Level Sets -numerical

Level Sets -numerical - 11/15/2010 Level Sets Numerical...

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Unformatted text preview: 11/15/2010 Level Sets Numerical Considerations Lagrangian Formulation r c • Consider the curve given by ø(x,y)=0. r V ( x, y ) • Suppose that the velocity is known for every point on the curve. • To move the points on the curve with this velocity, we needr to solve dc r = V ( x, y ) dt • For every point (x,y) on the curve. Math Modeling II, Fall 2010 (Dr. Xin Li) 1 11/15/2010 Eulerian Formulation • Now, for each t, there is an equation for the r c (t ) φ (t , x, y ) = 0 curve : . • Differentiate with respect to t gives: r • (1) φt + V ⋅ ∇φ = 0 r r V = (u, v) • Recall that if , then V ⋅ ∇φ = uφ x + vφ y • We refer to equation (1) as the level set equation • We need to solve (1) in the following sense: – We will use the initial curve for t=0 – Increase the time step by Δt φ (t + Δt , x, y ) – Solve for the value of on a grid in the xy‐plane – Even though V may be given only for points on the curve, we will assume that it is also given (or could be extended) as a vector field for the whole xy‐ plane (or for a region). Math Modeling II, Fall 2010 (Dr. Xin Li) 2 11/15/2010 Upwind Differencing • We need to evolve ø forward in time and move the curve (interface). • Assume a grid in the xy‐plane is given ‐ in digital image case, we have a very natural grid of the pixels • At time step tn, let øn= ø(tn) • Update ø in time = finding values of ø at every grid point after some time increment Spatial Discretization • First order accurate method for the time discretization is the forward Euler method given by n +1 n φ • • • • −φ + u nφ xn = 0 Δt How about the spatial derivatives of ø? Great care must be exercised! We will consider one‐dimensional spatial case Can be applied to higher in a dimension‐by‐ dimension manner Math Modeling II, Fall 2010 (Dr. Xin Li) 3 11/15/2010 Start Upwinding • At a specific spatial point x i φi n +1 − φi n Δt + ui (φ x ) in = 0 n ui > 0 • If , the values of ø are moving from left to xi right, so we need to look to the left of to determine what value of ø will land on the point at the end of a time step. Upwinding φx uin > 0 • So, D‐ø should be used to approximate when and D+ø should be used otherwise • Since D‐ and D+ are first order accurate, the combination of the forward time and upwind differencing is a consistent finite difference approximation to the partial differential equation (1) • According to Lax‐Richtmyer equivalence theorem, a finite difference approximation to a linear differential equation is convergent if it is consistent and stable. Math Modeling II, Fall 2010 (Dr. Xin Li) 4 11/15/2010 Stability • The stability condition is Δt max{ |u| |v| + Δx Δy }= α < 1 • Instead of upwinding, the spatial derivatives could be approximated with more accurate methods. • But we stay with upwind method • The first order upwind can be improved by using more accurate approximation Homework • Write done the steps needed for the implementation of the level sets method for solving equation (1) (do not need to implement in Matlab yet) Step 0: Ste up the grid in xy‐plane with uniform step‐length h; select time step k Step 1: Initialization of the level set function ø(0,x,y) at every grid point (x,y). If only a curve is given, we use signed distance function to initialize ø(0,x,y)=+or‐distance(<x,y>,C) Step 2: … Math Modeling II, Fall 2010 (Dr. Xin Li) 5 ...
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