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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