analyzing error of fit functions for ellipses paul l. rosin bmvc 1996
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Analyzing error of fit functions for ellipses
Paul L. Rosin
BMVC 1996
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Why?
• Ellipse fitting to pupil boundary
• RANSAC (Random sample consensus)
– Explore fits– Select best fit
• Selection based on error criterion
Pupil edge pixelsNoise pixels
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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
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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
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Gradient weighted AD (GWAD)
Inversely weight AD with its gradientEllipse boundary
Isovalue contours
- Reduced curvature bias
- Asymmetry exists
- Gradient inside > gradient outside
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Second order approximation
– Does not exist for points near high curvature sections
Ellipse boundary
Isovalue contours
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Pavlidis’ approximation
– Improvement over basic algebraic distance
Ellipse boundary Ellipse boundary
EOF1EOF8
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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
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Directional derivative weighted AD
– Wavy isovalue contours of GWAD are reduced
Ellipse boundary)( jXQ
rXj
C
Ellipse boundary
EOF2 EOF10
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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
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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
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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
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Focal bisector distance
– Reflection property: PF’ is a reflection of PF
– Very low curvature bias– Symmetric
Ellipse boundary
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Radial distance
– Comparison with focal bisector distance
C
T
EOF5 = XjT
Ellipse boundary
Ellipse boundary
EOF5 = XjT EOF13 = XjIj
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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
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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
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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
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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
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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
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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)
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Our work
• RANSAC consensus (selection)– Algebraic dist vs. Focal bisector dist
Selection using algebraic distance
Selection using focal bisector distance
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Thank you!!