Modeling Mean Curvature Flow using Cellular Automata Dr. Jeremy - - PowerPoint PPT Presentation

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Modeling Mean Curvature Flow using Cellular Automata Dr. Jeremy Scott LeCrone, Barbara Joy Smith, and Samantha Carol Zerger Kansas State University 22 July 2014 Dr. LeCrone, B. Smith, and S. Zerger Discrete Mean Curvature Flow 22 July 2014 1

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SLIDE 1

Modeling Mean Curvature Flow using Cellular Automata

  • Dr. Jeremy Scott LeCrone, Barbara Joy Smith, and Samantha Carol

Zerger

Kansas State University

22 July 2014

  • Dr. LeCrone, B. Smith, and S. Zerger

Discrete Mean Curvature Flow 22 July 2014 1 / 19

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SLIDE 2

Mean Curvature

H( x0) = ±

  • dT

ds

  • Given a curve Γ with a parametrization γ : R → R2, the curvature of Γ at

a point x0 = γ(t0) is given by the formula H( x0) = ±||γ′(t0) × γ′′(t0)|| ||γ′(t0)||3 , where the sign is determined if one fixes an orientation on the curve. Meanwhile, if f is known, the curvature takes the form H( x0) = −f ′′(x) (1 + f ′(x)2)

3 2

.

  • Dr. LeCrone, B. Smith, and S. Zerger

Discrete Mean Curvature Flow 22 July 2014 2 / 19

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SLIDE 3

Mean Curvature Flow

VΓ(t) = −HΓ(t) Evolves based on Geometric Evolution Equations Surface area is always going to be decreasing

  • Dr. LeCrone, B. Smith, and S. Zerger

Discrete Mean Curvature Flow 22 July 2014 3 / 19

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SLIDE 4

Cellular Automata

Finite dimensional grid of cells – such as an integer lattice. Each cell stores a finite number of states. Set of rules governing the evolution of these states. Rules are based on local conditions of neighboring cells. Goes through several generations.

  • Dr. LeCrone, B. Smith, and S. Zerger

Discrete Mean Curvature Flow 22 July 2014 4 / 19

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SLIDE 5

Theoretical Justification for Local Geometric Approximations

Theorem

The curvature at a point on a closed curve Γ ∈ C 2 can be calculated using rotations and translations so that the mean curvature at a point x0 ∈ Γ follows from lim

δ→0+

Af ,δ

δ3 3

= H( x0). Γ γ(t) γ′(t) n f ′(0) = 0 f (0) = 0 n

  • Dr. LeCrone, B. Smith, and S. Zerger

Discrete Mean Curvature Flow 22 July 2014 5 / 19

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SLIDE 6

Definitions

Bδ(0) ∩ Af δ Γ = graph(f ) Af ,δ Bδ(0) ∩ Aq δ q(x) Aq,δ Af := {(x, y) | y > f (x)} and Aq := {(x, y) | y > q(x)}. Af ,δ := |Bδ(0) ∩ Af | − 1 2|Bδ(0)| and Aq,δ := |Bδ(0) ∩ Aq| − 1 2|Bδ(0)|.

  • Dr. LeCrone, B. Smith, and S. Zerger

Discrete Mean Curvature Flow 22 July 2014 6 / 19

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SLIDE 7

Claim 1

Claim 1: lim

δ→0+

Aq,δ

δ3 3

= HΓ( x0).

  • Dr. LeCrone, B. Smith, and S. Zerger

Discrete Mean Curvature Flow 22 July 2014 7 / 19

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SLIDE 8

Claim 1

Claim 1: lim

δ→0+

Aq,δ

δ3 3

= HΓ( x0).

  • Aq,δ

δ3 3

= √ 2

3

f ′′(x)2

  • f ′′(x)2
  • f ′′(x)2δ2 + 1 + 1

3 + 3π 2δ + 3 √ 2f ′′(x) (

  • f ′′(x)2δ2 + 1 + 1)

3 2

− 3 δ sin−1

  • 2
  • f ′′(x)2δ2 + 1 + 1
  • lim

δ→0

Aq,δ

δ3 3

= −f ′′(0) = HΓ( x0)

  • Dr. LeCrone, B. Smith, and S. Zerger

Discrete Mean Curvature Flow 22 July 2014 7 / 19

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SLIDE 9

Claim 2

Claim 2: lim

δ→0+

Af ,δ − Aq,δ δ3 = 0.

  • Dr. LeCrone, B. Smith, and S. Zerger

Discrete Mean Curvature Flow 22 July 2014 8 / 19

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SLIDE 10

Claim 2

Claim 2: lim

δ→0+

Af ,δ − Aq,δ δ3 = 0.

  • |Af ,δ − Aq,δ|

δ3 ≤ δ

−δ

|f (x) − q(x)| dx δ3 = δ

−δ

|g(x)| dx δ3 ≤ δ

−δ

M|x3| dx δ3 = M δ3 δ

−δ

|x3| dx = Mδ 2

  • Dr. LeCrone, B. Smith, and S. Zerger

Discrete Mean Curvature Flow 22 July 2014 8 / 19

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SLIDE 11

Proof of Claim 2

Since f is twice differentiable, we can write f (x) = f (0) + f ′(0)x + 1 2f ′′(0)x2

  • q(x)

+ g(x), where g(x) = O(x3) as x → 0. That is, there exists some positive constant M and ǫ > 0 such that |g(x)| ≤ M|x3| for |x| < ǫ. Therefore, |f (x) − q(x)| x3 = |g(x)| x3 ≤ M for |x| < ǫ.

  • Dr. LeCrone, B. Smith, and S. Zerger

Discrete Mean Curvature Flow 22 July 2014 9 / 19

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SLIDE 12

Discrete Setting using Cellular Automata

  • Dr. LeCrone, B. Smith, and S. Zerger

Discrete Mean Curvature Flow 22 July 2014 10 / 19

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SLIDE 13

Defining the Interface

Neighbors The 8 adjacent cells surrounding a specific cell If at most 7 neighbors are alive around a particular cell it is declared an interface point.

  • Dr. LeCrone, B. Smith, and S. Zerger

Discrete Mean Curvature Flow 22 July 2014 11 / 19

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SLIDE 14

Template Circle Size

Box Radius Box Count π*radius2 Percent Error 3.5 37 38.4845

  • 3.8574

6 113 113.0973

  • .0861

7 149 153.9380

  • 3.2100

7.5 177 176.7146 .1615 9 253 254.4690

  • .5773

10.5 349 346.3606

  • 0.7620

radius = 6α radius = 7.5α radius = 10.5α

  • Dr. LeCrone, B. Smith, and S. Zerger

Discrete Mean Curvature Flow 22 July 2014 12 / 19

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SLIDE 15

Burning Algorithm

This important algorithm provides the most information about our curve because it relates to the curvature at a point. Extra live cells that we don’t want to consider Cells that are not alive that need to be considered

  • Dr. LeCrone, B. Smith, and S. Zerger

Discrete Mean Curvature Flow 22 July 2014 13 / 19

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SLIDE 16

Add/Remove Algorithm

Thought at first to make a “mean curvature function.” Decided to directly use ranges of burncounts and apply the add/remove algorithm (determined with previous knowledge) Modified to use two template circles +1

  • 3
  • 1
  • Dr. LeCrone, B. Smith, and S. Zerger

Discrete Mean Curvature Flow 22 July 2014 14 / 19

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SLIDE 17

Outline of Entire Algorithm

1 Initialize some surface and obstacles. 2 Identify interface points. 3 Place a template circle around an interface point. 4 Run through the burning algorithm. 5 Use this count to determine how many cells should be added or

removed.

6 Repeat 3-6 for every interface point. 7 Designate which cells are to be added or removed. 8 Change the cell values so the “to be removed” cells are now dead and

those “to be added” are now alive.

9 Repeat steps 2-8 (one generation) as long as desired.

  • Dr. LeCrone, B. Smith, and S. Zerger

Discrete Mean Curvature Flow 22 July 2014 15 / 19

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SLIDE 18

Batman!

Because everyone wants to see how Batman evolves... Initial Object After 1 step After 2 steps After 10 steps After 20 steps After 45 steps

  • Dr. LeCrone, B. Smith, and S. Zerger

Discrete Mean Curvature Flow 22 July 2014 16 / 19

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SLIDE 19

Embedding Obstacles

An obstacle matrix is overlayed on the larger cell matrix. Fix the obstacle on the lattice. The obstacle remains fixed, “immortal” in a way. Our algorithm removes these cells from the list of potential cells when determining which ones to add or remove. The evolution of the closed curve with one obstacle is expected to evolve toward and then wrap around the obstacle.

  • Dr. LeCrone, B. Smith, and S. Zerger

Discrete Mean Curvature Flow 22 July 2014 17 / 19

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SLIDE 20

Embedded Obstacles

Two obstacles After 30 steps After 50 steps After 80 steps After 100 steps After 122 steps

  • Dr. LeCrone, B. Smith, and S. Zerger

Discrete Mean Curvature Flow 22 July 2014 18 / 19

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SLIDE 21

Potential Research

Other methods to determine the curvature discretely looking at only the interface points. Determining the normal direction and velocity at an interface point.

Prioritizing which cells to add or remove based on the normal direction. Relationship between interface velocity and concentration of add/remove values.

Analysis in higher dimensions.

  • Dr. LeCrone, B. Smith, and S. Zerger

Discrete Mean Curvature Flow 22 July 2014 19 / 19