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August 15th, 2013, 02:40 PM   #1
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Derivation of IC approach to Lucal-Kanade algorithm

(where IC stand for Inverse Compositional, see below)

I'm reading this paper about Lucas-Kanade algorithm for image alignment. To make a long story short, they reduce the task of alignment to solving the following least-squares problem with respect to (page 13):



where
and - are template and image, both matrices of h*w (height by width of original image). From mathematical point of view they are functions whose domain is vector of coordinates and value is pixel intensity at that coordinates;
- is a warp function, that takes coordinates of pixel x and vector of model parameters p;
- gradient of the template along i and j axis, matrix of size h*2w (2 gradients of image size arranged horizontally);
- Jacobian of W(x, p) evaluated at (x, 0); effectively, this is a matrix of size 2w * nh, where n is a number of parameters and equals 6 in my case;
- increments to parameter vector p;

Just after first equation they suggest solution to this LSQ problem:



where H is the Hessian matrix with I replaced by T:



What I don't understand is how they derive second equation and what means. This is definitely not just matrix multiplication since matrices are incompatible. In some other places they used element-wise multiplication in similar equations, but if doesn't seem to be the case too. And since I don't understand derivation, I can't grasp it myself.

I'm really stuck on it, so any ideas of what happens here are highly appreciated.
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