Ellipse Fitting COMP 4900C Winter 2008.

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Presentation transcript:

Ellipse Fitting COMP 4900C Winter 2008

Ellipses in Images Perspective projection of circles form ellipses in the images.

Equations of Ellipse Let Then

Ellipse Fitting: Problem Statement Given a set of N image points find the parameter vector such that the ellipse fits best in the least square sense: Where is the distance from to the ellipse.

Euclidean Distance Fit is the point on the ellipse that is nearest to is normal to the ellipse at

Compute Distance Function Computing the distance function is a constrained optimization problem: subject to Using Lagrange multiplier, define: where Then the problem becomes: Set we have

Two Approximations 1. First-order approximation at 2. Assume is close to , then

Approximate Distance Function Solve for Substitute back

Ellipse Fitting with Euclidean Distance Given a set of N image points find the parameter vector such that This problem can be solved by using a numerical nonlinear optimization system.