Symposium on Geometry Processing SGP 2007 July 2007

Symposium on Geometry Processing – SGP 2007 July 2007, Barcelona, Spain Robust Statistical Estimation of Curvature on Discretized Surfaces Evangelos Kalogerakis Patricio Simari Derek Nowrouzezahrai Karan Singh

Introduction • Goal: A signal processing approach to obtain Maximum Likelihood (ML) estimates of surface derivatives. • Contributions: • automatic outlier rejection • adaptation to local features and noise • curvature-driven surface normal correction • major accuracy improvements 2

Motivation • Surface curvature plays a key role for many applications. • Surface derivatives are very sensitive to noise, sampling and mesh irregularities. • What is the most appropriate shape and size of the neighborhood around each point for a curvature operator? 3

Related Work (1/3) • Discrete curvature methods e. g. [Taubin 95], [Langer et al. 07] • Discrete approximations of Gauss-Bonnet theorem and Euler-Lagrange equation e. g. [Meyer et al. 03] • Normal Cycle theory [Cohen-Steiner & Morvan 02] • Local PCA e. g. [Yang et al. 06] • Patch Fitting methods e. g. [Cazals and Pouget 03], [Goldfeather and Interrante 04], [Gatzke and Grimm 06] • Per Triangle curvature estimation [Rusinkiewicz 04] 4

Related Work (2/3) 5

Related Work (3/3) 6

Curvature Tensor Fitting • Least Squares fit the components of covariant derivatives of normal vector field N: given normal variations ΔN along finite difference distances Δp around each point. • Least Squares fit the derivatives of curvature tensor 7

Sampling and Weighting (1/2) • Acquire all-pairs finite normal differences within an initial neighborhood. • Prior geometric weighting of the samples based on their geodesic distance from the center point. 8

Sampling and Weighting (2/2) • Iteratively re-weight samples based on their observed residuals. • Minimize cost function of residuals. 9

Statistical Curvature Estimation • Initial tensor guess based on one-ring neighborhood or 6 nearest point pair normal variations. 10

Automatic adaptation to noise 11

Structural Outlier Rejection • Typical behavior of algorithm near feature edges (curvature field discontinuities). Feature boundary 12

Normal re-estimation (1/2) • Estimated curvature tensors and final sample weights are used to correct noisy local frames. 13

Normal re-estimation (2/2) 14

Implementation • Typically we run 30 IRLS iterations. • Current implementation needs 20 sec for 10 K vertices, 20 min for 1 M vertices. 15

Error plots – Increasing Noise 16

Error plots – Increasing Resolution 17

Point cloud examples (1/2) 18

Point cloud examples (2/2) 19

Applications - NPR 20

Applications - Segmentation 21

Conclusions and Future Work • Robust statistical approach for surface derivative maximum likelihood estimates • Robust to outliers & locally adaptive to noise Ongoing/Future Work: • Automatic surface outlier detection • Curvature-driven surface reconstruction Special thanks to Eitan Grinspun, Guillaume Lavoué, Ryan Schmidt, Szymon Rusinkiewicz. Research funded by MITACS 22