]> Fitting Individual Background Optical Properties

Fitting Individual Background Optical Properties

If not known from some other experimental measurement, background optical properties can be obtained by fitting to the experimental data. While generalized routines for fitting the background optical properties are available, if only one set of optical properties ( $μ_{a}$ or $μ_{s}^{'}$ ) are needed, it is much faster to used the specialized fitting routines fitMua() and fitMusp().

These routines use the Matlab Optimization Toolbox, which is not part of the core Matlab package. If the Optimization Toolbox is not available at your local site, try the general fitting routines, which do not require additional toolboxes be installed.

Function Summary

Syntax:	mua = fitMua(SD, Medium, MeasList, data);
Inputs:	SD	SD structure
	Medium	Medium structure
	MeasList	Measurement List that corresponds to experimental data
	data	Experimental data to be fit against
Outputs:	mua	Best-fit optical absorption at each wavelength

Syntax:	musp = fitMusp(SD, Medium, MeasList, data);
Inputs:	SD	SD structure
	Medium	Medium structure
	MeasList	Measurement List that corresponds to experimental data
	data	Experimental data to be fit against
Outputs:	musp	Best-fit optical scattering at each wavelength

Detailed Descriptions

fitMua() and fitMusp() fit a theoretical curve to experimental data. The functions are implemented using fminbnd(), a Matlab function from the Optimization Toolbox which implements a bounded golden-mean minimization of a user-supplied cost function. Because the minimization is bounded and one-dimensional, convergence to the optimal solution is very fast. If the minimum lies outside the default bounds, the function will return one of the endpoints as an "optimal" solution. There is currently no way to change the bounds without editing the function directly.

The fit proceeds by minimizing the cost function

C = \sqrt{\sum_{i} \log^{2} | Φ_{i} / Φ_{i}^{0} (μ) |}

where the $Φ_{i}$ are the individual experimental measurements and and $Φ_{i}^{0} (μ)$ are the theoretical predictions, given optical properties $μ$ . The norm of the log of the ratio was chosen over the more common $χ^{2}$ cost functions so as to give equal weighting to points with large and small amplitudes, which seems to yield a better fit to the data. Given the the optical properties and the theoretical curve, the cost function also tries to determine the source and detector coupling coefficients (see fitSD()) that minimize the difference between theory and experiment. This is done before computing the cost function $C$ .