Dense Optical Flow - Polynomial Expansion by Gunner Farneback

It probably requires time to get into details of Polynomial Expansion in order to have a full understanding.

The follow paragraph could be very wrong. Need revisiting later to understand what's under-the-hood.
Farneback method uses Polynomial Expansion to approximate the neighbors of a pixel. The Expansion could be seen as a quadratic equation with Matrices and Vectors as variable and coefficients. This dense optical flow analysis produces a displacement field from two successive video frames. Each displacement vector in the field is estimated by minimizing the 'error' under a 'constraint'. The 'constraint' is an equation A(x)d(x)=delta-b(x) derived from the polynomial expansion. The 'error' is the weighted sum of differences in the pixel neighborhood (x + delta-x) between the images.
Image Pyramids is used to detect large displacements. Also uses Gaussian to smooth out the neighboring displacements. I suppose that's based on the assumption that pixels in proximity move in similar directions.

Sample (fback.cpp)

• The larger distance movement are not detected with fewer pyramid levels. And there is no noticeable gain in performance.
• Displacement vectors are drawn at 16 pixels, possibly to avoid cluttering.'
• Overall is slower than sparse as expected. It seems to have fewer noise. And it detects new objects by nature (Dense).

Resource
http://www.opencv.damienstewart.net/2009/05/25/optical-flow-gunnar-farneback/

Reading
Two Frame Motion Estimation Based on Polynomial Expansion, Farneback
Polynomial Expansion for Orientation and Motion Estimation, Farneback