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

2/22 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

3/22 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?

4/22 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]

5/22 Related Work (2/3)

6/22 Related Work (3/3)

7/22 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

8/22 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.

9/22 Sampling and Weighting (2/2) Iteratively re-weight samples based on their observed residuals. Minimize cost function of residuals.

10/22 Statistical Curvature Estimation Initial tensor guess based on one-ring neighborhood or 6 nearest point pair normal variations.

11/22 Automatic adaptation to noise

12/22 Structural Outlier Rejection Typical behavior of algorithm near feature edges (curvature field discontinuities). Feature boundary

13/22 Normal re-estimation (1/2) Estimated curvature tensors and final sample weights are used to correct noisy local frames.

14/22 Normal re-estimation (2/2)

15/22 Implementation Typically we run 30 IRLS iterations. Current implementation needs 20 sec for 10K vertices, 20 min for 1M vertices.

16/22 Error plots – Increasing Noise

17/22 Error plots – Increasing Resolution

18/22 Point cloud examples (1/2)

19/22 Point cloud examples (2/2)

20/22 Applications - NPR

21/22 Applications - Segmentation

22/22 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