- #1

Theraven1982

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## Homework Statement

I want to find certain coefficients [tex]\alpha_{uv}[/tex] 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 [tex]f(x,y)[/tex].

## Homework Equations

The error function:

[tex]

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}

[/tex]

with

[tex]

w(x,y)=P(f(x,y)\in M_1|f(x,y)),

[/tex]

which in turn is equal to

[tex]

\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}) }

[/tex]

## The Attempt at a Solution

[tex]

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

[/tex]

eventually gives:

[tex]

\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) }

[/tex]

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

Thanks in advance,

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