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

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

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

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

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

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

Semi-Differential Invariants ♦ Differential invariants use one point. n shape parameters n independent diff. invariants. up to n+2 th derivatives Numerically unstable! ♦ Semi-differential invariants involve n points. n curvatures + n 1 st derivs

Quadratics: Ellipse ♦ Two points involved ♦ Two independent invariants required shape classifiers

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

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

Invariants for Cubics cubical parabolasemi-cubical parabola

Simulations ParabolaEllipseHyperbolaCub. parSemi-cub. real min max mean ♦ Testing invariants (curvature & deriv. est. by finite differences) ♦ Shape recovery  Average error on shape parameter estimation ParabolaEllipseHyperbolaCub. Par.Semi-cub. 0.36%0.40%1.15%0.83%1.23% Summary over 100 different tests on randomly generated points for each curve Summary over 100 different shapes for each curve family

Simulations (cont’d) invariant data coniccubical parabola semi-cub. parabola conic (ellipse) (min) (max) (mean) 2.53 (stdev) cubical parabola semi-cub. parabola Each cell displays the summary over 100 values Data from one curve inapplicable for an invariant for a different class.

Recognition Tree Tactile data Parabola Sign EllipseHyperbola a a, b yes no yesno >0< 0 Cubical Parabola Semi-Cubical Parabola a, b no yes Cubic Spline ? …

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

Numerical Curvature Estimation ♦ Noisy tactile data  Curvature – inverse of radius of osculating circle  Derivative of curvature – finite difference ellipse signature curve 1 1 (cm) (1/cm) 2 ♦ A tentative approach courtesy of Liangchuan Mi for supplying raw data large errors!

Curvature Estimation – Local Fitting ♦ Curvature estimation  fit a quadratic curve to a few local data points  differentiate the curve fit (1) ♦ Curvature derivative estimation  generate multiple (s,  ) pairs in the neighborhood  fit and differentiate again  numerically estimate arc length s using curve fit (1)

Experiments ab real min max mean Summary over 80 different values for the ellipse ellipse signature curve (cm) (1/cm) 2

Experiments (cont’d) cubic spline signature curve but unstable invariant computation … Seemingly good curvature & derivative estimates,

Summary & Future Work ♦ Differential invariants for quadratic curves & certain cubic curves  Improvement on robustness to sensor noise  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)  Computable from local tactile data