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Projective Estimators for Point/Tangent Representations of Planar Curves Thomas Lewiner/ Marcos Craizer Dep. Matemática -PUC-Rio

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T 2 /22 Projective Estimators for Planar Curves - T. Lewiner & M. Craizer

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Projective Geometry Projective Geometry : Invariance by translations, rotations, shearing and projections. 3 /22 Projective Estimators for Planar Curves - T. Lewiner & M. Craizer

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Euclidean and Affine Geometry Euclidean Geometry: Invariance by translations and rotations. Affine Geometry: Invariance by translations, rotations and shearing. 4 /22 Projective Estimators for Planar Curves - T. Lewiner & M. Craizer

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Projective Transformations in homogeneous coordinates 5 /22 Projective Estimators for Planar Curves - T. Lewiner & M. Craizer Assuming det T=1 → 8 parameters

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Affine and Euclidean Transformations 6 /22 Projective Estimators for Planar Curves - T. Lewiner & M. Craizer Assuming det T=1 → 5 parameters Assuming det T=1 → 3 parameters

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Euclidean and Affine Invariants Euclidean length and curvature Affine length and curvature 7 /22 Projective Estimators for Planar Curves - T. Lewiner & M. Craizer Differential Invariant of order 2. Differential Invariant of order 4!

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Projective Length and Curvature 8 /22 Projective Estimators for Planar Curves - T. Lewiner & M. Craizer Differential Invariant of order 7!!

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Euclidean Estimators Example: The euclidean length of a polygon is the sum of the lengths of its sides. And the total curvature of a polygon is the sum of its external angles. 9 /22 Projective Estimators for Planar Curves - T. Lewiner & M. Craizer Plenty of estimators proposed in the literature.

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Parabolic Polygons SIBGRAPI 2006 10 /22 Projective Estimators for Planar Curves - T. Lewiner & M. Craizer Quadratic Spline

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Affine Estimators-Signature Estimator for the derivative of the affine curvature with respect to affine arc-length. Convergent and Affine Invariant 11 /22 Projective Estimators for Planar Curves - T. Lewiner & M. Craizer

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Projective length estimator Although convergent, this estimator is not projective invariant. 12 /22 Projective Estimators for Planar Curves - T. Lewiner & M. Craizer

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Curves with constant projective curvature: Spirals 13 /22 Projective Estimators for Planar Curves - T. Lewiner & M. Craizer

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General Spiral General spiral with zero projective curvature: Assuming det T=1 → 8 parameters 14 /22 Projective Estimators for Planar Curves - T. Lewiner & M. Craizer

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Adapting spirals to three point/tangent pairs Unknowns: 8 parameters + 2 projective lengths σ. Knowns: 3 points + 3 tangents → 9 data Underdetermined system !! We must estimate first the projective legths. Options: 1) Use the affine estimator for projective lengths. 2) Projective estimation by geometric methods (on going work). 15 /22 Projective Estimators for Planar Curves - T. Lewiner & M. Craizer

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Adapting spirals to three point/tangent pairs Projective Estimators for Planar Curves - T. Lewiner & M. Craizer 16/22 After estimating 2 projective lengths → 8 parameters and 9 data We drop the middle tangency condition

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Linear equations for the fitting problem Known projective lengths σ j 15 unknowns and 15 linear equations. Projective Estimators for Planar Curves - T. Lewiner & M. Craizer 17 /22

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Curvature Estimator From the Frenet thriedron at each sample, one can estimate the curvature k by: where 18 /22 Frenet Formulas

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Results Length estimator Curvature estimator with exact length Curvature estimator with estimated length 19 /22 Projective Estimators for Planar Curves - T. Lewiner & M. Craizer

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Convergence Length estimator Curvature estimator with exact length Curvature estimator with estimated length 20 /22 Projective Estimators for Planar Curves - T. Lewiner & M. Craizer

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Projective Invariance Length estimator of the transformed curve Curvature estimator of the transformed curve from exact lengths Curvature estimator of the transformed curve from estimated lengths 21 /22 Projective Estimators for Planar Curves - T. Lewiner & M. Craizer

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On-going works Estimators guaranteeing invariance more than convergence (projective spline). Direct geometric construction. Statistical scheme to cope with noise. Projective Estimators for Planar Curves - T. Lewiner & M. Craizer 22 /22

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