1 / 20

Semi-Differential Invariants for Algebraic Curve Recognition

This research explores a method using semi-differential invariants for recognizing algebraic curves, providing an innovative approach that is computationally efficient. By estimating curvature and derivatives with respect to arc length, this model-based tactile recognition technique helps identify and classify shapes accurately. Through experiments and simulations, the study demonstrates the effectiveness of utilizing invariant design for shape recognition and recovery, improving robustness to sensor noise.

andreh
Download Presentation

Semi-Differential Invariants for Algebraic Curve Recognition

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related