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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
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