accuracy of dense optical flow

Question

Currently I am learning dense optical flow by myself. To understand it, I conduct one experiment. I produce one image using Matlab. One box with a given grays value is placed under one uniform background and the box is translated two pixels in x and y directions in another image. The two images are input into the implementation of the algorithm called TV-L1. The generated motion vector outer of the box is not zero. Is the reason that the gradient outer of the box is zero? Is the values filled in from the values with large gradient value? In Horn and Schunck's paper, it reads

In parts of the image where the brightness gradient is zero, the velocity estimates will simply be averages of the neighboring velocity estimates. There is no local information to constrain the apparent velocity of motion of the brightness pattern in these areas.

The progress of this filling-in phenomena is similar to the propagation effects in the solution of the heat equation for a uniform flat plate, where the time rate of change of temperature is proportional to the Laplacian.

Is it not possible to obtain correct motion vectors for pixels with small gradients? Or the experiment is not practical. In practical applications, this doesn't happen.

Answer 1

Two pioneers in this field Lukas&Kanade (LK) and Horn&Schunch (HS) are developed methods for computing Optical Flow (OF). Both rely on brightness constancy assumption which feature location pixel values between two sequence frames not change. This constraint may be expressed as two equations: I(x+dx,y+dy,t+dt)=I(x,y,t) and ∂I/∂x dx+∂I/∂y dy+∂I/∂t dt=0 by using a Taylor series expansion I(x+dx,y+dy,t+dt) , we get (x+dx,y+dy,t+dt)=I(x,y,t)+∂I/∂x dx+∂I/∂y dy+∂I/∂t dt… letting ∂x/∂t=u and ∂y/∂t=v and combining these equations we get the OF constraint equation: ∂I/∂t=∂I/∂t u+∂I/∂t v . The OF equation has more than one solution, so the different techniques diverge here. LK equations are derived assuming that pixels in a neighborhood of each tracked feature move with the same velocity as the feature. In OpenCV, to catch large motions with a small window size (to keep the “same local velocity” assumption).

Answer 2

Yes, in so called homogenous image regions with very small gradients no information where a motion can dervided from exists. That's why the motion from your rectangle is propagated outer the border. If you give your background a texture this effect will be less dominant. I know such problem when it comes to estimate the ego-motion of a car. Then the streat makes a lot of problems cause of here homogenoutiy.