My Math Forum  

Go Back   My Math Forum > College Math Forum > Calculus

Calculus Calculus Math Forum

LinkBack Thread Tools Display Modes
August 15th, 2013, 01:40 PM   #1
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):

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.
ffriend is offline  

  My Math Forum > College Math Forum > Calculus

algorithm, approach, derivation, lucalkanade

Thread Tools
Display Modes

Similar Threads
Thread Thread Starter Forum Replies Last Post
Two questions - Approach? zaff9 Abstract Algebra 7 September 20th, 2013 11:50 PM
Hessian approach matthius Calculus 0 February 20th, 2012 07:26 AM
How to approach this problem JohnA Algebra 2 February 19th, 2012 09:29 AM
Different Geometry Approach tiba Math Books 0 February 19th, 2012 07:03 AM
How do I approach this? cmmcnamara Advanced Statistics 4 February 10th, 2010 05:49 AM

Copyright © 2019 My Math Forum. All rights reserved.