Analyzing error of fit functions for ellipses

1. Analyzing error of fit functions for ellipses Paul L. Rosin BMVC 1996

2. Noise pixels Pupil edge pixels Why? • Ellipse fitting to pupil boundary • RANSAC (Random sample consensus) • Explore fits • Select best fit • Selection based on error criterion

3. Overview • Ellipse Error of fit (EOF) functions • How far is a point from ellipse boundary? • Approx. to Euclidean dist (hard to compute!) • Ellipse fitting using Least Squares (LS) • Evaluation • Linearity, Curvature bias, Asymmetry X6 X5 e6 e5 e1 X1 X2 X4 e4 e2 e3 X3

4. Ellipse boundary Algebraic distance (AD) • Simple to compute • Closed form solution to LS ellipse exists • High curvature bias (skewed ellipses) • Super linear relationship with Euclidean dist (sensitive to outliers) Isovalue contours

5. Ellipse boundary Gradient weighted AD (GWAD) Inversely weight AD with its gradient Isovalue contours • Reduced curvature bias • Asymmetry exists • Gradient inside > gradient outside

6. Ellipse boundary Second order approximation • Does not exist for points near high curvature sections Isovalue contours

7. Ellipse boundary Ellipse boundary EOF8 EOF1 Pavlidis’ approximation • Improvement over basic algebraic distance

8. Ellipse boundary Reduced gradient weighted AD • Compromise between AD (p = 0) and GWAD (p = 1) • p is in the range (0, 1) • Curvature bias < AD • Asymmetry < GWAD

9. Ellipse boundary Ellipse boundary Directional derivative weighted AD • Wavy isovalue contours of GWAD are reduced C Xj r EOF2 EOF10

10. Ellipse boundary Combined conic and circular dist • Geometric mean of conic dist (AD) and circular dist • Reduced curvature bias • Asymmetry exists Circle Xc Xj Conic Conic ≈ Circle Isovalue contour Xk

11. 2a True ellipse: PF1 + PF2= 2a P Concentric ellipse: XjF1 + XjF2 = 2a’ F1 F2 Xj 2a’ Ellipse boundary Concentric ellipse estimation • Curvature bias significantly reduced

12. 2a True ellipse: PF1 + PF2= 2a P Concentric ellipse: XjF1 + XjF2 = 2a’ F1 F2 Xj 2a’ Ellipse boundary Concentric ellipse estimation • Geometric mean of EOF1(AD) and EOF12a • Low curvature bias • Asymmetry exists

13. Ellipse boundary Focal bisector distance • Reflection property: PF’ is a reflection of PF • Very low curvature bias • Symmetric

14. Ellipse boundary Ellipse boundary Radial distance • Comparison with focal bisector distance T EOF5 = XjT C EOF13 = XjIj EOF5 = XjT

15. Assessment • Linearity Pearson’s correlation coefficient Euclidean EOF ρ is in the range [0, 1], ideally ρ = 1 EOF2 ρ = 1 EOF EOF1 ρ < 1 Euclidean

16. Assessment • Linearity • Points on farther isovalue contours contribute more • Farther isovalue contours are longer Mean euclidean distance along an isovalue contour at Ei Modified Pearson’s correlation coefficient (more uniform sampling) Gaussian weighting according to distance d from ellipse boundary

17. Assessment • Curvature bias Local variation of euclidean distance along an isovalue contour at Ei Global curvature measure considering all isovalue contours Ei Low values of C imply low curvature bias, ideally C= 0

18. Assessment • Asymmetry Mean of euclidean distance along an outside isovalue contour at Ei Mean of euclidean distance along an inside isovalue contour at Ei Local assymetry w.r.t. isovalue contour at Ei Global assymetry measure considering all isovalue contours Ei Low values of A imply low asymmetry, ideally A= 0

19. Assessment • Combined measure • Overall goodness Weighted sum of square errors between euclidean distance and scaled EOF Global scaling factor S is determined by optimizing G

20. Results Normalized assessment measures w.r.t. EOF1 • EOF13 is the best! • Except EOF2 and EOF10, all have reasonable linearity • All have lower curvature bias than AD • Except EOF13, all have poor asymmetry (EOF2 and EOF10 are comparable)

21. Our work • RANSAC consensus (selection) • Algebraic dist vs. Focal bisector dist Selection using algebraic distance Selection using focal bisector distance

22. Thank you!!