[Insight-users] polynomial fitting for fcMRI processing

[J.S.Wijnhout at lumc.nl]

>Hi Jeroen,
>
>  part of what Luis adviced was using Levenberg-Marquardt to fit a
polynomial
>with sum of squared differences as a metric. It sounds to me pretty
much as
>least squares;) BTW, in non-linear cases you always end up with
numerical
>solutions to least squares, so it depends what kind of problem this is.

There is one big difference between the solution I provided and the
Levenberg-Marquardt.
LM is a very useful and generic algorithm, but it is intrinsically
numerical. The
least-squares solution I provided a link to, is not numerical. Just
because the function
is non-linear, doesn't mean you cannot solve the LeastSquares problem
exactly. In this
case, the polynomial least squares problem can be solved exactly. It
just takes a matrix
multiplication (the formulae for the matrix elements you'll have to
compute by hand from
the general formula). So this is probably much faster than
Levenberg-Marquardt, don't
you think?

best,
Jeroen


# Hi,
# 
# If you restrict yourself to polynomials, then why don't you use a 
# LeastSquares fitting algorithm?
# http://mathworld.wolfram.com/LeastSquaresFittingPolynomial.html
# You can work out the cubic case and obtain a closed-form formula for
the 
# fit. I don't know
# how the LevenbergMarquardt performance with respect to the
LeastSquares 
# fitting, but I guess
# that LeastSquares fitting is both faster and more accurate in this
case.
# 
# best,
# Jeroen
# 
# Karen Guan wrote:
# > Hi Luis,
# >
# > Thanks very much for the answer and the reference! However, after
the 
# > manual of itk::MultivariateLegendrePolynomial seems to suggest that 
# > this function works only for 2D and 3D data. Although it's possible
to 
# > add one dummy dimension (eg. all zero), I'm wondering whether
there's 
# > a function more suitable for 1D signal.
# >
# > Best,
# > - Karen
# >
# > On 6/6/07, *Luis Ibanez* <luis.ibanez at kitware.com 
# > <mailto:luis.ibanez at kitware.com>> wrote:
# >
# >
# >     Hi Karen,
# >
# >
# >     You can use the Multivariate Legendre Polynomials,
# >     and combine them with the Linear Kalman estimator,
# >     or with the Levenberg-Marquard optimizer.
# >
# >     
# >
#
http://www.itk.org/Insight/Doxygen/html/classitk_1_1MultivariateLegendre
Polynomial.html
# 
# >
# >     
# >
http://www.itk.org/Insight/Doxygen/html/classitk_1_1KalmanLinearEstimato
r.html
# 
# >
# >     
# >
#
<http://www.itk.org/Insight/Doxygen/html/classitk_1_1KalmanLinearEstimat
or.html>
# 
# >
# >     
# >
#
http://www.itk.org/Insight/Doxygen/html/classitk_1_1LevenbergMarquardtOp
timizer.html
# 
# >
# >
# >
# >     In both cases you will be using the sum of squared
# >     differences between your data and the polynomial,
# >     as the metric to minimize.
# >
# >
# >     Please look at the recent trail by Mathieu Malaterre
# >     on this topic in the users list.
# >
# >
# >
# >         Regards,
# >
# >
# >            Luis
# >
# >
# >     --------------------
# >     Karen Guan wrote:
# >     > Dear all,
# >     >
# >     > I'm working on fcMRI processing, and need polynomial fitting
# >     (3rd order,
# >     > ax3 + bx2 + cx +d) for each time course (with about 600 time
# >     points)
# >     > to remove B0 fluctuation or shifting. The entire data set
# >     > has 128 * 128 * 7 samples ( i.e. time courses).
# >     >
# >     > The questions are:
# >     > 1. Is there such an algorithm in ITK (including vnl/vcl)?
# >     > 2. If so, for fast processing of 1-D signal with 600 samples,
# >     what are
# >     > be best choices?
# >     >
# >     > I appreciate the help!
# >     >
# >     > - X.
# >     >
# >     >
# >     >
# >     >
# >     
# >
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# >
# >
# >
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