Laplacian Operator and Smoothing Xifeng Gao Acknowledgements for

Laplacian Operator and Smoothing Xifeng Gao Acknowledgements for the slides: Olga Sorkine-Hornung, Mario Botsch, and Daniele Panozzo Florida State University

Applications in Geometry Processing • Smoothing • Parameterization • Remeshing Florida State University

Laplacian Operator Laplacian operator function in Euclidean space gradient operator divergence operator Florida State University

Laplacian Operator Laplacian operator function in Euclidean space gradient operator 2 nd partial derivatives divergence operator Florida State University

Laplacian Operator Laplacian operator function in Euclidean space gradient operator divergence operator 2 nd partial derivatives Cartesian coordinates Florida State University

Laplacian Operator Laplacian operator gradient operator 2 nd partial derivatives Intuitive Explanation The Laplacian Δf(p) of a function f at a point p, is the rate at which the average value of f over spheres centered at p deviates from f(p) as the radius of the sphere grows. function in Euclidean space divergence operator Cartesian coordinates Florida State University

Laplace-Beltrami Operator • Extension of Laplace to functions on manifolds Laplace. Beltrami function on surface M gradient operator divergence operator Florida State University

Laplace-Beltrami Operator • For coordinate functions: Laplace. Beltrami P at function f gradient operator divergence operator mean curvature unit surface normal Florida State University

Discrete Curvatures • Mean curvature (sign defined according to normal) Florida State University

Surfaces, Parametric Form • Continuous surface n pu pv p(u, v) • Tangent plane v u Florida State University

Surfaces, Normal • Surface normal: n pu pv p(u, v) v u Florida State University

Surfaces, Curvature n pv pu t p Unit-length direction t in the tangent plane (if pu and pv are orthogonal): t j Tangent plane Florida State University

Surfaces, Curvature n pv pu t g p The curve g is the intersection of the surface with the plane through n and t. Normal curvature: kn(j)n = k(g(p))n = g” )p( t j Tangent plane Florida State University

Surfaces, Curvature • Mean curvature Florida State University

Discrete Laplace-Beltrami • Intuition for uniform discretization Florida State University

Discrete Laplace-Beltrami • Intuition for uniform discretization vi-1 vi vi+1 g Florida State University

Discrete Laplace-Beltrami vj 1 vj 2 vi vj 6 vj 3 vj 5 vj 4 Florida State University

Discrete Laplace-Beltrami • Uniform discretization: • Depends only on connectivity = simple and efficient • Bad approximation for irregular triangulations Florida State University

Discrete Laplace-Beltrami • Uniform discretization: • Cotangent weight: vi vj Wi vi bij aij vj Florida State University

Surface Smoothing – Motivation Florida State University

Curvature and Smoothness Florida State University

Curvature and Smoothness mean curvature plot Florida State University

Curvature and Smoothness mean curvature plot Florida State University

Curvature and Smoothness mean curvature plot Florida State University

Curvature and Smoothness • • Smoothing = reducing curvature? Smoothing = make curvature vary less? Florida State University

Example – smoothing curves • Laplace in 1 D = second derivative: Florida State University

Example – smoothing curves • Laplace in 1 D = second derivative: • In matrix-vector form for the whole curve Florida State University

Example – smoothing curves • Laplace in 1 D = second derivative: • In matrix-vector form for the whole curve Florida State University

Example – smoothing curves • Flow to reduce curvature: • • Scale factor 0 < l < 1 Matrix-vector form: • Drawbacks? Florida State University

Example – smoothing curves • Flow to reduce curvature: • • Scale factor 0 < l < 1 Matrix-vector form: • May shrink the shape; can be slow Florida State University

Filtering Curves Original curve Florida State University

Filtering Curves 1 st iteration; l=0. 5 Florida State University

Filtering Curves 2 nd iteration; l=0. 5 Florida State University

Filtering Curves 8 th iteration; l=0. 5 Florida State University

Filtering Curves 27 th iteration; l=0. 5 Florida State University

Filtering Curves 50 th iteration; l=0. 5 Florida State University

Filtering Curves 500 th iteration; l=0. 5 Florida State University

Filtering Curves 1000 th iteration; l=0. 5 Florida State University

Filtering Curves 5000 th iteration; l=0. 5 Florida State University

Filtering Curves 10000 th iteration; l=0. 5 Florida State University

Filtering Curves 50000 th iteration; l=0. 5 Florida State University

Filtering Surfaces (Demo) Florida State University

Thank you! https: //gaoxifeng. github. io Florida State University