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FittingGaussiansToDipoles.nb
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(************** Content-type: application/mathematica **************
CreatedBy='Mathematica 5.1'
Mathematica-Compatible Notebook
This notebook can be used with any Mathematica-compatible
application, such as Mathematica, MathReader or Publicon. The data
for the notebook starts with the line containing stars above.
To get the notebook into a Mathematica-compatible application, do
one of the following:
* Save the data starting with the line of stars above into a file
with a name ending in .nb, then open the file inside the
application;
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Data for notebooks contains only printable 7-bit ASCII and can be
sent directly in email or through ftp in text mode. Newlines can be
CR, LF or CRLF (Unix, Macintosh or MS-DOS style).
NOTE: If you modify the data for this notebook not in a Mathematica-
compatible application, you must delete the line below containing
the word CacheID, otherwise Mathematica-compatible applications may
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For more information on notebooks and Mathematica-compatible
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Notebook[{
Cell[TextData[{
StyleBox["Fitting Gaussians to Dipoles\n", "Title"],
StyleBox["Eugene d'Eon - NVIDIA Corporation\n", "Subsubtitle",
FontSlant->"Italic"],
StyleBox["\nThis notebook provides simple Mathematica code useful for \
finding a sum of Gaussians fit for a known diffusion profile. Here we work \
through examples using the dipole approximation found in [Jensen et al. 2001] \
(SIGGRAPH 2001), but the same code will work for multipole-based (or \
otherwise known) profiles.",
FontSlant->"Italic"]
}], "Text"],
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Cell["Dipole Definition", "Section"],
Cell["\<\
Definition of the dipole diffusion profile. Inputs are radial distance, r, \
index of refraction, eta, absorption cross-section, sigmaa, and scattering \
cross-section, sigmas. Eta is assumed to be greater than 1. Note that in \
[Jensen et al. 2001] there was an extra sigma term in the denominator of the \
second term. The correct formulae are found in [Jensen and Buhler 2002] and \
[Donner and Jensen 2005].\
\>", "Text"],
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Plotting the dipole for the green wavelength of marble (parameters taken from \
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useful for visually gauging accuracy of any fitting operation.\
\>", "Text"],
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Cell["Sums of Gaussians", "Section"],
Cell[CellGroupData[{
Cell["Gaussian definition", "Subsection"],
Cell["\<\
We define a gaussian, g, based on radius, r, and variance, v. (Variance is \
the square of the standard deviation). \
\>", "Text"],
Cell[BoxData[
\(g[r_,
v_] := \[ExponentialE]\^\(-\(r\^2\/\(2\ v\)\)\)\/\(2\ \[Pi]\ v\)\)], \
"Input"]
}, Open ]],
Cell[CellGroupData[{
Cell["Single Gaussian Fit", "Subsection"],
Cell["\<\
For fitting a single Gaussian function to the dipole function, we use \
FindMinimum with the error expressed as a summation in the form needed for \
Levenberg-Marquardt to work.\
\>", "Text"],
Cell[BoxData[
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Cell[CellGroupData[{
Cell[BoxData[
\(singleGaussFit =
Last[\[IndentingNewLine]FindMinimum[\[IndentingNewLine]\ \ Sum[\
\[IndentingNewLine]r \((singleGaussSum - R[r, eta, sigmaa, sigmas])\)\^2, {r,
0, 10, 10\/500}\[IndentingNewLine]], {{w1, .5}, {v1, .2}}\
\[IndentingNewLine]]\[IndentingNewLine]]\)], "Input"],
Cell[BoxData[
\({w1 \[Rule] 0.3395204000621387`,
v1 \[Rule] 0.12585927168766622`}\)], "Output"]
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Cell["\<\
Here we plot the dipole and the single Gaussian fit for comparison.\
\>", "Text"],
Cell[CellGroupData[{
Cell[BoxData[
\(\(Plot[{r\ R[r, eta, sigmaa, sigmas],
r\ ReplaceAll[singleGaussSum, singleGaussFit]}, {r, 0,
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