Construction of Manifolds via Compatible Sparse Representations - - PowerPoint PPT Presentation

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Construction of Manifolds via Compatible Sparse Representations - - PowerPoint PPT Presentation

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Construction of Manifolds via Compatible Sparse Representations Ruimin Wang, Ligang Liu , Zhouwang Yang, Kang Wang, Wen Shan, Jiansong Deng, Falai Chen University of Science and Technology of China Problem: Fitting Data with Smooth Surface

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Construction of Manifolds via Compatible Sparse Representations

Ruimin Wang, Ligang Liu, Zhouwang Yang, Kang Wang, Wen Shan, Jiansong Deng, Falai Chen University of Science and Technology of China

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Problem: Fitting Data with Smooth Surface

Point Cloud A Smooth Surface

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Problem: Fitting Data with Smooth Surface

  • Challenging: capturing sharp features
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Problem: Data Fitting

  • Input: A set of points 𝑦𝑗, 𝑧𝑗 , 𝑗 = 0, … , π‘œ
  • Output: A function which fits the point set

𝑧 = 𝑔(𝑦)

(𝑦𝑗, 𝑧𝑗) (𝑦0, 𝑧0) (π‘¦π‘œ, π‘§π‘œ)

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Data Fitting: What Function?

  • What type of functions for 𝑔(𝑦)?

𝑧 = 𝑔(𝑦)

(𝑦𝑗, 𝑧𝑗) (𝑦0, 𝑧0) (π‘¦π‘œ, π‘§π‘œ)

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Data Fitting: Function Space

  • Assuming: basis functions {𝑐𝑗 𝑦 , 𝑗 = 0, … , 𝑛}
  • Finding a member in a family of functions:

𝑔(𝑦) = 𝛽𝑗𝑐𝑗(𝑦)

𝑛 𝑙=0

i.e., representing 𝑔(𝑦) as a (coefficient) point 𝛽 = (𝛽0, 𝛽1,…, 𝛽𝑛) in 𝑆𝑛+1

  • Finding optimal (𝛽0, 𝛽1,…, 𝛽𝑛) by minimizing the fitting error:

min

𝛽 (𝑧𝑗 βˆ’ 𝑔(𝑦𝑗))2

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Data Fitting: Function Space

  • Basis functions {𝑐𝑗 𝑦 , 𝑗 = 0, … , 𝑛}

– Polynomial function basis {1, 𝑦, 𝑦2, … , 𝑦𝑛} – Trigonometric function basis {1, sin 𝑦 , cos 𝑦 , sin 2𝑦 , cos 2𝑦 , … } – Exponential function basis {1, 𝑓𝑦, 𝑓2𝑦, … , 𝑓𝑛𝑦} – …

  • If we choose enough number of basis (𝑛 = π‘œ), the fitting

error can be 0!

– the fitting function 𝑔(𝑦) is an interpolation

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Overfitting Problem

  • How to choose appropriate number of basis?

𝛽0 + 𝛽1𝑦 𝛽0 + 𝛽1𝑦 + 𝛽2𝑦2 𝛽0 + 𝛽1𝑦 + 𝛽2𝑦2 +𝛽3𝑦3 + 𝛽4𝑦4

High bias (underfitting) β€œJust right” High variance (overfitting)

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Sparse Representation

  • An over-complete dictionary (atom functions)

– Finding a β€˜best’ fit from larger family of functions

  • Choose as least number of basis as possible

– most of the elements of 𝛽 = (𝛽0, 𝛽1,…, 𝛽𝑛) are 0 – i.e., 𝛽 0 (number of non-zero elements) is less than some threshold πœ€

min

𝛽 (𝑧𝑗 βˆ’ 𝑔(𝑦𝑗))2

s.t. 𝛽 0 ≀ πœ€

min

𝛽 (𝑧𝑗 βˆ’ 𝑔(𝑦𝑗))2

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3D Surface Case

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Parameterization of Local Patch

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Representing Sharp Features?

  • Smooth functions cannot represent 𝐷0 sharp features

cusp dart crease

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Idea: 𝐷0 Atom Functions

  • Introduce 𝐷0 atom functions in the dictionary

– Shape functions representing non-smooth finite elements in FEM

  • Each atom function

– A bilinear quadrilateral element shape function defined on one edge A 𝐷0 atom function defined on the edge (in red) of a vertex (in green) with valence 5

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𝐷0 Atom Functions

  • A total of 55 shape functions for vertices with valence 3-7

– Add more atom functions for vertices with valence > 7

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Dictionary: Total Atom Functions

  • 120 polynomial functions with degree up to 14
  • 55 𝐷0 atom functions

A patch with sharp features Underfitting Overfitting Result by sparse fitting

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How to stitch local patches?

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Manifold Representation

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Previous Works on Manifold Construction

  • [Grimm and Hughes 1995]
  • [Ying and Zorin 2004]
  • [Gu et al. 2006]
  • [Wang et al. 2008]
  • [Della Vecchia and Juettler 2009]
  • [Tosun and Zorin 2011]
  • …
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Application 1: Approximating Subdivision Surface

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Problem

  • Construct manifolds to approximate subdivision

surfaces with sharp features

– Orange lines are specified as sharp features Input Mesh Manifold Surface

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Construction of the Charts

[Ying and Zorin 2004]

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Incompatible Local Patches

𝑀𝑗 π‘€π‘˜

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Global Fitting Error

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Global Fitting Error

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Optimization Solver

Forward error evaluation

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Optimization Solver

Forward error evaluation Backward Update

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Optimization Solver

Local sparse optimization and Global sparse optimization iteratively

Final Result

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Example

Control mesh Result Top 5 selected atoms (𝐷0 in red) Close-up

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Different Subdivision Rules

Control Mesh Different Geometry

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More Examples

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Application 2: Manifold from Curve Network

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Sampling Points on Curves

Input curve network Domain manifold Result manifold Sampled points Parameterization

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Results

Domain mesh Different manifold surfaces from different geometries

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Results

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Conclusions

  • A novel manifold construction method
  • Sparse representation for local geometry
  • Global compatibility
  • Representing sharp features
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Future Work

  • No guarantee to capture all geometric features

– Learning geometry features

  • Slow sparsity optimization

– Speed up

  • Other applications

– Surface reconstruction, denoising, and compression

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Thank you!

Ligang Liu, http://staff.ustc.edu.cn/~lgliu