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Refining Cellular Automatic Introduction Updating Rules to Model - - PowerPoint PPT Presentation

Nov 15, 2022 10 likes •241 views

Mean Curvature Flow W. Byrd W. Gerych K.Sternberg J. LeCrone Refining Cellular Automatic Introduction Updating Rules to Model Mean Normal Vectors Curvature Flow Amount of Life to Remove Current William B. Byrd, Walter L. Gerych,

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SLIDE 1

Mean Curvature Flow

  • W. Byrd
  • W. Gerych

K.Sternberg

  • J. LeCrone

Introduction Normal Vectors Amount of Life to Remove Current Evolution Real World Time Future Research Questions

Refining Cellular Automatic Updating Rules to Model Mean Curvature Flow

William B. Byrd, Walter L. Gerych, Kateri A. Sternberg Mentor: Jeremy S. LeCrone

Kansas State University

July 21, 2015

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SLIDE 2

Mean Curvature Flow

  • W. Byrd
  • W. Gerych

K.Sternberg

  • J. LeCrone

Introduction Normal Vectors Amount of Life to Remove Current Evolution Real World Time Future Research Questions

Introduction

  • Mean curvature flow is a type of surface flow where the

velocity of the surface along it’s normal direction is proportional to the mean curvature of the surface.

  • We are trying to determine efficient rules to make a cellular

automatic system mimic the behavior of a surface that undergoes mean curvature flow.

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SLIDE 3

Mean Curvature Flow

  • W. Byrd
  • W. Gerych

K.Sternberg

  • J. LeCrone

Introduction Normal Vectors Amount of Life to Remove Current Evolution Real World Time Future Research Questions

Introduction

A Cellular Automata is a collection of grid cells each with a finite number of states that evolves with each generation. The state of a cell in a certain generation is determined by the states of the neighboring cells in the previous generation and a set of evolution rules.

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SLIDE 4

Mean Curvature Flow

  • W. Byrd
  • W. Gerych

K.Sternberg

  • J. LeCrone

Introduction Normal Vectors Amount of Life to Remove Current Evolution Real World Time Future Research Questions

Previous Research

  • Proved that area calculations converge to the curvature.
  • Approximated curvature at a cell with area calculations.
  • Implemented burning algorithm to more accurately calculate

curvature

  • Added/Removed cells based on curvature calculations.
  • Allowed for obstacles to affect surface flow.
  • Cells that had fractional quantities of life
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SLIDE 5

Mean Curvature Flow

  • W. Byrd
  • W. Gerych

K.Sternberg

  • J. LeCrone

Introduction Normal Vectors Amount of Life to Remove Current Evolution Real World Time Future Research Questions

What We’ve Done

  • Determined an effective volumetric method of calculating

normal vectors to the surface.

  • Altered code to evolve the surface in the direction of the

normal vector.

  • Determined that separating cells by neighbor counts does

not improve accuracy

  • Uncovered data that could be implemented to improve the

accuracy of the evolution

  • Discussed theoretical relationship between real world time

and the evolution.

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SLIDE 6

Mean Curvature Flow

  • W. Byrd
  • W. Gerych

K.Sternberg

  • J. LeCrone

Introduction Normal Vectors Amount of Life to Remove Current Evolution Real World Time Future Research Questions

Theoretical Convergence

  • N =
  • R(

p − x y

  • )dA

2 3δ3

Theorem

limδ→0+ N is the outward unit normal vector at the point p.

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SLIDE 7

Mean Curvature Flow

  • W. Byrd
  • W. Gerych

K.Sternberg

  • J. LeCrone

Introduction Normal Vectors Amount of Life to Remove Current Evolution Real World Time Future Research Questions

Theoretical Convergence

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SLIDE 8

Mean Curvature Flow

  • W. Byrd
  • W. Gerych

K.Sternberg

  • J. LeCrone

Introduction Normal Vectors Amount of Life to Remove Current Evolution Real World Time Future Research Questions

Proof of Convergence

The Linear Convergence δ

−δ − √ δ2−x2 −

x y

  • dydx

2 3δ3

= − δ

−δ

  • x

√ δ2 − x2 − 1

2(δ2 − x2)

  • dx

2 3δ3

= −

  • − 1

2(δ2 − δ2)3/2 + 1 2(δ2 − δ2)3/2

(− δ3

2 + δ3 6 ) − ( δ3 2 − δ3 6 )

  • 2

3δ3

=

2 3δ3

  • 2

3δ3

= 1

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SLIDE 9

Mean Curvature Flow

  • W. Byrd
  • W. Gerych

K.Sternberg

  • J. LeCrone

Introduction Normal Vectors Amount of Life to Remove Current Evolution Real World Time Future Research Questions

The Convergence of Error

x-component Ex ≤ δ

−δ

|f (x)| |x|dydx

2 3δ3

= δ

−δ |xf (x)|dx 2 3δ3

Ex ≤ δ

−δ M|x3|dx 2 3δ3

= 2 δ

0 Mx3dx 2 3δ3

= 2M 1

4δ4 2 3δ3

= 3M 4 δ y-component Ey ≤ δ

−δ

|f (x)| |y|dydx

2 3δ3

= δ

−δ

|f (x)| ydydx

2 3δ3

= δ

−δ 1 2|f (x)|2dx 2 3δ3

Ey ≤ δ

−δ 1 2M2|x4|dx 2 3δ3

= δ

0 M2x4dx 2 3δ3

=

1 5M2δ5 2 3δ3

= 3M2 10 δ2

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SLIDE 10

Mean Curvature Flow

  • W. Byrd
  • W. Gerych

K.Sternberg

  • J. LeCrone

Introduction Normal Vectors Amount of Life to Remove Current Evolution Real World Time Future Research Questions

Error in Calculated Normal Vector

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SLIDE 11

Mean Curvature Flow

  • W. Byrd
  • W. Gerych

K.Sternberg

  • J. LeCrone

Introduction Normal Vectors Amount of Life to Remove Current Evolution Real World Time Future Research Questions

Determining the Life Lost

  • r(t) =

√ r 02 − 2t

  • Analyzed flow of discrete circle of radius 5 from

time t = 0 to time t = 12, in .1 increments.

  • Calculated difference in life between

generations

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SLIDE 12

Mean Curvature Flow

  • W. Byrd
  • W. Gerych

K.Sternberg

  • J. LeCrone

Introduction Normal Vectors Amount of Life to Remove Current Evolution Real World Time Future Research Questions

Separating by Neighbor Count

5 Neighbors 7 Neighbors

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SLIDE 13

Mean Curvature Flow

  • W. Byrd
  • W. Gerych

K.Sternberg

  • J. LeCrone

Introduction Normal Vectors Amount of Life to Remove Current Evolution Real World Time Future Research Questions

Different Neighbor Regressions

4 Neighbors 7 Neighbors

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SLIDE 14

Mean Curvature Flow

  • W. Byrd
  • W. Gerych

K.Sternberg

  • J. LeCrone

Introduction Normal Vectors Amount of Life to Remove Current Evolution Real World Time Future Research Questions

Evolution Comparison

Evolution of Circles Neighbor Count Evolution Discrete Circle

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SLIDE 15

Mean Curvature Flow

  • W. Byrd
  • W. Gerych

K.Sternberg

  • J. LeCrone

Introduction Normal Vectors Amount of Life to Remove Current Evolution Real World Time Future Research Questions

Finding a New Regression

Data For Regular Circles Original No Zeros Final

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SLIDE 16

Mean Curvature Flow

  • W. Byrd
  • W. Gerych

K.Sternberg

  • J. LeCrone

Introduction Normal Vectors Amount of Life to Remove Current Evolution Real World Time Future Research Questions

Inverted Circle

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SLIDE 17

Mean Curvature Flow

  • W. Byrd
  • W. Gerych

K.Sternberg

  • J. LeCrone

Introduction Normal Vectors Amount of Life to Remove Current Evolution Real World Time Future Research Questions

Fixing Inverted Circle Data

Data For Inverted Circles With Zeros No Zeros Final

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SLIDE 18

Mean Curvature Flow

  • W. Byrd
  • W. Gerych

K.Sternberg

  • J. LeCrone

Introduction Normal Vectors Amount of Life to Remove Current Evolution Real World Time Future Research Questions

Regression

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SLIDE 19

Mean Curvature Flow

  • W. Byrd
  • W. Gerych

K.Sternberg

  • J. LeCrone

Introduction Normal Vectors Amount of Life to Remove Current Evolution Real World Time Future Research Questions

How to Proceed

  • Isolate the areas where life should be growing and shrink the

dead region in those areas.

  • Could be done using the information gathered from the

shrinking circle.

  • Would accomplish one of our goals of ensuring that a

curve’s evolution is unaffected by whether or not the life is

  • n the inside or outside.
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SLIDE 20

Mean Curvature Flow

  • W. Byrd
  • W. Gerych

K.Sternberg

  • J. LeCrone

Introduction Normal Vectors Amount of Life to Remove Current Evolution Real World Time Future Research Questions

Current Evolution

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SLIDE 21

Mean Curvature Flow

  • W. Byrd
  • W. Gerych

K.Sternberg

  • J. LeCrone

Introduction Normal Vectors Amount of Life to Remove Current Evolution Real World Time Future Research Questions

Time vs Generations

  • A′ =
  • Γ(t)

k · n ds = −2π

  • Rate of change of area is constant
  • Allows us to calculate the time elapsed between

generations by calculating the loss of area

  • Different time steps between different

generations

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SLIDE 22

Mean Curvature Flow

  • W. Byrd
  • W. Gerych

K.Sternberg

  • J. LeCrone

Introduction Normal Vectors Amount of Life to Remove Current Evolution Real World Time Future Research Questions

Possible Future Research

  • Integrate real world time calculations into

evolution

  • Apply our ideas for new evolution rules and

compare them to discrete circles for accuracy

  • Redefine the interface to possibly increase

accuracy

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SLIDE 23

Mean Curvature Flow

  • W. Byrd
  • W. Gerych

K.Sternberg

  • J. LeCrone

Introduction Normal Vectors Amount of Life to Remove Current Evolution Real World Time Future Research Questions

Acknowledgements

This research was carried out as a part of the Summer Undergraduate Mathematics Research REU at Kansas State University funded by NSF award #1262877. The authors would like to thank Kansas State University for their hospitality.

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