ALGORITHMS FOR THE EVOLUTION OF SURFACES

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1 ALGORITHMS FOR THE EVOLUTION OF SURFACES MATTHEW STONE Abstract. We develop algorithms for simulating various curvature dependent motions of interfaces. These motions are described by a class of partial differential equations for which a notion of weak solution the viscosity solution has been developed that is unique and persists beyond topological changes that may occur during the evolution. A comparison principle that holds for these evolutions is the essential tool in the theory. It is also known that to establish the convergence to the viscosity solution of a consistent approximation scheme for such an evolution it is sufficient to show that it respects the comparison principle in which case it is called monotone. A familiar example that comes up in many applications is motion by mean curvature. We try to extend recent consistent and monotone numerical schemes for motion by mean curvature to evolution laws that have a more general dependence on the principal curvatures.. Introduction In [] Esedoglu Ruuth and Tsai ERT developed an algorithm for approximating the motion by certain functions of mean curvature of an interface. They adapt a similar method used by Merriman Bence and Osher MBO in [2] which alternated two steps: Convolution and simple thresholding. The MBO algorithm proceeded as follows: Let Σ R N be a region whose border denoted Σ is to be evolved via motion by mean curvature. For any time step δt > 0 the algorithm gives approximations { Σ n } to motion by mean curvature where Σ n is an approximation of the curve at time t = n δt. These approximations are generated inductively by the following with Σ 0 = Σ: Convolution: Form u : R N R as ux = G t Σn x where G t x is the N-dimensional Gaussian kernel x G t x = 2 e 4t 4πt N/2 2 Thresholding: Compute the next approximation with { Σ n+ = x : ux } 2 This algorithm while computationally inexpensive has its flaws. Most proently unless grid size is refined along with the time step the approximate motion generated by the algorithm will get stuck [][2].

2 2 MATTHEW STONE Thus Esedoglu Ruuth and Tsai replace the thresholding step with another efficient procedure: constructing the signed distance function to the interface and using formulas for approximations of the mean curvature in terms of convolutions with the Gaussian. Due to the Lipschitz continuity of signed distance functions this eliated the inaccuracies found in the MBO algorithm. This paper explores similar methods to the ERT algorithm investigating functions not only of mean curvature but of its components the principal curvatures. Due to certain problems encountered with maintaining the monotonicity of the algorithm our final goal is to approximate motion by fκ + κ 2 where f is any Lipschitz function that is increasing in each coordinate κ > κ 2 are the principal curvatures. The function x + is defined as 0 for x 0 and x otherwise. Similarly x is defined as 0 for x 0 and x otherwise... Things that need to be done. Prove theorems. 2. Expansions for the distance function In this section we first write down the Taylor expansion of the signed distance function d Σ x along a unital direction v in the neighborhood of a point p Σ on the smooth boundary Σ of a set Σ. We work in R 3 where we write x = x y z. This expansion allows us to obtain a Taylor expansion for the convolution of d Σ x y z with the Gaussian kernel G t v x y z which is defined as: G t v x = x v 2 e 4t. 4πt N/2 This will allow us to express the expansion coefficients in terms of the second fundamental form and thus using the convolution for every direction v will allow us to express the principle curvatures of Σ. 2.. Expansion for a smooth interface. First let us recall a few well known properties of the signed distance function. The following will be taken directly from []. For the following d s will denote dσ s d xs will denote 2 d Σ x s and so on. The taylor expansion at s = 0 is given by: 2. dx y z s = dx y z + s v = dx y z + d s x y z 0s + 2 d ssx y z 0s 2 + d xs x y z 0xs + d ys x y z 0ys + d zs x y z 0zs + 6 d sssx y z 0s d ssxx y z 0s 2 x + 2 d ssyx y z 0s 2 y + 2 d sszx y z 0s 2 z + d sxy x y z 0sxy + d sxz x y z 0sxz + d syz x y z 0syz + higher order terms We can substitute the expansion into the convolution integral G t v x y z σdx y z s σdσ R

3 ALGORITHMS FOR THE EVOLUTION OF SURFACES 3 to get a Taylor expansion for the convolution G t v dx y z s at x y z s = The terms we need are: c G t v x y z 0 = c s G t v x y z 0 = 0 s 2 G t v x y z 0 = 2t s 3 G t v x y z 0 = 0 Using these we arrive at the following expansion: Proposition 2.. Convolution of the signed distance function d with the Gaussian kernel G t v has the following expansion 2.2 d G t v x y z 0 = dx y z + 2 d ssx y z 0 2t + 2 d ssxx y z 0 2tx + 2 d ssyx y z 0 2ty + 2 d sszx y z 0 2tz + higher order terms Provided that each of x y z are Ot we get 2.3 d G t v x y z 0 = dx y z + d ss x y z 0t + Ot 2 Thus we find By its definition Finally giving G t v d d = d ss x y z 0t + Ot 2 d ss x y z 0 = D 2 d v v. G t v d d = D 2 d v v t + Ot 2 For any signed distance function d the eigenvalues of D 2 d are given by the principle curvatures with eigenvectors along the principle directions plus an eigenvalue of 0 with corresponding eigenvector of the normal to the surface. Thus v = D 2 d v v that is the greatest eigenvalue is either κ or 0 depending which is larger. That is v = D 2 d v v. Similarly v = D 2 d v v = κ 2. This leads to the conclusions that: 2.4 v = G t v d d = κ + t + Ot v = G t v d d = κ 2 t + Ot 2 Using these two equations for any Lipschitz function f 2.6 f t G t v d d v = t G t v d d = fκ + κ 2 + Ot v = This leads to the following algorithm:

4 4 MATTHEW STONE Algorithm 2.2. Given the initial set Σ 0 through its signed distance function d 0 x and a time step δt > 0 generate the sets Σ j via their signed distance functions d j x at subsequent discrete times t = jδt by alternating the following steps: Form the function 2.7 Ax := d j +δtf Mδt G Mδt v d d v = 2 Construct distance function d j+ by 2.8 d j+ x = RedistA. Mδt G Mδt v d d v = At the jth step of the algorithm the set Σ j can be recovered through the relation Σ j = {x : d j x > 0} So using Equation 2.6 gives the 0-level set of which has moved with the desired speed proving the consistency of the algorithm. Finally we must show the monotonicity of our algorithm assug f is Lipschitz and increasing: Proposition 2.3. If M L f then algorithm 2.7 and 2.8 is monotone for any choice of time step size δt > 0. Proof. Let Σ Σ 2 be two sets satisfying Σ Σ 2 and d x d 2 x be signed distance functions to Σ Σ 2 respectively. Firstly we have x d x d 2 x Using the same notation as in the algorithm let A x = d x+δtf A 2 x = d 2 x+δtf Then we calculate: 2.9 A 2 A = d 2 d + δt f Mδt G Mδt v d d v = Mδt G Mδt v d d v = Mδt G Mδt v d 2 d 2 v = Mδt G Mδt v d 2 d 2 v = [ f Mδt G Mδt v d 2 d 2 v = Mδt G Mδt v d d v = Mδt G Mδt v d 2 d 2 v = Mδt G Mδt v d d v = ]

5 ALGORITHMS FOR THE EVOLUTION OF SURFACES d 2 d v G Mδt v d 2 G Mδt v d G Mδt v d 2 d 2 G Mδt v d d and v = v = G Mδt v d 2 d 2 G Mδt v d d v = v = Substituting 2.0 into 2.9 using the fact that f is increasing in both variables: 2. A 2 A d 2 d + δt f [ f Mδt G Mδt v d 2 d 2 v = Mδt G Mδt v d 2 d v = Mδt G Mδt v d 2 d 2 v = Mδt G Mδt v d 2 d v = ] Using the fact that f is Lipschitz with constant L f and M L f : 2.2 A 2 A d 2 d δt 2Lf d 2 d 0 Mδt Thus we have that our algorithm is both consistent and monotone sufficient conditions to show the convergence of the algorithm to the desired motion as δt 0 References [] S. Esedoglu S. Ruuth and R. Tsai Diffusion generated motion using signed distance functions Journal of Computational Physics [2] B. Merriman J.K. Bence and S. Osher Motion of multiple junctions: a level set approach Journal of Computational Physics Department of Mathematics Yale University New Haven CT address: matthew.i.stone@yale.edu