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Semi-Differential Invariants for Recognition of Algebraic Curves

Semi-Differential Invariants for Recognition of Algebraic Curves. Yan-Bin Jia and Rinat Ibrayev. Department of Computer Science Iowa State University Ames, IA 50011-1040. July 13, 2004. estimate curvature  and derivative  w.r.t. arc length. s. Object.

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Semi-Differential Invariants for Recognition of Algebraic Curves

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  1. Semi-Differential Invariants for Recognition of Algebraic Curves Yan-Bin Jia and Rinat Ibrayev Department of Computer Science Iowa State University Ames, IA 50011-1040 July 13, 2004

  2. estimate curvature  and derivative  w.r.t. arc length s Object shapeparameters Model-Based Tactile Recognition Models: families of parametric shapes Tactile data • contact (x, y) Determine • Shape • Identify curve family • Estimate Each model: • Location of contact t on the object

  3. Shape Recognition through Touch Grimson & Lozano-Perez 1984; Fearing 1990; Allen & Michelman 1990; Moll & Erdmann 2002; etc. Vision & Algebraic Invariants Differential & Semi-differenitial Invariants Hilbert; Kriegman & Ponce 1990; Forsyth et al. 1991; Mundy & Zisserman 1992; Weiss 1993; Keren 1994; Civi et al. 2003; etc. Padjla & Van Gool 1992; Rivlin & Weiss 1995; Moons et al. 1995; Calabi et al. 1998; Keren et al. 2000; etc. Related Work

  4. cubical parabola: signature curve Signature Curve Plot curvature against its derivative along the curve: • Independent of rotation and translation • Used in model-based recognition Requiring global data What if just a few data points?

  5. curvature derivative constant Differential Invariants • Expressions of curvature and derivatives (w.r.t. arc length) • Computed from local geometry • Small amount of tactile data • Independent of position, orientation, and parameterization • Independent of point location on the shape • How to derive? Eliminate t from and Well, ideally so … invariant

  6. 0.5 0.5 0.5 1 rot, trans, and reparam. Parabola evaluated at one point shape classification signature curve • Only 1 parameter instead of 6 • Shape remains the same Invariant:

  7. Semi-Differential Invariants • Differential invariants use one point. nshape parameters nindependent diff. invariants. up to n+2th derivatives Numerically unstable! • Semi-differential invariantsinvolve n points. n curvatures + n 1st derivs

  8. Quadratics: Ellipse shape classifiers • Two independent invariants required • Two points involved

  9. distinguishes ellipses (+), hyperbolas (-), parabolas (0) Quadratics: Hyperbola • Invariants same as for ellipse • Different value expressions in terms of a, b

  10. Invariants in terms of Cubics • Eliminate parameter t directly? • High degree resultant polynomial in shape parameters • Computationally very expensive • Reparameterize with slope • Lower the resultant degree • Slope depends on rotation • Twoslopes related to change of tangential angle (measurable)

  11. Invariants for Cubics cubical parabola semi-cubical parabola

  12. Simulations • Testing invariants (curvature & deriv. est. by finite differences) Summary over 100 different tests on randomly generated points for each curve • Shape recovery • Average error on shape parameter estimation Summary over 100 different shapes for each curve family

  13. Simulations (cont’d) Data from one curve inapplicable for an invariant for a different class. Each cell displays the summary over 100 values

  14. Sign no yes Cubical Parabola yes no Cubic Spline? Semi-Cubical Parabola a, b … a, b Recognition Tree Tactile data no yes Parabola yes no a >0 < 0 Ellipse Hyperbola a, b a, b

  15. Locating Contact • Parameter value t determines the contact. • Solve for t after recognition. parabola:

  16. (cm) 1 (1/cm ) 2 (cm) 1 (1/cm) Numerical Curvature Estimation • Noisytactile data • A tentative approach Curvature – inverse of radius of osculating circle Derivative of curvature – finite difference signature curve ellipse courtesy ofLiangchuan Mi for supplying raw data large errors!

  17. Curvature Estimation – Local Fitting • Curvature estimation • fit a quadratic curve to a few local data points • differentiate the curve fit (1) • Curvature derivative estimation • numericallyestimate arc length s using curve fit (1) • generatemultiple (s, ) pairs in the neighborhood • fit and differentiate again

  18. (cm) (1/cm ) 2 1 (cm) 0.01 1 0.03 (1/cm) Experiments signature curve ellipse Summary over 80 different values for the ellipse

  19. Experiments (cont’d) signature curve cubic spline Seemingly good curvature & derivative estimates, but unstableinvariantcomputation …

  20. Summary & Future Work • Differential invariants for quadratic curves & certain cubic curves • Computable from local tactile data • Invariant to point locations on a shape (not just to transformation) • Discrimination of families of parametric curves • Unifying shape recognition, recovery, and localization • Numerical estimation of curvature and derivative Invariant design for more general shape classes (3D) Improvement on robustness to sensor noise

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