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

 Calculus Calculus Math Forum

 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 (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. Tags algorithm, approach, derivation, lucalkanade Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post zaff9 Abstract Algebra 7 September 21st, 2013 12:50 AM matthius Calculus 0 February 20th, 2012 08:26 AM JohnA Algebra 2 February 19th, 2012 10:29 AM tiba Math Books 0 February 19th, 2012 08:03 AM cmmcnamara Advanced Statistics 4 February 10th, 2010 06:49 AM

 Contact - Home - Forums - Cryptocurrency Forum - Top      