J. A. Sethian, "An analysis of flame propagation", Ph.D. Dissertation, Mathematics, University of Berkeley, 1982  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... 23] In the problems of interest to us in computer vision, the function of interest is ( a 1 (41) where a 0: The case a = 0, i.e. 1 is very important, and here equation (40) becomes an equation that has been studied in relation to problems in geometric optics [6] ame propagation [60], and shape morphology [10] as well as shape decomposition. Indeed, this is the di erential equation for the prairie re model described above. The part gives a di usive e ect, while the constant part gives a wave (hyperbolic) e ect which tends to create singularities and break a shape into its ....

....or shocks develop, one must be careful in de ning precisely what one means by solution to (61) This is precisely where the notion of viscosity solution arises and as we have seen above in Section 3. 2, the viscosity solution coincides with the entropy condition imposed by Sethian [60], interpreting the hyperbolic evolution law in the prairie re sense that once a particle is burnt, it remains burnt. See [64] for an extensive discussion on classical entropy conditions. Geometrically, it is very easy to see how discontinuities develop for the system (55,56) Indeed, ....

J. A. Sethian, An Analysis of Flame Propagation, Ph. D. Dissertation, University of California, 1982.

....Gaussian (smoothing) filter. In [7] n = 1, and in [27] n = 2. The function Psi(x; y; t) evolves in (2) according to the associated level set flow for planar curve evolution in the normal direction with speed a function of curvature which was introduced in the fundamental work of Osher Sethian [30, 31, 37, 38, 39]. As we have just seen, the Euclidean curve shortening part of this evolution, namely = kr Psikdiv (4) may be derived as a gradient flow for shrinking the perimeter as quickly as possible using only local information. As is explained in [7] the constant inflation term is added in (2) in ....

....Thus at such points lengths decrease and so one needs less energy in order to move. Consequently, it seems that such a metric is natural for attracting the deformable contour to an edge when OE has the form (3) We have implemented this snake model based on the algorithms of Osher Sethian [30, 31, 37, 38, 39] and Malladi et al. 27] 5 3 D Active Contour Models In this section, we will show how the 2D active contour model which we have presented may be easily extended to the 3D case. Here we use the corresponding surface evolution equations, gotten by modifying the Euclidean area by a function which ....

[Article contains additional citation context not shown here]

J. A. Sethian, An Analysis of Flame Propagation, Ph. D. Dissertation, University of California, 1982.

....image and G oe is a Gaussian (smoothing filter) filter. The function Psi(x; y; t) evolves in (2) according to the associated level set flow for planar curve evolution in the normal direction with speed a function of curvature which was introduced in the fundamental work of Osher Sethian [37, 38, 44, 45, 46]. It is important to note that as we have seen above, the Euclidean curve shortening part of this evolution, namely = kr Psikdiv( r Psi ) 4) is derived as a gradient flow for shrinking the perimeter as quickly as possible. As is explained in [10] the constant inflation term is added in ....

....and so derive a modified model of (2) given by 6 Notice that for OE as in (3) rOE will look like a doublet near an edge. Of course, one may choose other candidates for OE in order to pick out other features. We have implemented this snake model based on the algorithms of Osher Sethian [37, 38, 44, 45, 46] and Malladi et al. 31] We are also experimenting with some new code based on [35] Remarks 2. 1. Note that the metric ds OE has the property that it becomes small where OE is small and vice versa. Thus at such points lengths decrease and so one needs less energy in order to move. ....

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J. A. Sethian, An Analysis of Flame Propagation, Ph. D. Dissertation, University of California, 1982.

....these references in Chapter 3 and Chapter 4 by Suri et al. 1] The discussions on these references are out of the scope of this paper. The second class of deformable models is level sets. These deformable models were started by Osher and Sethian [67] which started from Sethian s Ph.D. Thesis [68]. The fundamental di erence between these two classes is: Parametric deformable curves (activecontours) are local methods based on an energy minimizing spline guided by external and image forces which pull or push the spline towards features such as lines and edges in the image. The classical ....

Sethian, J. A., An Analysis of Flame Propagation, Ph.D. Thesis, Department of Mathematics, UniversityofCalifornia,Berkeley, CA, 1982.

....Once the corner is developed, it is not clear how to continue the deformation, since the definition of the normal direction becomes ambiguous. A natural way to continue the deformation is to impose the so called entropy condition originally proposed in the area of interface propagation by Sethian [62]. In Section 3.3.5, we describe an entropy satisfying numerical scheme, proposed by Osher and Sethian [32] which implements geometric deformable contours. 3.3.3 Speed functions In this section, we provide a brief overview of three examples of speed functions used by geometric deformable ....

J. A. Sethian, An Analysis of Flame Propagation. Ph.D. thesis, Dept. of Mathematics, University of California, Berkeley, CA, 1982.

....of this process faces a number of fundamental difficulties, e.g. topological splitting, built up of error, capturing discontinuities, etc. Osher and Sethian, in application to flame propagation, proposed that curve evolution can be considered as the evolution of a surface, OE(x; y) [42, 49, 47]. The reaction diffusion space can then be generated by simulating OE t (fi 0 Gamma fi 1 )jrOEj = 0; 12) where is the curvature of the level set OE(x; y) 0. In addition, to capture and maintain discontinuities, shock capturing numerical schemes are required [19, 32, 41, 42, 48, 50] For ....

J. Sethian. An analysis of flame propagation. Ph.D. dissertation, University of California, Berekely, Berkeley, California, 1985.

....from the initial curve. This is known as the Huygens principle construction) Roughly speaking, we want to remove the tail from the swallowtail . In Fig. 6c, we show this alternate weak solution. Another way to characterize this weak solution is through the following entropy condition (see [26]) If the front is viewed as a burning flame, then once a particle is burnt it stays burnt. Careful adherence to this stipulation produces the Huygen s principle construction. Furthermore, this physically reasonable weak solution has an equally appealing mathematical quality: It is the formal ....

) Sethian, J.A., An analysis of flame propagation. Ph.D. Dissertation, University of California, Berkeley, California, June 1982; CPAM Rep. 79.

....swallow up all the variation until the moving front smooths into a circle. Thus, although corners initially correspond to a sharpening of the front and a singularity in the curvature, they also serve as a smoothing mechanism. We shall only outline the proof here; complete details may be found in [17]. Proposition 3 Let g(0) a(s ) b(s ) s [0,S ] be a simple, closed, piecewise C 2 , positively oriented initial curve. Let g(t ) be the entropy satisfying solution constructed from the ignition curves given in Equations (3.3,3.4) Then, as t , g(t ) approaches a circle. That is, let g ....

....of fluid constructions do not rely on discrete parameterizations of the moving front and can easily handle topological issues such as merging. Such techniques can be used in conjunction with Huyghens principle so that the entropy condition comes about in a natural way. For details, see [15] 3] [17]. Here, we shall briefly describe one such method for the case e=0, and demonstrate its use on a simple problem involving corner development. A square grid i , j of uniform mesh size is imposed on the domain, and a number f i j , 0f i j 1 is assigned to each cell, corresponding to the fraction of ....

) Sethian, J.A.: An analysis of flame propagation. PhD. Dissertation, University of California, Berkeley, California, June 1982; CPAM Rep. 79.

....this bound; by refining the mesh size, one can investigate both possible blow up in the curvature and the nature of the solution beyond the singularity. IV. Examples To begin, we use the above algorithm to follow a kidney bean shape expanding with constant velocity F (K ) 5. In Figure 1A (see [18]) we show the exact solution, found by solving the equations of motion, for a burnt region initially bounded by four joined semicircles. The position of the front is shown for various values of t . The concave region sharpens into a corner at t =1, which then opens up and smooths out. In Figure ....

) Sethian, J.A.: An analysis of flame propagation. PhD. Dissertation, University of California, Berkeley, California, June 1982; CPAM Rep. 79.

....(smoothing) filter. In [7] n = 1, and in [27] n = 2. The function Psi(x; y; t) evolves in (2) according to the associated level set flow for planar curve evolution in the normal direction with speed a function of curvature which was introduced in the fundamental work of Osher Sethian [30, 31, 37, 38, 39]. As we have just seen, the Euclidean curve shortening part of this evolution, namely Psi t = kr Psikdiv r Psi kr Psik (4) may be derived as a gradient flow for shrinking the perimeter as quickly as possible using only local information. As is explained in [7] the constant ....

....Thus at such points lengths decrease and so one needs less energy in order to move. Consequently, it seems that such a metric is natural for attracting the deformable contour to an edge when OE has the form (3) We have implemented this snake model based on the algorithms of Osher Sethian [30, 31, 37, 38, 39] and Malladi et al. 27] 5 3 D Active Contour Models In this section, we will show how the 2D active contour model which we have presented may be easily extended to the 3D case. Here we use the corresponding surface evolution equations, gotten by modifying the Euclidean area by a function which ....

[Article contains additional citation context not shown here]

J. A. Sethian, An Analysis of Flame Propagation, Ph. D. Dissertation, University of California, 1982.

....(4) This inflationary constant may be taken to be either positive (inward evolution) or negative in which case it would have an outward or expanding effect. For sufficiently large , this would cause the evolution to act as an expanding balloon or bubble [16, 58] The level set version of (3) [43, 44, 50, 51, 52] is given in terms of the evolving level set function Psi(x; y; z; t) by Psi t = OEkr Psikdiv( r Psi kr Psik ) rOE Delta r Psi: 5) Here again one may add a constant inflation term to the mean curvature to derive the level set version of (4) Psi t = OEkr Psik(div( r Psi kr Psik ) ....

....and will be reported in a future publication. 4 Numerical Implemenations We will now describe a numerical experiment to illustrate our methods. One of the most successful implementations of curvature flow equations is that based on the Osher Sethian level set formulations of the evolutions in [43, 44, 50, 51, 52, 38]. This formulation is global, and so effectively increases the problem dimension by one. More local implementations may be found in the recent work [1] Since we have been working with volumetric data, we have found it advantageous to use methods that do not increase the dimensionality but still ....

J. A. Sethian, An Analysis of Flame Propagation, Ph. D. Dissertation, University of California, 1982.

....image and G oe is a Gaussian (smoothing filter) filter. The function Psi(x; y; t) evolves in (2) according to the associated level set flow for planar curve evolution in the normal direction with speed a function of curvature which was introduced in the fundamental work of Osher Sethian [37, 38, 44, 45, 46]. It is important to note that as we have seen above, the Euclidean curve shortening part of this evolution, namely Psi t = kr Psikdiv( r Psi kr Psik ) 4) is derived as a gradient flow for shrinking the perimeter as quickly as possible. As is explained in [10] the constant inflation ....

....r Psi kr Psik ) rOE Delta r Psi: 8) Notice that for OE as in (3) rOE will look like a doublet near an edge. Of course, one may choose other candidates for OE in order to pick out other features. We have implemented this snake model based on the algorithms of Osher Sethian [37, 38, 44, 45, 46] and Malladi et al. 31] We are also experimenting with some new code based on [35] Remarks 2. 1. Note that the metric ds OE has the property that it becomes small where OE is small and vice versa. Thus at such points lengths decrease and so one needs less energy in order to move. ....

[Article contains additional citation context not shown here]

J. A. Sethian, An Analysis of Flame Propagation, Ph. D. Dissertation, University of California, 1982.

....intuition is that not all contours are shapes, but rather only those that can enclose physical material. A theory of contour deformation is derived from these principles, based on abstract conservation principles and Hamilton Jacobi theory. These principles are based on the work of Sethian [82, 86], the Osher Sethian level set formulation [65] the classical shock theory of Lax [53, 54] as well as curve evolution theory for a curve evolving as a function of the curvature and the relation to geometric smoothing of Gage Hamilton Grayson [32, 37] The result is a characterization of the ....

....this problem that we focus. In particular, we derive a formal framework for our theory from a mathematical model of deformations, and, in so doing, consider the notion of curve evolution via partial differential equations. The mathematical framework underlying this is based on the work of Sethian [82, 83, 85, 86, 84], and the fundamental Osher Sethian level set al..gorithm [65] which has already proven to be of enormous use in image processing and interface motion. We should also add that while the mathematics is different, the spirit of our work was prefigured in many ways by Koenderink [50] in his important ....

[Article contains additional citation context not shown here]

J. Sethian. An analysis of flame propagation. Ph.D. dissertation, University of California, Berekely, Berkeley, California, 1985.

....curve moving with normal speed equal to its curvature will shrink to a point, its shape becoming smoother and circular. More complicated phenomena are expected in higher dimensions, but no classification is available. This problem can be tackled by the use of the level set method (Osher Sethian [63, 56]) which consists in viewing the curve as the level set u = 0 of a function u constrained to solve the degenerate diffusion equation 1 u t = krukdiv ru kruk : Its singular behavior is reflected in the fact that unlike parabolic equations, it is possible to change the solution in the vicinity of ....

....(smoothing) filter. In [16] n = 1, and in [47] n = 2. The function Psi(x; y; t) evolves in (2) according to the associated level set flow for planar curve evolution in the normal direction with speed a function of curvature which was introduced in the fundamental work of Osher Sethian [55, 56, 63, 64, 65]. It is important to note that as we have seen above, the Euclidean curve shortening part of this evolution, namely Psi t = kr Psikdiv r Psi kr Psik # (4) is derived as a gradient flow for shrinking the perimeter as quickly as possible. As is explained in [16] the constant inflation ....

[Article contains additional citation context not shown here]

J. A. Sethian, An Analysis of Flame Propagation, Ph. D. Dissertation, University of California, 1982.

....initial data at t = 0. Equation (1. 3) occurs in many contexts as an evolution model: the propagation of wavefronts, flame fronts, and the evolution of material interfaces as in, for instance, the surface of an electronic chip subjected to etching and deposition processes during its manufacture [13, 14, 17, 18, 19, 20, 32, 33, 34, 37]. Hamilton Jacobi equation (1.3) is the integrated form of a conservation law for p = D z OE: D t p D z H(p) 0; 1.4) obtained by taking the gradient of OE in (1.3) Solutions to conservation laws can have discontinuities (e.g. shocks) which result in discontinuities in the surface gradient ....

J. A. Sethian. An Analysis of Flame Propagation. Ph.d. dissertation, University of California, Berkeley, 1982.

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Sethian, J.A., An Analysis of Flame Propagation, Ph.D. Dissertation, Dept. of Mathematics, University of California, Berkeley, CA, 1982.

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Sethian, J.A., \An Analysis of Flame Propagation", Ph.D. Dissertation, Dept. of Mathematics, University of California, Berkeley, 1982.

....which interprets the propagating front as a physical boundary between two regions. Two such techniques are Level Set techniques, introduced by Osher and Sethian [6] and Fast Marching Methods, introduced by Sethian in [12] Both grew out of the theory of curve and surface evolution developed in [9], 10] 11] which develops the notion of weak solutions and entropy limits for evolving interfaces, and links upwind numerical methodology for hyperbolic conservation laws to front Received November 17, 1997. 1980 Mathematics Subject Classification (1991 Revision) Primary 65M99; Secondary ....

....ext is known as the extension velocity . In many cases, construction of this extension velocity requires considerable effort, see [3] for fast techniques for doing so. 2. 3 Advantages of the Eulerian PDE Perspective The advantages of these perspectives include the following: ffl As discussed in [9], 10] 11] shocks and rarefactions can develop in the slope, corresponding to corners and fans in the evolving interface, and numerical techniques designed for hyperbolic conservation laws can be exploited to construct upwind schemes which produce the correct, physically reasonable entropy ....

Sethian J. A., An Analysis of Flame Propagation, Ph.D. Dissertation, Mathematics, University of California, Berkeley, 1982..

....and or oversmoothing. Level set methods, introduced by Osher and Sethian in [23] o er a highly robust and accurate method for tracking interfaces moving under complex motions. Their major virtue is that they naturally construct the fundamental weak solution to surface propagation posed by Sethian [32, 33]. They work in any number of space dimensions, handle topological merging and breaking naturally, and are easy to program. They approximate the equations of motion for the underlying propagating surface, which resemble Hamilton Jacobi equations with parabolic right hand sides. The central 2 ....

....where it changes topology. The key to constructing numerical schemes which correctly handle these mechanisms is to construct what are known as entropy satisfying approximations to the gradient term in both Eqn. 2 and Eqn. 5 2. 3 Shocks, Entropy Conditions, Curvature, and Viscosity As shown in [32, 33, 35], a propagating interface can develop corners and discontinuities as it evolves, which require the introduction of a weak solution in order to proceed. The correct weak solution 2 or conversely, always negative 5 Swallowtail(F = 1:0) F = 1: 0:25 Entropy Solution(F = 1:0) Figure 2: Cosine ....

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Sethian, J.A., \An Analysis of Flame Propagation", Ph.D. Dissertation, Dept. of Mathematics, University of California, Berkeley, 1982.

....in [13] approximate the solution of an initial value partial di erential equation. At the core, both techniques rely on viscosity solutions for Hamilton Jacobi equations, upwind schemes for hyperbolic conservation laws, and the earlier theory of curve and surface evolution developed by Sethian in [17, 18, 19]. They have been used in a large variety of applications, including problems in uid interface motion, combustion, dendritic solidi cation, etching and deposition in semi conductor manufacturing, robotic navigation and path planning, image segmentation in medical imaging scans, computation of ....

Sethian, J.A., An Analysis of Flame Propagation, Ph.D. Dissertation, Dept. of Mathematics, University of California, Berkeley, CA, 1982.

....see [27, 28] 5.1. Brief review of level set methods. Level set methods were introduced by Osher and Sethian [18] and o er highly robust and accurate methods for tracking interfaces moving under complex motions. They grew out of the theory of curve and surface evolution developed by Sethian in [23, 24, 25], which constructs the notion of weak solutions and entropy limits for evolving interfaces, and links upwind numerical methodology for hyperbolic conservation laws to front propagation problems. The resulting level set approach works in any number of space dimensions, handles topological merging ....

....t = 0) d; 28) 18 J. A. SETHIAN AND ANDREAS WIEGMANN then an initial value partial di erential equation can be obtained for the evolution of , namely t Fjr j = 0 (29) x; t = 0) given (30) This is known as the level set equation, introduced in [18] by Osher and Sethian. As discussed in [23, 24, 25], propagating fronts can develop shocks and rarefactions in the slope, corresponding to corners and fans in the evolving interface, and numerical techniques designed for hyperbolic conservation laws can be exploited to construct upwind schemes which produce the physically correct entropy ....

J. A. Sethian. An Analysis of Flame Propagation. PhD thesis, Dept. of Mathematics, University of California, Berkeley, CA, 11982.

....be thought of as solutions to the Eikonal equation. However, the solution that we want, corresponding to the shortest distance or rst arrival , is the one obtained through the Huygens construction. Another way to obtain this solution is through the notion of an entropy condition. As de ned in [23, 24], we imagine the boundary 5 curve as a source for a propagating ame, and the expanding ame satis es the requirement that once a point in the domain is ignited by the expanding front, it stays burnt. This construction yields the entropy satisfying Huygens construction given in Fig. 2b. Yet ....

....most cases, this length is small enough that, for all practical purposes, heap lookup is very fast. 24 Viscosity Solutions of Hamilton Jacobi Equations [8] H H H H H H H H H H H H H H Hj Corners, Shocks, Singularities and Entropy Conditions in Curve and Surface Evolution [24, 23] Tracking Interface Motion with Schemes for Conservation Laws [25] H H H H H H H H Hj Level Set Perspective t Fjr j = 0 Initial Value Problem [18] adaptivity NARROW BAND LEVEL SET ....

Sethian, J.A., An Analysis of Flame Propagation, Ph.D. Dissertation, Dept. of Mathematics, University of California, Berkeley, CA, 1982.

....near sharp corners and cusps, complexities in three dimensions, lack of robustness issues. Details of these methods are given in [3, 4, 42] Level set methods, developed by Osher and Sethian in [29] based on the theory and numerics of weak solutions to surface propagation formulated by Sethian [35, 36], o er o er a highly robust and accurate method for tracking interfaces moving under complex motions. They work in any number of space dimensions, handle topological merging and breaking naturally, and are easy to program. They approximate the equations of motion for the underlying propagating ....

Sethian, J.A., An Analysis of Flame Propagation, Ph.D. Dissertation, Mathematics, University of California, Berkeley, 1982.

....complex motions. Supported in part by the Applied Mathematics Subprogram of the O ce of Energy Research under contract DE AC03 76SF00098, and the National Science Foundation and DARPA under grant DMS 8919074. They grew out of the theory of curve and surface evolution developed by Sethian in [13, 14, 16], which constructs the notion of weak solutions and entropy limits for evolving interfaces, and links upwind numerical methodology for hyperbolic conservation laws to front propagation problems. The resulting level set approach works in any number of space dimensions, handles topological merging ....

....function , that is, let (x; t = 0) where x 2 R N is de ned by (x; t = 0) d; 1) then an initial value partial di erential equation can be obtained for the evolution of , namely t Fjr j = 0 (2) x; t = 0) given (3) This is known as the level set equation. As discussed in [13, 14, 16], propagating fronts can develop shocks and rarefactions in the slope, corresponding to corners and fans in the evolving interface, and numerical techniques designed for hyperbolic conservation laws can be exploited to construct upwind schemes which produce the correct, physically reasonable ....

Sethian, J.A., An Analysis of Flame Propagation, Ph.D. Dissertation, Mathematics, University of California, Berkeley, 1982.

....and imagine that this curve surface moves in its normal direction with a known speed function F . The goal is to track the motion of this interface as it evolves. We are only concerned with the motion of the interface in its normal direction, and shall ignore tangential motion. As shown in [27, 28, 31], a propagating interface can develop corners and discontinuities as it evolves, which require the introduction of a weak solution in order to proceed. The correct weak solution comes from enforcing an entropy condition for the propagating interface, similar to the one in gas dynamics. ....

.... by enforcing an entropy condition, similar to the one for a scalar hyperbolic conservation law, which selects the envelope obtained by Huygens principle as the Swallowtail(F = 1:0) F = 1: 0:25 Entropy Solution(F = 1:0) Figure 1: Cosine Curve Propagating with Unit Speed correct solution, see [27]. This weak solution corresponds to a decrease in total variation of the propagating front and is irreversible [28] For details, see [28] As a numerical technique, this suggests using the technology from hyperbolic conservation laws to solve the equations of motion, as described in [32] This ....

Sethian, J.A., An Analysis of Flame Propagation, Ph.D. Dissertation, Mathematics, University of California, Berkeley, 1982.

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J. A. Sethian, "An analysis of flame propagation", Ph.D. Dissertation, Mathematics, University of Berkeley, 1982

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) Sethian, J.A.: An analysis of flame propagation. PhD. Dissertation, University of California, Berkeley, California, June 1982; CPAM Rep. 79.

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) Sethian, J.A.: An analysis of flame propagation. PhD. Dissertation, University of California, Berkeley, California, June 1982; CPAM Rep. 79.

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J. A. Sethian, "An analysis of flame propagation", Ph.D. Dissertation, Mathematics, University of Berkeley, 1982

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J. A. Sethian, An Analysis of Flame Propagation. Ph.D. thesis, Dept. of Mathematics, University of California, Berkeley, CA, 1982.

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J. A. Sethian, "An analysis of flame propagation," Ph.D. dissertation, Univ. Calif., Berkeley, 1982.

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J. A. Sethian, An Analysis of Flame Propagation, Ph.D. Dissertation, University of California, Berkeley, June 1982.

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J. A. Sethian, "An analysis of flame propagation", Ph.D. Dissertation, Mathematics, University of Berkeley, 1982

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