# Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science (2nd edition)

About: This article is published in Kybernetes.The article was published on 2000-03-01. It has received 3822 citations till now. The article focuses on the topics: Computational mechanics & Computational geometry.

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TL;DR: Fiji is a distribution of the popular open-source software ImageJ focused on biological-image analysis that facilitates the transformation of new algorithms into ImageJ plugins that can be shared with end users through an integrated update system.

Abstract: Fiji is a distribution of the popular open-source software ImageJ focused on biological-image analysis. Fiji uses modern software engineering practices to combine powerful software libraries with a broad range of scripting languages to enable rapid prototyping of image-processing algorithms. Fiji facilitates the transformation of new algorithms into ImageJ plugins that can be shared with end users through an integrated update system. We propose Fiji as a platform for productive collaboration between computer science and biology research communities.

43,540 citations

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TL;DR: A set of automated procedures for obtaining accurate reconstructions of the cortical surface are described, which have been applied to data from more than 100 subjects, requiring little or no manual intervention.

9,599 citations

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TL;DR: The goal of this article is to review the state-of-the-art tracking methods, classify them into different categories, and identify new trends to discuss the important issues related to tracking including the use of appropriate image features, selection of motion models, and detection of objects.

Abstract: The goal of this article is to review the state-of-the-art tracking methods, classify them into different categories, and identify new trends. Object tracking, in general, is a challenging problem. Difficulties in tracking objects can arise due to abrupt object motion, changing appearance patterns of both the object and the scene, nonrigid object structures, object-to-object and object-to-scene occlusions, and camera motion. Tracking is usually performed in the context of higher-level applications that require the location and/or shape of the object in every frame. Typically, assumptions are made to constrain the tracking problem in the context of a particular application. In this survey, we categorize the tracking methods on the basis of the object and motion representations used, provide detailed descriptions of representative methods in each category, and examine their pros and cons. Moreover, we discuss the important issues related to tracking including the use of appropriate image features, selection of motion models, and detection of objects.

5,318 citations

### Cites background from "Level Set Methods and Fast Marching..."

...Explicit representations do not allow topology changes such as region split or merge [Sethian 1999]....

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TL;DR: An automated method for accurately measuring the thickness of the cerebral cortex across the entire brain and for generating cross-subject statistics in a coordinate system based on cortical anatomy is presented.

Abstract: Accurate and automated methods for measuring the thickness of human cerebral cortex could provide powerful tools for diagnosing and studying a variety of neurodegenerative and psychiatric disorders. Manual methods for estimating cortical thickness from neuroimaging data are labor intensive, requiring several days of effort by a trained anatomist. Furthermore, the highly folded nature of the cortex is problematic for manual techniques, frequently resulting in measurement errors in regions in which the cortical surface is not perpendicular to any of the cardinal axes. As a consequence, it has been impractical to obtain accurate thickness estimates for the entire cortex in individual subjects, or group statistics for patient or control populations. Here, we present an automated method for accurately measuring the thickness of the cerebral cortex across the entire brain and for generating cross-subject statistics in a coordinate system based on cortical anatomy. The intersubject standard deviation of the thickness measures is shown to be less than 0.5 mm, implying the ability to detect focal atrophy in small populations or even individual subjects. The reliability and accuracy of this new method are assessed by within-subject test-retest studies, as well as by comparison of cross-subject regional thickness measures with published values.

5,171 citations

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TL;DR: A new multiphase level set framework for image segmentation using the Mumford and Shah model, for piecewise constant and piecewise smooth optimal approximations, and validated by numerical results for signal and image denoising and segmentation.

Abstract: 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 overlaps it needs only log n level set functions for n phases in the piecewise constant cases it can represent boundaries with complex topologies, including triple junctionss 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.

2,649 citations

### Cites methods from "Level Set Methods and Fast Marching..."

...For more recent and general expositions on the level set method and applications, we refer the reader to Sethian (1999), Osher and Fedkiw (2001, to appear)....

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##### References

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TL;DR: There is a natural uncertainty principle between detection and localization performance, which are the two main goals, and with this principle a single operator shape is derived which is optimal at any scale.

Abstract: 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.

28,073 citations

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TL;DR: A tree is a graph with one and only one path between every two nodes, where at least one path exists between any two nodes and the length of each branch is given.

Abstract: We consider n points (nodes), some or all pairs of which are connected by a branch; the length of each branch is given. We restrict ourselves to the case where at least one path exists between any two nodes. We now consider two problems. Problem 1. Constrnct the tree of minimum total length between the n nodes. (A tree is a graph with one and only one path between every two nodes.) In the course of the construction that we present here, the branches are subdivided into three sets: I. the branches definitely assignec~ to the tree under construction (they will form a subtree) ; II. the branches from which the next branch to be added to set I, will be selected ; III. the remaining branches (rejected or not yet considered). The nodes are subdivided into two sets: A. the nodes connected by the branches of set I, B. the remaining nodes (one and only one branch of set II will lead to each of these nodes), We start the construction by choosing an arbitrary node as the only member of set A, and by placing all branches that end in this node in set II. To start with, set I is empty. From then onwards we perform the following two steps repeatedly. Step 1. The shortest branch of set II is removed from this set and added to

22,704 citations

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TL;DR: In this article, a constrained optimization type of numerical algorithm for removing noise from images is presented, where the total variation of the image is minimized subject to constraints involving the statistics of the noise.

15,225 citations

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General Electric

^{1}TL;DR: In this paper, a divide-and-conquer approach is used to generate inter-slice connectivity, and then a case table is created to define triangle topology using linear interpolation.

Abstract: We present a new algorithm, called marching cubes, that creates triangle models of constant density surfaces from 3D medical data. Using a divide-and-conquer approach to generate inter-slice connectivity, we create a case table that defines triangle topology. The algorithm processes the 3D medical data in scan-line order and calculates triangle vertices using linear interpolation. We find the gradient of the original data, normalize it, and use it as a basis for shading the models. The detail in images produced from the generated surface models is the result of maintaining the inter-slice connectivity, surface data, and gradient information present in the original 3D data. Results from computed tomography (CT), magnetic resonance (MR), and single-photon emission computed tomography (SPECT) illustrate the quality and functionality of marching cubes. We also discuss improvements that decrease processing time and add solid modeling capabilities.

13,231 citations