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A reaction–diffusion based level set method for image segmentation in three dimensions

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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.

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... 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: ...
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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
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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]. ...
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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
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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..