Laplacian Operator and Smoothingpanozzo/ustc/05 - Laplacian Operator.pdfFlorida State University...

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

Transcript of Laplacian Operator and Smoothingpanozzo/ustc/05 - Laplacian Operator.pdfFlorida State University...

Page 1: Laplacian Operator and Smoothingpanozzo/ustc/05 - Laplacian Operator.pdfFlorida State University Laplacian Operator and Smoothing Xifeng Gao Acknowledgements for the slides: Olga Sorkine-Hornung,

Florida State University

Laplacian Operator and Smoothing

Xifeng Gao

Acknowledgements for the slides: Olga Sorkine-Hornung, Mario Botsch, and Daniele Panozzo

Page 2: Laplacian Operator and Smoothingpanozzo/ustc/05 - Laplacian Operator.pdfFlorida State University Laplacian Operator and Smoothing Xifeng Gao Acknowledgements for the slides: Olga Sorkine-Hornung,

Florida State University

Applications in Geometry Processing

• Smoothing

• Parameterization

• Remeshing

Page 3: Laplacian Operator and Smoothingpanozzo/ustc/05 - Laplacian Operator.pdfFlorida State University Laplacian Operator and Smoothing Xifeng Gao Acknowledgements for the slides: Olga Sorkine-Hornung,

Florida State University

Laplacian Operator

Laplacianoperator

gradientoperator

divergenceoperator

function inEuclidean space

Page 4: Laplacian Operator and Smoothingpanozzo/ustc/05 - Laplacian Operator.pdfFlorida State University Laplacian Operator and Smoothing Xifeng Gao Acknowledgements for the slides: Olga Sorkine-Hornung,

Florida State University

Laplacianoperator

gradientoperator

2nd partialderivatives

divergenceoperator

function inEuclidean space

Laplacian Operator

Page 5: Laplacian Operator and Smoothingpanozzo/ustc/05 - Laplacian Operator.pdfFlorida State University Laplacian Operator and Smoothing Xifeng Gao Acknowledgements for the slides: Olga Sorkine-Hornung,

Florida State University

Laplacianoperator

gradientoperator

2nd partialderivatives

Cartesiancoordinatesdivergence

operator

function inEuclidean space

Laplacian Operator

Page 6: Laplacian Operator and Smoothingpanozzo/ustc/05 - Laplacian Operator.pdfFlorida State University Laplacian Operator and Smoothing Xifeng Gao Acknowledgements for the slides: Olga Sorkine-Hornung,

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

Page 7: Laplacian Operator and Smoothingpanozzo/ustc/05 - Laplacian Operator.pdfFlorida State University Laplacian Operator and Smoothing Xifeng Gao Acknowledgements for the slides: Olga Sorkine-Hornung,

Florida State University

Laplace-Beltrami Operator

function onsurface M

• Extension of Laplace to functions on manifolds

Laplace-Beltrami

gradientoperator

divergenceoperator

Page 8: Laplacian Operator and Smoothingpanozzo/ustc/05 - Laplacian Operator.pdfFlorida State University Laplacian Operator and Smoothing Xifeng Gao Acknowledgements for the slides: Olga Sorkine-Hornung,

Florida State University

Laplace-Beltrami Operator

• For coordinate functions:

mean curvature

unitsurface

normal

Laplace-Beltrami

gradientoperator

divergenceoperator

P at function f

Page 9: Laplacian Operator and Smoothingpanozzo/ustc/05 - Laplacian Operator.pdfFlorida State University Laplacian Operator and Smoothing Xifeng Gao Acknowledgements for the slides: Olga Sorkine-Hornung,

Florida State University

Discrete Curvatures

• Mean curvature (sign defined according to normal)

Page 10: Laplacian Operator and Smoothingpanozzo/ustc/05 - Laplacian Operator.pdfFlorida State University Laplacian Operator and Smoothing Xifeng Gao Acknowledgements for the slides: Olga Sorkine-Hornung,

Florida State University

Surfaces, Parametric Form

• Continuous surface

• Tangent plane

n

p(u,v)

pu

u

v

pv

Page 11: Laplacian Operator and Smoothingpanozzo/ustc/05 - Laplacian Operator.pdfFlorida State University Laplacian Operator and Smoothing Xifeng Gao Acknowledgements for the slides: Olga Sorkine-Hornung,

Florida State University

• Surface normal: n

p(u,v)

pu

u

v

pv

Surfaces, Normal

Page 12: Laplacian Operator and Smoothingpanozzo/ustc/05 - Laplacian Operator.pdfFlorida State University Laplacian Operator and Smoothing Xifeng Gao Acknowledgements for the slides: Olga Sorkine-Hornung,

Florida State University

n

p

pu pv

t

Unit-length direction t in the tangent plane (if pu and pv are orthogonal):

tj

Tangent plane

Surfaces, Curvature

Page 13: Laplacian Operator and Smoothingpanozzo/ustc/05 - Laplacian Operator.pdfFlorida State University Laplacian Operator and Smoothing Xifeng Gao Acknowledgements for the slides: Olga Sorkine-Hornung,

Florida State University

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

Page 14: Laplacian Operator and Smoothingpanozzo/ustc/05 - Laplacian Operator.pdfFlorida State University Laplacian Operator and Smoothing Xifeng Gao Acknowledgements for the slides: Olga Sorkine-Hornung,

Florida State University

• Mean curvature

Surfaces, Curvature

H =1

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=">AAADG3icnVJPaxQxHM2M1dbx31aPXkKXpduDy8witJdCUQ+9CBXctrCzHX6TzeyGTTJDkiksYT5HT34UT4KCeBMPfhsz02GtbVHwByGP98t7SV6SFpxpE4Y/Pf/O2t176xv3gwcPHz1+0tl8eqzzUhE6IjnP1WkKmnIm6cgww+lpoSiIlNOTdPG67p+cU6VZLt+bZUEnAmaSZYyAcVSy6YW9hdyPZyAEnNnt7SqI31BuILGxADMnwO3bqmqwErao8D5+EWcKiI0qGxesipk0Z3ZYw8SGjvttNY3PQRVzVgW9f5sOD2Uz9Vfuw9vsF4nst7Y7K/8dJ/37qa7K5Er3X5dJOt1wEDaFb4KoBV3U1lHS+R5Pc1IKKg3hoPU4CgszsaAMI5zW4ZSaFkAWMKNjByUIqie2edsK9xwzxVmu3JAGN2xwRWFBaL0UqVtZ56mv92rytt64NNnexDJZlIZKcrlRVnJsclx/FDxlihLDlw4AUcwdFpM5uLCM+06BSyG6fueb4Hg4iMJB9O5l9+BVm8cGeo62UB9FaBcdoEN0hEaIeBfeR++z98X/4H/yv/rfLpf6Xqt5hv4o/8cvQs8BlQ==</latexit><latexit sha1_base64="gvfm7MfJARVj8nptzczAm1+3bkc=">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</latexit><latexit sha1_base64="gvfm7MfJARVj8nptzczAm1+3bkc=">AAADG3icnVJPaxQxHM2M1dbx31aPXkKXpduDy8witJdCUQ+9CBXctrCzHX6TzeyGTTJDkiksYT5HT34UT4KCeBMPfhsz02GtbVHwByGP98t7SV6SFpxpE4Y/Pf/O2t176xv3gwcPHz1+0tl8eqzzUhE6IjnP1WkKmnIm6cgww+lpoSiIlNOTdPG67p+cU6VZLt+bZUEnAmaSZYyAcVSy6YW9hdyPZyAEnNnt7SqI31BuILGxADMnwO3bqmqwErao8D5+EWcKiI0qGxesipk0Z3ZYw8SGjvttNY3PQRVzVgW9f5sOD2Uz9Vfuw9vsF4nst7Y7K/8dJ/37qa7K5Er3X5dJOt1wEDaFb4KoBV3U1lHS+R5Pc1IKKg3hoPU4CgszsaAMI5zW4ZSaFkAWMKNjByUIqie2edsK9xwzxVmu3JAGN2xwRWFBaL0UqVtZ56mv92rytt64NNnexDJZlIZKcrlRVnJsclx/FDxlihLDlw4AUcwdFpM5uLCM+06BSyG6fueb4Hg4iMJB9O5l9+BVm8cGeo62UB9FaBcdoEN0hEaIeBfeR++z98X/4H/yv/rfLpf6Xqt5hv4o/8cvQs8BlQ==</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

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Page 15: Laplacian Operator and Smoothingpanozzo/ustc/05 - Laplacian Operator.pdfFlorida State University Laplacian Operator and Smoothing Xifeng Gao Acknowledgements for the slides: Olga Sorkine-Hornung,

Florida State University

Discrete Laplace-Beltrami

• Intuition for uniform discretization

H =1

Z2π

0

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∆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=">AAADG3icnVJPaxQxHM2M1dbx31aPXkKXpduDy8witJdCUQ+9CBXctrCzHX6TzeyGTTJDkiksYT5HT34UT4KCeBMPfhsz02GtbVHwByGP98t7SV6SFpxpE4Y/Pf/O2t176xv3gwcPHz1+0tl8eqzzUhE6IjnP1WkKmnIm6cgww+lpoSiIlNOTdPG67p+cU6VZLt+bZUEnAmaSZYyAcVSy6YW9hdyPZyAEnNnt7SqI31BuILGxADMnwO3bqmqwErao8D5+EWcKiI0qGxesipk0Z3ZYw8SGjvttNY3PQRVzVgW9f5sOD2Uz9Vfuw9vsF4nst7Y7K/8dJ/37qa7K5Er3X5dJOt1wEDaFb4KoBV3U1lHS+R5Pc1IKKg3hoPU4CgszsaAMI5zW4ZSaFkAWMKNjByUIqie2edsK9xwzxVmu3JAGN2xwRWFBaL0UqVtZ56mv92rytt64NNnexDJZlIZKcrlRVnJsclx/FDxlihLDlw4AUcwdFpM5uLCM+06BSyG6fueb4Hg4iMJB9O5l9+BVm8cGeo62UB9FaBcdoEN0hEaIeBfeR++z98X/4H/yv/rfLpf6Xqt5hv4o/8cvQs8BlQ==</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>

Page 16: Laplacian Operator and Smoothingpanozzo/ustc/05 - Laplacian Operator.pdfFlorida State University Laplacian Operator and Smoothing Xifeng Gao Acknowledgements for the slides: Olga Sorkine-Hornung,

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=">AAADG3icnVJPaxQxHM2M1dbx31aPXkKXpduDy8witJdCUQ+9CBXctrCzHX6TzeyGTTJDkiksYT5HT34UT4KCeBMPfhsz02GtbVHwByGP98t7SV6SFpxpE4Y/Pf/O2t176xv3gwcPHz1+0tl8eqzzUhE6IjnP1WkKmnIm6cgww+lpoSiIlNOTdPG67p+cU6VZLt+bZUEnAmaSZYyAcVSy6YW9hdyPZyAEnNnt7SqI31BuILGxADMnwO3bqmqwErao8D5+EWcKiI0qGxesipk0Z3ZYw8SGjvttNY3PQRVzVgW9f5sOD2Uz9Vfuw9vsF4nst7Y7K/8dJ/37qa7K5Er3X5dJOt1wEDaFb4KoBV3U1lHS+R5Pc1IKKg3hoPU4CgszsaAMI5zW4ZSaFkAWMKNjByUIqie2edsK9xwzxVmu3JAGN2xwRWFBaL0UqVtZ56mv92rytt64NNnexDJZlIZKcrlRVnJsclx/FDxlihLDlw4AUcwdFpM5uLCM+06BSyG6fueb4Hg4iMJB9O5l9+BVm8cGeo62UB9FaBcdoEN0hEaIeBfeR++z98X/4H/yv/rfLpf6Xqt5hv4o/8cvQs8BlQ==</latexit>

Page 17: Laplacian Operator and Smoothingpanozzo/ustc/05 - Laplacian Operator.pdfFlorida State University Laplacian Operator and Smoothing Xifeng Gao Acknowledgements for the slides: Olga Sorkine-Hornung,

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=">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</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=">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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>

Page 18: Laplacian Operator and Smoothingpanozzo/ustc/05 - Laplacian Operator.pdfFlorida State University Laplacian Operator and Smoothing Xifeng Gao Acknowledgements for the slides: Olga Sorkine-Hornung,

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=">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</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=">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</latexit>

Page 19: Laplacian Operator and Smoothingpanozzo/ustc/05 - Laplacian Operator.pdfFlorida State University Laplacian Operator and Smoothing Xifeng Gao Acknowledgements for the slides: Olga Sorkine-Hornung,

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

<latexit sha1_base64="D6dN/Y/AJFucU+2jjorPNAU8u84=">AAADr3icnVJda9swFFXsfXTeV7o97kUshCaMpnYYbC+Fsu0hD9toYWnK4sRTFCVRI8lGkgNB0b/bP9jT/s1kN3j9YpRdMD6ce8/RvZc7yRhVOgx/1zz/3v0HD3ceBY+fPH32vL774lSlucSkj1OWyrMJUoRRQfqaakbOMkkQnzAymCw/FvnBikhFU/FNrzMy4mgu6IxipB2V7NZ+fk7y1iqhbXgI45lE2ETWbL62aHtjY5XzxJzHVJiCsNaskvN9V2yD5qZXqjaFLEMSMUYYrLwq6qAbNONPhGmUmJgjvcCImS/Wllhyk1lnsN/tiaC5TEQrXiGZLWhbHMZzxDkam709exeHqvU4o9Y1rMemW8DEhI776zXdvnAnU9dW+WtV7t3b7C83Xvm3nfTfXV2Zt9L91zBJvRF2wjLgTRBtQQNs4zip/4qnKc45ERozpNQwCjM9MkhqihkplpMrkiG8RHMydFAgTtTIlPdmYdMxUzhLpfuEhiUbXFIYxJVa84mrLPaprucK8rbcMNez9yNDRZZrIvDFQ7OcQZ3C4njhlEqCNVs7gLCkrlmIF+7SsHYnHrgtRNdnvglOu50o7EQnbxtHH7b72AGvwGvQAhF4B45ADxyDPsDeG+/E++4N/cgf+GP/x0WpV9tqXoIr4dM/X2wyaA==</latexit><latexit sha1_base64="D6dN/Y/AJFucU+2jjorPNAU8u84=">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</latexit><latexit sha1_base64="D6dN/Y/AJFucU+2jjorPNAU8u84=">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</latexit><latexit sha1_base64="D6dN/Y/AJFucU+2jjorPNAU8u84=">AAADr3icnVJda9swFFXsfXTeV7o97kUshCaMpnYYbC+Fsu0hD9toYWnK4sRTFCVRI8lGkgNB0b/bP9jT/s1kN3j9YpRdMD6ce8/RvZc7yRhVOgx/1zz/3v0HD3ceBY+fPH32vL774lSlucSkj1OWyrMJUoRRQfqaakbOMkkQnzAymCw/FvnBikhFU/FNrzMy4mgu6IxipB2V7NZ+fk7y1iqhbXgI45lE2ETWbL62aHtjY5XzxJzHVJiCsNaskvN9V2yD5qZXqjaFLEMSMUYYrLwq6qAbNONPhGmUmJgjvcCImS/Wllhyk1lnsN/tiaC5TEQrXiGZLWhbHMZzxDkam709exeHqvU4o9Y1rMemW8DEhI776zXdvnAnU9dW+WtV7t3b7C83Xvm3nfTfXV2Zt9L91zBJvRF2wjLgTRBtQQNs4zip/4qnKc45ERozpNQwCjM9MkhqihkplpMrkiG8RHMydFAgTtTIlPdmYdMxUzhLpfuEhiUbXFIYxJVa84mrLPaprucK8rbcMNez9yNDRZZrIvDFQ7OcQZ3C4njhlEqCNVs7gLCkrlmIF+7SsHYnHrgtRNdnvglOu50o7EQnbxtHH7b72AGvwGvQAhF4B45ADxyDPsDeG+/E++4N/cgf+GP/x0WpV9tqXoIr4dM/X2wyaA==</latexit>

L(vi) =1

Wi

X

j∈N(i)

wij(vj − vi)

<latexit sha1_base64="WDLmJi/CTl2G8heA5y+pqse+hfI=">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</latexit><latexit sha1_base64="WDLmJi/CTl2G8heA5y+pqse+hfI=">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</latexit><latexit sha1_base64="WDLmJi/CTl2G8heA5y+pqse+hfI=">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</latexit><latexit sha1_base64="WDLmJi/CTl2G8heA5y+pqse+hfI=">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</latexit>

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)

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Page 20: Laplacian Operator and Smoothingpanozzo/ustc/05 - Laplacian Operator.pdfFlorida State University Laplacian Operator and Smoothing Xifeng Gao Acknowledgements for the slides: Olga Sorkine-Hornung,

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Surface Smoothing – Motivation

Page 21: Laplacian Operator and Smoothingpanozzo/ustc/05 - Laplacian Operator.pdfFlorida State University Laplacian Operator and Smoothing Xifeng Gao Acknowledgements for the slides: Olga Sorkine-Hornung,

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Curvature and Smoothness

Page 22: Laplacian Operator and Smoothingpanozzo/ustc/05 - Laplacian Operator.pdfFlorida State University Laplacian Operator and Smoothing Xifeng Gao Acknowledgements for the slides: Olga Sorkine-Hornung,

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Curvature and Smoothness

mean curvature plot

Page 23: Laplacian Operator and Smoothingpanozzo/ustc/05 - Laplacian Operator.pdfFlorida State University Laplacian Operator and Smoothing Xifeng Gao Acknowledgements for the slides: Olga Sorkine-Hornung,

Florida State University

Curvature and Smoothness

mean curvature plot

Page 24: Laplacian Operator and Smoothingpanozzo/ustc/05 - Laplacian Operator.pdfFlorida State University Laplacian Operator and Smoothing Xifeng Gao Acknowledgements for the slides: Olga Sorkine-Hornung,

Florida State University

Curvature and Smoothness

mean curvature plot

Page 25: Laplacian Operator and Smoothingpanozzo/ustc/05 - Laplacian Operator.pdfFlorida State University Laplacian Operator and Smoothing Xifeng Gao Acknowledgements for the slides: Olga Sorkine-Hornung,

Florida State University

Curvature and Smoothness

• Smoothing =

reducing curvature?

• Smoothing =

make curvature vary less?

Page 26: Laplacian Operator and Smoothingpanozzo/ustc/05 - Laplacian Operator.pdfFlorida State University Laplacian Operator and Smoothing Xifeng Gao Acknowledgements for the slides: Olga Sorkine-Hornung,

Florida State University

Example – smoothing curves

• Laplace in 1D = second derivative:

Page 27: Laplacian Operator and Smoothingpanozzo/ustc/05 - Laplacian Operator.pdfFlorida State University Laplacian Operator and Smoothing Xifeng Gao Acknowledgements for the slides: Olga Sorkine-Hornung,

Florida State University

Example – smoothing curves

• Laplace in 1D = second derivative:

• In matrix-vector form for the whole curve

Page 28: Laplacian Operator and Smoothingpanozzo/ustc/05 - Laplacian Operator.pdfFlorida State University Laplacian Operator and Smoothing Xifeng Gao Acknowledgements for the slides: Olga Sorkine-Hornung,

Florida State University

Example – smoothing curves

• Laplace in 1D = second derivative:

• In matrix-vector form for the whole curve

Page 29: Laplacian Operator and Smoothingpanozzo/ustc/05 - Laplacian Operator.pdfFlorida State University Laplacian Operator and Smoothing Xifeng Gao Acknowledgements for the slides: Olga Sorkine-Hornung,

Florida State University

Example – smoothing curves

• Flow to reduce curvature:

• Scale factor 0 < l < 1

• Matrix-vector form:

• Drawbacks?

Page 30: Laplacian Operator and Smoothingpanozzo/ustc/05 - Laplacian Operator.pdfFlorida State University Laplacian Operator and Smoothing Xifeng Gao Acknowledgements for the slides: Olga Sorkine-Hornung,

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

Page 31: Laplacian Operator and Smoothingpanozzo/ustc/05 - Laplacian Operator.pdfFlorida State University Laplacian Operator and Smoothing Xifeng Gao Acknowledgements for the slides: Olga Sorkine-Hornung,

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Original curve

Filtering Curves

Page 32: Laplacian Operator and Smoothingpanozzo/ustc/05 - Laplacian Operator.pdfFlorida State University Laplacian Operator and Smoothing Xifeng Gao Acknowledgements for the slides: Olga Sorkine-Hornung,

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1st iteration; l=0.5

Filtering Curves

Page 33: Laplacian Operator and Smoothingpanozzo/ustc/05 - Laplacian Operator.pdfFlorida State University Laplacian Operator and Smoothing Xifeng Gao Acknowledgements for the slides: Olga Sorkine-Hornung,

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2nd iteration; l=0.5

Filtering Curves

Page 34: Laplacian Operator and Smoothingpanozzo/ustc/05 - Laplacian Operator.pdfFlorida State University Laplacian Operator and Smoothing Xifeng Gao Acknowledgements for the slides: Olga Sorkine-Hornung,

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8th iteration; l=0.5

Filtering Curves

Page 35: Laplacian Operator and Smoothingpanozzo/ustc/05 - Laplacian Operator.pdfFlorida State University Laplacian Operator and Smoothing Xifeng Gao Acknowledgements for the slides: Olga Sorkine-Hornung,

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27th iteration; l=0.5

Filtering Curves

Page 36: Laplacian Operator and Smoothingpanozzo/ustc/05 - Laplacian Operator.pdfFlorida State University Laplacian Operator and Smoothing Xifeng Gao Acknowledgements for the slides: Olga Sorkine-Hornung,

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50th iteration; l=0.5

Filtering Curves

Page 37: Laplacian Operator and Smoothingpanozzo/ustc/05 - Laplacian Operator.pdfFlorida State University Laplacian Operator and Smoothing Xifeng Gao Acknowledgements for the slides: Olga Sorkine-Hornung,

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500th iteration; l=0.5

Filtering Curves

Page 38: Laplacian Operator and Smoothingpanozzo/ustc/05 - Laplacian Operator.pdfFlorida State University Laplacian Operator and Smoothing Xifeng Gao Acknowledgements for the slides: Olga Sorkine-Hornung,

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1000th iteration; l=0.5

Filtering Curves

Page 39: Laplacian Operator and Smoothingpanozzo/ustc/05 - Laplacian Operator.pdfFlorida State University Laplacian Operator and Smoothing Xifeng Gao Acknowledgements for the slides: Olga Sorkine-Hornung,

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5000th iteration; l=0.5

Filtering Curves

Page 40: Laplacian Operator and Smoothingpanozzo/ustc/05 - Laplacian Operator.pdfFlorida State University Laplacian Operator and Smoothing Xifeng Gao Acknowledgements for the slides: Olga Sorkine-Hornung,

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10000th iteration; l=0.5

Filtering Curves

Page 41: Laplacian Operator and Smoothingpanozzo/ustc/05 - Laplacian Operator.pdfFlorida State University Laplacian Operator and Smoothing Xifeng Gao Acknowledgements for the slides: Olga Sorkine-Hornung,

Florida State University

50000th iteration; l=0.5

Filtering Curves

Page 42: Laplacian Operator and Smoothingpanozzo/ustc/05 - Laplacian Operator.pdfFlorida State University Laplacian Operator and Smoothing Xifeng Gao Acknowledgements for the slides: Olga Sorkine-Hornung,

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Filtering Surfaces (Demo)

Page 43: Laplacian Operator and Smoothingpanozzo/ustc/05 - Laplacian Operator.pdfFlorida State University Laplacian Operator and Smoothing Xifeng Gao Acknowledgements for the slides: Olga Sorkine-Hornung,

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Thank you!

https://gaoxifeng.github.io