Mesh Denoising via L0 MinimizationRelated Work [Vollmer et al. 1999] [Desbrun et al. 1999] [Kim et...

50
Mesh Denoising via L 0 Minimization Lei He Scott Schaefer Texas A&M University

Transcript of Mesh Denoising via L0 MinimizationRelated Work [Vollmer et al. 1999] [Desbrun et al. 1999] [Kim et...

Page 1: Mesh Denoising via L0 MinimizationRelated Work [Vollmer et al. 1999] [Desbrun et al. 1999] [Kim et al. 2005] [Nealen et al. 2006] [Clarenz et al. 2000] [Bajaj and Xu 2003] [Hildebrandt

Mesh Denoising via L0 Minimization

Lei He Scott Schaefer

Texas A&M University

Page 2: Mesh Denoising via L0 MinimizationRelated Work [Vollmer et al. 1999] [Desbrun et al. 1999] [Kim et al. 2005] [Nealen et al. 2006] [Clarenz et al. 2000] [Bajaj and Xu 2003] [Hildebrandt

Surface Denoising

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Related Work

[Desbrun et al. 1999] [Vollmer et al. 1999] [Nealen et al. 2006] [Kim et al. 2005]

[Clarenz et al. 2000] [Bajaj and Xu 2003] [Hildebrandt et al. 2004] [Tasdizen et al. 2002]

[Yagou et al. 2002] [Fleishman et al. 2003] [Jones et al. 2003] [Zheng et al. 2012]

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Related Work

[Desbrun et al. 1999] [Vollmer et al. 1999] [Nealen et al. 2006] [Kim et al. 2005]

[Clarenz et al. 2000] [Bajaj and Xu 2003] [Hildebrandt et al. 2004] [Tasdizen et al. 2002]

[Yagou et al. 2002] [Fleishman et al. 2003] [Jones et al. 2003] [Zheng et al. 2012]

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Related Work

[Desbrun et al. 1999] [Vollmer et al. 1999] [Nealen et al. 2006] [Kim et al. 2005]

[Clarenz et al. 2000] [Bajaj and Xu 2003] [Hildebrandt et al. 2004] [Tasdizen et al. 2002]

[Yagou et al. 2002] [Fleishman et al. 2003] [Jones et al. 2003] [Zheng et al. 2012]

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Motivation

[Xu et al. 2011]

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L0 Norm

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L0 Minimization for Images

[Xu et al. 2011]

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L0 Minimization for Images

auxiliary variables

[Xu et al. 2011]

> 1, for L=0,1,2,…

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L0 Minimization for Images

• Fix , update locally

• Fix , update globally

• Increment

[Xu et al. 2011]

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L0 Minimization for Images

[Xu et al. 2011]

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L0 Minimization for Images

piecewise constant images! [Xu et al. 2011]

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L0 Minimization for Images

[Xu et al. 2011]

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L0 Minimization for Surfaces

?

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L0 Minimization for Surfaces

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L0 Minimization for Surfaces

• Properties of

• a discrete linear operator

• measure planarity

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Discrete Differential Operator

• When are planar,

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Discrete Differential Operator

A

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Discrete Differential Operator

• When are planar,

• Vertex-based cotangent operator

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Discrete Differential Operator

• When are planar,

• Vertex-based cotangent operator

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Discrete Differential Operator

• When are planar,

• Vertex-based cotangent operator

[Pinkall and Polthier 1993]

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Discrete Differential Operator

input surface vertex-based

cotangent operator

cotangent

edge operator

area-based

edge operator

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Discrete Differential Operator

• When are planar,

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Discrete Differential Operator

• When are planar,

• Edge-based cotangent operator

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Discrete Differential Operator

• When are planar,

• Edge-based cotangent operator

[Bergou et al. 2006]

Page 26: Mesh Denoising via L0 MinimizationRelated Work [Vollmer et al. 1999] [Desbrun et al. 1999] [Kim et al. 2005] [Nealen et al. 2006] [Clarenz et al. 2000] [Bajaj and Xu 2003] [Hildebrandt

Discrete Differential Operator

input surface vertex-based

cotangent operator

cotangent

edge operator

area-based

edge operator

Page 27: Mesh Denoising via L0 MinimizationRelated Work [Vollmer et al. 1999] [Desbrun et al. 1999] [Kim et al. 2005] [Nealen et al. 2006] [Clarenz et al. 2000] [Bajaj and Xu 2003] [Hildebrandt

Discrete Differential Operator

Page 28: Mesh Denoising via L0 MinimizationRelated Work [Vollmer et al. 1999] [Desbrun et al. 1999] [Kim et al. 2005] [Nealen et al. 2006] [Clarenz et al. 2000] [Bajaj and Xu 2003] [Hildebrandt

Discrete Differential Operator

• When are planar,

Page 29: Mesh Denoising via L0 MinimizationRelated Work [Vollmer et al. 1999] [Desbrun et al. 1999] [Kim et al. 2005] [Nealen et al. 2006] [Clarenz et al. 2000] [Bajaj and Xu 2003] [Hildebrandt

Discrete Differential Operator

• When are planar,

Page 30: Mesh Denoising via L0 MinimizationRelated Work [Vollmer et al. 1999] [Desbrun et al. 1999] [Kim et al. 2005] [Nealen et al. 2006] [Clarenz et al. 2000] [Bajaj and Xu 2003] [Hildebrandt

Discrete Differential Operator

• When are planar,

Page 31: Mesh Denoising via L0 MinimizationRelated Work [Vollmer et al. 1999] [Desbrun et al. 1999] [Kim et al. 2005] [Nealen et al. 2006] [Clarenz et al. 2000] [Bajaj and Xu 2003] [Hildebrandt

Discrete Differential Operator

• When are planar,

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Discrete Differential Operator

• When are planar,

Page 33: Mesh Denoising via L0 MinimizationRelated Work [Vollmer et al. 1999] [Desbrun et al. 1999] [Kim et al. 2005] [Nealen et al. 2006] [Clarenz et al. 2000] [Bajaj and Xu 2003] [Hildebrandt

Discrete Differential Operator

input surface vertex-based

cotangent operator

cotangent

edge operator

area-based

edge operator

Page 34: Mesh Denoising via L0 MinimizationRelated Work [Vollmer et al. 1999] [Desbrun et al. 1999] [Kim et al. 2005] [Nealen et al. 2006] [Clarenz et al. 2000] [Bajaj and Xu 2003] [Hildebrandt

Regularization

ground truth noisy input without

regularization

With

regularization

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Regularization

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Regularization

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Regularization

ground truth noisy input without

regularization

with

regularization

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Optimization

> 1, for L=0,1,2,…

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Optimization

> 1, for L=0,1,2,…

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Parameters

input mesh

increase

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Parameters

ground truth noisy input

decrease

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Results

[Desbrun et al. 1999] [Vollmer et al. 1999] [Nealen et al. 2006] [Kim et al. 2005]

[Clarenz et al. 2000] [Bajaj and Xu 2003] [Hildebrandt et al. 2004] [Tasdizen et al. 2002]

[Yagou et al. 2002] [Fleishman et al. 2003] [Jones et al. 2003] [Zheng et al. 2012]

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Results

[Desbrun et al. 1999] [Vollmer et al. 1999] [Nealen et al. 2006] [Kim et al. 2005]

[Clarenz et al. 2000] [Bajaj and Xu 2003] [Hildebrandt et al. 2004] [Tasdizen et al. 2002]

[Yagou et al. 2002] [Fleishman et al. 2003] [Jones et al. 2003] [Zheng et al. 2012]

PMC

BNF BF MF

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Results

noisy input BF PMC MF BNF our result

Page 45: Mesh Denoising via L0 MinimizationRelated Work [Vollmer et al. 1999] [Desbrun et al. 1999] [Kim et al. 2005] [Nealen et al. 2006] [Clarenz et al. 2000] [Bajaj and Xu 2003] [Hildebrandt

Results

noisy input BF PMC MF BNF our result

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Results

noisy input BF PMC MF BNF our result

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Results

noisy input BF PMC MF BNF our result

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Results

noisy input BF PMC MF BNF our result

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Results

noisy input our result

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A failure case

ground truth our result noisy input


Silver et al. v. Pueblo et al.

Silver et al. v. Pueblo et al.

et al. 2015 Cultural... · Bain (1986) Morton et al. (1986) Bowles et al. (1988) Graham & Noonan (2010) Kremers et al. (2012) Crance et al. (2014) Musser et al. (2014) Topic Geographic

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US10199543B2 - Bridgelux 10199543... · 2020-01-03 · Aldaz et al. Uruno et al. Shum et al Aoyagi Shum Komoto Yamamoto et al. Coman et al. Schubert Lin et al. Tyan et al. Cheng et

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(12) United States Patent (io) · Sevcik et al. Ryanetal..... 144/154.5 Govindaraj et al. Kennedy et al. Malone Woods et al. Takahashi et al. Dick et al. Ceroll et al. Caron et al.

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EXHIBIT E - cases.justia.com · 811996 Konrad 1211996 Koppolu et al. 211997 Duscher et al. 311997 Koppolu et al. 511997 Koppolu et al. 711997 Ashe et al. 511998 Koppolu et al. 911998

EXHIBIT E - cases.justia.com · 811996 Konrad 1211996 Koppolu et al. 211997 Duscher et al. 311997 Koppolu et al. 511997 Koppolu et al. 711997 Ashe et al. 511998 Koppolu et al. 911998

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