INFORMATIK

Laplacian Surface EditingOlga Sorkine Daniel Cohen-Or Yaron Lipman

Tel Aviv University

Marc Alexa

TU Darmstadt

Christian Rössl Hans-Peter Seidel

Max-Planck Institut für Informatik

INFORMATIKDifferential coordinates

Intrinsic surface representation Allows various surface editing operations:

– Detail-preserving mesh editing

INFORMATIKDifferential coordinates

Intrinsic surface representation Allows various surface editing operations:

– Detail-preserving mesh editing– Coating transfer

INFORMATIKDifferential coordinates

Intrinsic surface representation Allows various surface editing operations:

– Detail-preserving mesh editing– Coating transfer– Mesh transplanting

INFORMATIKWhat is it?

Differential coordinates are defined by the discrete Laplacian operator:

For highly irregular meshes: cotangent weights [Desbrun et al. 99]

( )

1( )

j N ii

Ld

i i i jδ v v v

average of the neighbors

INFORMATIKWhy differential coordinates?

They represent the local detail / local shape description– The direction approximates the normal– The size approximates the mean curvature

( )

1

N iid

i iv

δ v v 1

( )ds

len

i

v

v v

( ) 0

1lim ( )

( )lends H

len

i i i

v

v v v n

INFORMATIKWhy differential coordinates?

Local detail representation – enables detail preservation through various modeling tasks

Representation with sparse matrices Efficient linear surface reconstruction

INFORMATIKOverall framework

Compute differential representation

Pose modeling constraints

Reconstruct the surface – in least-squares sense

( )L V

,i i i C v u

2 2arg min ( ) i

i Ci

VV L V

uv

INFORMATIKOverall framework

ROI is bounded by a belt (static anchors) Manipulation through handle(s)

INFORMATIKRelated work

Multi-resolution: [Zorin el al. 97], [Kobbelt et al. 98], [Guskov et al. 99], [Boier-Martin et al. 04], [Botsch and Kobbelt 04] 2

Laplacian smoothing: Taubin [SIGGRAPH 95] Laplacian Morphing: Alexa [TVC 03] Image editing: Perez et al. [SIGGRAPH 03] Mesh Editing: Yu et al. [SIGGRAPH 04]

INFORMATIKProblem: invariance to transformations

The basic Laplacian operator is translation-invariant, but not rotation- and scale-invariant

Reconstruction attempts to preserve the original global orientation of the details

INFORMATIKInvariance – solutions

Explicit transformation of the differential coordinates prior to surface reconstruction

– Lipman, Sorkine, Cohen-Or, Levin, Rössl and Seidel, “Differential Coordinates for Interactive Mesh Editing“, SMI 2004

• Estimation of rotations from naive reconstruction

– Yu, Zhou, Xu, Shi, Bao, Guo and Shum, “Mesh Editing With Poisson-Based Gradient Field Manipulation“, SIGGRAPH 2004

• Propagation of handle transformation to the rest of the ROI

INFORMATIKEstimation of rotations

[Lipman et al. 2004] estimate rotation of local frames– Reconstruct the surface with the original Laplacians– Estimate the normals of underlying smooth surface– Rotate the Laplacians and reconstruct again

INFORMATIKExplicit assignment of rotations

Disadvantages:– Heuristic estimation of the rotations– Speed depends on the support of the smooth normal estimation

operator; for highly detailed surfaces it must be large

almost a height field not a height field

INFORMATIKImplicit definition of transformations

The idea: solve for local transformations AND the edited surface simultaneously!

22

1

arg min ( ) ( )n

ii jV

ji

ij C

LV T

δv v u

Transformationof the local frame

INFORMATIKDefining the transformations Ti

How to formulate Ti ?

– Based on the local (1-ring) neighborhood

– Linear dependence on the unknown vi’

1

2 2

1

kk

ii

i

i

i

i

i i

i

T

T

T

v v

v

v

v

v

22

1

arg min ( ) ( )n

ii jV

ji

ij C

LV T

δv v u

Members of the 1-ring of i-th vertex

iT

INFORMATIKDefining the transformations Ti

First attempt: define Ti simply by solving

2

1

arg minj j

i

i

k

i ij

iT

TT

v v

1 21 2

| | |

| | |

| | |

| | |k ki i i i iiiT

vv v vv v

INFORMATIKDefining the transformations Ti

Plug the expressions for Ti into the least-squares reconstruction formula:

22

1

arg min ( )n

i ji

i ij

jV C

V TL

v vδ u

Linear combination

of the unknown vi’

INFORMATIKConstraining Ti

Trivial solution for Ti will result in membrane surface reconstruction

To preserve the shape of the details we constrain Ti to rotations, uniform scales and translations

11 12 13 14

21 22 23 24

31 32 33 34

41 42 43 44

i

t t t t

t t t tT

t t t t

t t t t

Linear constraints on tlm

so that Ti is rotation+scale+translation

??

INFORMATIKConstraining Ti – 2D case

Easy in 2D:

0 0 cos sin

0 0 sin cos

0 0 1 0 0 1 0 0 1

x x

i y y

s d w a t

T s d a w t

INFORMATIKConstraining Ti – 3D case

Not linear in 3D:

Linearize by dropping the quadratic term

rotation +exp

uniform scaleTs H s I H

h h

is 3 3 skew-symmetric, H H x h x

INFORMATIKAdjusting Ti

Due to linearization, Ti scale the space along the h axis by cos

When is large, this causes anisotropy

Possible correction:– Compute Ti , remove the scaling component and reconstruct the

surface again from the corrected i

– Apply our technique from [Lipman et al. 04] first, and then the current technique – with small .

INFORMATIKSome results

INFORMATIKSome results

INFORMATIKSome results

INFORMATIKSome results

INFORMATIKSome results

Video...

INFORMATIKDetail transfer and mixing

“Peel“ the coating of one surface and transfer to another

INFORMATIKDetail transfer and mixing

Correspondence:

– Parameterization onto a common domain and elastic warp to align the features, if needed

INFORMATIKDetail transfer and mixing

Detail peeling:

i i i

i iSmoothing by

[Desbrun et al.99]

INFORMATIKDetail transfer and mixing

Changing local frames:

ii

INFORMATIKDetail transfer and mixing

Reconstruction of target surface from :

target i i

target

INFORMATIKExamples

INFORMATIKExamples

INFORMATIKMixing Laplacians

Taking weighted average of i and ‘i

INFORMATIKMesh transplanting

The user defines– Part to transplant– Where to transplant– Spatial orientation and scale

Topological stitching Geometrical stitching via Laplacian mixing

INFORMATIKMesh transplanting

Details gradually change in the transition area

INFORMATIKMesh transplanting

Details gradually change in the transition area

INFORMATIKConclusions

Differential coordinates are useful for applications that need to preserve local details

Reconstruction by linear least-squares – smoothly distributes the error across the domain

Linearization of 3D rotations was needed in order to solve for optimal local transformations – can we do better?

INFORMATIKAcknowledgments

German Israel Foundation (GIF) Israel Science Foundation (founded by the Israel

Academy of Sciences and Humanities) Israeli Ministry of Science

Bunny, Dragon, Feline courtesy of Stanford University Octopus courtesy of Mark Pauly

INFORMATIK

Thank you!

INFORMATIKGradual transition