed to express these independent sets as a single but richer-shaped level set. Several benchmark examples show that the proposed method significantly improves ROA estimations, especially for non-symmetric or unbounded ROA without a significant computational burden.

A new level set based multi-material topology optimization method using alternating active-phase algorithm

This paper proposes a new level set based multi-material topology optimization method, where a difference-set-based multi-material level set (DS-MMLS) model is developed for topology description and an alternating active-phase algorithm is implemented. Based on the alternating active-phase algorithm, a multi-material topology optimization problem with N + 1 phases is split into N(N + 1)/2 binary-phase topology optimization sub-problems. Compared with the initial multi-material problem, each sub-problem involves fewer design variables and volume constraints. In the DS-MMLS model, N + 1 phases are represented by the sequential difference set of N level set functions. Based on this model, the topological evolution of two active phases can be easily achieved by updating a single level set function in a fixed domain, which contributes a great convenience to the implementation of the alternating active-phase algorithm with level set method. Therefore, the proposed method can be easily extended to topology optimization problems with more material phases. To demonstrate its effectiveness, some 2D and 3D numerical examples with different material phases are presented. The results reveal that the proposed method is effective for multi-material topology optimization problems.

Multi-region segmentation by a single level set generalization applied to stroke CT images

This article proposes a level set approach to segment images with N regions by using a single level set function. Many works use several level set fronts or hierarchical approach to solve this problem. We propose a general formulation of the level set propagation function based on the knowledge for two regions. A modified likelihood function is proposed, and each background probability is maximized. A single propagation function is achieved from N logarithmic components. Experiments are performed on synthetic images with normal probability and computed tomography images of patients with hemorrhagic stroke. Our approach is compared with other ones known in the literature, and the level set was superior in 10 metrics out of 13 evaluated, with an accuracy of 99.67% and FSIM 93.96%. These results confirm the effectiveness of the proposed method.

Inhomogeneous Brain Magnetic Resonance Images Segmentation Using a Novel Double Level Set Method

Magnetic resonance (MR) image segmentation plays an important role in the clinical diagnosis and pathological analysis of brain diseases, and has become a focus in the field of medical image processing. However, MR image segmentation is also a complex task because it is easily corrupted by inhomogeneous intensity and noise during the process of imaging. In this paper, we use double level set function to replace single level set of the data energy fitting model and propose a model based on Legendre polynomial and Heaviside function, which is used to segment brain magnetic resonance images. The double level set method (DLSM) can extract simultaneously the white matter (WM) and gray matter (GM) of brain tissue and ensure the robustness of level set initialization. Moreover, the bias field caused by intensity inhomogeneity is represented by a set of smooth basis functions, which can satisfy its property of slow variety. Finally, compared with the local intensity clustering model and multiplicative intrinsic component optimization model, both visual and objective results can prove the superior of the proposed DLSM model, and the computational speed is faster.

Design of shell-infill structures by a multiscale level set topology optimization method

Shell-infill structures are widely used in additive manufacturing (AM) to retain the external appearance and reduce the printing costs. However, shell-infill structures are generally designed by a mono-scale topology optimization method. In this case, these shell-infill structures are not optimal designs due to their incompatible shell and infill layouts. In order to obtain the optimized shell and infill simultaneously, this paper presents a multiscale level set topology optimization method for designing shell-infill structures. In macroscale optimization, two distinct level sets of a single level set function (LSF) are used to represent the interface of the shell and the infill, respectively. The thickness of the shell is assumed to be uniform, which is guaranteed by a level set reinitialization scheme. In microscale optimization, the pattern of the microstructure is optimized to achieve the optimal macro structural performance. The numerical homogenization method is applied to evaluate the effective elasticity matrix of the microstructural infill. The shell-coated macro structure and the micro infill are optimized concurrently to achieve the optimal shell-infill design with prescribed volume fractions. Both 2D and 3D examples are investigated to demonstrate the effectiveness of the presented method.

Converting SVDD scores into probability estimates: Application to outlier detection

To enable post-processing, the output of a support vector data description (SVDD) should be transformed into a calibrated probability, as it can be done for SVM. But standard SVDD only estimate a single level set and do not provide such probabilities. We present a method for estimating these probabilities from SVDD scores. The first step of our approach uses a generalization of the SVDD model that estimate simultaneously various coherent level sets. Then we introduce two calibration mechanisms for converting these level sets into probabilities. A synthetic dataset and datasets from the UCI repository are used to compare the performance of our method against a robust kernel density estimator in an outlier detection task, illustrating the interest of our approach.

Topology optimization of multi-material negative Poisson’s ratio metamaterials using a reconciled level set method

Metamaterials are defined as a family of rationally designed artificial materials which can provide extraordinary effective properties compared with their nature counterparts. This paper proposes a level set based method for topology optimization of both single and multiple-material Negative Poisson’s Ratio (NPR) metamaterials. For multi-material topology optimization, the conventional level set method is advanced with a new approach exploiting the reconciled level set (RLS) method. The proposed method simplifies the multi-material topology optimization by evolving each individual material with a single level set function and reconciling the result level set field with the Merriman–Bence–Osher (MBO) operator. The NPR metamaterial design problem is recast as a variational problem, where the effective elastic properties of the spatially periodic microstructure are formulated as the strain energy functionals under uniform displacement boundary conditions. The adjoint variable method is utilized to derive the shape sensitivities by combining the general linear elastic equation with a weak imposition of Dirichlet boundary conditions. The design velocity field is constructed using the steepest descent method and integrated with the level set method. Both single and multiple-material mechanical metamaterials are achieved in 2D and 3D with different Poisson’s ratios and volumes. Benchmark designs are fabricated with multi-material 3D printing at high resolution. The effective auxetic properties of the achieved designs are verified through finite element simulations and characterized using experimental tests as well.

On Geodesic Curvature Flow with Level Set Formulation Over Triangulated Surfaces

The geodesic curvature flow is an important concept in Riemannian geometry. The flow with level set formulation has many applications in image processing, computer vision, material sciences, etc. The existing discretizations on triangulated surfaces are based on either finite volume method or finite element method with piecewise linear function space, which are suitable for vertex-based two-phase problems. The contour (zero level set) in existing methods passes through triangles of the mesh. However, in some graphic applications, such as mesh segmentation (to divide a whole mesh into several sub-meshes without ambiguous triangular stripes), the cutting contour is needed to be along the edges of the mesh. Moreover, multi-phase segmentation by a single level set function is a difficult problem for a long time. In this paper, we try to tackle these two problems. We propose a new discretization which has simpler formulation and more sparse coefficient matrix. We prove the existence and uniqueness, regularization behavior and maximum–minimum principle of our discrete flow. Therein the maximum–minimum principal has not been presented before. Lots of experiments show that, the limit of the flow would be a piecewise constant solution with ’discontinuity set’ to be the closed geodesics of the surface. We therefore propose a constrained discrete geodesic curvature flow, which is also analyzed theoretically. The linear system of the constrained flow can be equivalently reformulated into a much smaller one (especially in the narrow band algorithm), which dramatically reduces the computation cost. Combined with a narrow band algorithm, the constrained flow with topologically correct initializations (easy to be got by simple existing methods or manual inputs) yields a multi-phase segmentation method by a single level set function. We test our two flows in closed curve evolution and multi-region segmentation applications. The numerical experiments are given to demonstrate the effectiveness.

A multi-object image segmentation C–V model based on region division and gradient guide

The Chan–Vese (C–V) model is an ineffective method for processing images in which the intensity is inhomogeneous. This is especially true for multi-object segmentation, in which the target may be missed or excessively segmented. In addition, for images with rich texture information, the processing speed of the C–V is slow. To overcome these problems, this paper proposes an effective multi-object C–V segmentation model based on region division and gradient guide. First, a rapid initial contour search is conducted using Otsu’s method. This contour line becomes the initial contour for our multi-object segmentation C–V model based on a gradient guide. To achieve the multi-object segmentation the image is then converted to a single level set whose evolution is controlled using an adaptive gradient. The feasibility of the proposed model is analyzed theoretically, and a number of simulation experiments are conducted to validate its effectiveness.

Simultaneous extraction of endocardial and epicardial contours of the left ventricle by distance regularized level sets.

Segmentation of the cardiac left ventricle (LV) is still an open problem and is challenging due to the poor contrast between tissues around the epicardium and image artifacts. To extract the endocardium and epicardium of the cardiac left ventricle accurately, the authors propose a two-layer level set approach for segmentation of the LV from cardiac magnetic resonance short-axis images. In the proposed method, the endocardium and epicardium are represented by two specified level contours of a level set function. Segmentation of the LV is formulated as a problem of optimizing the level set function such that these two level contours best fit the epicardium and endocardium, subject to a distance regularization (DR) term to preserve a smoothly varying distance between them. The DR term introduces a desirable interaction between the two level contours of a single level set function, which contributes to preserve the anatomical geometry of the epicardium and endocardium of the LV. In addition, the proposed method has an intrinsic ability to deal with intensity inhomogeneity in MR images, which is a common image artifact in MRI. Their method is quantitatively validated by experiments on the datasets for the MICCAI 2009 grand challenge on left ventricular segmentation and the MICCAI 2013 challenge workshop on segmentation: algorithms, theory and applications (SATA). To overcome discontinuity of 2D segmentation results at some adjacent slices for a few cases, the authors extend distance regularized two-layer level set to 3D to refine the segmentation results. The corresponding metrics for their method are better than the methods in the MICCAI 2009 challenge. Their method was ranked at the first place in terms of Hausdorff distance and the second place in terms of Dice similarity coefficient in the MICCAI 2013 challenge. Experimental results demonstrate the advantages of their method in terms of segmentation accuracy and consistency with the heart anatomy.

Multi-Region Active Contours with a Single Level Set Function.

Segmenting an image into an arbitrary number of coherent regions is at the core of image understanding. Many formulations of the segmentation problem have been suggested over the past years. These formulations include, among others, axiomatic functionals, which are hard to implement and analyze, and graph-based alternatives, which impose a non-geometric metric on the problem. We propose a novel method for segmenting an image into an arbitrary number of regions using an axiomatic variational approach. The proposed method allows to incorporate various generic region appearance models, while avoiding metrication errors. In the suggested framework, the segmentation is performed by level set evolution. Yet, contrarily to most existing methods, here, multiple regions are represented by a single non-negative level set function. The level set function evolution is efficiently executed through the Voronoi Implicit Interface Method for multi-phase interface evolution. The proposed approach is shown to obtain accurate segmentation results for various natural 2D and 3D images, comparable to state-of-the-art image segmentation algorithms.

Transient filling modelling at meso-level for RTM process using a single phase LSM

A single phase level set method for capturing the propagating front within textile unit-cells is presented. The computational domain is embedded within an artificial Cartesian grid in order to avoid the difficulties associated with the meshing step. The flow is modelled by using the Stokes/Brinkman equations. Hence, the necessity of describing the yarn boundaries is avoided. The propagating front is represented by using a level set formulation where the extra step of reinitialization is avoided. The flow equations are solved only in the filled part of the domain. The velocity in the unfilled part is obtained through a harmonic continuation. Numerical tests on simple configurations in 2D and 3D space are presented in order to demonstrate the proposed method.

Urinary bladder segmentation in CT urography (CTU) using CLASS

The authors are developing a computerized system for bladder segmentation on CTU, as a critical component for computer aided diagnosis of bladder cancer. A challenge for bladder segmentation is the presence of regions without contrast (NC) and filled with intravenous contrast (C). The authors have designed a Conjoint Level set Analysis and Segmentation System (CLASS) specifically for this application. CLASS performs a series of image processing tasks: preprocessing, initial segmentation, 3D and 2D level set segmentation, and postprocessing, designed according to the characteristics of the bladder in CTU. The NC and the C regions of the bladder were segmented separately in CLASS. The final contour is obtained in the postprocessing stage by the union of the NC and C contours. With Institutional Review Board (IRB) approval, the authors retrospectively collected 81 CTU scans, in which 40 bladders contained lesions, 26 contained diffuse wall thickening, and 15 were considered to be normal. The bladders were segmented by CLASS and the performance was assessed by rating the quality of the contours on a 10-point scale (1 = "very poor," 5 = "fair," 10 = "perfect"). For 30 bladders, 3D hand-segmented contours were obtained and the segmentation accuracy of CLASS was evaluated and compared to that of a single level set method in terms of the average minimum distance, average volume intersection ratio, average volume error and Jaccard index. Of the 81 bladders, the average quality rating for CLASS was 6.5 ± 1.3. Thirty nine bladders were given quality ratings of 7 or above. Only five bladders had ratings under 5. The average minimum distance, average volume intersection ratio, average volume error, and average Jaccard index for CLASS were 3.5 ± 1.3 mm, (79.0 ± 8.2)%, (16.1 ± 16.3)%, and (75.7 ± 8.4)%, respectively, and for the single level set method were 5.2 ± 2.6 mm, (78.8 ± 16.3)%, (8.3 ± 33.1)%, (71.0 ± 15.4)%, respectively. The results demonstrate the potential of CLASS for segmentation of the bladder.

Multi-agent stochastic level set method in image segmentation

A stochastic structure for single and multi-agent level set method is investigated in this article in an attempt to overcome local optima problems in image segmentation. Like other global optimization methods that take advantage of random operators and multi-individual search algorithms, the best agent in this proposed algorithm plays the role of leader in order to enable the algorithm to find the global solution. To accomplish this, the procedure employs a set of stochastic partial differential equations (SPDE), each one of which evolves based on its own stochastic dynamics. The agents are then compelled to simultaneously converge to the best available topology. Moreover, the stochastic dynamics of each agent extends the stochastic level set approach by using a multi source structure. Each source is a delta function centered on a point of evolving front. Lastly, while the computational costs of these methods are higher than the region-based level set method, the probability of finding the global solution is significantly increased.

HOMOGENIZATION OF MATERIALS HAVING INCLUSIONS SURROUNDED BY LAYERS MODELED BY THE EXTENDED FINITE ELEMENT METHOD

We study the homogenization of materials having three phases: matrix, inclusions, and inclusion coatings. To model the inclusions and their coatings we investigate two weakly discontinuous enrichment functions within the extended finite element method (XFEM) framework described by a single level set function. In both formulations the inclusion and coating shapes are independent of each other. The first approach, denoted as a V-type enrichment, combines several weak discontinuities by stacking the corresponding inclusion and coating enriched degrees of freedom in a single node (one on top of the other) while the second, denoted as a zigzag-type enrichment, only adds one additional degree of freedom per each direction. The XFEM approach is extremely efficient and avoids excessive remeshing compared to standard FEM, in particular when the coatings are very thin as in the case of aggregates surrounded by the interface transition zone (ITZ) in concrete materials. Comprehensive verification studies are presented including two-dimensional continuum problems and homogenization of concrete. Herein we mainly focus on the microscopic material response via homogenization of the unit cell. Nonetheless, once the homogenized material properties are obtained, the application to a full multiscale analysis is straightforward. While both enrichment types are different possible extensions to XFEM applied to such three-phase materials, both methods seem to work well and provide significant reduction in degrees of freedom and computation time as compared to standard FEM.

Joint Inversion Through A Level Set Formulation

Geophysical data processing is a highly quantitative field that involves modelling, inversion and visualization. In most cases a geophysical experiment is conducted to collect data that are sensitive to a particular physical property of the earth. The data is processed and inverted to generate an earth model of the physical property in question. To better understand the structure of the earth, different experiments are conducted using a variety of imaging modalities. For example, from seismic, gravity and electromagnetic experiments we may obtain information about the earth's elastic, density and conductivity characteristics. Usually the data of each experiment are inverted separately to generate an ensemble of earth models. However, since the inversion process of each geophysical modality is typically carried out independently, most inversion algorithms do not utilize the information obtained through other modalities. In this research we propose to jointly invert the data obtained by two physical experiments since the information contained in each model can be used to correct the other model. In many of the cases the two models share the important structures, therefore, edges occur in the same locations. In order to exploit this information, we propose using a level set formulation of the problems. Assuming that both models take two known discrete values, we can then use a single level set function for both models together. This can be later extended to multi-level set functions and with unknown values. By using this formulation we are able to improve inversion results of both problems.

Automatic Image Segmentation Method Using Sequential Level Set

为了实现图像的完全分割,基于无须重新初始化的水平集方法提出了一种接力水平集方法.该方法在待分割图像中自动交替地创建嵌套子区域和相应的初始水平集函数,使水平集函数在其中演化并收敛,然后重复这个过程直到子区域面积为0.与原始算法及经典的基于区域的水平集方法相比,该方法具有如下优点:1) 自动完成,无须交互式的初始化;2) 多次分割图像,能够比原始算法检测到更多的边缘;3) 对于非匀质的图像,能够取得比经典的基于区域的水平集方法更好的分割效果;4) 提供一个开放的分割算法框架,其他单水平集方法稍作修改后也可替换这里所使用的单水平集方法.实验结果表明,此算法对人造图像和医学影像实现了无须交互的完全分割,对非匀质图像分割表现出更好的鲁棒性.;Based on the level set method without re-initialization, a sequential level set method is proposed to realize full image segmentation. The proposed method automatically and alternatively creates nested sub-regions or the corresponding initial level set functions in the image to be segmented, and then makes the level set function evolved to be convergence in the corresponding sub-region. This step is sequentially repeated until the sub-region vanishes. Compared with the original method and a representative region-based level set method, the proposed method has many advantages as follows: 1) It is automatically executed and does not need the interactive initialization anymore; 2) It segments image more than once and detects more boundaries than the original method; 3) It can get better performance on non-homogenous images than the representative region-based level set method; 4) It is an open image segmentation framework in which the single level set method is used can be replaced by other single level set methods after some modification. Experimental results indicate that the proposed method could fully segment the synthetic and medical images without interactive step and additionally works more robust on non-homogenous images.