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August 15th, 2013, 02:40 PM  #1 
Newbie Joined: Jul 2013 Posts: 2 Thanks: 0  Derivation of IC approach to LucalKanade algorithm
(where IC stand for Inverse Compositional, see below) I'm reading this paper about LucasKanade algorithm for image alignment. To make a long story short, they reduce the task of alignment to solving the following leastsquares 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 elementwise 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. 

Tags 
algorithm, approach, derivation, lucalkanade 
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