C A R S T E N S P R U N G E R S H AW N WA L K E R

GEOMETRIC FLOWS WITH APPLICATIONS TO CRYSTAL GROWTH AND GRAPHICS

PROCESSING

ABOUT ME

•  Grew up in Baton Rouge, LA

•  University of Michigan – Ann Arbor •  Majoring in pure mathematics and computer science

•  Ahead: graduate school in math

INTRODUCTION

•  A geometric flow is when a surface evolves in space and time based on its geometry

•  Usually associated with some extrinsic or intrinsic curvature

•  Examples: •  Mean curvature flow •  Anisotropic MCF •  Willmore flow

Source: http://www.birs.ca/workshops/2011/11w5010/files/Dziuk.pdf

FRAMEWORK

•  The finite element method •  Surfaces represented by triangulated grids •  Heat equation PDE discretization •  Investigate different methods of

computing the “weakened” curvature on a triangulated grid

http://people.sc.fsu.edu/~jburkardt/latex/mesh_2014_fsu/mesh_2014_fsu.html

MEAN CURVATURE FLOW DISCRETIZATION

•  We deform the surface Γ in time so that 𝑋↓𝑡 =−𝜅𝜈

•  Multiplying by a test function 𝑉 and integrating by parts ∫Γ↑▒𝑋↓𝑡 ⋅𝑉 +∫Γ↑▒𝛻↓Γ 𝑋:𝛻↓Γ 𝑉 =0,  ∀𝑉

•  Discretize time with semi-implicit Euler method ∫Γ↑▒𝑋↑𝑛+1 ⋅𝑉 +Δ𝑡∫Γ↑▒𝛻↓Γ 𝑋↑𝑛+1 :𝛻↓Γ 𝑉 =∫Γ↑▒𝑋↑𝑛 ⋅𝑉 ,  

∀𝑉

MEAN CURVATURE FLOW WITH VOLUME CONSTRAINT

Frame 1, frame 4, and frame 20, with Δ𝑡= 10↑−3 , axes are spatial

ANISOTROPIC MEAN CURVATURE FLOW WITH VOLUME CONSTRAINT

Frame 1, frame 77, and frame 150, with Δ𝑡= 10↑−4 , axes are spatial

IMAGE FILTERING OF NOISY CUBE

Plots of the mean curvature on the surface of a cube with no noise, noise added, and after 10 iterations of smoothing

𝜔↓𝑖 = 1/𝐴𝑟𝑒𝑎(𝑆𝑡(𝑉↓𝑖 )) ∑𝑇↓𝑗 ∈𝑆𝑡( 𝑉↓𝑖 )↑▒𝐴𝑟𝑒𝑎(𝑇↓𝑗 )𝜈↓𝑗                                ∫Γ↑▒𝜅 𝑉 =∫Γ↑▒(𝛻↓Γ 𝜔)𝑉 

DEMONSTRATION

CONCLUSIONS

•  Shape differential calculus •  The finite element approach •  Anisotropic mean curvature flow requires a time step

restriction •  Future work on graphics

processing

APPLICATIONS

•  Surface tension and minimal surfaces •  Crystal growth:

•  3-D scanning:

http://www.crydev.net/viewtopic.php?f=315&t=106840

REFLECTION

• Glimpse into applied math •  Solidified my interests in pure math and computing

•  I definitely want to be a researcher

•  Thanks: •  Shawn Walker •  CCT REU staff •  NSF