# LEVEL SET METHODS AND FAST MARCHING METHODS PDF

ing interface, while level set methods, introduced by Osher and Sethian in Fast Marching and level set methods is given in a recent cumulativeintro-. Your browser doesn't seem to have a PDF viewer, please download the The fast marching methods and narrow band level set method are. interface, while level set methods, introduced by Osher and Sethian in [20], ap- mary of current work on Fast Marching and level set methods is given in a.

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Level set methods, introduced by Osher and Sethian [56], rely in part on the Both sets of techniques, that is, level set methods and fast marching methods. Level Sets: Smart handling of propagating contours The Fast Marching algorithm. Alive . J.A. Sethian, ”Level Set Methods and Fast Marching Methods”, . ABSTRACT. In recent years, level set methods have been used in a vari- ety of settings for problems in computer vision and image processing. A related.

With this formulation, the final algorithms can be constructed. Overview of Interface Propagation In order to understand interface propagation, the interface itself must first be understood. When given an arbitrary region, such as the one in Figure 1, the interface is defined as the curve, or surface, separating the area inside of the region from the area outside of the region. The motion of the interface must be understood in order to propagate it properly.

Two different types of motion are distinguished: To further define the motion of the interface, a description of the velocity is required. Three different velocity components are formulated: It is more convenient to ignore the separate directions of these velocity components and simply use a scalar-valued function F to Figure 1: Definition of an interface.

Figure 2: Definition of a level set function. Figure 1 shows F. The function F is often referred to as the speed function, and can be formed from the previously mentioned components.

This equation is known as the Eikonal equation. Given a coordinate pair x, y , T will give the time at which the interface reaches x, y. This function, in effect, is what the fast marching method will produce — it describes the time at which the interface will arrive at any given point. This implies that an interface described by the boundary value formulation is strictly expanding or contracting. Since the interface can only move in one direction, the efficient Fast Marching Method can be used.

The initial value formulation does not impose this restriction. That is, using the initial value formulation, an interface may propagate back to points it has already propagated to. It follows that the speed function F is no longer necessarily strictly greater or strictly less than 0 for all time. In order to facilitate this more general definition, a level set function will be required.

The level set function used is two dimensions higher than the surface: With the level set function, the interface at time T can be 'embedded' as the 0 level set, where the other level sets negative and positive are at times less than and greater than T, respectively.

This is illustrated in Fig 2, where a circle propagating outwards is shown as a zero level set embedded into a level set function. While the initial value formulation allows for a more general way to move an interface, the Fast Marching Method can no longer be used.

Instead, the less efficient narrow band level set method can be used.

## Level set methods and fast marching methods

When working with interface propagation, information from the next time step is unknown. In other words, numerical schemes used by algorithms to solve interface propagation problems must use upwind values appropriately. The Entropy Condition There is no guarantee of any level of smoothness in an interface when it moves at a constant speed.

This fact is illustrated in Figure 3, where the singular point leaves overlapping pieces near the center. The earlier formulation of an interface as a curve which separates two regions may have seemed trivial, but now becomes an important distinction geometrically. The possibility of curve overlap is inevitable, and since the interface should separate two regions, it is clear that leaving overlap pieces is not the desired solution to the propagation.

Figure 4 shows that these overlapping pieces will need to be 'clipped'. It follows that there will be no overlap: However, there is a flaw with this solution: No new information should be added during the propagation. Thus, this solution will not produce the singularities that are simply a property of the original interface after undergoing expansion. This condition imposed on the creation of new information is known as the entropy condition.

## A fast marching level set method for monotonically advancing fronts.

As with the traditional definition of entropy, in this case it describes a constraint on the addition of randomness; no new information can be added during interface propagation. The most straightforward way of propagating an interface such that it satisfies the entropy condition requires the use of Huygens' principle see [1] for more detail on this. Waves emanate from the interface and no new information is introduced. Propagating these waves will not produce the overlap while propagating Figure 3: A smooth curve can quickly develop a singular point.

Figure 4: A clip is required to remove the dashed overlapping pieces at the bottom. Figure 5: When pieces come apart during propagation, forming a gap, a rarefaction fan must be built. Sethian showed that this scheme is equivalent to using only grid points which are reached as first arrivals when expanding an interface. In short, if a grid point only allows the interface to traverse it once, then the problem of having overlap is no longer a possibility — the overlap would have to be placed on grid points that have already been traversed.

Furthermore, no smoothing is involved and the entropy condition is satisfied. Conversely to the overlap issue, a gap can form when an interface expands, as shown in Figure 5. This problem can be resolved in the same way — the first arrival solution will give a rarefaction fan that joins the two pieces of the fan.

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Viscosity Solutions If a smoothing term was added to the speed function F, the expanded interface clearly would violate the entropy condition. It would smooth out the singular points which would have been produced from the original information. It follows that if the solution to a propagating interface is formulated using a smoothing term according to curvature of the interface, the entropy condition is violated.

This limit is known as the viscous limit, and this entropy satisfying solution is known as the viscosity solution. As an aside, it is known as the viscosity solution because of the diffusive fluid viscosity term used in hyperbolic conservation laws.

Hamilton-Jacobi Equations Recall both the initial value formulation and the boundary value formulation: Furthermore, if the Hamilton-Jacobi equation was written with a smoothing term, the smoothed version of the interface propagation that was mentioned earlier is achieved: In this way, the Hamilton-Jacobi equation can be used to rewrite the original formulations. Indeed, this equation is more desirable for purposes of numerical approximation.

It follows that there will be no overlap: the interface simply smooths the singularity out into the curve. However, there is a flaw with this solution: the desired solution to the expansion not only must have no overlap, it must be derived only from the original interface information.

No new information should be added during the propagation. Thus, this solution will not produce the singularities that are simply a property of the original interface after undergoing expansion. This condition imposed on the creation of new information is known as the entropy condition.

As with the traditional definition of entropy, in this case it describes a constraint on the addition of randomness; no new information can be added during interface propagation. The most straightforward way of propagating an interface such that it satisfies the entropy condition requires the use of Huygens' principle see [1] for more detail on this. Waves emanate from the interface and no new information is introduced. Propagating these waves will not produce the overlap while propagating Figure 3: A smooth curve can quickly develop a singular point.

Figure 4: A clip is required to remove the dashed overlapping pieces at the bottom. Figure 5: When pieces come apart during propagation, forming a gap, a rarefaction fan must be built.

Sethian showed that this scheme is equivalent to using only grid points which are reached as first arrivals when expanding an interface. In short, if a grid point only allows the interface to traverse it once, then the problem of having overlap is no longer a possibility — the overlap would have to be placed on grid points that have already been traversed.

Furthermore, no smoothing is involved and the entropy condition is satisfied. Conversely to the overlap issue, a gap can form when an interface expands, as shown in Figure 5. This problem can be resolved in the same way — the first arrival solution will give a rarefaction fan that joins the two pieces of the fan. Viscosity Solutions If a smoothing term was added to the speed function F, the expanded interface clearly would violate the entropy condition.

It would smooth out the singular points which would have been produced from the original information. It follows that if the solution to a propagating interface is formulated using a smoothing term according to curvature of the interface, the entropy condition is violated.

This limit is known as the viscous limit, and this entropy satisfying solution is known as the viscosity solution. As an aside, it is known as the viscosity solution because of the diffusive fluid viscosity term used in hyperbolic conservation laws.

In this way, the Hamilton-Jacobi equation can be used to rewrite the original formulations. Indeed, this equation is more desirable for purposes of numerical approximation. Using the Hamilton-Jacobi equations, an entropy-satisfying scheme which properly uses upwind values can be formulated. If a numerical approximation to a flux function produces the entropy satisfying solution, a scheme using that flux function will parallel solving the interface propagation problems initially formulated.

There are many numerical flux functions that will give a flux function approximation, and this is one such function.

## A fast marching level set method for monotonically advancing fronts.

It was originally developed by Engquist and Osher[3]. In order to see why this flux function, let alone any other flux function is useful, some more work must be done. Equation B So, if a scheme is in conservation form, the solution to the wave equation for the next time can be approximated easily. An additional property is required — the scheme to be used for propagating interfaces must satisfy the entropy condition.

In order to ensure this, the scheme must be monotonic.

The flux function gEO comes from a scheme which is in conservation form, and monotonic. This is why it is particularly desirable. With the formulation of the flux function gEO, a numerical approximation for time values in the interface propagation problem can now be given.

Figure 6: The Flux function can be used to compute the solution to a wave equation at the next time step. A General Algorithm for Convex Speed Functions Recall that the initial value formulation and boundary value formulation can be written as a general Hamilton-Jacobi equation.

This equation is the same hyperbolic conservation law that was seen earlier, and it can be approximated by using a numerical flux function. Using Equation A with the scheme that has been formulated, a basic level set method can be created. Higher order schemes can be built in order to increase accuracy, but they will not be discussed here.

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The narrow band level set method can add some flexibility to this constraint, but the principle is the same: the amount of error in the level set method approximation will become very large if care is not taken with respect to time and space steps. Approximating Curvature If a given interface needs to be propagated according to curvature, the simpler algorithm given before will not suffice. Instead, a new formulation comes next that requires the calculation of curvature of the zero level set is needed.

A General Level Set Method The general level set method described here combines three speed functions: one for the expansion speed desired, one for speed accumulated from curvature and one which moves the interface according to underlying properties of the surface on which the interface lies. It is apparent already that there is no time step involved here — the fast marching method does not need one.

However, without a time step a different method of propagation is required. Where the level set method propagates according to time and space step, the fast marching method propagates according to the smallest known time value, some T x, y.

During each iteration, the grid point with the earliest time in the min-heap is put into the set of known grid points, and a fringe of trial values adjacent to that grid point receive updated time values and are added to the min-heap. Propagating this way is advantage to algorithm efficiency and will be outlined later in the run time analysis.

There are two ways of thinking of this. This quadratic was formulated from a flux function which is intended to both upwind properly and satisfy the entropy condition. To upwind properly over a discrete grid, for a given point P, intuitively it seems that only known values adjacent to P should be used. Journal of Scientific Computing , Crossref Non-Euclidean, convolutional learning on cortical brain surfaces. Journal of Geophysical Research: Solid Earth , The Visual Computer , Crossref A new multi-material level set topology optimization method with the length scale control capability.

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After solving the quadratic formulated for Figure 7a. The schemes are of use in a variety of applications, including problems in shape offsetting, computing distances from complex curves and surfaces, shape-from-shading, photolithographic development, computing first arrivals in seismic travel times, construction of shortest geodesics on surfaces, optimal path planning around obstacles, and visibility and reflection calculations.