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

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

The Attempt at a Solution

$$\frac{\partial E}{\partial \alpha_{s,t}}=0$$

eventually gives:

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

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 ;)