Determining Planar Translation and Rotation by Optical Flow David Hong NCSSM, Mini-term 2008

Example Problem 1

Example Problem 2

Example Problem 3

Problem (Formal Statement)

Motivation Many Applications: – Special Effects (Movie) – Video Compression – Pattern Recognition – Image Stabilization (Digital Cameras) – Dead-reckoning (Mobile Robotics)

State of the Art Block Motion Estimation Demir-Ertűrk (2007) Optical Flow Lucas-Kanade (1985) Optical Flow with Smoothness Constraint Horn-Schunck (1980)

X Y u(x,y,t) U(X,Y) x y Idea

Then the floor-coordinate is (X 0,Y 0 ) and the sensor-coordinate is (x,y) at time t. Let us consider a point on the plane. From there, we can see:

Idea Differentiating on time gives us:

Idea Expressing (x,y) in terms of (X 0,Y 0 ) and the mouse position (X s,Y s,Θ s ) gives us:

Idea Putting the two together, we get: This is underdetermined!

Idea We choose a convenient unit so Δx= Δy=Δt=1 Discretizing, we get:

Idea This is normally over-determined! Now, using all points, we get: Use least-squares!

Algorithm u u’

Improvement by Iteration u u’ u’’ u’’’ It is more difficult with rotation! Lagrange Interpolation!

Improvement by Iteration u u’

Performance of Algorithm Surface:

Performance of Algorithm Good Surface:Bad Surface: Algorithm Fails!

Performance of Algorithm Surface: First iteration: Second iteration:

Performance of Algorithm Surface:

Performance of Algorithm Surface:

Future Work Smoothness Constraint Pre-processing the image Condition for Convergence Phase Correlation

Thank You!