Robust Statistical Estimation of Curvature on Discretized Surfaces

Evangelos KalogerakisPatricio Simari

  Derek Nowrouzezahrai  Karan Singh

Symposium on Geometry Processing – SGP 2007July 2007, Barcelona, Spain

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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

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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?

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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 PCAe.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]

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Related Work (2/3)

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Related Work (3/3)

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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

, ,u u v vN u N v N u N v

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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.

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Sampling and Weighting (2/2)• Iteratively re-weight samples based on their

observed residuals.• Minimize cost function of residuals.

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Statistical Curvature Estimation• Initial tensor guess based on one-ring neighborhood

or 6 nearest point pair normal variations.

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Automatic adaptation to noise

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Structural Outlier Rejection• Typical behavior of algorithm near feature edges

(curvature field discontinuities).

Feature boundary

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Normal re-estimation (1/2)• Estimated curvature tensors and final sample

weights are used to correct noisy local frames.

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Normal re-estimation (2/2)

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Implementation• Typically we run 30 IRLS iterations.• Current implementation needs 20 sec for 10K

vertices, 20 min for 1M vertices.

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Error plots – Increasing Noise

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Error plots – Increasing Resolution

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Point cloud examples (1/2)

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Point cloud examples (2/2)

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Applications - NPR

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Applications - Segmentation

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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