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

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

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 e1e1 e2e2 e3e3 e4e4 e5e5 e6e6 X1X1 X2X2 X3X3 X4X4 X5X5 X6X6

4. 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) Ellipse boundary Isovalue contours

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

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

7. Pavlidis’ approximation –Improvement over basic algebraic distance Ellipse boundary EOF 1 EOF 8

8. 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 Ellipse boundary

9. Directional derivative weighted AD –Wavy isovalue contours of GWAD are reduced Ellipse boundary r XjXj C EOF 2 EOF 10

10. Combined conic and circular dist –Geometric mean of conic dist (AD) and circular dist –Reduced curvature bias –Asymmetry exists XjXj Conic Circle XcXc XkXk Conic ≈ Circle Isovalue contour Ellipse boundary

11. Concentric ellipse estimation –Curvature bias significantly reduced True ellipse: PF 1 + PF 2 = 2a F1F1 F2F2 P XjXj 2a 2a’ Concentric ellipse: X j F 1 + X j F 2 = 2a’ Ellipse boundary

12. Concentric ellipse estimation 2a F1F1 F2F2 P XjXj 2a’ True ellipse: PF 1 + PF 2 = 2a Concentric ellipse: X j F 1 + X j F 2 = 2a’ –Geometric mean of EOF 1 (AD) and EOF 12a –Low curvature bias –Asymmetry exists Ellipse boundary

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

14. Radial distance –Comparison with focal bisector distance C T EOF 5 = X j T Ellipse boundary EOF 5 = X j T EOF 13 = X j I j

15. Assessment Linearity Pearson’s correlation coefficient EOF Euclidean ρ is in the range [0, 1], ideally ρ = 1 EOF 1 ρ < 1 EOF 2 ρ = 1 EOF Euclidean

16. Assessment Linearity –Points on farther isovalue contours contribute more –Farther isovalue contours are longer Mean euclidean distance along an isovalue contour at E i 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 E i Global curvature measure considering all isovalue contours E i Low values of C imply low curvature bias, ideally C = 0

18. Assessment Asymmetry Mean of euclidean distance along an outside isovalue contour at E i Mean of euclidean distance along an inside isovalue contour at E i Local assymetry w.r.t. isovalue contour at E i Global assymetry measure considering all isovalue contours E i 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. EOF 1 EOF 13 is the best! Except EOF 2 and EOF 10, all have reasonable linearity All have lower curvature bias than AD Except EOF 13, all have poor asymmetry (EOF 2 and EOF 10 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!!