My Math Forum Derivation of IC approach to Lucal-Kanade algorithm

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 August 15th, 2013, 02:40 PM #1 Newbie   Joined: Jul 2013 Posts: 2 Thanks: 0 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 $\Delta p$ (page 13): $\sum_x \left[ T(W(x, 0)) + \nabla T \frac{\partial W}{\partial x} \Delta p - I(W(x, p)) \right]^2$ where $T$ and $I$ - 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; $W(x, p)$ - is a warp function, that takes coordinates of pixel x and vector of model parameters p; $\nabla T$ - gradient of the template along i and j axis, matrix of size h*2w (2 gradients of image size arranged horizontally); $\frac{\partial W}{\partial x}$ - 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; $\Delta p$ - increments to parameter vector p; Just after first equation they suggest solution to this LSQ problem: $\Delta p = H^{-1} \sum_x \left[ \nabla T \frac{\partial W}{\partial x} \right]^T \left[ I(W(x, p)) - T(x) \right]$ where H is the Hessian matrix with I replaced by T: $H = \sum_x \left[ \nabla T \frac{\partial W}{\partial x} \right]^T \left[ \nabla T \frac{\partial W}{\partial x} \right]$ What I don't understand is how they derive second equation and what $\nabla T \frac{\partial W}{\partial x}$ 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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