Noise-Adaptive Shape Reconstruction from Raw Point Sets 60 Simon - PowerPoint PPT Presentation

mm 40 60 80 100 120 40 Noise-Adaptive Shape Reconstruction from Raw Point Sets 60 Simon Giraudot, David Cohen-Steiner & Pierre Alliez 80 Simon Giraudot EUROGRAPHICS Symposium on Geometry Processing Titane Inria Sophia Antipolis - M´ editerran´ ee July 5, 2013

Focus Robust shape reconstruction mm 40 60 80 100 120 Resilience to raw, defect-laden point sets 40 60 80 Simon Giraudot - Noise-Adaptive Shape Reconstruction July 5, 2013- 2

Focus Robust shape reconstruction mm 40 60 80 100 120 Resilience to raw, defect-laden point sets 40 60 ◮ Variable noise (sensor noise, acquisition noise, registration noise) ◮ Outliers (structured, unstructured, background noise) 80 ◮ Missing data Simon Giraudot - Noise-Adaptive Shape Reconstruction July 5, 2013- 2

Previous Work mm 40 60 80 100 120 Noise Outliers Variable noise Laplace operator Data clustering Automatic MLS scale [Kazhdan et al., 2006] 40 [Song, 2010] [Wang et al., 2009] Scale space Spectral methods Adjusting statistical estimator [Digne, 2010] [Kolluri et al., 2004] [Unnikrish- nan et al., 2010] 60 Delaunay triangu- Robust distances Adaptive bandwidth lation properties [Chazal et al., 2011] [Mellado et al., 2012] [Dey & Goswami, 2006] 80 Simon Giraudot - Noise-Adaptive Shape Reconstruction July 5, 2013- 3

Previous Work Growing Least Squares for the Continuous Analysis of mm 40 60 80 100 120 Manifolds in Scale-Space 1 40 3 2 1 3 2 60 [N. Mellado, G. Guennebaud, P. Barla, P. Reuter, C. Schlick, 2012] ◮ Analysis of geometric variation ν 80 Simon Giraudot - Noise-Adaptive Shape Reconstruction July 5, 2013- 4

Previous Work Growing Least Squares for the Continuous Analysis of mm 40 60 80 100 120 Manifolds in Scale-Space 1 40 3 2 1 3 2 60 [N. Mellado, G. Guennebaud, P. Barla, P. Reuter, C. Schlick, 2012] ◮ Analysis of geometric variation ν ◮ Threshold for ∂ν 80 ∂ t Simon Giraudot - Noise-Adaptive Shape Reconstruction July 5, 2013- 4

Problem Statement mm 40 60 80 100 120 Input: raw point set P sampling submanifold of known dimension k , possibly defect-laden with: 40 60 80 Simon Giraudot - Noise-Adaptive Shape Reconstruction July 5, 2013- 5

Problem Statement mm 40 60 80 100 120 Input: raw point set P sampling submanifold of known dimension k , possibly defect-laden with: 40 ◮ Variable noise ◮ Outliers ◮ Missing data 60 80 Simon Giraudot - Noise-Adaptive Shape Reconstruction July 5, 2013- 5

Problem Statement mm 40 60 80 100 120 Input: raw point set P sampling submanifold of known dimension k , possibly defect-laden with: 40 ◮ Variable noise ◮ Outliers ◮ Missing data 60 Output: smooth closed shape defined by implicit function 80 Simon Giraudot - Noise-Adaptive Shape Reconstruction July 5, 2013- 5

Overview mm 40 60 80 100 120 Point set 40 60 80 Simon Giraudot - Noise-Adaptive Shape Reconstruction July 5, 2013- 6

Overview mm 40 60 80 100 120 1 — Unsigned noise-adaptive distance function 40 60 80 Simon Giraudot - Noise-Adaptive Shape Reconstruction July 5, 2013- 7

Overview mm 40 60 80 100 120 2 — Sign guess 40 60 80 Simon Giraudot - Noise-Adaptive Shape Reconstruction July 5, 2013- 8

Overview mm 40 60 80 100 120 3 — Signed implicit function 40 60 80 Simon Giraudot - Noise-Adaptive Shape Reconstruction July 5, 2013- 9

mm 40 60 80 100 120 40 1 Adaptive Distance Function 60 80 Simon Giraudot - Noise-Adaptive Shape Reconstruction July 5, 2013- 10

Non-Adaptive Distance Function mm 40 60 80 100 120 Unsigned distance function to a measure [Chazal et al., 2011] µ, m : R n → R , q �→ 1 � d 2 � q − y � 2 d µ ( y ) m 40 B ( q , r µ, m ( q )) 60 80 Simon Giraudot - Noise-Adaptive Shape Reconstruction July 5, 2013- 11

Non-Adaptive Distance Function mm 40 60 80 100 120 Unsigned distance function to a measure [Chazal et al., 2011] µ, m : R n → R , q �→ 1 � d 2 � q − y � 2 d µ ( y ) m 40 B ( q , r µ, m ( q )) µ m 60 || q-y || 2 q r µ,m (q) B(q,r µ,m (q)) 80 Simon Giraudot - Noise-Adaptive Shape Reconstruction July 5, 2013- 11

Non-Adaptive Distance Function mm 40 60 80 100 120 Unsigned distance function to a measure [Chazal et al., 2011] µ, m : R n → R , q �→ 1 � d 2 � q − y � 2 d µ ( y ) m 40 B ( q , r µ, m ( q )) µ Note: scale parameter m m 60 || q-y || 2 q r µ,m (q) B(q,r µ,m (q)) 80 Simon Giraudot - Noise-Adaptive Shape Reconstruction July 5, 2013- 11

Non-Adaptive Distance Function mm 40 60 80 100 120 Unsigned distance function to a measure [Chazal et al., 2011] µ, m : R n → R , q �→ 1 � d 2 � q − y � 2 d µ ( y ) m 40 B ( q , r µ, m ( q )) µ Note: scale parameter m m 60 ◮ User-specified || q-y || 2 q r µ,m (q) B(q,r µ,m (q)) 80 Simon Giraudot - Noise-Adaptive Shape Reconstruction July 5, 2013- 11

Non-Adaptive Distance Function mm 40 60 80 100 120 Unsigned distance function to a measure [Chazal et al., 2011] µ, m : R n → R , q �→ 1 � d 2 � q − y � 2 d µ ( y ) m 40 B ( q , r µ, m ( q )) µ Note: scale parameter m m 60 ◮ User-specified || q-y || 2 q ◮ Depends on point set properties r µ,m (q) B(q,r µ,m (q)) 80 Simon Giraudot - Noise-Adaptive Shape Reconstruction July 5, 2013- 11

Non-Adaptive Distance Function mm 40 60 80 100 120 Unsigned distance function to a measure [Chazal et al., 2011] µ, m : R n → R , q �→ 1 � d 2 � q − y � 2 d µ ( y ) m 40 B ( q , r µ, m ( q )) µ Note: scale parameter m m 60 ◮ User-specified || q-y || 2 q ◮ Depends on point set properties r µ,m (q) ◮ Global: not noise-adaptive B(q,r µ,m (q)) 80 Simon Giraudot - Noise-Adaptive Shape Reconstruction July 5, 2013- 11

Non-Adaptive Distance Function mm 40 60 80 100 120 40 1 0 . 8 60 0 . 6 K = 6 0 . 4 0 . 2 0 0 . 8 0 . 6 K = 70 0 . 4 80 0 . 2 0 Simon Giraudot - Noise-Adaptive Shape Reconstruction July 5, 2013- 12

α Case of an Ambiant Noise Uniform measure in d -dimensional space mm 40 60 80 100 120 µ 40 m q 60 80 Simon Giraudot - Noise-Adaptive Shape Reconstruction July 5, 2013- 13

α Case of an Ambiant Noise Uniform measure in d -dimensional space mm 40 60 80 100 120 2 d 2 µ, m ( q ) = c · m d µ 40 m q 60 80 Simon Giraudot - Noise-Adaptive Shape Reconstruction July 5, 2013- 13

α Case of an Ambiant Noise Uniform measure in d -dimensional space mm 40 60 80 100 120 2 d 2 µ, m ( q ) = c · m d µ 40 m 1 d for q fixed d µ, m ( q ) ∝ m q 60 80 Simon Giraudot - Noise-Adaptive Shape Reconstruction July 5, 2013- 13

Case of an Ambiant Noise Uniform measure in d -dimensional space mm 40 60 80 100 120 2 d 2 µ, m ( q ) = c · m d µ 40 m 1 d for q fixed d µ, m ( q ) ∝ m q d µ, m ( q ) decreasing for α > 1 m α d 60 d µ,m (q) α m 80 m Simon Giraudot - Noise-Adaptive Shape Reconstruction July 5, 2013- 13

α Case of a Submanifold Uniform measure on k -submanifold mm 40 60 80 100 120 µ 40 q m h 60 80 Simon Giraudot - Noise-Adaptive Shape Reconstruction July 5, 2013- 14

α Case of a Submanifold Uniform measure on k -submanifold mm 40 60 80 100 120 2 d 2 k + h 2 µ, m ( q ) = c · m µ 40 q m h 60 80 Simon Giraudot - Noise-Adaptive Shape Reconstruction July 5, 2013- 14

α Case of a Submanifold Uniform measure on k -submanifold mm 40 60 80 100 120 2 d 2 k + h 2 µ, m ( q ) = c · m µ 40 1 k for q fixed d µ, m ( q ) ∝ m q m h 60 80 Simon Giraudot - Noise-Adaptive Shape Reconstruction July 5, 2013- 14

Case of a Submanifold Uniform measure on k -submanifold mm 40 60 80 100 120 2 d 2 k + h 2 µ, m ( q ) = c · m µ 40 1 k for q fixed d µ, m ( q ) ∝ m q m h d µ, m ( q ) increasing for α < 1 m α k 60 d µ,m (q) m α 80 m Simon Giraudot - Noise-Adaptive Shape Reconstruction July 5, 2013- 14

Discrete Noisy Case mm 40 60 80 100 120 Scale m = 10 nearest neighbors 40 60 80 Simon Giraudot - Noise-Adaptive Shape Reconstruction July 5, 2013- 15

Discrete Noisy Case mm 40 60 80 100 120 Scale m = 10 nearest neighbors 40 60 ◮ Apparent dimension = 2 80 Simon Giraudot - Noise-Adaptive Shape Reconstruction July 5, 2013- 15

Discrete Noisy Case mm 40 60 80 100 120 Scale m = 10 nearest neighbors 40 60 ◮ Apparent dimension = 2 ◮ Ambiant noise in 2 D 80 Simon Giraudot - Noise-Adaptive Shape Reconstruction July 5, 2013- 15

Discrete Noisy Case Scale m = 10 nearest neighbors mm 40 60 80 100 120 40 ◮ Apparent dimension = 2 ◮ Ambiant noise in 2 D 60 ◮ d µ, m ( q ) decreasing for α > 1 2 m α d µ,m (q) m α 80 m Simon Giraudot - Noise-Adaptive Shape Reconstruction July 5, 2013- 15

Discrete Noisy Case Scale m = 10 nearest neighbors Scale m = 30 nearest neighbors mm 40 60 80 100 120 40 ◮ Apparent dimension = 2 ◮ Ambiant noise in 2 D 60 ◮ d µ, m ( q ) decreasing for α > 1 2 m α d µ,m (q) m α 80 m Simon Giraudot - Noise-Adaptive Shape Reconstruction July 5, 2013- 15