L-Tangent Norm: a Low Computational Cost Criterion for Choosing - - PowerPoint PPT Presentation

L-Tangent Norm: a Low Computational Cost Criterion for Choosing Regularization Weights and its use for Range Surface Reconstruction

• F. Brunet1,2,3
• A. Bartoli1
• R. Malgouyres3
• N. Navab2

1LASMEA

CNRS/Univ. Blaise Pascal Clermont-Ferrand FRANCE

2CAMPAR

TU München Munich GERMANY

3LAIC

Université d'Auvergne Clermont-Ferrand FRANCE

{brunet,remy.malgouyres}@laic.u-clermont1.fr adrien.bartoli@lasmea.univ-bpclermont.fr navab@cs.tum.edu

3DPVT2008, Atlanta, June 20th

3DPVT 2008, Atlanta, June 20th Surface Reconstruction and Regularization

• F. Brunet et al.
• 1. Introduction
• 2. Previous work
• 3. The L-Tangent Norm
• 4. Experimental results
• 5. Conclusion

Presentation Outline

3DPVT 2008, Atlanta, June 20th Surface Reconstruction and Regularization

• F. Brunet et al.

Surface reconstruction

Approximating a set of range (2.5D) data points by a smooth surface

Introduction Previous work The L-Tangent norm Experimental results Conclusion

3DPVT 2008, Atlanta, June 20th Surface Reconstruction and Regularization

• F. Brunet et al.

The regularization problem

Find a good compromise between

• verfitting and underfitting

Introduction Previous work The L-Tangent norm Experimental results Conclusion

3DPVT 2008, Atlanta, June 20th Surface Reconstruction and Regularization

• F. Brunet et al.

Surface model

• Parametric surface model linear in its control points Pi,j
• In this talk: 1D curves and 2.5D B-spline surfaces

𝑔 𝑦, 𝑧; 𝐪 = 𝑄𝑗,𝑘 𝐶𝑗 𝑦 𝐶

𝑘 𝑧 𝑘 𝑗

Introduction Previous work The L-Tangent norm Experimental results Conclusion

3DPVT 2008, Atlanta, June 20th Surface Reconstruction and Regularization

• F. Brunet et al.

A minimization problem

Data term Error between the surface and the data points

ℰ𝑒 𝑞 = 1 𝑜 𝑔 𝑦𝑗,𝑧𝑗; 𝐪 − 𝑨𝑗 2

𝑜 𝑗=1

Mean squared residual

𝐪∗ = arg min

𝐪∈ℝℎ ℰ𝑒 𝐪 + 𝜇 1−𝜇ℰ𝑠 𝐪

Introduction Previous work The L-Tangent norm Experimental results Conclusion

Σ 2

3DPVT 2008, Atlanta, June 20th Surface Reconstruction and Regularization

• F. Brunet et al.

A minimization problem

Smoothness term Measures the surface smoothness

Bending energy

𝐪∗ = arg min

𝐪∈ℝℎ ℰ𝑒 𝐪 + 𝜇 1−𝜇ℰ𝑠 𝐪

Introduction Previous work The L-Tangent norm Experimental results Conclusion

ℰ𝑠 𝐪 = 𝑒 2 𝜖2𝑔 𝜖𝑦2−𝑒𝜖𝑧𝑒 𝑦, 𝑧; 𝑞

2 2 𝑒=0

d𝑦d𝑧

Ω

3DPVT 2008, Atlanta, June 20th Surface Reconstruction and Regularization

• F. Brunet et al.

A Linear Least Squares problem

Introduction Previous work The L-Tangent norm Experimental results Conclusion

𝐪∗ = arg min

𝐪∈ℝℎ 𝑁𝐪 − 𝐳 2 2 ⇔ 𝐪∗ = 𝑁†𝐳

ℰ𝑒 𝐪 ℰ𝑠 𝐪

𝜇

M

3DPVT 2008, Atlanta, June 20th Surface Reconstruction and Regularization

• F. Brunet et al.

A minimization problem

Regularization parameter Controls the trade-off between the closeness of the surface to the data and its smoothness

𝐪∗ = arg min

𝐪∈ℝℎ ℰ𝑒 𝐪 + 𝜇 1−𝜇ℰ𝑠 𝐪

Introduction Previous work The L-Tangent norm Experimental results Conclusion

λ 1

3DPVT 2008, Atlanta, June 20th Surface Reconstruction and Regularization

• F. Brunet et al.

A minimization problem

Goal: automatically select the regularization parameter

𝐪∗ = arg min

𝐪∈ℝℎ ℰ𝑒 𝐪 + 𝜇 1−𝜇ℰ𝑠 𝐪

Introduction Previous work The L-Tangent norm Experimental results Conclusion

λ 1

3DPVT 2008, Atlanta, June 20th Surface Reconstruction and Regularization

• F. Brunet et al.
• 1. Introduction
• 2. Previous work
• 3. The L-Tangent Norm
• 4. Experimental results
• 5. Conclusion

Outline

Introduction Previous work The L-Tangent norm Experimental results Conclusion

3DPVT 2008, Atlanta, June 20th Surface Reconstruction and Regularization

• F. Brunet et al.

Cross-Validation

Introduction Previous work The L-Tangent norm Experimental results Conclusion

• Introduced by [Wahba 1979]

3DPVT 2008, Atlanta, June 20th Surface Reconstruction and Regularization

• F. Brunet et al.

Cross-Validation

• Introduced by [Wahba 1979]
• A well reconstructed surface is one that generalizes

Introduction Previous work The L-Tangent norm Experimental results Conclusion

3DPVT 2008, Atlanta, June 20th Surface Reconstruction and Regularization

• F. Brunet et al.

Cross-Validation

Introduction Previous work The L-Tangent norm Experimental results Conclusion

λ=0.05

3DPVT 2008, Atlanta, June 20th Surface Reconstruction and Regularization

• F. Brunet et al.

Cross-Validation

Introduction Previous work The L-Tangent norm Experimental results Conclusion

λ=0.05

3DPVT 2008, Atlanta, June 20th Surface Reconstruction and Regularization

• F. Brunet et al.

Cross-Validation

Introduction Previous work The L-Tangent norm Experimental results Conclusion

λ=0.05

3DPVT 2008, Atlanta, June 20th Surface Reconstruction and Regularization

• F. Brunet et al.

Cross-Validation

Introduction Previous work The L-Tangent norm Experimental results Conclusion

λ=0.05

3DPVT 2008, Atlanta, June 20th Surface Reconstruction and Regularization

• F. Brunet et al.

Cross-Validation

Introduction Previous work The L-Tangent norm Experimental results Conclusion

λ=0.05

3DPVT 2008, Atlanta, June 20th Surface Reconstruction and Regularization

• F. Brunet et al.

Cross-Validation

Introduction Previous work The L-Tangent norm Experimental results Conclusion

λ=0.05

3DPVT 2008, Atlanta, June 20th Surface Reconstruction and Regularization

• F. Brunet et al.

Cross-Validation

Introduction Previous work The L-Tangent norm Experimental results Conclusion

λ=0.05

3DPVT 2008, Atlanta, June 20th Surface Reconstruction and Regularization

• F. Brunet et al.

Cross-Validation

Introduction Previous work The L-Tangent norm Experimental results Conclusion

λ=0.05

3DPVT 2008, Atlanta, June 20th Surface Reconstruction and Regularization

• F. Brunet et al.

Cross-Validation

Introduction Previous work The L-Tangent norm Experimental results Conclusion

λ=0.05 λ=0.45

3DPVT 2008, Atlanta, June 20th Surface Reconstruction and Regularization

• F. Brunet et al.

Cross-Validation

Introduction Previous work The L-Tangent norm Experimental results Conclusion

λ=0.05 λ=0.45 λ=0.9

3DPVT 2008, Atlanta, June 20th Surface Reconstruction and Regularization

• F. Brunet et al.

Cross-Validation

Introduction Previous work The L-Tangent norm Experimental results Conclusion

λ=0.05 λ=0.45 λ=0.9 λ=0.99

3DPVT 2008, Atlanta, June 20th Surface Reconstruction and Regularization

• F. Brunet et al.

Cross-Validation

Introduction Previous work The L-Tangent norm Experimental results Conclusion

λ=0.05 λ=0.45 λ=0.9 λ=0.99

0 λ 1 λ*

3DPVT 2008, Atlanta, June 20th Surface Reconstruction and Regularization

• F. Brunet et al.

Problems with Cross-Validation

• Computation time
• Numerical instability
• Pathological cases

Introduction Previous work The L-Tangent norm Experimental results Conclusion

λ λ

• Closed form expression

3DPVT 2008, Atlanta, June 20th Surface Reconstruction and Regularization

• F. Brunet et al.

The L-Curve

Introduction Previous work The L-Tangent norm Experimental results Conclusion

• Introduced by [Lawson and Hanson, 1974]

3DPVT 2008, Atlanta, June 20th Surface Reconstruction and Regularization

• F. Brunet et al.

The L-Curve

• Introduced by [Lawson and Hanson, 1974]
• Search a trade-off between underfitting and overfitting
• The data term shouldn’t be too large
• The regularization term shouldn’t be too large

Introduction Previous work The L-Tangent norm Experimental results Conclusion

3DPVT 2008, Atlanta, June 20th Surface Reconstruction and Regularization

• F. Brunet et al.

The L-Curve

Introduction Previous work The L-Tangent norm Experimental results Conclusion

λ=0.01

0 λ 1 0 λ 1

Data term Smoother

3DPVT 2008, Atlanta, June 20th Surface Reconstruction and Regularization

• F. Brunet et al.

The L-Curve

Introduction Previous work The L-Tangent norm Experimental results Conclusion

λ=0.01

0 λ 1 0 λ 1

Data term Smoother

Data term Smoother

L-Curve

3DPVT 2008, Atlanta, June 20th Surface Reconstruction and Regularization

• F. Brunet et al.

The L-Curve

Introduction Previous work The L-Tangent norm Experimental results Conclusion

λ=0.01 λ=0.02

0 λ 1 0 λ 1

L-Curve Data term Smoother

Data term Smoother

3DPVT 2008, Atlanta, June 20th Surface Reconstruction and Regularization

• F. Brunet et al.

The L-Curve

Introduction Previous work The L-Tangent norm Experimental results Conclusion

λ=0.01 λ=0.02 λ=0.05

0 λ 1 0 λ 1

L-Curve Data term Smoother

Data term Smoother

3DPVT 2008, Atlanta, June 20th Surface Reconstruction and Regularization

• F. Brunet et al.

The L-Curve

Introduction Previous work The L-Tangent norm Experimental results Conclusion

λ=0.01 λ=0.02 λ=0.05 λ=0.1

0 λ 1 0 λ 1

L-Curve Data term Smoother

Data term Smoother

3DPVT 2008, Atlanta, June 20th Surface Reconstruction and Regularization

• F. Brunet et al.

The L-Curve

Introduction Previous work The L-Tangent norm Experimental results Conclusion

λ=0.01 λ=0.02 λ=0.05 λ=0.1 λ=0.3

0 λ 1 0 λ 1

L-Curve Data term Smoother

Data term Smoother

3DPVT 2008, Atlanta, June 20th Surface Reconstruction and Regularization

• F. Brunet et al.

The L-Curve

Introduction Previous work The L-Tangent norm Experimental results Conclusion

λ=0.01 λ=0.02 λ=0.05 λ=0.1 λ=0.3 λ=0.5

0 λ 1 0 λ 1

L-Curve Data term Smoother

Data term Smoother

3DPVT 2008, Atlanta, June 20th Surface Reconstruction and Regularization

• F. Brunet et al.

The L-Curve

Introduction Previous work The L-Tangent norm Experimental results Conclusion

λ=0.01 λ=0.02 λ=0.05 λ=0.1 λ=0.3 λ=0.7 λ=0.5

0 λ 1 0 λ 1

L-Curve Data term Smoother

Data term Smoother

3DPVT 2008, Atlanta, June 20th Surface Reconstruction and Regularization

• F. Brunet et al.

The L-Curve

Introduction Previous work The L-Tangent norm Experimental results Conclusion

λ=0.01 λ=0.02 λ=0.05 λ=0.1 λ=0.3 λ=0.7 λ=0.9 λ=0.5

0 λ 1 0 λ 1

L-Curve Data term Smoother

Data term Smoother

3DPVT 2008, Atlanta, June 20th Surface Reconstruction and Regularization

• F. Brunet et al.

The L-Curve

Introduction Previous work The L-Tangent norm Experimental results Conclusion

λ=0.01 λ=0.99 λ=0.02 λ=0.05 λ=0.1 λ=0.3 λ=0.7 λ=0.9 λ=0.5

0 λ 1 0 λ 1

L-Curve Data term Smoother

Data term Smoother

3DPVT 2008, Atlanta, June 20th Surface Reconstruction and Regularization

• F. Brunet et al.

The L-Curve

𝜇 1 𝜇∗ 𝜇∗

The regularization parameter is chosen as the corner of the L-Curve (the maximal curvature)

Data term Smoother Curvature

L-Curve Curvature of the L-Curve

Introduction Previous work The L-Tangent norm Experimental results Conclusion

3DPVT 2008, Atlanta, June 20th Surface Reconstruction and Regularization

• F. Brunet et al.

The L-Curve

• Computations are fast
• There usually are several maxima

Introduction Previous work The L-Tangent norm Experimental results Conclusion

3DPVT 2008, Atlanta, June 20th Surface Reconstruction and Regularization

• F. Brunet et al.
• 1. Introduction
• 2. Previous work
• 3. The L-Tangent Norm
• 4. Experimental results
• 5. Conclusion

Outline

Introduction Previous work The L-Tangent norm Experimental results Conclusion

3DPVT 2008, Atlanta, June 20th Surface Reconstruction and Regularization

• F. Brunet et al.

The L-Tangent Norm

Data term Smoother

0 λ 1 0 λ 1

λ=0.01

1.8 10

Introduction Previous work The L-Tangent norm Experimental results Conclusion

3DPVT 2008, Atlanta, June 20th Surface Reconstruction and Regularization

• F. Brunet et al.

The L-Tangent Norm

0 λ 1 0 λ 1

λ=0.01 λ=0.15

1.8 10

Introduction Previous work The L-Tangent norm Experimental results Conclusion

Data term Smoother

3DPVT 2008, Atlanta, June 20th Surface Reconstruction and Regularization

• F. Brunet et al.

The L-Tangent Norm

0 λ 1 0 λ 1

λ=0.01 λ=0.15 λ=0.29

1.8 10

Introduction Previous work The L-Tangent norm Experimental results Conclusion

Data term Smoother

3DPVT 2008, Atlanta, June 20th Surface Reconstruction and Regularization

• F. Brunet et al.

The L-Tangent Norm

0 λ 1 0 λ 1

λ=0.01 λ=0.15 λ=0.29 λ=0.43

1.8 10

Introduction Previous work The L-Tangent norm Experimental results Conclusion

Data term Smoother

3DPVT 2008, Atlanta, June 20th Surface Reconstruction and Regularization

• F. Brunet et al.

The L-Tangent Norm

0 λ 1 0 λ 1

λ=0.01 λ=0.15 λ=0.29 λ=0.43 λ=0.57

1.8 10

Introduction Previous work The L-Tangent norm Experimental results Conclusion

Data term Smoother

3DPVT 2008, Atlanta, June 20th Surface Reconstruction and Regularization

• F. Brunet et al.

The L-Tangent Norm

0 λ 1 0 λ 1

λ=0.01 λ=0.15 λ=0.29 λ=0.43 λ=0.57 λ=0.71

1.8 10

Introduction Previous work The L-Tangent norm Experimental results Conclusion

Data term Smoother

3DPVT 2008, Atlanta, June 20th Surface Reconstruction and Regularization

• F. Brunet et al.

The L-Tangent Norm

0 λ 1 0 λ 1

λ=0.01 λ=0.15 λ=0.29 λ=0.43 λ=0.57 λ=0.71 λ=0.85

1.8 10

Introduction Previous work The L-Tangent norm Experimental results Conclusion

Data term Smoother

3DPVT 2008, Atlanta, June 20th Surface Reconstruction and Regularization

• F. Brunet et al.

The L-Tangent Norm

0 λ 1 0 λ 1

λ=0.01 λ=0.99 λ=0.15 λ=0.29 λ=0.43 λ=0.57 λ=0.71 λ=0.85

1.8 10

Introduction Previous work The L-Tangent norm Experimental results Conclusion

Data term Smoother

3DPVT 2008, Atlanta, June 20th Surface Reconstruction and Regularization

• F. Brunet et al.

The L-Tangent Norm

0 λ 1 0 λ 1

λ=0.01 λ=0.99 λ=0.15 λ=0.29 λ=0.43 λ=0.57 λ=0.71 λ=0.85

1.8 10

Normalization

1 1

Introduction Previous work The L-Tangent norm Experimental results Conclusion

Data term Smoother

3DPVT 2008, Atlanta, June 20th Surface Reconstruction and Regularization

• F. Brunet et al.

The L-Tangent Norm

Data term Smoother

0 λ 1 0 λ 1

λ=0.01 λ=0.99 λ=0.15 λ=0.29 λ=0.43 λ=0.57 λ=0.71 λ=0.85

1.8 10

• Norm. data term
• Norm. smoother

1 1

Introduction Previous work The L-Tangent norm Experimental results Conclusion

3DPVT 2008, Atlanta, June 20th Surface Reconstruction and Regularization

• F. Brunet et al.

The L-Tangent Norm

Data term Smoother

0 λ 1 0 λ 1

λ=0.01 λ=0.99 λ=0.15 λ=0.29 λ=0.43 λ=0.57 λ=0.71 λ=0.85

1.8 10

• Norm. data term

Norm. smoother

0 λ 1 λ* 1 1

Introduction Previous work The L-Tangent norm Experimental results Conclusion

3DPVT 2008, Atlanta, June 20th Surface Reconstruction and Regularization

• F. Brunet et al.

The L-Tangent Norm

Introduction Previous work The L-Tangent norm Experimental results Conclusion

3DPVT 2008, Atlanta, June 20th Surface Reconstruction and Regularization

• F. Brunet et al.

The L-Tangent Norm

Introduction Previous work The L-Tangent norm Experimental results Conclusion

3DPVT 2008, Atlanta, June 20th Surface Reconstruction and Regularization

• F. Brunet et al.

The L-Tangent Norm

Introduction Previous work The L-Tangent norm Experimental results Conclusion

3DPVT 2008, Atlanta, June 20th Surface Reconstruction and Regularization

• F. Brunet et al.

The L-Tangent Norm

Introduction Previous work The L-Tangent norm Experimental results Conclusion

3DPVT 2008, Atlanta, June 20th Surface Reconstruction and Regularization

• F. Brunet et al.
• 1. Introduction
• 2. Previous work
• 3. The L-Tangent Norm
• 4. Experimental results
• 5. Conclusion

Outline

Introduction Previous work The L-Tangent norm Experimental results Conclusion

3DPVT 2008, Atlanta, June 20th Surface Reconstruction and Regularization

• F. Brunet et al.

Experimental results

Synthetic data

Introduction Previous work The L-Tangent norm Experimental results Conclusion

3DPVT 2008, Atlanta, June 20th Surface Reconstruction and Regularization

• F. Brunet et al.

Experimental results

Real range images

Introduction Previous work The L-Tangent norm Experimental results Conclusion

3DPVT 2008, Atlanta, June 20th Surface Reconstruction and Regularization

• F. Brunet et al.

Experimental results

Computation time (seconds) Ratio Cross Validation L-Tangent Norm

Evaluation of the criterion for a single λ

• Cross Validation vs. L-Tangent Norm
• Synthetic data

Introduction Previous work The L-Tangent norm Experimental results Conclusion

3DPVT 2008, Atlanta, June 20th Surface Reconstruction and Regularization

• F. Brunet et al.

Experimental results

Computation time (seconds)

Total time of the surface reconstruction

• Cross Validation vs. L-Tangent Norm
• Real range images

Introduction Previous work The L-Tangent norm Experimental results Conclusion

Range image

3DPVT 2008, Atlanta, June 20th Surface Reconstruction and Regularization

• F. Brunet et al.

Experimental results

Comparison of the regularization parameters

…L-Tangent Norm …Cross Validation Regularization parameters obtained with…

Introduction Previous work The L-Tangent norm Experimental results Conclusion

3DPVT 2008, Atlanta, June 20th Surface Reconstruction and Regularization

• F. Brunet et al.

Experimental results

Integral Relative Error (IRE)

Generated surface (ground truth) Reconstructed surface

Introduction Previous work The L-Tangent norm Experimental results Conclusion

Quality of the reconstructed surfaces

• 200 randomly generated surfaces (ground truth)
• Comparison ground truth vs. reconstruction
• =

3DPVT 2008, Atlanta, June 20th Surface Reconstruction and Regularization

• F. Brunet et al.

Experimental results

Introduction Previous work The L-Tangent norm Experimental results Conclusion

Quality of the reconstructed surfaces

• 200 randomly generated surfaces (ground truth)
• Comparison ground truth vs. reconstruction

IRE

3DPVT 2008, Atlanta, June 20th Surface Reconstruction and Regularization

• F. Brunet et al.

Experimental results

Introduction Previous work The L-Tangent norm Experimental results Conclusion

Quality of the reconstructed surfaces

• Real range images

Error Map

• =

Generated surface (ground truth) Reconstructed surface

3DPVT 2008, Atlanta, June 20th Surface Reconstruction and Regularization

• F. Brunet et al.

Experimental results

Introduction Previous work The L-Tangent norm Experimental results Conclusion

Quality of the reconstructed surfaces

• Real range images
• Compact representation: 40x40 control points

3DPVT 2008, Atlanta, June 20th Surface Reconstruction and Regularization

• F. Brunet et al.
• 1. Introduction
• 2. Previous work
• 3. The L-Tangent Norm
• 4. Experimental results
• 5. Conclusion

Outline

Introduction Previous work The L-Tangent norm Experimental results Conclusion

3DPVT 2008, Atlanta, June 20th Surface Reconstruction and Regularization

• F. Brunet et al.

Conclusion

• Novel approach to choose the regularization parameter
• Low computational cost, easy optimization process
• Approximation of the Cross-Validation
• Automatically provides a compact and analytical form
• f large datasets

What’s next?

• Prove that the LTN is an approximation of the CV
• Multiple regularization terms

Introduction Previous work The L-Tangent norm Experimental results Conclusion

Thanks for your attention!