Use Of Level Sets Research Articles

Level set methods for gradient-free optimization of metasurface arrays

Global optimization techniques are increasingly preferred over human-driven methods in the design of electromagnetic structures such as metasurfaces, and careful construction and parameterization of the physical structure is critical in ensuring computational efficiency and convergence of the optimization algorithm to a globally optimal solution. While many design variables in physical systems take discrete values, optimization algorithms often benefit from a continuous design space. This work demonstrates the use of level set functions as a continuous basis for designing material distributions for metasurface arrays and introduces an improved parameterization which is termed the periodic level set function. We explore the use of alternate norms in the definition of the level set function and define a new pseudo-inverse technique for upsampling basis coefficients with these norms. The level set method is compared to the fragmented parameterization and shows improved electromagnetic responses for two dissimilar cost functions: a narrowband objective and a broadband objective. Finally, we manufacture an optimized level set metasurface and measure its scattering parameters to demonstrate real-world performance.

Modelling laser-fluid coupling and cavitation in laser lithotripsy

The absorption of laser by fluid and the resulting vapor bubbles plays a significant role in laser lithotripsy and many other industrial applications. In this talk, we present a coupled photo-thermal-mechanical model of laser interaction in compressible multiphase fluid flows. The model couples the three-dimensional compressible Navier–Stokes equation with a nonlocal laser absorption equation. The computational framework is an extension of the recently developed FIVER (“a Finite Volume method with Exact multi-material Riemann solver”) framework, which features the use of level set and embedded boundary methods for tracking material (fluid–fluid and fluid–solid) interfaces, and the solution of local, one-dimensional multi-material Riemann problems for enforcing interface conditions. We will start with a brief review of the key model equations and numerical algorithms adopted by FIVER, as well as recent efforts on verification and validation. Next, physical models of laser absorption and laser-induced heating will be presented. Numerical models and methods for simulating phase change and tracking the vapor–liquid interface will be introduced and assessed. The performance of this computational framework will be demonstrated using several simplified model problems and a numerical experiment of cavitation in Holmium:YAG laser lithotripsy.

Coupled finite-element/topology optimization of continua using the Newton-Raphson method

We show that conventional finite element technology can be applied to topology optimization of 2D and 3D continua. Nodal design variables, classical Newton-Raphson solution for the first-order KKT equations and the screened-Poisson equation are sufficient to produce smooth results without the use of level-sets or phase-field technologies. Side-constraints are addressed with variable transformation and a single Lagrange multiplier enforces the volume constraint. We also use our mesh-division Algorithm specialized to topology optimization as a form of producing compatible meshes. By means of 2D and 3D numerical experimentation, we make the case for this approach to density-based compliance minimization. Verification examples are shown, with the corresponding compliance and volume evolution exhibiting a sound behavior.

K-means algorithm with level set for brain tumor segmentation

<p>Brain is a complicated structure consisting of millions of millions cells so that, it’s difficult to identify any diseases without using any computerized technology. Magnetic resonance imaging (mri) is one of the main assessments of brain tumors. One of the most important steps on medical image processing is segmentation. Segmenting brain mri images, which provide accurate information for the diagnosis and therapy decisions of brain tumors. We proposed to segment brain tumor mri images into three parts (wm (white matter), gm (gray matter), and background). The first algorithm is for applying median filtering on brain mri image for removing the noise from the image for achieving accurate results. The second algorithm is for applying k-means algorithm for accuracy in time consuming and for clustering into regions and the third algorithm indicate the detecting the boundary of the image with the use of level set. By comparison, our proposed method, its efficiency to segment perfectly more than other previous used algorithms especially on time consuming.</p>

Heuristic and Eulerian interface capturing approaches for shallow water type flow and application to granular flows

Determining the wet–dry boundary and avoiding the related spurious thin-layer problem when solving the depth-averaged shallow-water (SW) equations (or similar granular flow models) remains an outstanding challenge, though it has been the focus of much research effort. In this paper, we introduce the use of level set and phase field based methods to address this issue and related problems. We also propose new heuristic methods to address this problem. We implemented all of these methods in TITAN2D, which is a parallel adaptive mesh refinement toolkit designed for numerical simulation of granular flows. Results of the methods for flow over a simple inclined plane and Colima volcano are used to illustrate the methods. For the inclined plane, we compared the results with experimental data and for Colima volcano they are compared to field data. Our approaches successfully captured the interface of the flow and solved the accuracy and stability problems related to the thin layer problem in SW numerical solution. The comparison of results shows that although all of the methods can be used to address this problem, each of them has its own advantages/disadvantages and methods have to be chosen carefully for each problem.

Cellular automata-based left ventricle reconstruction from magnetic resonance images

Despite advanced medical imaging technologies being available, image segmentation still continues to represent a major bottleneck in practical applications due to high noise level and lack of contrast. This paper presents a semi-automatic algorithm that uses stochastic resonance as a pre-processing step for enhancing the contrast of input magnetic resonance (MR) images followed by a cellular automata (CA)-based mechanism for surface reconstruction of the left ventricle (LV) from a few selected MR slices. The main contribution, in this work, is a new formulation for preventing the CA method to leak into surrounding areas of similar intensity. Another contribution is the use of level sets for segmenting the rest of the slices of a subject automatically between the selected slices segmented by the CA to build the LV surface. Both qualitative and quantitative evaluations performed on York and MICCAI Grand Challenge workshop database of MR images show promising segmentation accuracy when compared with the ground truth, reflecting the potential of the proposed method.

An adaptive spectral/DG method for a reduced phase-space based level set approach to geometrical optics on curved elements

We present an adaptive spectral/discontinuous Galerkin (DG) method on curved elements to simulate high-frequency wavefronts within a reduced phase-space formulation of geometrical optics. Following recent work, the approach is based on the use of level sets defined by functions satisfying the Liouville equations in reduced phase-space and, in particular, it relies on the smoothness of these functions to represent them by rapidly convergent spectral expansions in the phase variables. The resulting (hyperbolic) system of equations for the coefficients in these expansions are then amenable to a high-order accurate treatment via DG approximations. In the present work, we significantly expand on the applicability and efficiency of the approach by incorporating mechanisms that allow for its use in scattering simulations and for a reduced overall computational cost. With regards to the former we demonstrate that the incorporation of curved elements is necessary to attain any kind of accuracy in calculations that involve scattering off non-flat interfaces. With regards to efficiency, on the other hand, we also show that the level-set formulation allows for a space p-adaptive scheme that under-resolves the level-set functions away from the wavefront without incurring in a loss of accuracy in the approximation of its location. As we show, these improvements enable simulations that are beyond the capabilities of previous implementations of these numerical procedures.

A modified level set approach to 2D modeling of dynamic recrystallization

The macroscopic properties of metallic materials depend on the state of the grain microstructure. Recrystallization acts as one of the most important mechanisms in the evolution of the microstructure and hence also of the macroscopic properties. This paper presents a mesoscale model of microstructure evolution due to recrystallization, based on a level set formulation employed in a finite element setting. The use of level sets to represent grains and grain boundaries in polycrystal microstructures is a relatively recent development in computational materials science and the present contribution suggests new methodologies such as interface reconstruction, allowing for example boundary conditions to be prescribed along grain boundary interfaces and distinct localization and representation of grain boundary junctions. Polycrystal plasticity is modeled by considering the evolution of dislocation density in the individual crystals. The influence of grain boundaries on dislocation accumulation is captured in the model, causing the formation of dislocation density gradients within the grains. The model is used in simulations of dynamic recrystallization, taking pure copper as example material. It is shown that the proposed model captures the salient features of dynamic recrystallization during thermomechanical materials processing.

A hybrid method based on level set and 3D region growing for segmentation of the thoracic aorta

This study sought to develop a completely automatic method for image segmentation of the thoracic aorta. We used a total of 4682 images from 10 consecutive patients. The proposed method is based on the use of level set and region growing, automatically initialized using the Hough transform. The results obtained were compared to those of manual segmentation as performed by an external expert radiologist. Concordance between the developed method and manual segmentation ranged from 92.79 to 95.77% in the descending regions of the aorta and from 90.68 to 96.54% in the ascending regions, with a mean value of 93.83% being obtained for total segmentation.

Curve and boundaries measurement using B-splines and virtual images

This paper presents a digital image correlation method able to detect and measure a contour or silhouette in a digital image, with a sub-pixel precision. A virtual image that describes a simple contour or silhouette roughly similar to the physical object to detect is created. The matching with the physical image of the object is then performed based on digital image correlation. We look for the displacement field which transforms the physical image which is the closest to the virtual one. In order to keep a problem of finite dimension, this field is decomposed on a B-spline basis. The algorithm is close to the digital image correlation method; an efficient implementation is proposed that makes use of levelsets. The full method is tested over both synthesized and real images, one of a star shaped toy and one on an ill-defined image of a high temperature metallic specimen on which no other boundary measurement was possible. The influence of the arbitrary choice of the virtual image contour or silhouette arbitrary parameters is discussed with respect to the precision of the method. This one is shown to be sub-pixel in any case.

Level set propagation for mixed-mode crack advance

In the extended finite element method framework, the advantage of being able to represent cracks independently from mesh rests on the use of level sets. Propagation simulation is connected to the update of these level sets, following the resolution of differential equations of evolution. In this document, we analyse these equations so that fast and robust mixed-mode propagation methods can be put forward. The proposed methodology can also manage the separation of a crack front into several fronts as well as the merging of several fronts during the propagation process. An example of 3D non-planar propagation is proposed at the end to show the efficiency of our choices.

A multiscale strategy for structural optimization

AbstractThis paper presents a multiscale strategy dedicated to structural optimization. The applications concern the study of geometric details (such as holes, surface profiles, etc) within the structures with frictional contacts. The first characteristic of the method is that it uses a micro–macro approach. This approach is based on a domain decomposition into substructures and interfaces, which involves the resolution of independent ‘micro’ problems in each substructure and transfers ‘macro’ information only through the interfaces. The second characteristic is the use of a multiresolution strategy in order to reduce the computation cost for problems with evolving design parameters. The last characteristic is the capability to model the geometry of details without remeshing thanks to two features: the use of a local enrichment method , and the use of level set functions to easily modify the boundary of the detail during the optimization process. Copyright © 2008 John Wiley & Sons, Ltd.

Using trapezoids for representing granular objects: Applications to learning and OWA aggregation

We discuss the role and benefits of using trapezoidal representations of granular information. We focus on the use of level sets as a tool for implementing many operations on trapezoidal sets. We point out the simplification that the linearity of the trapezoid brings by requiring us to perform operations on only two level sets. We investigate the classic learning algorithm in the case when our observations are granule objects represented as trapezoidal fuzzy sets. An important issue that arises is the adverse effect that very uncertain observations have on the quality of our estimates. We suggest an approach to addressing this problem using the specificity of the observations to control its effect. We next consider the OWA aggregation of information represented as trapezoids. An important problem that arises here is the ordering of the trapezoidal fuzzy sets needed for the OWA aggregation. We consider three approaches to accomplish this ordering based on the location, specificity and fuzziness of the trapezoids. From these three different approaches three fundamental methods of ordering are developed. One based on the mean of the 0.5 level sets, another based on the length of the 0.5 level sets and a third based on the difference in lengths of the core and support level sets. Throughout this work particular emphasis is placed on the simplicity of working with trapezoids while still retaining a rich representational capability.

A FEM‐BEM approach using level‐sets in electrical capacitance tomography

PurposeFor the spatial reconstruction of a two phase flow, as it might occur in a pipe, the main problem has always been the blurring of the resulting images.Design/methodology/approachIn this paper, we present a method where blurring is implicitly avoided by the use of level sets. The level set method describes the iteratively evolving interface between different materials. The underlying field problem is solved with the boundary element method formulated in the region, where the degrees of freedom are present and the finite element method in all other regions.FindingsFinally reconstruction results of an electrical capacitance tomography sensor are presented to show the validity of the method.Originality/valuePresents a method where blurring is avoided by the use of level sets.

A Level Set Model for Image Classification

We present a supervised classification model based on a variational approach. This model is devoted to find an optimal partition composed of homogeneous classes with regular interfaces. The originality of the proposed approach concerns the definition of a partition by the use of level sets. Each set of regions and boundaries associated to a class is defined by a unique level set function. We use as many level sets as different classes and all these level sets are moving together thanks to forces which interact in order to get an optimal partition. We show how these forces can be defined through the minimization of a unique fonctional. The coupled Partial Differential Equations (PDE) related to the minimization of the functional are considered through a dynamical scheme. Given an initial interface set (zero level set), the different terms of the PDE's are governing the motion of interfaces such that, at convergence, we get an optimal partition as defined above. Each interface is guided by internal forces (regularity of the interface), and external ones (data term, no vacuum, no regions overlapping). Several experiments were conducted on both synthetic and real images.

An adaptive approach to defuzzification based on level sets

We look at the process of defuzzification used in fuzzy logic controllers. We provide a parametrized formulation for this procedure based upon the use of level sets which we call generalized level set defuzzification. We show that the commonly used defuzzification procedures, mean of maxima and center of area, are special cases of this general procedure and are simply distinguished by the choice of the parameter. We provide a simple algorithm for adaptive choice of the best defuzzification procedure.

PROOF OF A FUZZY REGULAR LANGUAGE THEOREM BY USING LEVEL SETS—A NOTE

Fuzzy languages and level sets are defined and illustrated by examples. Properties of level sets are presented. A fuzzy regular language theorem may be proved through the use of the properties of level sets. Level sets offer what appears to be a useful tool in theorem proving in fuzzy languages, pictorial databases, natural language translation and processing, and many others. Through the use of level sets, results obtained in formal languages may be used in fuzzy languages and related areas.

ON PROGRAMMING WITH FUZZY CONSTRAINT SETS

Two solution concepts for a FMP problem are suggested. The first one makes use of level sets of the fuzzy set of feasible alternatives. The second solution is based on the concept of Pareto maximum in vector optimization. It is shown that both solutions are equivalent in a sense that they give the same fuzzy value of a function maximized. It is suggested that if a decision‐maker is to choose a single element, then his choice must be based not only on the membership value of this element in the solution fuzzy set but also on the corresponding value of the function maximized. In this respect the situation is similar to that typical for vector optimization. The approach suggested in this paper is further used for analysing games with fuzzy sets of strategies of the players. A fuzzy equilibrium solution is introduced, which can provide a base for an agreement between the players.