Florida State University
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
Laplacianoperator
gradientoperator
divergenceoperator
function inEuclidean space
Florida State University
Laplacianoperator
gradientoperator
2nd partialderivatives
divergenceoperator
function inEuclidean space
Laplacian Operator
Florida State University
Laplacianoperator
gradientoperator
2nd partialderivatives
Cartesiancoordinatesdivergence
operator
function inEuclidean space
Laplacian Operator
Florida State University
Laplacianoperator
gradientoperator
2nd partialderivatives
Cartesiancoordinatesdivergence
operator
function inEuclidean space
Intuitive Explanation
The Laplacian Δf(p) of a function f
at a point p, is the rate at whichthe average value of f overspheres centered at p deviates
from f(p) as the radius of thesphere grows.
Laplacian Operator
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Laplace-Beltrami Operator
function onsurface M
• Extension of Laplace to functions on manifolds
Laplace-Beltrami
gradientoperator
divergenceoperator
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Laplace-Beltrami Operator
• For coordinate functions:
mean curvature
unitsurface
normal
Laplace-Beltrami
gradientoperator
divergenceoperator
P at function f
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Discrete Curvatures
• Mean curvature (sign defined according to normal)
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Surfaces, Parametric Form
• Continuous surface
• Tangent plane
n
p(u,v)
pu
u
v
pv
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• Surface normal: n
p(u,v)
pu
u
v
pv
Surfaces, Normal
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n
p
pu pv
t
Unit-length direction t in the tangent plane (if pu and pv are orthogonal):
tj
Tangent plane
Surfaces, Curvature
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n
p
pu pv
t
g
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)
tj
Tangent plane
Surfaces, Curvature
Florida State University
• Mean curvature
Surfaces, Curvature
H =1
2π
Z2π
0
kn(ϕ)dϕ<latexit sha1_base64="h2moCiTDtSTrEHB2y1n3YVpd2Ys=">AAACIXicbZDLSsNAFIYn9VbrrerSzWAp1E1JilA3QtFNlxXsBZoYJpNJO3QyCTOTQgl5FFc+iitBQXQlvozTNgtt/WHg4z/ncOb8XsyoVKb5ZRQ2Nre2d4q7pb39g8Oj8vFJT0aJwKSLIxaJgYckYZSTrqKKkUEsCAo9Rvre5HZe70+JkDTi92oWEydEI04DipHSlltutuE1tAOBcGplacOOaWZTrh6W6KZmlk5cXrOnSMRjeuHnkLnlilk3F4LrYOVQAbk6bvnT9iOchIQrzJCUQ8uMlZMioShmJCtV7USSGOEJGpGhRo5CIp10cWEGq9rxYRAJ/biCC7f0ayJFoZSz0NOdIVJjuVqbm//VhokKrpyU8jhRhOPloiBhUEVwHhf0qSBYsZkGhAXVn4V4jHRcSoda0ilYqzevQ69Rt8y6dXdZad3keRTBGTgHNWCBJmiBNuiALsDgETyDV/BmPBkvxrvxsWwtGPnMKfgj4/sHvhuj3Q==</latexit><latexit sha1_base64="h2moCiTDtSTrEHB2y1n3YVpd2Ys=">AAACIXicbZDLSsNAFIYn9VbrrerSzWAp1E1JilA3QtFNlxXsBZoYJpNJO3QyCTOTQgl5FFc+iitBQXQlvozTNgtt/WHg4z/ncOb8XsyoVKb5ZRQ2Nre2d4q7pb39g8Oj8vFJT0aJwKSLIxaJgYckYZSTrqKKkUEsCAo9Rvre5HZe70+JkDTi92oWEydEI04DipHSlltutuE1tAOBcGplacOOaWZTrh6W6KZmlk5cXrOnSMRjeuHnkLnlilk3F4LrYOVQAbk6bvnT9iOchIQrzJCUQ8uMlZMioShmJCtV7USSGOEJGpGhRo5CIp10cWEGq9rxYRAJ/biCC7f0ayJFoZSz0NOdIVJjuVqbm//VhokKrpyU8jhRhOPloiBhUEVwHhf0qSBYsZkGhAXVn4V4jHRcSoda0ilYqzevQ69Rt8y6dXdZad3keRTBGTgHNWCBJmiBNuiALsDgETyDV/BmPBkvxrvxsWwtGPnMKfgj4/sHvhuj3Q==</latexit><latexit sha1_base64="h2moCiTDtSTrEHB2y1n3YVpd2Ys=">AAACIXicbZDLSsNAFIYn9VbrrerSzWAp1E1JilA3QtFNlxXsBZoYJpNJO3QyCTOTQgl5FFc+iitBQXQlvozTNgtt/WHg4z/ncOb8XsyoVKb5ZRQ2Nre2d4q7pb39g8Oj8vFJT0aJwKSLIxaJgYckYZSTrqKKkUEsCAo9Rvre5HZe70+JkDTi92oWEydEI04DipHSlltutuE1tAOBcGplacOOaWZTrh6W6KZmlk5cXrOnSMRjeuHnkLnlilk3F4LrYOVQAbk6bvnT9iOchIQrzJCUQ8uMlZMioShmJCtV7USSGOEJGpGhRo5CIp10cWEGq9rxYRAJ/biCC7f0ayJFoZSz0NOdIVJjuVqbm//VhokKrpyU8jhRhOPloiBhUEVwHhf0qSBYsZkGhAXVn4V4jHRcSoda0ilYqzevQ69Rt8y6dXdZad3keRTBGTgHNWCBJmiBNuiALsDgETyDV/BmPBkvxrvxsWwtGPnMKfgj4/sHvhuj3Q==</latexit><latexit sha1_base64="h2moCiTDtSTrEHB2y1n3YVpd2Ys=">AAACIXicbZDLSsNAFIYn9VbrrerSzWAp1E1JilA3QtFNlxXsBZoYJpNJO3QyCTOTQgl5FFc+iitBQXQlvozTNgtt/WHg4z/ncOb8XsyoVKb5ZRQ2Nre2d4q7pb39g8Oj8vFJT0aJwKSLIxaJgYckYZSTrqKKkUEsCAo9Rvre5HZe70+JkDTi92oWEydEI04DipHSlltutuE1tAOBcGplacOOaWZTrh6W6KZmlk5cXrOnSMRjeuHnkLnlilk3F4LrYOVQAbk6bvnT9iOchIQrzJCUQ8uMlZMioShmJCtV7USSGOEJGpGhRo5CIp10cWEGq9rxYRAJ/biCC7f0ayJFoZSz0NOdIVJjuVqbm//VhokKrpyU8jhRhOPloiBhUEVwHhf0qSBYsZkGhAXVn4V4jHRcSoda0ilYqzevQ69Rt8y6dXdZad3keRTBGTgHNWCBJmiBNuiALsDgETyDV/BmPBkvxrvxsWwtGPnMKfgj4/sHvhuj3Q==</latexit>
∆Mp = −
1
π
Z2π
0
γ00
dϕ<latexit sha1_base64="gvfm7MfJARVj8nptzczAm1+3bkc=">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</latexit><latexit sha1_base64="gvfm7MfJARVj8nptzczAm1+3bkc=">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</latexit><latexit sha1_base64="gvfm7MfJARVj8nptzczAm1+3bkc=">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</latexit><latexit sha1_base64="gvfm7MfJARVj8nptzczAm1+3bkc=">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</latexit>
∆Mp = −2Hn<latexit sha1_base64="L/xv12jK+WJ/bCySEdUpnVwPrkY=">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</latexit><latexit sha1_base64="L/xv12jK+WJ/bCySEdUpnVwPrkY=">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</latexit><latexit sha1_base64="L/xv12jK+WJ/bCySEdUpnVwPrkY=">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</latexit><latexit sha1_base64="L/xv12jK+WJ/bCySEdUpnVwPrkY=">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</latexit>
kn(ϕ)n = γ00
<latexit sha1_base64="KiIsw+1xy+hT6EzSxTxlM4D9+H0=">AAADTXicnVLPaxQxFM5Mra1jtVs9egkupduDy8wi6KVQ1EMvQgvdtrCzHd5kM7thk8yQZApLmD/QkwdP/hWeBAURM9Nh7S+k+CDk43t5X977eGnBmTZh+NXzVx6sPlxbfxQ83njydLOz9exE56UidEhynquzFDTlTNKhYYbTs0JRECmnp+n8fZ0/vaBKs1wem0VBxwKmkmWMgHFUsuWl2/EHyg0kNhZgZgS4/VhVDVbCFhXew68GBzKYJ7IXX4AqZmxX7sVTEALO7c5OFdxDIM4UEBtVNi5YFTNpzu2ghokNHfdXa9L+cC9R11Vz9Zbqg7vkrza+1N91pf/u6tq8y7r/GibpdMN+2AS+DaIWdFEbh0nnSzzJSSmoNISD1qMoLMzYgjKMcFqbU2paAJnDlI4clCCoHttmGyq87ZgJznLljjS4YYMrFRaE1guRupe1n/pmribvyo1Kk70dWyaL0lBJLj/KSo5NjuvVwhOmKDF84QAQxVyzmMzAmWXcAgbOhejmzLfByaAfhf3o6HV3/13rxzp6gV6iHorQG7SPDtAhGiLiffK+eT+8n/5n/7v/y/99+dT32prn6FqsrP0ByRgUtA==</latexit><latexit sha1_base64="KiIsw+1xy+hT6EzSxTxlM4D9+H0=">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</latexit><latexit sha1_base64="KiIsw+1xy+hT6EzSxTxlM4D9+H0=">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</latexit><latexit sha1_base64="KiIsw+1xy+hT6EzSxTxlM4D9+H0=">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</latexit>
Florida State University
Discrete Laplace-Beltrami
• Intuition for uniform discretization
H =1
2π
Z2π
0
kn(ϕ)dϕ<latexit sha1_base64="h2moCiTDtSTrEHB2y1n3YVpd2Ys=">AAACIXicbZDLSsNAFIYn9VbrrerSzWAp1E1JilA3QtFNlxXsBZoYJpNJO3QyCTOTQgl5FFc+iitBQXQlvozTNgtt/WHg4z/ncOb8XsyoVKb5ZRQ2Nre2d4q7pb39g8Oj8vFJT0aJwKSLIxaJgYckYZSTrqKKkUEsCAo9Rvre5HZe70+JkDTi92oWEydEI04DipHSlltutuE1tAOBcGplacOOaWZTrh6W6KZmlk5cXrOnSMRjeuHnkLnlilk3F4LrYOVQAbk6bvnT9iOchIQrzJCUQ8uMlZMioShmJCtV7USSGOEJGpGhRo5CIp10cWEGq9rxYRAJ/biCC7f0ayJFoZSz0NOdIVJjuVqbm//VhokKrpyU8jhRhOPloiBhUEVwHhf0qSBYsZkGhAXVn4V4jHRcSoda0ilYqzevQ69Rt8y6dXdZad3keRTBGTgHNWCBJmiBNuiALsDgETyDV/BmPBkvxrvxsWwtGPnMKfgj4/sHvhuj3Q==</latexit><latexit sha1_base64="h2moCiTDtSTrEHB2y1n3YVpd2Ys=">AAACIXicbZDLSsNAFIYn9VbrrerSzWAp1E1JilA3QtFNlxXsBZoYJpNJO3QyCTOTQgl5FFc+iitBQXQlvozTNgtt/WHg4z/ncOb8XsyoVKb5ZRQ2Nre2d4q7pb39g8Oj8vFJT0aJwKSLIxaJgYckYZSTrqKKkUEsCAo9Rvre5HZe70+JkDTi92oWEydEI04DipHSlltutuE1tAOBcGplacOOaWZTrh6W6KZmlk5cXrOnSMRjeuHnkLnlilk3F4LrYOVQAbk6bvnT9iOchIQrzJCUQ8uMlZMioShmJCtV7USSGOEJGpGhRo5CIp10cWEGq9rxYRAJ/biCC7f0ayJFoZSz0NOdIVJjuVqbm//VhokKrpyU8jhRhOPloiBhUEVwHhf0qSBYsZkGhAXVn4V4jHRcSoda0ilYqzevQ69Rt8y6dXdZad3keRTBGTgHNWCBJmiBNuiALsDgETyDV/BmPBkvxrvxsWwtGPnMKfgj4/sHvhuj3Q==</latexit><latexit sha1_base64="h2moCiTDtSTrEHB2y1n3YVpd2Ys=">AAACIXicbZDLSsNAFIYn9VbrrerSzWAp1E1JilA3QtFNlxXsBZoYJpNJO3QyCTOTQgl5FFc+iitBQXQlvozTNgtt/WHg4z/ncOb8XsyoVKb5ZRQ2Nre2d4q7pb39g8Oj8vFJT0aJwKSLIxaJgYckYZSTrqKKkUEsCAo9Rvre5HZe70+JkDTi92oWEydEI04DipHSlltutuE1tAOBcGplacOOaWZTrh6W6KZmlk5cXrOnSMRjeuHnkLnlilk3F4LrYOVQAbk6bvnT9iOchIQrzJCUQ8uMlZMioShmJCtV7USSGOEJGpGhRo5CIp10cWEGq9rxYRAJ/biCC7f0ayJFoZSz0NOdIVJjuVqbm//VhokKrpyU8jhRhOPloiBhUEVwHhf0qSBYsZkGhAXVn4V4jHRcSoda0ilYqzevQ69Rt8y6dXdZad3keRTBGTgHNWCBJmiBNuiALsDgETyDV/BmPBkvxrvxsWwtGPnMKfgj4/sHvhuj3Q==</latexit><latexit sha1_base64="h2moCiTDtSTrEHB2y1n3YVpd2Ys=">AAACIXicbZDLSsNAFIYn9VbrrerSzWAp1E1JilA3QtFNlxXsBZoYJpNJO3QyCTOTQgl5FFc+iitBQXQlvozTNgtt/WHg4z/ncOb8XsyoVKb5ZRQ2Nre2d4q7pb39g8Oj8vFJT0aJwKSLIxaJgYckYZSTrqKKkUEsCAo9Rvre5HZe70+JkDTi92oWEydEI04DipHSlltutuE1tAOBcGplacOOaWZTrh6W6KZmlk5cXrOnSMRjeuHnkLnlilk3F4LrYOVQAbk6bvnT9iOchIQrzJCUQ8uMlZMioShmJCtV7USSGOEJGpGhRo5CIp10cWEGq9rxYRAJ/biCC7f0ayJFoZSz0NOdIVJjuVqbm//VhokKrpyU8jhRhOPloiBhUEVwHhf0qSBYsZkGhAXVn4V4jHRcSoda0ilYqzevQ69Rt8y6dXdZad3keRTBGTgHNWCBJmiBNuiALsDgETyDV/BmPBkvxrvxsWwtGPnMKfgj4/sHvhuj3Q==</latexit>
∆Mp = −
1
π
Z2π
0
γ00
dϕ<latexit sha1_base64="gvfm7MfJARVj8nptzczAm1+3bkc=">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</latexit><latexit sha1_base64="gvfm7MfJARVj8nptzczAm1+3bkc=">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</latexit><latexit sha1_base64="gvfm7MfJARVj8nptzczAm1+3bkc=">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</latexit><latexit sha1_base64="gvfm7MfJARVj8nptzczAm1+3bkc=">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</latexit>
∆Mp = −2Hn<latexit sha1_base64="L/xv12jK+WJ/bCySEdUpnVwPrkY=">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</latexit><latexit sha1_base64="L/xv12jK+WJ/bCySEdUpnVwPrkY=">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</latexit><latexit sha1_base64="L/xv12jK+WJ/bCySEdUpnVwPrkY=">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</latexit><latexit sha1_base64="L/xv12jK+WJ/bCySEdUpnVwPrkY=">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</latexit>
kn(ϕ)n = γ00
<latexit sha1_base64="KiIsw+1xy+hT6EzSxTxlM4D9+H0=">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</latexit><latexit sha1_base64="KiIsw+1xy+hT6EzSxTxlM4D9+H0=">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</latexit><latexit sha1_base64="KiIsw+1xy+hT6EzSxTxlM4D9+H0=">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</latexit><latexit sha1_base64="KiIsw+1xy+hT6EzSxTxlM4D9+H0=">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</latexit>
Florida State University
Discrete Laplace-Beltrami
• Intuition for uniform discretization
vivi+1
vi-1
g
∆Mp = −
1
π
Z2π
0
γ00
dϕ<latexit sha1_base64="gvfm7MfJARVj8nptzczAm1+3bkc=">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</latexit><latexit sha1_base64="gvfm7MfJARVj8nptzczAm1+3bkc=">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</latexit><latexit sha1_base64="gvfm7MfJARVj8nptzczAm1+3bkc=">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</latexit><latexit sha1_base64="gvfm7MfJARVj8nptzczAm1+3bkc=">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</latexit>
Florida State University
Discrete Laplace-Beltrami
vi
vj1 vj2
vj3
vj4vj5
vj6 ∆Mp = −2Hn<latexit sha1_base64="L/xv12jK+WJ/bCySEdUpnVwPrkY=">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</latexit><latexit sha1_base64="L/xv12jK+WJ/bCySEdUpnVwPrkY=">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</latexit><latexit sha1_base64="L/xv12jK+WJ/bCySEdUpnVwPrkY=">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</latexit><latexit sha1_base64="L/xv12jK+WJ/bCySEdUpnVwPrkY=">AAADQnicnVLPa9RAGJ3EqjX+6FaPXgaXpduDS7IIeikU9dCLUMHtFjbb8GV2sjvszCTMTArLkH/Nk/+BJ++eBAXx6sFJGra1LSp+EPJ437w33zy+tOBMmzD85Pk3Nm7eur15J7h77/6Drc72wyOdl4rQEcl5ro5T0JQzSUeGGU6PC0VBpJyO0+Wruj8+pUqzXL4zq4JOBcwlyxgB46hk2xvHryk3kNhYgFkQ4PZNVTVYCVtUeA8/HR7IoLeUe/EchIATu7NTBb2/y+JMAbFRZeOCVTGT5sQOa5jY0HHnXrP4FFSxYP9k6mZpfv21+/A6+2Ui+63t7tp/10n/PNVFmVzr/usxSacbDsKm8FUQtaCL2jpMOh/jWU5KQaUhHLSeRGFhphaUYYTTOpxS0wLIEuZ04qAEQfXUNitQ4Z5jZjjLlfukwQ0bXFBYEFqvROpO1nnqy72avK43KU32YmqZLEpDJTm7KCs5Njmu9wnPmKLE8JUDQBRzw2KyABeWcVsXuBSiy2++Co6GgygcRG+fdfdftnlsosfoCeqjCD1H++gAHaIRIt5777P31fvmf/C/+N/9H2dHfa/VPEK/lf/zF5hlEUE=</latexit>
h = 1<latexit sha1_base64="KQU7OL1RkfABT7Pd0XH3qkoujic=">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</latexit><latexit sha1_base64="KQU7OL1RkfABT7Pd0XH3qkoujic=">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</latexit><latexit sha1_base64="KQU7OL1RkfABT7Pd0XH3qkoujic=">AAAC7nicjVJbixMxFM7MelnHW3d99CVYCjOIZWYquC+FRRFkEVlhu7vQaUMmzbTpzo0kUyjZ/AefBAXx1f/jk//GTGdcteuDB5Lz5TuXnJyTuEyZkL7/w7J3bty8dXv3jnP33v0HDzt7+6eiqDihI1KkBT+PsaApy+lIMpnS85JTnMUpPYsvXtX2sxXlghX5iVyXdJLhec4SRrA0FNqzdhZwCAOn9xZV7goxb+hGCcdEBVpdvnOZd6kjUWVILSOWq5rQWq3QUnvPjLfT22yvh01MlGG54JmSXLtH0xMk4RGSnlZXhhmV2lB62viHWg20dnq/wgWbZ3gaouDpFQx/w4FWbnNAQavDVg+8/0+o1XYSjTpdv+9vBF4HQQu6oJVj1PkezQpSZTSXJMVCjAO/lBOFuWQkpaaAqBK0xOQCz+nYwBxnVEzUZlwa9gwzg0nBzcol3LDOHxEKZ0Kss9h41n0T27aa/JdtXMnkYKJYXlaS5qS5KKlSKAtYzx7OGKdEpmsDMOHMFAvJAptOSfNDHNOFYPvN18Fp2A/8fvD+effwZduPXfAYPAEuCMALcAjegGMwAsRaWh+sT9Znu7Q/2l/sr42rbbUxj8BfYn/7CSMU6ms=</latexit><latexit sha1_base64="KQU7OL1RkfABT7Pd0XH3qkoujic=">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</latexit>
Florida State University
Discrete Laplace-Beltrami
• Uniform discretization:
• Depends only on connectivity = simple and efficient
• Bad approximation for irregular triangulations
Lu(vi) = (1
|N(i)|
X
j∈N(i)
vj)− vi
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vi<latexit sha1_base64="0dZJqAMaJguUcHrqfZI9QXK+plw=">AAAC5nicjVLNa9swFJfdfXTeR7P1uItZCNiMBdsZbJdCaRmMMkoHTVuIEyErcqLWso0kB4Kq+06DDcau+5922v/Sw+TY67Z0hz14vJ9+70NP7ykpMypkEPyw7I1bt+/c3bzn3H/w8NFW5/GTE1FUHJMhLrKCnyVIkIzmZCipzMhZyQliSUZOk4v92n+6IFzQIj+Wy5KMGZrlNKUYSUPBzlXvHay8BaT+jhenHGEVanV56FH/UseiYlCdxzRXNaG1WsBz7b8w0U6tvTc7TUrMkJxzpiTX3sHkGEr3AEpfq2vHlEhtKD1p4iOtBlo7vV/pgs4YmkQwfH4No99woJXXHGDY2qi1A///C2q1XkTDTjfoBytxb4KwBV3QyhHsfI+nBa4YySXOkBCjMCjlWCEuKc6IaSCuBCkRvkAzMjIwR4yIsVqtSbs9w0zdtOBGc+muWOePDIWYEEuWmMh6bmLdV5P/8o0qmb4eK5qXlSQ5bi5Kq8yVhVvv3J1STrDMlgYgzKlp1sVzZCYlzc9wzBTC9TffBCdRPwz64fuX3d29dh6b4Cl4BjwQgldgF7wFR2AIsJVYH6xP1md7bn+0v9hfm1DbanO2wV9if/sJMvnp1Q==</latexit><latexit sha1_base64="0dZJqAMaJguUcHrqfZI9QXK+plw=">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</latexit><latexit sha1_base64="0dZJqAMaJguUcHrqfZI9QXK+plw=">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</latexit><latexit sha1_base64="0dZJqAMaJguUcHrqfZI9QXK+plw=">AAAC5nicjVLNa9swFJfdfXTeR7P1uItZCNiMBdsZbJdCaRmMMkoHTVuIEyErcqLWso0kB4Kq+06DDcau+5922v/Sw+TY67Z0hz14vJ9+70NP7ykpMypkEPyw7I1bt+/c3bzn3H/w8NFW5/GTE1FUHJMhLrKCnyVIkIzmZCipzMhZyQliSUZOk4v92n+6IFzQIj+Wy5KMGZrlNKUYSUPBzlXvHay8BaT+jhenHGEVanV56FH/UseiYlCdxzRXNaG1WsBz7b8w0U6tvTc7TUrMkJxzpiTX3sHkGEr3AEpfq2vHlEhtKD1p4iOtBlo7vV/pgs4YmkQwfH4No99woJXXHGDY2qi1A///C2q1XkTDTjfoBytxb4KwBV3QyhHsfI+nBa4YySXOkBCjMCjlWCEuKc6IaSCuBCkRvkAzMjIwR4yIsVqtSbs9w0zdtOBGc+muWOePDIWYEEuWmMh6bmLdV5P/8o0qmb4eK5qXlSQ5bi5Kq8yVhVvv3J1STrDMlgYgzKlp1sVzZCYlzc9wzBTC9TffBCdRPwz64fuX3d29dh6b4Cl4BjwQgldgF7wFR2AIsJVYH6xP1md7bn+0v9hfm1DbanO2wV9if/sJMvnp1Q==</latexit>
Florida State University
• Uniform discretization:
• Cotangent weight:
Discrete Laplace-Beltrami
Wivi
vj
aijbij
vi
vj
Lu(vi) =1
|N(i)|
X
j∈N(i)
vj − vi
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L(vi) =1
Wi
X
j∈N(i)
wij(vj − vi)
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Wi = 1, wij =1
|N(i)|<latexit sha1_base64="jT6sg1195pcpFvEm8tA//pFCyq4=">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</latexit><latexit sha1_base64="jT6sg1195pcpFvEm8tA//pFCyq4=">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</latexit><latexit sha1_base64="jT6sg1195pcpFvEm8tA//pFCyq4=">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</latexit><latexit sha1_base64="jT6sg1195pcpFvEm8tA//pFCyq4=">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</latexit>
wij =1
2(cotαij + cotβij)
<latexit sha1_base64="2Mr6Yt5XX5s8hKUcYGJQqy07xj8=">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</latexit><latexit sha1_base64="2Mr6Yt5XX5s8hKUcYGJQqy07xj8=">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</latexit><latexit sha1_base64="2Mr6Yt5XX5s8hKUcYGJQqy07xj8=">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</latexit><latexit sha1_base64="2Mr6Yt5XX5s8hKUcYGJQqy07xj8=">AAAEQ3icnVNdi9NAFJ1N/FjjV1cffRksZVN0u0kR9KWwqA99UFnBbgubbphMp+20k0mYTCplmt/mk7/AJ3+AT4KC+Co4SUP2o4ssDoQczr3nzL03uUHMaCId5+uWYV67fuPm9i3r9p279+7Xdh4cJVEqMOnhiEViEKCEMMpJT1LJyCAWBIUBI/1g/iqP9xdEJDTiH+QyJsMQTTgdU4ykpvwdY/DRV3SWwY43FggrN1PtzMaRhB5i8RStg09gwQREromm1ej7FHag+xRu6FfvbNpcZVbjjb3waVNnVSEtyrwkDX018yhXeWKWqbWDTp7t5YJc6acb2tJ2U17qtGzVLVSrXBYjgRgjDFZeFbXfthrea8LyZrwQySlGTL3NsgKLUMW6G7jX7nKrMfe57S2QiKe0yTveBIUhOlG7u9lVHKrSvVi3Tbk8Ue0c+srR3KnXqLzhSqa6rOJln36vy+zPFl75N7X031Wd67fS/Vczfq3utJziwE3glqAOynPo1754owinIeESM5Qkx64Ty6FCQlLMSD6cNCExwnM0IccachSSZKiKHchgQzMjOI6EfriEBWudUSgUJskyDHRmPs/kYiwnL4sdp3L8Yqgoj1NJOF5fNE4ZlBHMFwqOqCBYsqUGCAuqi4V4qv80LPXaWXoK7sWeN8FRu+U6Lff9s/rBy3Ie2+AReAxs4ILn4AB0wSHoAWx8Mr4ZP4yf5mfzu/nL/L1ONbZKzUNw7ph//gKZBGs2</latexit>
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
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Curvature and Smoothness
• Smoothing =
reducing curvature?
• Smoothing =
make curvature vary less?
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Example – smoothing curves
• Laplace in 1D = second derivative:
Florida State University
Example – smoothing curves
• Laplace in 1D = second derivative:
• In matrix-vector form for the whole curve
Florida State University
Example – smoothing curves
• Laplace in 1D = 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?
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Example – smoothing curves
• Flow to reduce curvature:
• Scale factor 0 < l < 1
• Matrix-vector form:
• May shrink the shape; can be slow
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Original curve
Filtering Curves
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1st iteration; l=0.5
Filtering Curves
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2nd iteration; l=0.5
Filtering Curves
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8th iteration; l=0.5
Filtering Curves
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27th iteration; l=0.5
Filtering Curves
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50th iteration; l=0.5
Filtering Curves
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500th iteration; l=0.5
Filtering Curves
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1000th iteration; l=0.5
Filtering Curves
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5000th iteration; l=0.5
Filtering Curves
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10000th iteration; l=0.5
Filtering Curves
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50000th iteration; l=0.5
Filtering Curves
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Filtering Surfaces (Demo)
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Thank you!
https://gaoxifeng.github.io