Hierarchical Lucas and Kanada with multi-scale

The Lucas–Kanade method is a widely used differential method for optical flow estimation.

Assumptions –

1. flow is essentially constant in a local neighbourhood – solves the basic optical flow equations for all the pixels in that neighbourhood, by the least squares criterion.
2. motion is linear :
   When it is non linear – we will have to do iterative refinement.
3. motion is small:
   When motion is large we will have to do a coarse to fine method to reduce resolution and solve each pixel at a lower resolution.

The optical flow equations become solvable when the momentum matrix has two eigenvalues(λ1 and λ2)  with large values(same as with harris corner detection method), which usually happens along corners or edges that have many various gradients (enabling easy unique location identification).

Hierarchical multi-scaling with gaussian pyramids:

• Compute a pyramid scale of the image.(see next clause for alias issues).
• start by using the iterative method on coarsest level.
• Compare the current level to the next level and upscale to the next level with a very close accuracy.
• Repeat the iterative method.Computing a pyramid of the image can cause alias issues in subsampling if you are not using a gaussian filter before subsampling. If the subsampling is done without a filter, then an aliasing problem occurs and when comparing and upscaling the pixels won’t be the more or less the same.The solution to this problem, is computing the pyramid with a gaussian filter and also saving the DOG(band pass) images as a byproduct, in order to reconstruct the pyramid when doing the upscaling.