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Suppose we have 2d data f(l,p), l[1,25000] p[1,5000] and we want to
estimate a 2d polynomial D.26 with these data. The
system of equations looks like

(D29) 
The convention used in the Doris software is that we first normalize
the data to avoid numerical instabilities (see source utilities.[hc]). The maximum coordinates are that of the original
master (stored in the result file of the master image, typically 25000
lines and 5000 range pixels). The coordinates are rescaled to the
interval [2,2].

(D30) 
(Another way, perhaps a better one (?), would be to make the data
zeromean, unit standard deviation) The estimated coefficients thus
correspond to the normalized data. For evaluation, the data has to be
normalized by the same factors a,b. Normally the information from the
master.originalwindow.linelo etc. are used, e.g., for the
coregistration and the reference surface polynomial. These numbers
can be found in the master result file after the step readfiles at place number
of lines original of datafile. A function normalize is called
to do the normalization, so it is easy to change the implementation to
a different normalization. It has been noticed that for higher order
polynomials the normalization factor is very important to obtain a
stable estimate.
Next: Evaluation of polynomials
Up: Polynomials
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Leijen
20090414