Analyzing error of fit functions for ellipses

Paul L. Rosin

BMVC 1996

Why?

• Ellipse fitting to pupil boundary

• RANSAC (Random sample consensus)

– Explore fits– Select best fit

• Selection based on error criterion

Pupil edge pixelsNoise pixels

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

e1

e2

e3

e4

e5

e6

X1X2

X3

X4

X5X6

N

jj

Pe

1

2min

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

Gradient weighted AD (GWAD)

Inversely weight AD with its gradientEllipse boundary

Isovalue contours

- Reduced curvature bias

- Asymmetry exists

- Gradient inside > gradient outside

Second order approximation

– Does not exist for points near high curvature sections

Ellipse boundary

Isovalue contours

Pavlidis’ approximation

– Improvement over basic algebraic distance

Ellipse boundary Ellipse boundary

EOF1EOF8

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

Directional derivative weighted AD

– Wavy isovalue contours of GWAD are reduced

Ellipse boundary)( jXQ

rXj

C

Ellipse boundary

EOF2 EOF10

Combined conic and circular dist

– Geometric mean of conic dist (AD) and circular dist

– Reduced curvature bias– Asymmetry exists

Xj

Conic

CircleXc

Xk

Conic ≈ Circle Isovalue contour

Ellipse boundary

Concentric ellipse estimation

– Curvature bias significantly reduced

True ellipse: PF1 + PF2 = 2a

F1 F2

P

Xj

2a

2a’

Concentric ellipse: XjF1 + XjF2 = 2a’

Ellipse boundary

Concentric ellipse estimation2a

F1 F2

P

Xj

2a’

True ellipse: PF1 + PF2 = 2a

Concentric ellipse: XjF1 + XjF2 = 2a’

– Geometric mean of EOF1(AD) and EOF12a

– Low curvature bias– Asymmetry exists

Ellipse boundary

Focal bisector distance

– Reflection property: PF’ is a reflection of PF

– Very low curvature bias– Symmetric

Ellipse boundary

Radial distance

– Comparison with focal bisector distance

C

T

EOF5 = XjT

Ellipse boundary

Ellipse boundary

EOF5 = XjT EOF13 = XjIj

Assessment

• Linearity

Pearson’s correlation coefficient

EOF Euclidean

ρ is in the range [0, 1], ideally ρ = 1

EOF1

ρ < 1

EOF2 ρ = 1EOF

Euclidean

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

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

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

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

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)

Our work

• RANSAC consensus (selection)– Algebraic dist vs. Focal bisector dist

Selection using algebraic distance

Selection using focal bisector distance

Thank you!!