- #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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