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Minimizing error function -> sloving linear system of equations

  1. Feb 6, 2009 #1
    1. The problem statement, all variables and given/known data
    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].



    2. Relevant 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]
    3. 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,
     
    Last edited: Feb 6, 2009
  2. jcsd
  3. Feb 6, 2009 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    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.
     
  4. Feb 6, 2009 #3
    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 ;)
     
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