IntroductionAlgorithms

Experiments

Skeleton Based As-Rigid-As-PossibleVolume Modeling

Shaoting Zhang, Andrew Nealen and Dimitris Metaxas

Computer Science Department, Rutgers University

Shaoting Zhang, Andrew Nealen and Dimitris Metaxas Skeleton Based As-Rigid-As-Possible Volume Modeling

IntroductionAlgorithms

Experiments

Introduction

As-rigid-as-possible (ARAP) shapemodeling is a popular technique toobtain natural deformations. Therehave been many excellent methods.

We are interested in the volumepreservation.

VGL is a good approach. However,we do not want to break themanifoldness of ARAP surfacemodeling or sacrificing the speed.

We do it by leveraging the skeletoninformation.

Olga Sorkine, et al,: Laplacian Surface Editing. SGP2004

Olga Sorkine and Marc Alexa: As-Rigid-As-Possible Surface Modeling. Eurographics SGP2007Kun Zhou, et al,: Large mesh deformation using the volumetric graph Laplacian. SIGGRAPH 2005

Shaoting Zhang, Andrew Nealen and Dimitris Metaxas Skeleton Based As-Rigid-As-Possible Volume Modeling

IntroductionAlgorithms

Experiments

Introduction

As-rigid-as-possible (ARAP) shapemodeling is a popular technique toobtain natural deformations. Therehave been many excellent methods.

We are interested in the volumepreservation.

VGL is a good approach. However,we do not want to break themanifoldness of ARAP surfacemodeling or sacrificing the speed.

We do it by leveraging the skeletoninformation.

Olga Sorkine, et al,: Laplacian Surface Editing. SGP2004Olga Sorkine and Marc Alexa: As-Rigid-As-Possible Surface Modeling. Eurographics SGP2007

Kun Zhou, et al,: Large mesh deformation using the volumetric graph Laplacian. SIGGRAPH 2005

Shaoting Zhang, Andrew Nealen and Dimitris Metaxas Skeleton Based As-Rigid-As-Possible Volume Modeling

IntroductionAlgorithms

Experiments

Introduction

As-rigid-as-possible (ARAP) shapemodeling is a popular technique toobtain natural deformations. Therehave been many excellent methods.

We are interested in the volumepreservation.

VGL is a good approach. However,we do not want to break themanifoldness of ARAP surfacemodeling or sacrificing the speed.

We do it by leveraging the skeletoninformation.

Olga Sorkine, et al,: Laplacian Surface Editing. SGP2004Olga Sorkine and Marc Alexa: As-Rigid-As-Possible Surface Modeling. Eurographics SGP2007

Kun Zhou, et al,: Large mesh deformation using the volumetric graph Laplacian. SIGGRAPH 2005

Shaoting Zhang, Andrew Nealen and Dimitris Metaxas Skeleton Based As-Rigid-As-Possible Volume Modeling

IntroductionAlgorithms

Experiments

Introduction

As-rigid-as-possible (ARAP) shapemodeling is a popular technique toobtain natural deformations. Therehave been many excellent methods.

We are interested in the volumepreservation.

VGL is a good approach. However,we do not want to break themanifoldness of ARAP surfacemodeling or sacrificing the speed.

We do it by leveraging the skeletoninformation.

Olga Sorkine, et al,: Laplacian Surface Editing. SGP2004Olga Sorkine and Marc Alexa: As-Rigid-As-Possible Surface Modeling. Eurographics SGP2007Kun Zhou, et al,: Large mesh deformation using the volumetric graph Laplacian. SIGGRAPH 2005

Shaoting Zhang, Andrew Nealen and Dimitris Metaxas Skeleton Based As-Rigid-As-Possible Volume Modeling

IntroductionAlgorithms

Experiments

Introduction

As-rigid-as-possible (ARAP) shapemodeling is a popular technique toobtain natural deformations. Therehave been many excellent methods.

We are interested in the volumepreservation.

VGL is a good approach. However,we do not want to break themanifoldness of ARAP surfacemodeling or sacrificing the speed.

We do it by leveraging the skeletoninformation.

Olga Sorkine, et al,: Laplacian Surface Editing. SGP2004Olga Sorkine and Marc Alexa: As-Rigid-As-Possible Surface Modeling. Eurographics SGP2007Kun Zhou, et al,: Large mesh deformation using the volumetric graph Laplacian. SIGGRAPH 2005

Shaoting Zhang, Andrew Nealen and Dimitris Metaxas Skeleton Based As-Rigid-As-Possible Volume Modeling

IntroductionAlgorithms

Experiments

Linear LSERotation and edge length constraintsSkeleton and volume constraints

Linear LSE with C 0 continuity

Laplacian coordinates representeach point as the weighteddifference between such point andits neighborhoods.

Given original coordinates V , theconnectivity, and m control points,the reconstructed object V ′ can beobtained by minimizing:‖LV ′ − δ‖2

2 + Σmi=1‖v ′

ci− vci‖2

2[LIc

]V ′ =

[δVc

]When rotations are large, thedeformation may not be natural.

Shaoting Zhang, Andrew Nealen and Dimitris Metaxas Skeleton Based As-Rigid-As-Possible Volume Modeling

IntroductionAlgorithms

Experiments

Linear LSERotation and edge length constraintsSkeleton and volume constraints

Linear LSE with C 0 continuity

Laplacian coordinates representeach point as the weighteddifference between such point andits neighborhoods.

Given original coordinates V , theconnectivity, and m control points,the reconstructed object V ′ can beobtained by minimizing:‖LV ′ − δ‖2

2 + Σmi=1‖v ′

ci− vci‖2

2

[LIc

]V ′ =

[δVc

]When rotations are large, thedeformation may not be natural.

Shaoting Zhang, Andrew Nealen and Dimitris Metaxas Skeleton Based As-Rigid-As-Possible Volume Modeling

IntroductionAlgorithms

Experiments

Linear LSERotation and edge length constraintsSkeleton and volume constraints

Linear LSE with C 0 continuity

Laplacian coordinates representeach point as the weighteddifference between such point andits neighborhoods.

Given original coordinates V , theconnectivity, and m control points,the reconstructed object V ′ can beobtained by minimizing:‖LV ′ − δ‖2

2 + Σmi=1‖v ′

ci− vci‖2

2[LIc

]V ′ =

[δVc

]

When rotations are large, thedeformation may not be natural.

Shaoting Zhang, Andrew Nealen and Dimitris Metaxas Skeleton Based As-Rigid-As-Possible Volume Modeling

IntroductionAlgorithms

Experiments

Linear LSERotation and edge length constraintsSkeleton and volume constraints

Linear LSE with C 0 continuity

Laplacian coordinates representeach point as the weighteddifference between such point andits neighborhoods.

Given original coordinates V , theconnectivity, and m control points,the reconstructed object V ′ can beobtained by minimizing:‖LV ′ − δ‖2

2 + Σmi=1‖v ′

ci− vci‖2

2[LIc

]V ′ =

[δVc

]When rotations are large, thedeformation may not be natural.

Shaoting Zhang, Andrew Nealen and Dimitris Metaxas Skeleton Based As-Rigid-As-Possible Volume Modeling

IntroductionAlgorithms

Experiments

Linear LSERotation and edge length constraintsSkeleton and volume constraints

Linear LSE with C 0 continuity

Laplacian coordinates representeach point as the weighteddifference between such point andits neighborhoods.

Given original coordinates V , theconnectivity, and m control points,the reconstructed object V ′ can beobtained by minimizing:‖LV ′ − δ‖2

2 + Σmi=1‖v ′

ci− vci‖2

2[LIc

]V ′ =

[δVc

]When rotations are large, thedeformation may not be natural.

Shaoting Zhang, Andrew Nealen and Dimitris Metaxas Skeleton Based As-Rigid-As-Possible Volume Modeling

IntroductionAlgorithms

Experiments

Linear LSERotation and edge length constraintsSkeleton and volume constraints

Rotation and edge length constraints

Iterate two steps to recoverrotations:

Step1: Initial guess from solvingnaive LSE.

Step2: Find optimal rotations, thenupdate the linear system (edgelength preserving).

Robustness, simplicity, efficiency.

However, there is no volumepreserving constraint.

Olga Sorkine and Marc Alexa: As-Rigid-As-Possible Surface Modeling. Eurographics SGP2007Olga Sorkine: Least-Squares Rigid Motion Using SVD

Shaoting Zhang, Andrew Nealen and Dimitris Metaxas Skeleton Based As-Rigid-As-Possible Volume Modeling

IntroductionAlgorithms

Experiments

Linear LSERotation and edge length constraintsSkeleton and volume constraints

Rotation and edge length constraints

Iterate two steps to recoverrotations:

Step1: Initial guess from solvingnaive LSE.

Step2: Find optimal rotations, thenupdate the linear system (edgelength preserving).

Robustness, simplicity, efficiency.

However, there is no volumepreserving constraint.

Olga Sorkine and Marc Alexa: As-Rigid-As-Possible Surface Modeling. Eurographics SGP2007Olga Sorkine: Least-Squares Rigid Motion Using SVD

Shaoting Zhang, Andrew Nealen and Dimitris Metaxas Skeleton Based As-Rigid-As-Possible Volume Modeling

IntroductionAlgorithms

Experiments

Linear LSERotation and edge length constraintsSkeleton and volume constraints

Rotation and edge length constraints

Iterate two steps to recoverrotations:

Step1: Initial guess from solvingnaive LSE.

Step2: Find optimal rotations, thenupdate the linear system (edgelength preserving).

Robustness, simplicity, efficiency.

However, there is no volumepreserving constraint.

Olga Sorkine and Marc Alexa: As-Rigid-As-Possible Surface Modeling. Eurographics SGP2007Olga Sorkine: Least-Squares Rigid Motion Using SVD

Shaoting Zhang, Andrew Nealen and Dimitris Metaxas Skeleton Based As-Rigid-As-Possible Volume Modeling

IntroductionAlgorithms

Experiments

Linear LSERotation and edge length constraintsSkeleton and volume constraints

Rotation and edge length constraints

Iterate two steps to recoverrotations:

Step1: Initial guess from solvingnaive LSE.

Step2: Find optimal rotations, thenupdate the linear system (edgelength preserving).

Robustness, simplicity, efficiency.

However, there is no volumepreserving constraint.

Olga Sorkine and Marc Alexa: As-Rigid-As-Possible Surface Modeling. Eurographics SGP2007Olga Sorkine: Least-Squares Rigid Motion Using SVD

Shaoting Zhang, Andrew Nealen and Dimitris Metaxas Skeleton Based As-Rigid-As-Possible Volume Modeling

IntroductionAlgorithms

Experiments

Linear LSERotation and edge length constraintsSkeleton and volume constraints

Rotation and edge length constraints

Iterate two steps to recoverrotations:

Step1: Initial guess from solvingnaive LSE.

Step2: Find optimal rotations, thenupdate the linear system (edgelength preserving).

Robustness, simplicity, efficiency.

However, there is no volumepreserving constraint.

Olga Sorkine and Marc Alexa: As-Rigid-As-Possible Surface Modeling. Eurographics SGP2007Olga Sorkine: Least-Squares Rigid Motion Using SVD

Shaoting Zhang, Andrew Nealen and Dimitris Metaxas Skeleton Based As-Rigid-As-Possible Volume Modeling

IntroductionAlgorithms

Experiments

Linear LSERotation and edge length constraintsSkeleton and volume constraints

Skeleton and volume constraints

Use volumetric mesh? Manifoldness,computational complexity, etc.

Use both the skeleton and edge lengthconstraint to roughly preserve the volume,without breaking the manifoldness ofARAP or increasing the computationcomplexity. L | 0

Ic | 0Ls

[V ′

V ′s

]=

δVc

δs

One-way coupling property.

Shaoting Zhang, Andrew Nealen and Dimitris Metaxas Skeleton Based As-Rigid-As-Possible Volume Modeling

IntroductionAlgorithms

Experiments

Linear LSERotation and edge length constraintsSkeleton and volume constraints

Skeleton and volume constraints

Use volumetric mesh? Manifoldness,computational complexity, etc.

Use both the skeleton and edge lengthconstraint to roughly preserve the volume,without breaking the manifoldness ofARAP or increasing the computationcomplexity.

L | 0Ic | 0

Ls

[V ′

V ′s

]=

δVc

δs

One-way coupling property.

α

β

Wij

Shaoting Zhang, Andrew Nealen and Dimitris Metaxas Skeleton Based As-Rigid-As-Possible Volume Modeling

IntroductionAlgorithms

Experiments

Linear LSERotation and edge length constraintsSkeleton and volume constraints

Skeleton and volume constraints

Use volumetric mesh? Manifoldness,computational complexity, etc.

Use both the skeleton and edge lengthconstraint to roughly preserve the volume,without breaking the manifoldness ofARAP or increasing the computationcomplexity. L | 0

Ic | 0Ls

[V ′

V ′s

]=

δVc

δs

One-way coupling property.

α

β

Wij

Shaoting Zhang, Andrew Nealen and Dimitris Metaxas Skeleton Based As-Rigid-As-Possible Volume Modeling

IntroductionAlgorithms

Experiments

Linear LSERotation and edge length constraintsSkeleton and volume constraints

Skeleton and volume constraints

Use volumetric mesh? Manifoldness,computational complexity, etc.

Use both the skeleton and edge lengthconstraint to roughly preserve the volume,without breaking the manifoldness ofARAP or increasing the computationcomplexity. L | 0

Ic | 0Ls

[V ′

V ′s

]=

δVc

δs

One-way coupling property.

α

β

Wij

Shaoting Zhang, Andrew Nealen and Dimitris Metaxas Skeleton Based As-Rigid-As-Possible Volume Modeling

IntroductionAlgorithms

Experiments

Linear LSERotation and edge length constraintsSkeleton and volume constraints

Mesh editing framework

Manually define theskeleton.

Evenly generate skeletonpoints, and connect themwith surface verticesautomatically.

Manually select anchorpoints (bottom) and controlpoints (top).

Interactively deform theshape.

Shaoting Zhang, Andrew Nealen and Dimitris Metaxas Skeleton Based As-Rigid-As-Possible Volume Modeling

IntroductionAlgorithms

Experiments

Linear LSERotation and edge length constraintsSkeleton and volume constraints

Mesh editing framework

Manually define theskeleton.

Evenly generate skeletonpoints, and connect themwith surface verticesautomatically.

Manually select anchorpoints (bottom) and controlpoints (top).

Interactively deform theshape.

Shaoting Zhang, Andrew Nealen and Dimitris Metaxas Skeleton Based As-Rigid-As-Possible Volume Modeling

IntroductionAlgorithms

Experiments

Linear LSERotation and edge length constraintsSkeleton and volume constraints

Mesh editing framework

Manually define theskeleton.

Evenly generate skeletonpoints, and connect themwith surface verticesautomatically.

Manually select anchorpoints (bottom) and controlpoints (top).

Interactively deform theshape.

Shaoting Zhang, Andrew Nealen and Dimitris Metaxas Skeleton Based As-Rigid-As-Possible Volume Modeling

IntroductionAlgorithms

Experiments

Linear LSERotation and edge length constraintsSkeleton and volume constraints

Mesh editing framework

Manually define theskeleton.

Evenly generate skeletonpoints, and connect themwith surface verticesautomatically.

Manually select anchorpoints (bottom) and controlpoints (top).

Interactively deform theshape.

Shaoting Zhang, Andrew Nealen and Dimitris Metaxas Skeleton Based As-Rigid-As-Possible Volume Modeling

IntroductionAlgorithms

Experiments

Linear LSERotation and edge length constraintsSkeleton and volume constraints

Mesh editing framework

Manually define theskeleton.

Evenly generate skeletonpoints, and connect themwith surface verticesautomatically.

Manually select anchorpoints (bottom) and controlpoints (top).

Interactively deform theshape.

Shaoting Zhang, Andrew Nealen and Dimitris Metaxas Skeleton Based As-Rigid-As-Possible Volume Modeling

IntroductionAlgorithms

Experiments

Linear LSERotation and edge length constraintsSkeleton and volume constraints

Mesh editing framework

Manually define theskeleton.

Evenly generate skeletonpoints, and connect themwith surface verticesautomatically.

Manually select anchorpoints (bottom) and controlpoints (top).

Interactively deform theshape.

Shaoting Zhang, Andrew Nealen and Dimitris Metaxas Skeleton Based As-Rigid-As-Possible Volume Modeling

IntroductionAlgorithms

Experiments

Experimental settingsResults

Experimental settings

The C++ implementation was run on a Intel Core2 Quad 2.40GHzCPU with 8G RAM.

We compare the linear LSE, ARAP surface modeling and ourmethod.

We tested on the cactus model (620 vertices, 1, 236 polygons) andthe horse model (2, 482 vertices, 4, 960 polygons).

The relative root mean square errors of edge lengths and volumemagnitudes are reported.

Shaoting Zhang, Andrew Nealen and Dimitris Metaxas Skeleton Based As-Rigid-As-Possible Volume Modeling

IntroductionAlgorithms

Experiments

Experimental settingsResults

Results

Model RRMS-E RE-V Times

(b) 0.126 0.453 0.017(c) 0.074 0.131 0.024(d) 0.075 0.056 0.025

Shaoting Zhang, Andrew Nealen and Dimitris Metaxas Skeleton Based As-Rigid-As-Possible Volume Modeling

IntroductionAlgorithms

Experiments

Experimental settingsResults

Results

Model RRMS-E RE-V Times

(b) 0.068 0.356 0.117(c) 0.040 0.125 0.121

Shaoting Zhang, Andrew Nealen and Dimitris Metaxas Skeleton Based As-Rigid-As-Possible Volume Modeling

IntroductionAlgorithms

Experiments

Experimental settingsResults

Conclusions

We proposed an approach to approximately preserve thevolume without breaking the manifoldness of traditionalARAP or increasing the computational complexity.

Our method is easy-to-implement and may be useful tosystems relying on ARAP techniques.

Limitations: Skeleton generation; complex skeletons; selfintersection.

Shaoting Zhang, Andrew Nealen and Dimitris Metaxas Skeleton Based As-Rigid-As-Possible Volume Modeling

IntroductionAlgorithms

Experiments

Experimental settingsResults

Thanks for listening.

Shaoting Zhang, Andrew Nealen and Dimitris Metaxas Skeleton Based As-Rigid-As-Possible Volume Modeling