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From: luca on 9 Jul 2010 05:30
Hi,

in a paper i am reading, there is the following derivation:


Suppose you have a vector v=[x y 1] and a 3x3 real matrix G. They
define a "group action" w : SL(3)xP^2 -> P^2.
For all G in SL(3), we have that w(G):P^2->P^2 is an automorphism.
So,
if
G=[g11 g12 g13; g21 g22 g23; g31 g32 g33]= [g1; g2; g3] (g1,g2 and g3
are the first, second and third row respectively):


p' = w(G)(p) = [<g1, p>/<g3, p> <g2, p>/<g3, p> 1]


where < x, y > is the dot-product of the vectors x and y.


Now, i have to calculate the jacobian of w, Jw with respect to Z in
SL(3), computed at Z=I.
The paper says that:


Jw is a 3x9 matrix of the form


p^T 0 -x*p^T
0 p^T -y*p^T
0 0 0


where 0 is the null row vector (1x3) and p^T is the transpose of the
column vector p.


How do i arrive at this solution?


Than you,
Luca


Ps.
Here G is an homography. They use the following parameterization:


G(x) = exp(A(x))


where A(x) = sum i=1..8 (x_i * A_i)


A_i = 3x3 matrices with a null trace (a canonical basis of the Lie
algebra sl(3))
x = (x1,..,x8) is a vector of reals of 8 dimension.


This is a non-standard parameterization of an homography and since i
do not know lie algebras, lie groups and so on
i don't understand how Jw is computed.


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