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. Brunet 1,2,3 A. Bartoli 1 R. Malgouyres 3 N. Navab 2 1 LASMEA 2 CAMPAR 3 LAIC CNRS/Univ. Blaise Pascal TU München Université d'Auvergne Clermont-Ferrand Munich Clermont-Ferrand FRANCE GERMANY FRANCE {brunet,remy.malgouyres}@laic.u-clermont1.fr adrien.bartoli@lasmea.univ-bpclermont.fr navab@cs.tum.edu 3DPVT2008, Atlanta, June 20 th

Presentation Outline 1. Introduction 2. Previous work 3. The L-Tangent Norm 4. Experimental results 5. Conclusion F. Brunet et al. Surface Reconstruction and Regularization 3DPVT 2008, Atlanta, June 20 th

 Introduction  Previous work  The L-Tangent norm  Experimental results  Conclusion Surface reconstruction Approximating a set of range (2.5D) data points by a smooth surface F. Brunet et al. Surface Reconstruction and Regularization 3DPVT 2008, Atlanta, June 20 th

 Introduction  Previous work  The L-Tangent norm  Experimental results  Conclusion The regularization problem Find a good compromise between overfitting and underfitting F. Brunet et al. Surface Reconstruction and Regularization 3DPVT 2008, Atlanta, June 20 th

 Introduction  Previous work  The L-Tangent norm  Experimental results  Conclusion Surface model • Parametric surface model linear in its control points P i,j • In this talk: 1D curves and 2.5D B-spline surfaces 𝑔 𝑦 , 𝑧 ; 𝐪 = 𝑄 𝑗 , 𝑘 𝐶 𝑗 𝑦 𝐶 𝑘 𝑧 𝑗 𝑘 F. Brunet et al. Surface Reconstruction and Regularization 3DPVT 2008, Atlanta, June 20 th

 Introduction  Previous work  The L-Tangent norm  Experimental results  Conclusion A minimization problem 𝐪 ∗ = arg min 𝜇 𝐪∈ℝ ℎ ℰ 𝑒 𝐪 + 1 −𝜇 ℰ 𝑠 𝐪 Data term Error between the surface and the data points Mean squared residual Σ 2 𝑜 ℰ 𝑒 𝑞 = 1 𝑜 𝑔 𝑦 𝑗 , 𝑧 𝑗 ; 𝐪 − 𝑨 𝑗 2 𝑗 =1 F. Brunet et al. Surface Reconstruction and Regularization 3DPVT 2008, Atlanta, June 20 th

 Introduction  Previous work  The L-Tangent norm  Experimental results  Conclusion A minimization problem 𝐪 ∗ = arg min 𝜇 𝐪∈ℝ ℎ ℰ 𝑒 𝐪 + 1 −𝜇 ℰ 𝑠 𝐪 Smoothness term Measures the surface smoothness Bending energy 2 2 𝜖 2 𝑔 ℰ 𝑠 𝐪 = 𝑒 𝜖𝑦 2 −𝑒 𝜖𝑧 𝑒 𝑦 , 𝑧 ; 𝑞 d 𝑦 d 𝑧 2 Ω 𝑒 =0 F. Brunet et al. Surface Reconstruction and Regularization 3DPVT 2008, Atlanta, June 20 th

 Introduction  Previous work  The L-Tangent norm  Experimental results  Conclusion A Linear Least Squares problem 𝐪 ∗ = arg min 2 ⇔ 𝐪 ∗ = 𝑁 † 𝐳 𝐪∈ℝ ℎ 𝑁𝐪 − 𝐳 2 M ℰ 𝑒 𝐪 ℰ 𝑠 𝐪 𝜇 F. Brunet et al. Surface Reconstruction and Regularization 3DPVT 2008, Atlanta, June 20 th

 Introduction  Previous work  The L-Tangent norm  Experimental results  Conclusion A minimization problem 𝐪 ∗ = arg min 𝜇 𝐪∈ℝ ℎ ℰ 𝑒 𝐪 + 1 −𝜇 ℰ 𝑠 𝐪 Regularization parameter Controls the trade-off between the closeness of the surface to the data and its smoothness 0 λ 1 F. Brunet et al. Surface Reconstruction and Regularization 3DPVT 2008, Atlanta, June 20 th

 Introduction  Previous work  The L-Tangent norm  Experimental results  Conclusion A minimization problem 𝐪 ∗ = arg min 𝜇 𝐪∈ℝ ℎ ℰ 𝑒 𝐪 + 1 −𝜇 ℰ 𝑠 𝐪 Goal: automatically select the regularization parameter 0 λ 1 F. Brunet et al. Surface Reconstruction and Regularization 3DPVT 2008, Atlanta, June 20 th

 Introduction  Previous work  The L-Tangent norm  Experimental results  Conclusion Outline 1. Introduction 2. Previous work 3. The L-Tangent Norm 4. Experimental results 5. Conclusion F. Brunet et al. Surface Reconstruction and Regularization 3DPVT 2008, Atlanta, June 20 th

 Introduction  Previous work  The L-Tangent norm  Experimental results  Conclusion Cross-Validation • Introduced by [Wahba 1979] F. Brunet et al. Surface Reconstruction and Regularization 3DPVT 2008, Atlanta, June 20 th

 Introduction  Previous work  The L-Tangent norm  Experimental results  Conclusion Cross-Validation • Introduced by [Wahba 1979] • A well reconstructed surface is one that generalizes F. Brunet et al. Surface Reconstruction and Regularization 3DPVT 2008, Atlanta, June 20 th

 Introduction  Previous work  The L-Tangent norm  Experimental results  Conclusion Cross-Validation λ =0.05 F. Brunet et al. Surface Reconstruction and Regularization 3DPVT 2008, Atlanta, June 20 th

 Introduction  Previous work  The L-Tangent norm  Experimental results  Conclusion Cross-Validation λ =0.05 F. Brunet et al. Surface Reconstruction and Regularization 3DPVT 2008, Atlanta, June 20 th

 Introduction  Previous work  The L-Tangent norm  Experimental results  Conclusion Cross-Validation λ =0.05 F. Brunet et al. Surface Reconstruction and Regularization 3DPVT 2008, Atlanta, June 20 th

 Introduction  Previous work  The L-Tangent norm  Experimental results  Conclusion Cross-Validation λ =0.05 F. Brunet et al. Surface Reconstruction and Regularization 3DPVT 2008, Atlanta, June 20 th

 Introduction  Previous work  The L-Tangent norm  Experimental results  Conclusion Cross-Validation λ =0.05 F. Brunet et al. Surface Reconstruction and Regularization 3DPVT 2008, Atlanta, June 20 th

 Introduction  Previous work  The L-Tangent norm  Experimental results  Conclusion Cross-Validation λ =0.05 F. Brunet et al. Surface Reconstruction and Regularization 3DPVT 2008, Atlanta, June 20 th

 Introduction  Previous work  The L-Tangent norm  Experimental results  Conclusion Cross-Validation λ =0.05 F. Brunet et al. Surface Reconstruction and Regularization 3DPVT 2008, Atlanta, June 20 th

 Introduction  Previous work  The L-Tangent norm  Experimental results  Conclusion Cross-Validation λ =0.05 F. Brunet et al. Surface Reconstruction and Regularization 3DPVT 2008, Atlanta, June 20 th

 Introduction  Previous work  The L-Tangent norm  Experimental results  Conclusion Cross-Validation λ =0.05 λ =0.45 F. Brunet et al. Surface Reconstruction and Regularization 3DPVT 2008, Atlanta, June 20 th

 Introduction  Previous work  The L-Tangent norm  Experimental results  Conclusion Cross-Validation λ =0.05 λ =0.45 λ =0.9 F. Brunet et al. Surface Reconstruction and Regularization 3DPVT 2008, Atlanta, June 20 th

 Introduction  Previous work  The L-Tangent norm  Experimental results  Conclusion Cross-Validation λ =0.05 λ =0.45 λ =0.9 λ =0.99 F. Brunet et al. Surface Reconstruction and Regularization 3DPVT 2008, Atlanta, June 20 th

 Introduction  Previous work  The L-Tangent norm  Experimental results  Conclusion Cross-Validation λ =0.05 λ =0.45 λ =0.9 λ =0.99 0 λ 1 λ * F. Brunet et al. Surface Reconstruction and Regularization 3DPVT 2008, Atlanta, June 20 th

 Introduction  Previous work  The L-Tangent norm  Experimental results  Conclusion Problems with Cross-Validation • Closed form expression • Computation time • Numerical instability • Pathological cases λ λ F. Brunet et al. Surface Reconstruction and Regularization 3DPVT 2008, Atlanta, June 20 th

 Introduction  Previous work  The L-Tangent norm  Experimental results  Conclusion The L-Curve • Introduced by [Lawson and Hanson, 1974] F. Brunet et al. Surface Reconstruction and Regularization 3DPVT 2008, Atlanta, June 20 th

 Introduction  Previous work  The L-Tangent norm  Experimental results  Conclusion 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 F. Brunet et al. Surface Reconstruction and Regularization 3DPVT 2008, Atlanta, June 20 th

 Introduction  Previous work  The L-Tangent norm  Experimental results  Conclusion The L-Curve λ =0.01 Data term Smoother 0 λ 1 0 λ 1 F. Brunet et al. Surface Reconstruction and Regularization 3DPVT 2008, Atlanta, June 20 th

 Introduction  Previous work  The L-Tangent norm  Experimental results  Conclusion The L-Curve λ =0.01 Data term Smoother L-Curve Smoother 0 λ 1 0 λ 1 Data term F. Brunet et al. Surface Reconstruction and Regularization 3DPVT 2008, Atlanta, June 20 th

 Introduction  Previous work  The L-Tangent norm  Experimental results  Conclusion The L-Curve λ =0.01 λ =0.02 Data term Smoother L-Curve Smoother 0 λ 1 0 λ 1 Data term F. Brunet et al. Surface Reconstruction and Regularization 3DPVT 2008, Atlanta, June 20 th

 Introduction  Previous work  The L-Tangent norm  Experimental results  Conclusion The L-Curve λ =0.01 λ =0.02 λ =0.05 Data term Smoother L-Curve Smoother 0 λ 1 0 λ 1 Data term F. Brunet et al. Surface Reconstruction and Regularization 3DPVT 2008, Atlanta, June 20 th