Indirect Fourier transform
Indirect Fourier transform (IFT) is a solution of ill-posed given by Fourier transform of noisy data (as from biological small-angle scattering) proposed by Glatter. IFT is used instead of direct Fourier transform of noisy data, since a direct FT would give large systematic errors.
Transform is computed by linear fit to a subfamily of functions corresponding to constraints on a reasonable solution. If a result of the transform is distance distribution function, it is common to assume that the function is non-negative, and is zero at P(0) = 0 and P(Dmax)≥;0, where Dmax is a maximum diameter of the particle. It is approximately true, although it disregards inter-particle effects.
IFT is also performed in order to regularize noisy data.
Fourier transformation in small angle scattering
see Lindner et al. for a thorough introduction 
The intensity I per unit volume V is expressed as:
where is the scattering length density. We introduce the correlation function by:
That is, taking the fourier transformation of the correlation function gives the intensity.
The probability of finding, within a particle, a point at a distance from a given point is given by the distance probability function . And the connection between the correlation function and the distance probability function is given by:
where is the scattering length of the point . That is, the correlation function is weighted by the scattering length. For X-ray scattering, the scattering length is directly proportional to the electron density .
Distance distribution function p(r)
See main article on distribution functions.
We introduce the distance distribution function also called the pair distance distribution function (PDDF). It is defined as:
The function can be considered as a probability of the occurrence of specific distances in a sample weighted by the scattering length density . For diluted samples, the function is not weightened by the scattering length density, but by the excess scattering length density , i.e. the difference between the scattering length density of position in the sample and the scattering length density of the solvent. The excess scattering length density is also called the contrast. Since the contrast can be negative, the function may contain negative values. That is e.g. the case for alkyl groups in fat when dissolved in H2O.
Introduction to indirect fourier transformation
In indirect fourier transformation, a Dmax is defined and an initial distance distribution function is expressed as a sum of N cubic spline functions evenly distributed on the interval (0,Dmax):
where are scalar coefficients. The relation between the scattering intensity I(q) and the PDDF pi(r) is:
Inserting the expression for pi(r) (1) into (2) and using that the transformation from p(r) to I(q) is linear gives:
where is given as:
The 's are unchanged under the linear Fourier transformation and can be fitted to data, thereby obtaining the coifficients . Inserting these new coefficients into the expression for gives a final PDDF . The coefficients are chosen to minimize the reduced of the fit, given by:
where is the number of datapoints, is number of free parameters and is the standard deviation (the error) on data point . However, the problem is ill posed and a very oscillating function would also give a low . Therefore, the smoothness function is introduced:
The larger the oscillations, the higher . Instead of minimizing , the Lagrangian is minimized, where the Lagrange multiplier is called the smoothness parameter. It seems reasonably to call the method indirect fourier transformation, since a direct formation is not performed, but is done in three steps: .
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