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A reaction–diffusion based level set method for image segmentation in three dimensions
Abstract
The image segmentation of computed tomography data for three-dimensional biological structures remains challenging because of the limitations of existing numerical techniques and computer resources. The work represents the structures as the zero-level contour of a level set function whose value is constrained to a narrow band ranging. A cost functional composed of fitting energy for extracting the local intensity and diffusion energy for regularization is minimized within a framework of optimization. To avoid the re-initialization procedure and accelerate the convergence when updating the level set function, a reaction–diffusion technique is developed to replace the upwind algorithm by finite element analysis. Numerical examples demonstrate elegant biological structures with clear and smooth interfaces can be generated within a few iteration steps because the time step 100-fold larger than the allowable value of Courant–Friedrichs–Lewy stability condition can be applied in the proposed method.
... In this study, an elegant structure, including a slender hyoid, a long upper beak, a short lower beak, and an encephalocoele filled with viscoelastic brain substances, was established to investigate the woodpecker's non-impact-injury mechanisms. Initially, a surface model was obtained via a reaction-diffusion based imaging process on the micro-CT data (Zhang et al., 2019). Then, this model was discretized into a mesh of 1.1 million hexahedral elements. ...
... The original model was obtained by using the image separation technique (Haralick and Shapiro, 1985;Marion, 2013), which can partition the raw data into non-overlapped geometric regions based on their intensity or textural information. In contrast to conventional image processing techniques such as region growing (Nishino et al., 1986;Adams and Bischof, 1994), thresholding (Otsu, 1979), and edge detection (Chan and Vese, 2001), this reaction-diffusion based level set method (Zhang et al., 2019) is capable of obtaining a precise numerical model featured with smoother and more closed interfaces. It will be briefly introduced in this section and the readers can refer to our previous work (Zhang et al., 2019) for details. ...
... In contrast to conventional image processing techniques such as region growing (Nishino et al., 1986;Adams and Bischof, 1994), thresholding (Otsu, 1979), and edge detection (Chan and Vese, 2001), this reaction-diffusion based level set method (Zhang et al., 2019) is capable of obtaining a precise numerical model featured with smoother and more closed interfaces. It will be briefly introduced in this section and the readers can refer to our previous work (Zhang et al., 2019) for details. ...
Numerical investigation into the impact-resistance of complex biological organs remains challenging because of the difficulties in obtaining accurate models and precise material properties. In this work, the elegance of a woodpecker's head, including a slender hyoid connected by a spherical hinge and two revolute hinges, a long upper beak, a short lower beak, and an encephalocoele filled with viscoelastic brain substances, was obtained via a reaction-diffusion based imaging process on the micro-CT data. The material heterogeneity was fully considered in subsequent finite element analysis in LS-Dyna via categorizing the intensity into 53 groups and interpolating their properties from available data of rhamphotheca, hyoid, skull, and beak. Compared to a non-hyoid model, we found the hyoid helps to significantly alleviate the strain in the brain and restrain opposite velocity for maintaining structural stability, especially after impact. Numerical investigation also indicates that a longer upper beak is favorable in flatting the curve of impact force and improve structural crashworthiness.
... Although the sensitivity in the SIMP method is simple and easy to calculate, the generated gray elements lead to the unsmooth solid-void interfaces [21]. The level set method [43][44][45] can geometrically represent smooth boundaries according to the zero-level set. However, from the perspective of computational cost, the implicit level set equation needs to be updated by solving the PDE or reaction-diffusion equation. ...
... Besides, the elegant geometric representation can be obtained directly without postprocessing or material interpolation. Compared to our previous work, we found the solid-void boundaries based on the relative density distribution instead of the level set function [45]. Let us assume that Fig. 3(a) represents the density field over a rectangular design domain. ...
The bi-directional evolutionary structural optimization (BESO) method effectively uses basic strategies of removing and adding material based on element sensitivity. However, challenges remain in generating smooth boundaries to improve the finite element analysis accuracy and achieve structural aesthetics. This work develops a body-fitted triangular/tetrahedral mesh generation algorithm to yield smooth boundaries in the BESO method. The optimization problem is regularized by adding a diffusion term in the objective function. We found that the first has the best regularization effect of Lorentzian, Tikhonov, Perona–Malik, Huber, and Tukey functions. The void elements are excluded from spatial optimization to save computation costs and computer memory. Numerical examples show that the proposed method converges quickly, only taking dozens of iterations to converge. Also, the smooth boundaries of the optimized structures in 2D/3D scenarios are naturally obtained from the proposed method, not from smoothing post-processing. Compared with the optimization toolbox in Abaqus, the example of the automotive control arm demonstrates smoother boundaries and lower average mean compliance.
... However, the original SIMP method can suffer from drawbacks such as instability problems, checkerboard patterns, mesh dependency, and grayscales. To prevent those drawbacks, the level set method, in particular, the one via solving the reaction diffusion equation (RDE), has been proposed by Yamada et al. (2010) and has since emerged as an attracting alternative for various topology optimization problems (Choi et al. 2011;Otomori et al. 2012;Yamada et al. 2013;Fujii et al. 2013;Lu et al. 2013;Yaji et al. 2014;Isakari et al. 2014;Jing et al. 2015;Yaji et al. 2015;Otomori et al. 2015;Isakari et al. 2017;Emmendoerfer et al. 2019;Nguyen et al. 2020;Zhang et al. 2020;Kim et al. 2020;Gao et al. 2021;Kubo et al. 2021;Li et al. 2021;Zhuang et al. 2021;Jung et al. 2021;Ho-Nguyen-Tan and Kim 2022). The enhanced smoothness and stability of the RDE-based level set method can be attributed to the introduction of a fictitious interfacial energy. ...
A time-saving finite element method (FEM) based approach for structural topology optimization with exact boundary representation is proposed in this work. The optimization process consists of two loops. The first loop adopts a fixed and fairly coarse mesh. Afterwards, the second loop reconstructs the material domain and hence the boundary representation becomes exact. A novelty of this work is that our two-loop approach is realized with the domain reconstruction (not just remeshing). The convergence of the second loop is only made possible by imposing the volume constraint in an exact fashion. The proposed approach can save a substantial amount of computational time while allowing the exact representation of boundary (no grayscale throughout the second loop). For the two-loop approach, its computational time can be reduced to merely 13.6% of that for the single loop approach. The optimized structure is also found independent of mesh size. In addition, the two-loop approach resolves the issue of a deteriorated connectivity of the reconstructed domain Ω once the constrained volume is set extremely small.
... Active contour models have been employed in the segmentation of these brain regions (Anandh, Sujatha, & Ramakrishnan, 2014;Ramaniharan et al., 2016). Level set methods that utilize curves or surfaces and partial differential equations are known to be effective for MR image segmentation (Zhang, Xie, Li, & Zhou, 2020). ...
In this study, an attempt has been made to analyze the effect of the Extreme Learning Machine (ELM) classifier and its variants in the differentiation of Healthy Controls (HC) and Alzheimer's Disease (AD) using structural MR images. For this, sub-anatomic brain structures, namely Corpus Callosum (CC) and Lateral Ventricles (LV) are segmented and characterized using morphometric features. Significant features from these regions are subjected to ELM, online sequential ELM, and Self-adaptive Differential Evolution ELM (SaDE-ELM) classifiers for the differentiation of HC and AD. The activation functions and the number of hidden neurons in ELM classifiers are tuned by evaluating the performance using standard metrics. Results indicate that ELM and its variant classifiers are able to identify AD using morphometric features from CC and LV. ELM classifiers achieve an accuracy greater than 90% using the sigmoid activation function for both regions. Among the ELM variants, SaDE-ELM attains a maximum sensitivity of 97% and 94% using LV and CC features respectively with the lowest number of hidden neurons. The extracted features from LV demonstrate higher discriminative power than CC in AD classification. However, a specificity greater than 95% is observed in ELM for CC features. As the proposed approach is able to characterize and differentiate the morphometric alterations in CC and LV due to AD, the study seems to be clinically significant for the differentiation of HC and AD.
... In addition, the geometrical complexity of the optimized structure could be flexibly controlled by incorporating fictitious interface energy. Based on these advantages, the reaction diffusionbased level set method (RDLS) has been applied in multi-material structures [49], image processing [50], and further developed within the body-fitted mesh. Specifically, the bodyfitted mesh was utilized in RDLS for structural optimization to obtain ultra-smooth boundaries [24]. ...
This work uses the zero-level contour of a parameterized level set function, a linear combination of cubic B-spline basis functions, to express the structural profile in structural topology optimization. Together with mean compliance, diffusion energy is minimized under a volume constraint to control the structural complexity. The design variables, namely the coefficients of cubic B-spline basis functions, are updated by solving the reaction–diffusion equation within a finite element analysis framework. The bisectional algorithm accurately calculates the Lagrangian multiplier of the volume constraint in each iteration. In addition to expressing the optimized structure smoothly, the proposed method is highly efficient. For instance, it only takes 20 iterations to solve the cantilever and MBB beams in 2D. For 3D optimization, we obtain several elegant bridge designs using nearly one million elements, demonstrating the great potential of the proposed method for practical applications.
... Numerous segmentation assortments are obtainable in the literature based on threshold, region, fuzzy, edge. A proficient strategy is necessary for various images [2,27]. ...
Nowadays, the contour models (CMs) are widely used in image segmentation. Among these CMs, the Chan and Vese model depending on level set is the current regional based model, considering the regularity of intensity in every region. If the contour is not initialized correctly, the conventional level set (LS) model frequently sticks in local minima. This is becoming more important in the medical images context. In this manuscript, a multi-level set model with an ideal energy active contour is proposed, which is anticipated to realize the performance of acceptable segmentation, regardless of the contour’s initial choice. The active contour models are utilized to identify an object outline from the image. The active CMs with energy based segmentation methods minimizes the energy related with active contour. This work makes the appropriate energy minimized difficult to solve by using meta-heuristic optimization algorithm and makes a proficient execution of the approach by Chaotic Multi Verse Improved Harris Hawks Optimization (CMV-IHHO) technique. Here, the proposed approach is compared with six existing approaches. The existing methods such as DA, Symmetry Analysis, Fuzzy C-Means, Rough Fuzzy C-Means, K-Means Level Set, Random Forest, and Support vector machine method. The accuracy of the proposed method is 0.91%, 5.84%, 15.63%, 8.30%, 10.97%, 15.77%, and 5.14% better than the existing approaches. The sensitivity of the proposed method is 3.37%, 5.74%, 22.66%, 4.54%, 17.94%, 4.54% and 15% better than the existing methods.
... Otomori et al. [16] presented a simple 2D MATLAB code for the reaction diffusion-based level set topology optimization. Zhang et al. [17] applied this method in the 3D image segmentation field. Afterward, Zhuang et al. [18] proposed a reaction diffusion-based level set method using body-fitted mesh for structural topology optimization. ...
... Among the various PDEs representing natural phenomena, we deal with reaction-diffusion type equations. The reaction-diffusion model has been applied and used in various fields such as biology [3,4,5], chemistry [6,7,8], image segmentation [9,10,11], image inpainting [12,13,14], medical [15,16,17], and so on. In this paper, we use second order reaction-diffusion type equations: ...
This is a draft. You can see the paper "Learning finite difference methods for reaction-diffusion type equations with FCNN"
https://doi.org/10.1016/j.camwa.2022.08.006
... Based on the psychological research on the visual perception of texture by the human eye, Yao et al. [12] proposed six texture attributes that can constitute a texture visual model, which are contrast, granularity, directionality, line type, roughness, and uniformity. Zhang et al. [13] found that the gradient cooccurrence matrix method, which combines the boundary operator and the gray distribution, can better describe the texture characteristics. This method first convolves the five special convolution kernels in pairs and then convolves the oil painting on the template obtained from the convolution and extracts texture features. ...
The oil painting retrieval technology based on the reaction diffusion equation has attracted widespread attention in the fields of oil painting processing and pattern recognition. The description and extraction of oil painting information and the classification method of oil paintings are two important processes in content-based oil painting retrieval. Inspired by the restoration and decomposition functional model of equal oil painting, we propose a reaction diffusion equation model. The new model contains two reaction diffusion equations with different principal parts. One principal part is total variation diffusion, which is used to remove noise. The other main part is thermal diffusion, which is used to modify the source term of the denoising reaction-diffusion equation to achieve the effect of protecting the texture of the oil painting. The interaction of the two reaction-diffusion equations finally achieves denoising while maintaining the boundaries and textures. Under the framework of the above reaction diffusion equation model, we introduce Laplace flow to replace the original total variation flow, so that the new denoising reaction diffusion equation combines the isotropic diffusion and total variation flow of the thermal reaction diffusion equation to achieve the effect of adaptive theoretical research. Using regularization methods and methods, we, respectively, get the well-posedness of the two model solutions, which provides the necessary preparation for numerical calculations. Based on the statistical theory and classification principles of support vector machines, combined with the characteristics of oil painting classification, the research and analysis are carried out from the three important aspects of kernel function, training algorithm, and multiclass classifier algorithm that affect the classification effect and speed. Numerical experiments show that the given filter model has a better processing effect on images with different types and different degrees of noise pollution. On this basis, an oil painting classification system based on texture features is designed, combined with an improved gray-level cooccurrence matrix algorithm and a multiclass support vector machine classification model, to extract, train, and classify oil paintings. Experiments with three types of oil paintings prove that the system can achieve a good oil painting classification effect. Different from the original model, the new model is based on the framework of reaction-diffusion equations. In addition, the new model has good effects in removing step effects, maintaining boundaries and denoising, especially in maintaining texture.
1. Introduction
Generally speaking, oil painting texture processing technology takes digital oil painting as the object, including oil painting texture acquisition, oil painting texture analysis, and oil painting texture understanding. Oil painting analysis is a link between oil painting acquisition and oil painting understanding, which directly affects the degree of computer comprehension [1]. The problems of oil painting restoration and oil painting segmentation belong to the research category of oil painting analysis and have always been two important and widely studied issues in oil painting processing [2]. With the increasing demand and breakthroughs in physical technology, people pay close attention to and explore the essence of oil painting processing and try to use strict mathematical theory to classify and improve the existing oil painting processing models [3]. Currently, digital oil painting processing technology has three main tools: random theory, wavelet analysis theory, and partial differential reaction diffusion equation theory. Among them, the stochastic model is directly applied to oil painting based on Bayesian estimation and Markov random field, and many reasonable models have been established, while wavelet analysis is based on Fourier analysis to transform oil painting information into the frequency domain and then build a model on the basis of it, which indirectly acts on oil painting texture recognition [4].
The noise and detailed texture of the oil painting are in the high-frequency area of the oil painting, and most of the information in the oil painting is stored in the edge part. In the process of removing the oil painting noise, the texture detail information of the oil painting will be mistakenly regarded as the noise information of the oil painting and be filtered out, resulting in the loss of image information and blurring of the oil painting [5]. Obtaining high-resolution oil paintings often requires high-cost and complex systems. In order to obtain finer details of the imaging scene under given system parameters, image interpolation methods can be used [6]. The usual nearest neighbor interpolation method, bilinear interpolation method, bicubic interpolation method, cubic convolution interpolation method, etc. enlarge the original oil painting, which improves the visual effect of the oil painting to a certain extent, but the interpolation precision of these algorithms is not high enough. The visual effect of oil painting is not improved enough [7]. The model parameters and missing pixels are estimated through nonlinear optimization methods, and the model parameters and missing pixels are estimated at the same time using nonlinear optimization methods. Therefore, the protection of oil painting details and texture information and the removal of oil painting noise information have become an irreconcilable contradiction. The goal of oil painting denoising is to keep the edge and texture details as much as possible while removing noise [8]. However, traditional oil painting denoising algorithms cannot balance the above two contradictions. It is imperative to find an algorithm that can well balance the contradiction between noise elimination and edge preservation information. The oil painting denoising algorithm based on the reaction diffusion equation can selectively smooth the oil painting and better balance the contradiction between the two [9]. This paper uses a combination of theoretical analysis and simulation experiments, combined with fractional calculus and fidelity terms, to study the oil painting denoising model based on the reaction-diffusion equation.
By analyzing the dependence of the oil painting boundary on the position and structure of the oil painting, we construct a boundary mapping function with strong antinoise ability. The corresponding reaction diffusion equation not only has the characteristics of anisotropic diffusion but also conducts heat in the homogeneous area inside the oil painting. The average curvature diffusion is carried out in the near-boundary area, and finally, the effect of adaptive denoising is achieved. In terms of theoretical research, we first use the fixed-point method to prove the well-posedness of the new model, and then study the asymptotic state of the solution, and the conclusion shows that the new model limit state of the recovery result is the local mean value of the initial oil painting. Compared with the TV model, the new model conducts thermal diffusion in the internal homogeneous region, effectively avoiding the step effect. Compared with the PM model, the new model is well-posed, so the solution is more stable, and due to the diffusion mode and diffusion speed of the new model, the model has stronger adaptive ability, better denoising in the internal homogeneous area, and more precise maintenance of the border near the border. This paper studies a variety of texture features in the oil painting content description and extraction method. Several numerical experiments verify the robustness of the proposed index against changes in translation, scale, and rotation. The application of image denoising shows the effectiveness of the proposed index. Starting from the characteristics of texture and the practical application of oil painting content retrieval technology, the gray-level cooccurrence matrix algorithm in the statistical method is analyzed emphatically. Aiming at the shortcomings of this method that it has a large amount of redundant calculations and requires a large amount of storage space, several existing improved algorithms based on gray-level cooccurrence matrix algorithms are further studied, including the sum-and-difference statistical method algorithms.
2. Related Work
Many researchers have done further research along this route. Xing et al. [10] studied statistical information, and for the first time experimented and proposed that the deficit moment, contrast, and entropy have the greatest recognition ability. The two-dimensional cooccurrence matrix proposed by the predecessors discarded the color information, and Zhang et al. [11] proposed a three-dimensional form of the cooccurrence matrix to solve this problem. Based on the psychological research on the visual perception of texture by the human eye, Yao et al. [12] proposed six texture attributes that can constitute a texture visual model, which are contrast, granularity, directionality, line type, roughness, and uniformity. Zhang et al. [13] found that the gradient cooccurrence matrix method, which combines the boundary operator and the gray distribution, can better describe the texture characteristics. This method first convolves the five special convolution kernels in pairs and then convolves the oil painting on the template obtained from the convolution and extracts texture features. Chi et al. [14] proposed a two-stage self-supervised texture segmentation method. The method first obtains the initial segmentation map through unsupervised automatic regression clustering and then performs self-supervised wavelet classification on this basis. Wen et al. [15] introduced a statistical texture analysis method based on texture element mode, which can provide spatial structure information of oil painting texture and has the monotonic invariance of pixel gray value. Zhao et al. [16] proposed a texture analysis method based on BVLC moments and BDIP moments. BVLC can display rough and smooth characteristics, and BDIP can extract troughs and edges very well. They are processed directly on the color space, which can effectively combine color features. The frequency domain analysis method of texture has also been developed rapidly.
As a remedy, nonlinear technology has successfully coordinated the processing of oil painting denoising and edge enhancement and has become a very effective technical method. Among them, nonlinear diffusion technology has been a hot topic in recent years, including the reaction diffusion equation method and geometric method, among which good results have been achieved in oil painting enhancement, oil painting segmentation and edge detection, texture generation, and mathematical morphology. The high-quality results concerned are computer vision and pattern recognition [17]. The method based on the Gabor filter conforms to the characteristics of the human visual perception system and the physiological vision of the human eye and is an important development direction for texture oil painting analysis. Gabor wavelet is a joint space-frequency method, and its texture model is based on a narrow-band texture field model, which can achieve local optima in both the frequency domain and the spatial domain. After the theoretical structure of wavelet transform was established, many scholars began to study how to use wavelet transform to represent texture features. Some scholars use statistics (variance and average) extracted from wavelet subbands as texture features, which have good retrieval results [18]. It includes the use of wavelet frame transform package theory to analyze the texture of oil paintings to obtain the characteristics of translation invariance and stability. In order to explore the characteristics of medium waves, scholars use tree-shaped wavelet transform to further improve the accuracy of classification. The combination of wavelet transform and other technologies achieves better performance.
The model uses directional diffusion with protective edge characteristics instead of isotropic diffusion with Gaussian smooth kernel, laying the theoretical foundation of nonlinear diffusion method for oil painting processing, and opening up a new field of oil painting processing application research. Based on the same framework, the introduction of total variation further demonstrates the importance of PDE models and theories in oil painting processing. The use of nonlinear diffusion methods to obtain continuous models makes mesh selection and isotropic diffusion easier to deal with; nonlinear diffusion methods are also allowed to merge or separate known filtering methods and form new models; further PDE models make algorithm analysis and synthesis more natural and flexible to achieve high accuracy and stability [19–21]. Gaussian filtering, median filtering, and their improvements all have the advantage of a small amount of calculation. This is indispensable in the real-time processing of computer vision. For the need to restore high-quality oil paintings, nonlinear anisotropic diffusion model is an important method [22, 23].
3. Construction of an Adaptive Extraction Model of Oil Painting Texture Features Based on Reaction-Diffusion Equations
3.1. Distribution of the Solution Set of the Reaction-Diffusion Equation
There are two important frameworks for oil painting restoration and decomposition models based on reaction-diffusion equations. The variational model is based on energy functionals, and the reaction-diffusion equation model is based on fluid diffusion theory. First, we review the basic frameworks of denoising models based on variational methods and some classic models.
The value on the main diagonal of the gray-level cooccurrence matrix is used to measure the smoothness of the texture. The closer the gray pair value is to the main diagonal value, the smoother the texture is. The eigenvalues DIS, CON, INV, and IDM reflect the relationship between the value of the gray pair and the value of the main diagonal, so they can express the smoothness of the texture. DIS and CON are inversely proportional to INV and IDM, respectively. The CON value measures the size of the texture change.
Figure 1 shows the spatial distribution of the solution set of the reaction diffusion equation. According to the known training samples, the characteristic parameters are selected, and the discriminant function is established to classify each object. This method must have prior knowledge of the classification area to establish the discriminant function and obtain the training classifier.
... Unlike the traditional upwind scheme [57][58][59], the restraint to the time step due to the Courant-Friedrichs-Lewy (CFL) stability condition is invalid in the RDLS method [60,61]. Thus, the large time step can be used to reduce the number of iterations [62]. The fictitious interface energy term, together with the interception of the level set function in {-1,1} regularizes the level set function, making the level set function re-initialization unnecessary [53]. ...
The level set method can express smooth boundaries in structural topology optimization with the level set function's zero-level contour. However, most applications still use rectangular/hexahedral mesh in finite element analysis, which results in zigzag interfaces between the void and solid phases. We propose a reaction diffusion-based level set method using the adaptive triangular/tetrahedral mesh for structural topology optimization in this work. Besides genuinely expressing smooth boundaries, such a body-fitted mesh can increase finite element analysis accuracy. Unlike the traditional upwind algorithm, the proposed method breaks through the constraint of Courant-Friedrichs-Lewy stability condition with an updating scheme based on finite element analysis. Numerical examples for minimum mean compliance and maximum output displacement at specified positions, in both 2D and 3D, converge within dozens of iterations and present elegant structures.
In recent years, Physics-informed neural networks (PINNs) have been widely used to solve partial differential equations alongside numerical methods because PINNs can be trained without observations and deal with continuous-time problems directly. In contrast, optimizing the parameters of such models is difficult, and individual training sessions must be performed to predict the evolutions of each different initial condition. To alleviate the first problem, observed data can be injected directly into the loss function part. To solve the second problem, a network architecture can be built as a framework to learn a finite difference method. In view of the two motivations, we propose Five-point stencil CNNs (FCNNs) containing a five-point stencil kernel and a trainable approximation function for reaction-diffusion type equations including the heat, Fisher's, Allen–Cahn, and other reaction-diffusion equations with trigonometric function terms. We show that FCNNs can learn finite difference schemes using few data and achieve the low relative errors of diverse reaction-diffusion evolutions with unseen initial conditions. Furthermore, we demonstrate that FCNNs can still be trained well even with using noisy data.
A novel mathematical approach is proposed to find the exact Lagrangian multiplier Λ for volume constraint during topology optimization via reaction-diffusion equation (RDE). With such a Λ, exact volume constraint is made possible. The mathematical trick is to split the density function (or the level-set function) and their RDEs into two parts, respectively. Both of them become independent of Λ, whose exact value is then determined by superposing the two parts to satisfy the volumetric requirement at the current optimization step. Thanks to the exactness, our proposed volume constraint method can achieve convergence for even complex optimization problems such as the minimum compliance problem with only material domain being reconstructed and remeshed, while the existing volume constraint method (the augmented Lagrangian method for constrained problems) fails to do so. For the latter, its Λ is determined from the previous optimization step, and hence the volume constraint at the current step is not exact. Apart from the superior robustness, other advantages of our proposed method include (1) exact volume control within a given error tolerance, (2) less total computational time due to fewer optimization steps needed and (3) a smaller compliance of the optimized structure after removing both grey area and mesh outside the material domain. Exact volume constraint during the whole optimization process is different from merely adjusting the final volume by choosing a level set other than zero. The latter one is merely an engineering trick that does not affect the convergence.
Multi-level image thresholding segmentation divides an image into multiple non-overlapping regions. This paper presents a novel two-dimensional (2D) histogram-based segmentation method to improve the efficiency of multi-level image thresholding segmentation. In the proposed method, a new non-local means 2D histogram and a novel variant of gravitational search algorithm (exponential Kbest gravitational search algorithm) have been used to find the optimal thresholds. Further, for the optimization, a 2D Rényi entropy has been redefined for multi-level thresholding. The proposed method has been tested on the Berkeley Segmentation Dataset and Benchmark (BSDS300) in terms of both subjective and objective assessments. The experimental results affirm that the proposed method outperforms the other 2D histogram-based image thresholding segmentation methods on majority of performance parameters.
The level set method [1] is a popular technique for tracking moving interfaces in several disciplines including computer vision and fluid dynamics. However, despite its high flexibility, the original level set method is limited by two important numerical issues. Firstly, the level set method does not implicitly preserve the level set function as a distance function, which is necessary to estimate accurately geometric features s.a. the curvature or the contour normal. Secondly, the level set algorithm is slow because the time step is limited by the standard CFL condition, which is also essential to the numerical stability of the iterative scheme. Recent advances with graph cut methods [2], [3] and continuous convex relaxation methods [4][6] provide powerful alternatives to the level set method for image processing problems because they are fast, accurate and guaranteed to find the global minimizer independently to the initialization. These recent techniques use binary functions to represent the contour rather than distance functions, which are usually considered for the level set method. However, the binary function cannot provide the distance information, which can be essential for some applications s.a. the surface reconstruction problem from scattered points and the cortex segmentation problem in medical imaging. In this paper, we propose a fast algorithm to preserve distance functions in level set methods. Our algorithm is inspired by recent efficient `1 optimization techniques, which will provide an efficient and easy to implement algorithm. It is interesting to note that our algorithm is not limited by the CFL condition and it naturally preserves the level set function as a distance function during the evolution, which avoids the classical re-distancing problem in level set methods. We apply the proposed algorithm to carry out image segmentation, where our methods proves to be 5 to 6 times faster than standard distance preserving level set techniques. We also present two applications where preserving a distance function is essential. Nonetheless, our method stays generic and can be applied to any level set methods that require the distance information.
This paper describes a computational approach to edge detection. The success of the approach depends on the definition of a comprehensive set of goals for the computation of edge points. These goals must be precise enough to delimit the desired behavior of the detector while making minimal assumptions about the form of the solution. We define detection and localization criteria for a class of edges, and present mathematical forms for these criteria as functionals on the operator impulse response. A third criterion is then added to ensure that the detector has only one response to a single edge. We use the criteria in numerical optimization to derive detectors for several common image features, including step edges. On specializing the analysis to step edges, we find that there is a natural uncertainty principle between detection and localization performance, which are the two main goals. With this principle we derive a single operator shape which is optimal at any scale. The optimal detector has a simple approximate implementation in which edges are marked at maxima in gradient magnitude of a Gaussian-smoothed image. We extend this simple detector using operators of several widths to cope with different signal-to-noise ratios in the image. We present a general method, called feature synthesis, for the fine-to-coarse integration of information from operators at different scales. Finally we show that step edge detector performance improves considerably as the operator point spread function is extended along the edge.
Level set methods have been widely used in image processing and computer vision. In conventional level set formulations, the level set function typically develops irregularities during its evolution, which may cause numerical errors and eventually destroy the stability of the evolution. Therefore, a numerical remedy, called reinitialization, is typically applied to periodically replace the degraded level set function with a signed distance function. However, the practice of reinitialization not only raises serious problems as when and how it should be performed, but also affects numerical accuracy in an undesirable way. This paper proposes a new variational level set formulation in which the regularity of the level set function is intrinsically maintained during the level set evolution. The level set evolution is derived as the gradient flow that minimizes an energy functional with a distance regularization term and an external energy that drives the motion of the zero level set toward desired locations. The distance regularization term is defined with a potential function such that the derived level set evolution has a unique forward-and-backward (FAB) diffusion effect, which is able to maintain a desired shape of the level set function, particularly a signed distance profile near the zero level set. This yields a new type of level set evolution called distance regularized level set evolution (DRLSE). The distance regularization effect eliminates the need for reinitialization and thereby avoids its induced numerical errors. In contrast to complicated implementations of conventional level set formulations, a simpler and more efficient finite difference scheme can be used to implement the DRLSE formulation. DRLSE also allows the use of more general and efficient initialization of the level set function. In its numerical implementation, relatively large time steps can be used in the finite difference scheme to reduce the number of iterations, while ensuring sufficient-
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numerical accuracy. To demonstrate the effectiveness of the DRLSE formulation, we apply it to an edge-based active contour model for image segmentation, and provide a simple narrowband implementation to greatly reduce computational cost.
An algorithm for automatic and accurate segmentation of multi-dimensional images is presented in this paper. It improves the classical watershed transform whose results are inaccurate when applied on noisy or anisotropic data. This algorithm combines a watershed-like region growing with a very simple marker selection step. It is particularly well suited for accurate segmentation of complex objects, such as the brain in 3D Magnetic Resonance (MR) images of the head since it provides an accurate and fully 3D segmentation in a reasonable computation time. Comparative results of the segmentation obtained by this algorithm and by the classical watershed transform are shown in the case of 3D MR images. Applications of this technique to 3D visualisation and brain sulcii identification are also presented. (C) 1997 Elsevier Science B.V.
In this paper, we present methods for edge segmentation of satellite images; we used seven techniques for this category; Sobel operator technique, Prewitt technique, Kiresh technique, Laplacian technique, Canny technique, Roberts technique and Edge Maximization Technique (EMT) and they are compared with one another so as to choose the best technique for edge detection segment image. These techniques applied on one satellite images to choose base guesses for segmentation or edgedetection image.
A theory of edge detection is presented. The analysis proceeds in two parts. (1) Intensity changes, which occur in a natural image over a wide range of scales, are detected separately at different scales. An appropriate filter for this purpose at a given scale is found to be the second derivative of a Gaussian, and it is shown that, provided some simple conditions are satisfied, these primary filters need not be orientation-dependent. Thus, intensity changes at a given scale are best detected by finding the zero values of delta 2G(x,y)*I(x,y) for image I, where G(x,y) is a two-dimensional Gaussian distribution and delta 2 is the Laplacian. The intensity changes thus discovered in each of the channels are then represented by oriented primitives called zero-crossing segments, and evidence is given that this representation is complete. (2) Intensity changes in images arise from surface discontinuities or from reflectance or illumination boundaries, and these all have the property that they are spatially. Because of this, the zero-crossing segments from the different channels are not independent, and rules are deduced for combining them into a description of the image. This description is called the raw primal sketch. The theory explains several basic psychophysical findings, and the operation of forming oriented zero-crossing segments from the output of centre-surround delta 2G filters acting on the image forms the basis for a physiological model of simple cells (see Marr & Ullman 1979).
Accurate segmentation of images is one of the most important objectives in image analysis. The two conventional methods of image segmentation, region based segmentation and boundary finding, often suffer from a variety of limitations. Many methods have been proposed to overcome the limitations but the solutions tend to be problem specific. Here we present a new region growing method with the capability of finding the boundary of a relatively bright/dark region in a textured background. The method relies on a measure of contrast of the region which represents the variation of the region gray level as a function of its evolving boundary during the growing process. It helps to identify the best external boundary of the region. The application of a reverse test using a gradient measure then yields the highest gradient boundary for the region being grown. A number of experiments have been performed both on synthetic and real images to evaluate the new approach. The proposed scheme can be ca...
In this paper, we present a new variational formulation for geometric active contours that forces the level set function to be close to a signed distance function, and therefore completely eliminates the need of the costly re-initialization procedure. Our variational formulation consists of an internal energy term that penalizes the deviation of the level set function from a signed distance function, and an external energy term that drives the motion of the zero level set toward the desired image features, such as object boundaries. The resulting evolution of the level set function is the gradient flow that minimizes the overall energy functional. The proposed variational level set formulation has three main advantages over the traditional level set formulations. First, a significantly larger time step can be used for numerically solving the evolution partial differential equation, and therefore speeds up the curve evolution. Second, the level set function can be initialized with general functions that are more efficient to construct and easier to use in practice than the widely used signed distance function. Third, the level set evolution in our formulation can be easily implemented by simple finite difference scheme and is computationally more efficient. The proposed algorithm has been applied to both simulated and real images with promising results.
Intensity inhomogeneities often occur in real-world images and may cause considerable difficulties in image segmentation. In order to overcome the difficulties caused by intensity inhomogeneities, we propose a region-based active contour model that draws upon intensity information in local regions at a controllable scale. A data fitting energy is defined in terms of a contour and two fitting functions that locally approximate the image intensities on the two sides of the contour. This energy is then incorporated into a variational level set formulation with a level set regularization term, from which a curve evolution equation is derived for energy minimization. Due to a kernel function in the data fitting term, intensity information in local regions is extracted to guide the motion of the contour, which thereby enables our model to cope with intensity inhomogeneity. In addition, the regularity of the level set function is intrinsically preserved by the level set regularization term to ensure accurate computation and avoids expensive reinitialization of the evolving level set function. Experimental results for synthetic and real images show desirable performances of our method.
We review machine learning methods employing positive definite kernels. These methods formulate learning and estimation problems in a reproducing kernel Hilbert space (RKHS) of functions defined on the data domain, expanded in terms of a kernel. Working in linear spaces of function has the benefit of facilitating the construction and analysis of learning algorithms while at the same time allowing large classes of functions. The latter include nonlinear functions as well as functions defined on nonvectorial data. We cover a wide range of methods, ranging from binary classifiers to sophisticated methods for estimation with structured data. Comment: Published in at http://dx.doi.org/10.1214/009053607000000677 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)
The shortcoming of geometric active contours is that they are extremely sensitive to positions of initialization. Initial contours of the distance regularized level set evolution model must be set inside or outside target completely. In addition, the model is prone to problems of falling into false boundary, leaking from weak edge and poor anti-noise ability. In this paper, we propose a robust active contour model driven by adaptive functions (including an adaptive edge indicator function and adaptive sign function)and fuzzy c-means energy. Utilize the adaptive edge indicator function which is composed of image intensity information to substitute traditional edge indicator function in the area term. Active contours can expand or shrink from initialization automatically, which improves the disadvantages of poor robustness to initialization and unidirectional movement. Due to the adaptive sign function and fuzzy c-means energy, the problems of slow convergence and leaking from weak edge have been solved. Moreover, a novel distance regularized term (mainly a potential function and evolution speed function)is proposed to make evolution more stable. Experimental results have proved that our model can segment images with intensity inhomogeneity effectively. Compared with other classic models, the proposed model not only shortens time spent and improves segmentation accuracy, but also shows a better robustness to initialization.
We propose an efficient and robust algorithm to reconstruct the volumes of multi-labeled objects from sets of cross sections without overlapping regions, artificial gaps, or mismatched interfaces. The algorithm can handle cross sections wherein different regions have different labels. The present study represents a multicomponent extension of our previous work (Li et al. (2015), [1]), wherein we modified the original Cahn–Hilliard (CH) equation by adding a fidelity term to keep the solution close to the single-labeled slice data. The classical CH equation possesses desirable properties, such as smoothing and conservation. The key idea of the present work is to employ a multicomponent CH system to reconstruct multicomponent volumes without self-intersections. We utilize the linearly stabilized convex splitting scheme introduced by Eyre with the Fourier-spectral method so that we can use a large time step and solve the discrete equation quickly. The proposed algorithm is simple and produces smooth volumes that closely preserve the original volume data and do not self-intersect. Numerical results demonstrate the effectiveness and robustness of the proposed method.
Active contour models have been widely used for image segmentation purposes. However, they may fail to delineate objects of interest depicted on images with intensity inhomogeneity. To resolve this issue, a novel image feature, termed as local edge entropy, is proposed in this study to reduce the negative impact of inhomogeneity on image segmentation. An active contour model is developed on the basis of this feature, where an edge entropy fitting (EEF) energy is defined with the combination of a redesigned regularization term. Minimizing the energy in a variational level set formulation can successfully drive the motion of an initial contour curve towards optimal object boundaries. Experiments on a number of test images demonstrate that the proposed model has the capability of handling intensity inhomogeneity with reasonable segmentation accuracy.
It had been known that the famous region scalable-fitting model can segment the images with intensity inhomogeneity effectively, but it largely depends on the position of initial contour. In this paper, an active contour model which combines region-scalable fitting energy and optimized Laplacian of Gaussian (LoG) energy is proposed for image segmentation. We first present a LoG energy term optimized by an energy functional which can smooth the homogeneous regions and enhance edge information at the same time. Then, we integrate the optimized LoG energy term with the region-scalable fitting energy term which makes use of local region information to drive the curve towards the boundaries. With the addition of LoG term, the proposed model is insensitive to the positions of initial contour and realizes an accurate segmentation result. Experiments on some synthetic and real images have proved that the proposed model not only has a good robustness of initialization, but also has a higher segmentation accuracy and efficiency than other major region-based models.
We describe a fast and efficient numerical algorithm for the process of three-dimensional narrow volume reconstruction from scattered data in three dimensions. The present study is an extension of previous research [Li et al., Surface embedding narrow volume reconstruction from unorganized points, Comput. Vis. Image Underst. 121 (2014) 100–107]. In the previous work, we modified the original Allen–Cahn equation by multiplying a control function to restrict the evolution within a narrow band around the given surface data set. The key idea of the present work is to perform the computations only on a narrow band around the given surface data set. In this way, we can significantly reduce the storage memory and CPU time. The proposed numerical method, based on operator splitting techniques, can employ a large time step size and exhibits unconditional stability. We perform a number of numerical experiments in order to demonstrate the efficiency of this method.
In this paper, a novel level set segmentation model integrating the intensity and texture terms is proposed to segment complicated two-phase nature images. Firstly, an intensity term based on the global division algorithm is proposed, which can better capture intensity information of image than the Chan–Vese model (CV). Particularly, the CV model is a special case of the proposed intensity term under a certain condition. Secondly, a texture term based on the adaptive scale local variation degree (ASLVD) algorithm is proposed. The ASLVD algorithm adaptively incorporates the amplitude and frequency components of local intensity variation, thus, it can extract the non-stationary texture feature accurately. Finally, the intensity term and the texture term are jointly incorporated into level set and used to construct effective image segmentation model named as the Intensity-Texture model. Since the intensity term and the texture term are complementary for image segmentation, the Intensity-Texture model has strong ability to accurately segment those complicated two-phase nature images. Experimental results demonstrate the effectiveness of the proposed Intensity-Texture model.
The active contour models have been popularly employed for image segmentation for almost a decade now. Among these active contour models, the level set based Chan and Vese algorithm is a popular region-based model that inherently utilizes intensity homogeneity in each region under consideration. However, the Chan and Vese model often suffers from the possibility of getting trapped in a local minimum, if the contour is not properly initialized. This problem assumes greater importance in the context of medical images where the intensity variations may assume large varieties of local and global profiles. In this work we propose a robust version of the Chan and Vese algorithm which is expected to achieve satisfactory segmentation performance, irrespective of the initial choice of the contour. This work formulates the fitting energy minimization problem to be solved using a metaheuristic optimization algorithm and makes a successful implementation of our algorithm using particle swarm optimization (PSO) technique. Our algorithm has been developed for two-phase level set implementation of the Chan and Vese model and it has been successfully utilized for both scalar-valued and vector-valued images. Extensive experimentations utilizing different varieties of medical images demonstrate how our proposed method could significantly improve upon the quality of segmentation performance achieved by Chan and Vese algorithm with varied initializations of contours.
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 also a generalization of an active contour model without edges based 2-phase segmentation, developed by the authors earlier in T. Chan and L. Vese (1999. In Scale-Space'99, M. Nilsen et al. (Eds.), LNCS, vol. 1682, pp. 141–151) and T. Chan and L. Vese (2001. IEEE-IP, 10(2):266–277). The multiphase level set formulation is new and of interest on its own: by construction, it automatically avoids the problems of vacuum and overlap; it needs only log n level set functions for n phases in the piecewise constant case; it can represent boundaries with complex topologies, including triple junctions; in the piecewise smooth case, only two level set functions formally suffice to represent any partition, based on The Four-Color Theorem. Finally, we validate the proposed models by numerical results for signal and image denoising and segmentation, implemented using the Osher and Sethian level set method.
This paper presents a novel reaction-diffusion (RD) method for implicit active contours, which is completely free of the costly re-initialization procedure in level set evolution (LSE). A diffusion term is introduced into LSE, resulting in an RD-LSE equation, to which a piecewise constant solution can be derived. In order to have a stable numerical solution of the RD based LSE, we propose a two-step splitting method (TSSM) to iteratively solve the RD-LSE equation: first iterating the LSE equation, and then solving the diffusion equation. The second step regularizes the level set function obtained in the first step to ensure stability, and thus the complex and costly re-initialization procedure is completely eliminated from LSE. By successfully applying diffusion to LSE, the RD-LSE model is stable by means of the simple finite difference method, which is very easy to implement. The proposed RD method can be generalized to solve the LSE for both variational level set method and PDE-based level set method. The RD-LSE method shows very good performance on boundary anti-leakage. The extensive and promising experimental results on synthetic and real images validate the effectiveness of the proposed RD-LSE approach.
This paper presents a structural topology optimization method based on a reaction–diffusion equation. In our approach, the design sensitivity for the topology optimization is directly employed as the reaction term of the reaction–diffusion equation. The distribution of material properties in the design domain is interpolated as the density field which is the solution of the reaction–diffusion equation, so free generation of new holes is allowed without the use of the topological gradient method. Our proposed method is intuitive and its implementation is simple compared with optimization methods using the level set method or phase field model. The evolution of the density field is based on the implicit finite element method. As numerical examples, compliance minimization problems of cantilever beams and force maximization problems of magnetic actuators are presented to demonstrate the method’s effectiveness and utility.
The coarsening kinetics of a two-phase mixture with a large diffusional mo-bility disparity between the two phases is studied using a variable-mobility Cahn-Hilliard equation. The semi-implicit spectral numerical technique was employed, and a number of interpolation functions are considered for describing the change in diffu-sion mobility across the interface boundary from one phase to another. The coarsening rate of domain size was measured using both structure and pair correlation functions as well as the direct computation of particle sizes in real space for the case that the coarsening phase consists of dispersed particles. We discovered that the average size (R) versus time (t) follows the R 10/3 ∝t law, in contrast to the conventional LSW theory, R 3 ∝ t, and the interface-diffusion dominated two-phase coarsening, R 4 ∝ t.
A new region-based active contour model that embeds the image local information is proposed in this paper. By introducing the local image fitting (LIF) energy to extract the local image information, our model is able to segment images with intensity inhomogeneities. Moreover, a novel method based on Gaussian filtering for variational level set is proposed to regularize the level set function. It can not only ensure the smoothness of the level set function, but also eliminate the requirement of re-initialization, which is very computationally expensive. Experiments show that the proposed method achieves similar results to the LBF (local binary fitting) energy model but it is much more computationally efficient. In addition, our approach maintains the sub-pixel accuracy and boundary regularization properties.
This paper proposes a new topology optimization method, which can adjust the geometrical complexity of optimal configurations, using the level set method and incorporating a fictitious interface energy derived from the phase field method. First, a topology optimization problem is formulated based on the level set method, and the method of regularizing the optimization problem by introducing fictitious interface energy is explained. Next, the reaction–diffusion equation that updates the level set function is derived and an optimization algorithm is then constructed, which uses the finite element method to solve the equilibrium equations and the reaction–diffusion equation when updating the level set function. Finally, several optimum design examples are shown to confirm the validity and utility of the proposed topology optimization method.
A variety of numerical techniques are available for tracking moving interfaces. In this review, we concentrate on techniques that result from the link between the partial differential equations that describe moving interfaces and numerical schemes designed for approximating the solutions to hyperbolic conservation laws. This link gives rise to computational techniques for tracking moving interfaces in two and three space dimensions under complex speed laws. We discuss the evolution of these techniques, the fundamental numerical approximations, involved, implementation details, and applications. In particular, we review some work on three aspects of materials sciences: semiconductor process simulations, seismic processing, and optimal structural topology design.
In this paper, we propose a new model for active contours to detect objects in a given image, based on techniques of curve evolution, Mumford--Shah functional for segmentation and level sets. Our model can detect objects whose boundaries are not necessarily defined by gradient. We minimize an energy which can be seen as a particular case of the minimal partition problem. In the level set formulation, the problem becomes a "mean-curvature flow"-like evolving the active contour, which will stop on the desired boundary. However, the stopping term does not depend on the gradient of the image, as in the classical active contour models, but is instead related to a particular segmentation of the image. We will give a numerical algorithm using finite differences. Finally, we will present various experimental results and in particular some examples for which the classical snakes methods based on the gradient are not applicable. Also, the initial curve can be anywhere in the image, and interior contours are automatically detected.
A novel hybrid filter based on gradient approximation by upwind scheme is proposed to restore images corrupted by impulsive and Gaussian noises, and simultaneously to preserve the details. In this work, the gradient is approximated by the hybrid upwind scheme, and then the impulses are separated according to the characteristic classification. The remaining pixels are processed by Gaussian filter (GF) with their corresponding weights that are acquired from the hybrid upwind scheme and are distinct in edges and smooth domains. The simulation results demonstrate that the proposed filter achieves better performance in restoring the images corrupted by different noise while the details are preserved.
In this paper, we present a scheme of improvement on the region-scalable fitting (RSF) model proposed by Li et al. (Minimization of region-scalable fitting energy for image segmentation, IEEE Transactions on Image Processing 17(10) (2008) 1940–1949) in terms of robustness to initialization and noise. First, the Gaussian kernel for the RSF energy is replaced with a “mollifying” kernel with compact support. Second, the RSF energy is redefined as a weighted energy integral, where the weight is local entropy deriving from a grey level distribution of image. The total energy functional is then incorporated into a variational level set formulation with two extra internal energy terms. The new RSF model not only handles better intensity inhomogeneity, but also allows for more flexible initialization and more robustness to noise compared to the original RSF model.
This paper presents an upwind finite-difference method for the numerical approximation of viscosity solutions of a Hamilton-Jacobi-Bellman
(HJB) equation governing a class of optimal feedback control problems. The method is based on an explicit finite-difference
scheme, and it is shown that the method is stable under certain constraints on the step lengths of the discretization. Numerical
results, performed to verify the usefulness of the method, show that the method gives accurate approximate solutions to both
the control and the state variables.
Nonlinear diffusion filtering in image processing is usually
performed with explicit schemes. They are only stable for very small
time steps, which leads to poor efficiency and limits their practical
use. Based on a discrete nonlinear diffusion scale-space framework we
present semi-implicit schemes which are stable for all time steps. These
novel schemes use an additive operator splitting (AOS), which guarantees
equal treatment of all coordinate axes. They can be implemented easily
in arbitrary dimensions, have good rotational invariance and reveal a
computational complexity and memory requirement which is linear in the
number of pixels. Examples demonstrate that, under typical accuracy
requirements, AOS schemes are at least ten times more efficient than the
widely used explicit schemes
A semi-implicit time discretization of gradient flows is given and analyzed. It is shown that the scheme is unconditionally gradient stable and that the discrete equations are uniquely solvable for all time steps. The key feature of the method is a separation of the contractive and expansive terms of the equation across the time step. The Cahn-Hilliard equation is used as an example. 1 Introduction. This paper will concentrate on a numerical method for solving the initial value problem defined by the system of real ordinary differential equations and auxillary conditions du dt = GammarF (u); u(0) = u 0 : (1) It is assumed that u(t) 2 C 1 (R + ; R p ), F (u) 2 C 2 (R p ; R), rF (u) is the gradient of F , and 8 ? ? ? ? ? ! ? ? ? ? ? : F (u) 0 8u 2 R p F (u) !1 as kuk !1 hJ(rF )(u) u; ui 8u 2 R p (2) Department of Mathematics, University of Utah, Salt Lake City, UT 84112, eyre@math.utah.edu 1 where J(rF )(u) is the Jacobian of rF (u) and 2 R, and where hDelta; ...
The courant-friedrichs-lewy (cfl) condition. AMC 10
- C De Moura
- C Kubrusly
De Moura, C., Kubrusly, C., 2013. The courant-friedrichs-lewy (cfl) condition. AMC 10,
12.
Multigrid narrow band surface reconstruction via level set functions
- J Ye
- I Yanovsky
- B Dong
- R Gandlin
- A Brandt
- S Osher
Ye, J., Yanovsky, I., Dong, B., Gandlin, R., Brandt, A., Osher, S., 2012. Multigrid narrow
band surface reconstruction via level set functions. In: Int. Symp. Visual Comput..