Author

# Changfeng Gui

Other affiliations: University of Connecticut, Courant Institute of Mathematical Sciences, University of Minnesota ...read more

Bio: Changfeng Gui is an academic researcher from University of Texas at San Antonio. The author has contributed to research in topics: Uniqueness & Allen–Cahn equation. The author has an hindex of 26, co-authored 82 publications receiving 6667 citations. Previous affiliations of Changfeng Gui include University of Connecticut & Courant Institute of Mathematical Sciences.

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##### Papers

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20 Jun 2005

TL;DR: 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.

Abstract: 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.

2,005 citations

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TL;DR: A new variational level set formulation in which the regularity of the level set function is intrinsically maintained during thelevel set evolution called distance regularized level set evolution (DRLSE), which eliminates the need for reinitialization and thereby avoids its induced numerical errors.

Abstract: 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 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.

1,947 citations

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383 citations

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TL;DR: In this article, it was shown that the problem has interior spike solutions at a local maximum point of the distance function d(P, ∂Ω), P e Ω, where Ω is a bounded smooth domain in RN, ǫ > 0 is a small parameter, and ƒ is a superlinear, subcritical nonlinearity.

243 citations

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TL;DR: In this paper, the authors considered the general semilinear equation (1.6) and showed the existence of critical points of the modified functionals at certain energy level and in certain neighborhood.

Abstract: In this paper, we consider the general semilinear equation (1.6) {epsilon}{sup 2}{Delta}{sub u}-V(x)u+f(u)=O, x {epsilon} R{sup N}. We assume that V(x) satisfies (V1) V(x) is locally Hoelder continuous in R{sup N} and V(x){ge}V{sub O}>O; (V2) there exist k disjoint bounded regions {Omega}{sub 1}, {Omega}{sub 2}...,{Omega}{sub k} such that (1.7) M{sub i}:=inf/{partial_derivative}{Omega}{sub i}/V(x) > {omega}{sub i}: = inf/{Omega}{sub i}V(x), i=1,2,...,k. We also assume that f(u) satisfies (f1) f(u) {epsilon} C{sup 1}(R), f(u){equivalent_to}O for u {le} O and f(u){equivalent_to} O for u > O; (f2) f(u)/u is nondecreasing in u; (f3) O {le} f{sub u}(u) {le} a{sub 1} + a{sub 2}u{sup p-1} for some positive constants a{sub 1}, a{sub 2} and 1 < p < N+2/N-2 (we use the convention now and later that N+2/N-2 should be replaced by {infinity} when N = 1,2); (f4) there exists {beta} {epsilon} (O,1/2) such that F(u) {le} {beta}uf(u), u{ge}O where F(u) = {integral}{sub O}{sup u} f(t) dt. This paper is organized as follows. In Section 2, we scale the equation (1.6) properly and present some results for generalized Palais Smale sequences of a family of modified functionals. In Section 3, we show the existence of critical points of the modified functionals at certain energy levelmore » and in certain neighborhood. Finally, we show the concentration property of these critical points and therefore obtain the solution for equation (1.6). This also finishes the proof of the main theorem. I have learned from the referee that he has just received a closely related paper by del Pino and Felmer, but which is based on different methods. 27 refs.« less

213 citations

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TL;DR: A new variational level set formulation in which the regularity of the level set function is intrinsically maintained during thelevel set evolution called distance regularized level set evolution (DRLSE), which eliminates the need for reinitialization and thereby avoids its induced numerical errors.

Abstract: 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 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.

1,947 citations

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TL;DR: This work proposes a region-based active contour model that draws upon intensity information in local regions at a controllable scale to cope with intensity inhomogeneity and shows desirable performances of this model.

Abstract: 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.

1,630 citations

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TL;DR: A novel region-based method for image segmentation, which is able to simultaneously segment the image and estimate the bias field, and the estimated bias field can be used for intensity inhomogeneity correction (or bias correction).

Abstract: Intensity inhomogeneity often occurs in real-world images, which presents a considerable challenge in image segmentation. The most widely used image segmentation algorithms are region-based and typically rely on the homogeneity of the image intensities in the regions of interest, which often fail to provide accurate segmentation results due to the intensity inhomogeneity. This paper proposes a novel region-based method for image segmentation, which is able to deal with intensity inhomogeneities in the segmentation. First, based on the model of images with intensity inhomogeneities, we derive a local intensity clustering property of the image intensities, and define a local clustering criterion function for the image intensities in a neighborhood of each point. This local clustering criterion function is then integrated with respect to the neighborhood center to give a global criterion of image segmentation. In a level set formulation, this criterion defines an energy in terms of the level set functions that represent a partition of the image domain and a bias field that accounts for the intensity inhomogeneity of the image. Therefore, by minimizing this energy, our method is able to simultaneously segment the image and estimate the bias field, and the estimated bias field can be used for intensity inhomogeneity correction (or bias correction). Our method has been validated on synthetic images and real images of various modalities, with desirable performance in the presence of intensity inhomogeneities. Experiments show that our method is more robust to initialization, faster and more accurate than the well-known piecewise smooth model. As an application, our method has been used for segmentation and bias correction of magnetic resonance (MR) images with promising results.

1,201 citations

10 Jan 2003

TL;DR: This article summarizes various aspects and results for some general formulations of the classical chemotaxis models also known as Keller-Segel models and offers possible generalizations of these results to more universal models.

Abstract: This article summarizes various aspects and results for some general formulations of the classical chemotaxis models also known as Keller-Segel models. It is intended as a survey of results for the most common formulation of this classical model for positive chemotactical movement and offers possible generalizations of these results to more universal models. Furthermore it collects open questions and outlines mathematical progress in the study of the Keller-Segel model since the first presentation of the equations in 1970.

1,138 citations

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04 Oct 2007

TL;DR: In this article, the authors propose a model for solving the model elliptic problems and model parabolic problems. But their model is based on Equations with Gradient Terms (EGS).

Abstract: Preliminaries.- Model Elliptic Problems.- Model Parabolic Problems.- Systems.- Equations with Gradient Terms.- Nonlocal Problems.

935 citations