Multiphase Level Set Research Articles

Design of piezoelectric actuators using a multiphase level set method of piecewise constants

This paper presents a multiphase level set method of piecewise constants for shape and topology optimization of multi-material piezoelectric actuators with in-plane motion. First, an indicator function which takes level sets of piecewise constants is used to implicitly represent structural boundaries of the multiple phases in the design domain. Compared with standard level set methods using n scalar functions to represent 2 n phases, each constant value in the present method denotes one material phase and 2 n phases can be represented by 2 n pre-defined constants. Thus, only one indicator function including different constant values is required to identify all structural boundaries between different material phases by making use of its discontinuities. In the context of designing smart actuators with in-plane motions, the optimization problem is defined mathematically as the minimization of a smooth energy functional under some specified constraints. Thus, the design optimization of the smart actuator is transferred into a numerical process by which the constant values of the indicator function are updated via a semi-implicit scheme with additive operator splitting (AOS) algorithm. In such a way, the different material phases are distributed simultaneously in the design domain until both the passive compliant host structure and embedded piezoelectric actuators are optimized. The compliant structure serves as a mechanical amplifier to enlarge the small strain stroke generated by piezoelectric actuators. The major advantage of the present method is to remove numerical difficulties associated with the solution of the Hamilton–Jacobi equations in most conventional level set methods, such as the CFL condition, the regularization procedure to retain a signed distance level set function and the non-differentiability related to the Heaviside and the Delta functions. Two widely studied examples are chosen to demonstrate the effectiveness of the present method.

Clustering Initiated Multiphase Active Contours and Robust Separation of Nuclei Groups for Tissue Segmentation.

Computer assisted or automated histological grading of tissue biopsies for clinical cancer care is a long-studied but challenging problem. It requires sophisticated algorithms for image segmentation, tissue architecture characterization, global texture feature extraction, and high-dimensional clustering and classification algorithms. Currently there are no automatic image-based grading systems for quantitative pathology of cancer tissues. We describe a novel approach for tissue segmentation using fuzzy spatial clustering, vector-based multiphase level set active contours and nuclei detection using an iterative kernel voting scheme that is robust even in the case of clumped touching nuclei. Early results show that we can reach a 91% detection rate compared to manual ground truth of cell nuclei centers across a range of prostate cancer grades.

A Variational Framework for Multiregion Pairwise-Similarity-Based Image Segmentation

Variational cost functions that are based on pairwise similarity between pixels can be minimized within level set framework resulting in a binary image segmentation. In this paper we extend such cost functions and address multi-region image segmentation problem by employing a multi-phase level set framework. For multi-modal images cost functions become more complicated and relatively difficult to minimize. We extend our previous work, proposed for background/foreground separation, to the segmentation of images in more than two regions. We also demonstrate an efficient implementation of the curve evolution, which reduces the computational time significantly. Finally, we validate the proposed method on the Berkeley Segmentation Data Set by comparing its performance with other segmentation techniques.

Multi-region segmentation of SAR image by a multiphase level set approach

In this letter, a multiphase level set approach unifying region and boundary-based information for multi-region segmentation of Synthetic Aperture Radar (SAR) image is presented. An energy functional that is applicable for SAR image segmentation is defined. It consists of two terms describing the local statistic characteristics and the gradient characteristics of SAR image respectively. A multiphase level set model that explicitly describes the different regions in one image is proposed. The purpose of such a multiphase model is not only to simplify the way of denoting multi-region by level set but also to guarantee the accuracy of segmentation. According to the presented multiphase model, the curve evolution equations with respect to edge curves are deduced. The multi-region segmentation is implemented by the numeric solution of the partial differential equations. The performance of the approach is verified by both simulation and real SAR images. The experiments show that the proposed algorithm reduces the speckle effect on segmentation and increases the boundary alignment accuracy, thus correctly divides the multi-region SAR image into different homogenous regions.

A variational formulation for a level set representation of multiphase flow and area preserving curvature flow

Variational descriptions for various multiphase level set formulations involving cur- vature flow are discussed. A representation of n phases using ni1 level set functions is introduced having the advantage that constraints preventing overlaps or vacuum are not needed. The repre- sentation is then used in conjunction with our variational formulation to deduce a novel level set based algorithm for multiphase flow. In addition, a similar variational formulation is applied to area preserving curvature flow. In this flow, the area (or volume in 3D) enclosed by each level set is preserved. Each algorithm has been implemented numerically and the results of such computations are shown.

Automatic segmentation of high-throughput RNAi fluorescent cellular images.

High-throughput genome-wide RNA interference (RNAi) screening is emerging as an essential tool to assist biologists in understanding complex cellular processes. The large number of images produced in each study make manual analysis intractable; hence, automatic cellular image analysis becomes an urgent need, where segmentation is the first and one of the most important steps. In this paper, a fully automatic method for segmentation of cells from genome-wide RNAi screening images is proposed. Nuclei are first extracted from the DNA channel by using a modified watershed algorithm. Cells are then extracted by modeling the interaction between them as well as combining both gradient and region information in the Actin and Rac channels. A new energy functional is formulated based on a novel interaction model for segmenting tightly clustered cells with significant intensity variance and specific phenotypes. The energy functional is minimized by using a multiphase level set method, which leads to a highly effective cell segmentation method. Promising experimental results demonstrate that automatic segmentation of high-throughput genome-wide multichannel screening can be achieved by using the proposed method, which may also be extended to other multichannel image segmentation problems.

A fast multiphase level set algorithm for solving the Chan‐Vese model

AbstractIn this work, we specifically solve the C‐V active contour model by multiphase level set methods. We first develop a fast algorithm based on calculating the variational energy of the C‐V model without the length term. We check whether the energy decreases or not when we move a point to another segmented region. Then we draw a connection between this algorithm and the topological derivative, a concept emerged from the shape optimization field. Furthermore, to include the length term of the C‐V model, a preprocessing step is taken by using nonlinear diffusion. Numerical experiments have demonstrated the efficiency and the robustness of our algorithm. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Modeling the growth and interaction of multiple dendrites in solidification using a level set method

A level set method is presented to study the growth and interaction of multiple dendrites in solidification. The method couples thermal and solute diffusion with propagation of multiple interfaces. A single signed distance function is used to track the solid–liquid interface with the aid of markers, the value of which is the orientation angle, for identification of different crystals. The problem of evolving multiple crystal interfaces is reduced to two tasks: (1) tracking one level set variable (signed distance function) and (2) determination of the marker for a newly solidified finite element nodal point. Tracking a single level set variable is implemented by solving the level set equation with interface velocity computed from an extended Stefan equation using the marker information (crystal orientation). Determination of the marker for a newly solidified finite element nodal point is implemented by using an algorithm modified from the fast marching technique. Both of these two steps are computationally efficient and the approach is suitable for incorporating effects of multiple crystals. Convergence and accuracy of this approach are demonstrated by using different grid spacings and comparing with results obtained from the multi-phase level set method. A parametric study is performed to investigate the effects of solidification speed and thermal gradient on the resulting solidification microstructure pattern. Numerical results of columnar-to-equiaxed transition (CET) qualitatively agree with an analytical estimation and are similar to previous numerical results obtained using a phase field method. A convergence study is performed to determine the appropriate grid spacing for numerical simulation. At lower surface tension, CET occurs at a lower thermal gradient for a giving solidification speed. Secondary dendrite formation is more apparent with lower surface tension. The differences and similarities between the three-dimensional and two-dimensional growth results are analyzed. Randomness in crystal orientation and required undercooling for nucleation are modeled and found to have a great effect on the microstructure pattern. The efficiency of the present approach is finally demonstrated with an example that includes the growth of hundreds of crystals with consideration of randomness effects.

Brain MRI Segmentation with Multiphase Minimal Partitioning: A Comparative Study

This paper presents the implementation and quantitative evaluation of a multiphase three-dimensional deformable model in a level set framework for automated segmentation of brain MRIs. The segmentation algorithm performs an optimal partitioning of three-dimensional data based on homogeneity measures that naturally evolves to the extraction of different tissue types in the brain. Random seed initialization was used to minimize the sensitivity of the method to initial conditions while avoiding the need for a priori information. This random initialization ensures robustness of the method with respect to the initialization and the minimization set up. Postprocessing corrections with morphological operators were applied to refine the details of the global segmentation method. A clinical study was performed on a database of 10 adult brain MRI volumes to compare the level set segmentation to three other methods: “idealized” intensity thresholding, fuzzy connectedness, and an expectation maximization classification using hidden Markov random fields. Quantitative evaluation of segmentation accuracy was performed with comparison to manual segmentation computing true positive and false positive volume fractions. A statistical comparison of the segmentation methods was performed through a Wilcoxon analysis of these error rates and results showed very high quality and stability of the multiphase three-dimensional level set method.

A level set simulation of dendritic solidification of multi-component alloys

A level set method combining features of front tracking methods and fixed domain methods is presented to model microstructure evolution in the solidification of multi-component alloys. Phase boundaries are tracked by solving the multi-phase level set equations. Diffused interfaces are constructed from these tracked phase boundaries using the level set functions. Based on the assumed diffused interfaces, volume-averaging techniques are applied for energy, species and momentum transport. Microstructure evolution in multi-component alloy systems is predicted using realistic material parameters. The methodology avoids the difficulty of parameter identification needed in other diffused interface models, and allows easy application to various practical alloy systems. Techniques including fast marching, narrow band computing and adaptive meshing are utilized to speed up computations. Several numerical examples are considered to validate the method and examine its potential for modeling solidification of practical alloy systems. These examples include two- and three-dimensional solidification of a binary alloy in an undercooled melt, a study of planar/cellular/dendritic transition in the solidification of a Ni–Cu alloy, and eutectic and peritectic solidification of an Fe–C system. Adaptive mesh refinement in the rapidly varying interface region makes the method practical for coupling the microstructure evolution at the meso-scale with buoyancy driven flow in the macro-scale, which is shown in the solidification of a Ni–Al–Ta ternary alloy.

Segmentation of tissue boundary evolution from brain MR image sequences using multi-phase level sets

In this paper, we focus on the automated extraction of the cerebrospinal fluid-tissue boundary, particularly around the ventricular surface, from serial structural MRI of the brain acquired in imaging studies of aging and dementia. This is a challenging segmentation problem because of the common occurrence of peri-ventricular lesions which locally alter the appearance of white matter. We examine a level set approach which evolves a 4D description of the ventricular surface over time. This has the advantage of allowing constraints on the contour in the temporal direction, improving the consistency of the extracted object over time. The 3D MR images of the entire brain are first aligned using global rigid registration. We then follow the approach proposed by Chan and Vese which is based on the Mumford and Shah model and implemented using the Osher and Sethian level set method. We have extended this to the 4D case to propagate a 4D contour toward the tissue boundaries through the evolution of a 5D implicit function. For convergence we use region-based information provided by the image rather than the gradient of the image. This model is then adapted to allow intensity contrast changes between time frames in the MRI sequence. Results on time sequences of 3D brain MR images are presented and discussed.

Motion Competition: A Variational Approach to Piecewise Parametric Motion Segmentation

We present a novel variational approach for segmenting the image plane into a set of regions of parametric motion on the basis of two consecutive frames from an image sequence. Our model is based on a conditional probability for the spatio-temporal image gradient, given a particular velocity model, and on a geometric prior on the estimated motion field favoring motion boundaries of minimal length. Exploiting the Bayesian framework, we derive a cost functional which depends on parametric motion models for each of a set of regions and on the boundary separating these regions. The resulting functional can be interpreted as an extension of the Mumford-Shah functional from intensity segmentation to motion segmentation. In contrast to most alternative approaches, the problems of segmentation and motion estimation are jointly solved by continuous minimization of a single functional. Minimizing this functional with respect to its dynamic variables results in an eigenvalue problem for the motion parameters and in a gradient descent evolution for the motion discontinuity set. We propose two different representations of this motion boundary: an explicit spline-based implementation which can be applied to the motion-based tracking of a single moving object, and an implicit multiphase level set implementation which allows for the segmentation of an arbitrary number of multiply connected moving objects. Numerical results both for simulated ground truth experiments and for real-world sequences demonstrate the capacity of our approach to segment objects based exclusively on their relative motion.

A level-set based variational method for design and optimization of heterogeneous objects

A heterogeneous object is referred to as a solid object made of different constituent materials. The object is of a finite collection of regions of a set of prescribed material classes of continuously varying material properties. These properties have a discontinuous change across the interface of the material regions. In this paper, we propose a level-set based variational approach for the design of this class of heterogeneous objects. Central to the approach is a variational framework for a well-posed formulation of the design problem. In particular, we adapt the Mumford–Shah model which specifies that any point of the object belongs to either of two types: inside a material region of a well-defined gradient or on the boundary edges and surfaces of discontinuities. Furthermore, the set of discontinuities is represented implicitly, using a multi-phase level set model. This level-set based variational approach yields a computational system of coupled geometric evolution and diffusion partial differential equations. Promising features of the proposed method include strong regularity in the problem formulation and inherent capabilities of geometric and material modeling, yielding a common framework for optimization of the heterogeneous objects that incorporates dimension, shape, topology, and material properties. The proposed method is illustrated with several 2D examples of optimal design of multi-material structures and materials.

A Multiphase Level Set Framework for Image Segmentation Using the Mumford and Shah Model

We propose a new multiphase level set framework for image segmentation using the Mumford and Shah model, for piecewise constant and piecewise smooth optimal approximations. The proposed method is a...