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Robust Statistical Estimation of Curvature on Discretized Surfaces

Explore a signal processing approach to achieve accurate maximum likelihood estimates of surface derivatives using adaptation to local features, outlier rejection, and curvature-driven corrections.

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Robust Statistical Estimation of Curvature on Discretized Surfaces

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

  2. 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. 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. Related Work (1/3) • Discrete curvature methodse.g. [Taubin 95], [Langer et al. 07] • Discrete approximationsof 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 methodse.g. [Cazals and Pouget 03], [Goldfeather and Interrante 04], [Gatzke and Grimm 06] • Per Triangle curvature estimation [Rusinkiewicz 04]

  5. Related Work (2/3)

  6. Related Work (3/3)

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

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

  11. Automatic adaptation to noise

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

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

  14. Normal re-estimation (2/2)

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

  16. Error plots – Increasing Noise

  17. Error plots – Increasing Resolution

  18. Point cloud examples (1/2)

  19. Point cloud examples (2/2)

  20. Applications - NPR

  21. Applications - Segmentation

  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, GuillaumeLavoué, Ryan Schmidt, Szymon Rusinkiewicz. Research funded by MITACS

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