Minimizing error function -> sloving linear system of equations

[Theraven1982]

Homework Statement

I want to find certain coefficients \alpha_{uv} by minimizing a error function. In the end, I want to make a function of this, so iteration is not a problem. I want to estimate these coefficients to find the best estimate for f(x,y).

Homework Equations

The error function:
<br /> E(\alpha)=\sum_{x,y}w(x,y)\Big( f(x,y)-\sum_{u,v=-N}^{N}\alpha_{u,v}f(x+u, y+v)\Big)^{2}<br />
with
<br /> w(x,y)=P(f(x,y)\in M_1|f(x,y)),<br />
which in turn is equal to

<br /> \frac{1}{\sigma\sqrt{2\pi}} \frac{exp\Big[ -\frac{1}{2\sigma^2} \Big( f(x,y) - \sum_{u,v=-N}^{N}\alpha_{u,v}f(x+u, y+v)\Big)^2\Big]}{ \sum_{i=1}^{2}P(f(x,y)|f(x,y)\in M_{i}) }<br />

The Attempt at a Solution

<br /> \frac{\partial E}{\partial \alpha_{s,t}}=0<br />

eventually gives:

<br /> \sum_{u,v=-N}^{N}\alpha_{u,v}=\frac{ \sum_{x,y}w(x,y)f(x+s, y+t)f(x,y) }{ \sum_{x,y}w(x,y)f(x+s, y+t)f(x+u, y+v) }<br />

So, now we have a linear system of equations. I don't know how to solve this in a efficient way.
Thanks in advance,

 

[HallsofIvy]

That's a very wide question! There are many different ways of solving a linear system- which is most efficient depends on the coefficients. Gaussian eliminaion with pivoting is probably the best general method.

 

[Theraven1982]

I was afraid of that ;). I'll find my linear algebra book ;).
Is there anyone who has experience in solving these equations in Matlab? There are probably functions for this purpose ; again, any kick in the right direction is welcome ;)