Fitting

Experimental data analysis frequently leads to the following set of m simultaneous equations for the n (< m) unknowns cj (an overdetermined system):

File:Hepa img354.gif

Here the cj are the unknowns and the fj(ui), b File:Hepa img355.gif are known. If we introduce

the (m,n) matrix  | A = (fj(ui))

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the (n,1) matrix  | File:Hepa img356.gif
the (m,1) matrix  | File:Hepa img357.gif

the problem to solve becomes

File:Hepa img358.gif

where the sign  means that we want to find the vector x in the range of A which is closest to b according to some norm (  Branham90, Flowers95).

As an example we choose the fitting of a second-order polynomial. With fi(uj) = uji-1, the matrix A in the above equation becomes

File:Hepa img360.gif

and Ax = b can be solved e.g. by QR decomposition: QRx = b becomes x = R-1 QTb.

File:Hepa img361.gif

As a second example we look at the fitting of a second-order surface

File:Hepa img362.gif

through the  neighbours of a point in an image. The coordinates u,v and the given values b are:

File:Hepa img364.gif

The coefficients of the second-order polynomial File:Hepa img365.gif can be found by solving Ax = z with the least squares condition, where

File:Hepa img366.gif

Using the pseudoinverse, one gets x = A+b.

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