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(D_max)≥;0, where D_max 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:

$I(\mathbf{q}) = \frac{1}{V}\int_V\int_V\rho(\mathbf{r})\rho(\mathbf{r}')e^{-i\mathbf{q}(\mathbf{r}-\mathbf{r}')}\text{d}\mathbf{r}\text{d}\mathbf{r}',$

where $\rho(\mathbf{r})$ is the scattering length density. We introduce the correlation function $\gamma(\mathbf{r})$ by:

$I(\mathbf{q}) = \int_V\gamma(\mathbf{r})e^{-i\mathbf{q}\cdot\mathbf{r}}\text{d}\mathbf{r}$

That is, taking the fourier transformation of the correlation function gives the intensity.

The probability of finding, within a particle, a point $i$ at a distance $r$ from a given point $j$ is given by the distance probability function $\gamma_0(r)$. And the connection between the correlation function $\gamma(r)$ and the distance probability function $\gamma_0(r)$ is given by:

$\gamma(r) = b_i\cdot bj\gamma_0(r)V$,

where $b_k$ is the scattering length of the point $k$. That is, the correlation function is weighted by the scattering length. For X-ray scattering, the scattering length $b$ is directly proportional to the electron density $\rho_e$.

Distance distribution function p(r)

See main article on distribution functions.

We introduce the distance distribution function $p(r)$ also called the pair distance distribution function (PDDF). It is defined as:

$p(r) = \gamma(r)\cdot r^2.$

The $p(r)$ function can be considered as a probability of the occurrence of specific distances in a sample weighted by the scattering length density $\rho(\mathbf{r})$. For diluted samples, the $p(r)$ function is not weightened by the scattering length density, but by the excess scattering length density $\Delta\rho(\mathbf{r})$, i.e. the difference between the scattering length density of position $r$ 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 $p(r)$ 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

This is an brief outline of the method introduced by Otto Glatter (Glatter, 1977). Another approach is given by Moore (Moore, 1980).

In indirect fourier transformation, a D_max is defined and an initial distance distribution function $p_i(r)$ is expressed as a sum of N cubic spline functions $\phi_i(r)$ evenly distributed on the interval (0,D_max):

$p_i(r) = \sum_{i=1}^N c_i\phi_i(r),$

(1)

where $c_i$ are scalar coefficients. The relation between the scattering intensity I(q) and the PDDF p_i(r) is:

$I(q) = 4\pi\int_0^\infty p(r)\frac{\sin(qr)}{qr}\text{d}r.$

(2)

Inserting the expression for p_i(r) (1) into (2) and using that the transformation from p(r) to I(q) is linear gives:

$I(q) = 4\pi\sum_{i=1}^N c_i\psi_i(r),$

where $\psi_i(r)$ is given as:

$\psi_i(r)=\int_0^\infty\phi_i(r)\frac{\sin(qr)}{qr}$

The $c_i$'s are unchanged under the linear Fourier transformation and can be fitted to data, thereby obtaining the coifficients $c_i^{fit}$. Inserting these new coefficients into the expression for $p_i(r)$ gives a final PDDF $p_f(r)$. The coefficients $c_i^{fit}$ are chosen to minimize the reduced $\chi^2$ of the fit, given by:

$\chi^2 = \frac{1}{M-P}\sum_{k=1}^{M}\frac{[I_{experiment}(q_k)-I_{fit}(q_k)]^2}{\sigma^2(q_k)}$

where $M$ is the number of datapoints, $P$ is number of free parameters and $\sigma(q_k)$ is the standard deviation (the error) on data point $k$. However, the problem is ill posed and a very oscillating function would also give a low $\chi^2$. Therefore, the smoothness function $S$ is introduced:

$S = \sum_{i=1}^{N-1}(c_{i+1}-c_i)^2$.

The larger the oscillations, the higher $S$. Instead of minimizing $\chi^2$, the Lagrangian $L = \chi^2 + \alpha S$ is minimized, where the Lagrange multiplier $\alpha$ 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: $p_i(r) \rightarrow \text{fitting} \rightarrow p_f(r)$.

Applications

There are recent proposals at automatic determination of constraint parameters using Bayesian reasoning or heuristics.

Alternative approaches

The distance distribution function $p(r)$ can also be obtained by IFT with an approach using maximum entropy (e.g. Jaynes, 1983; Skilling, 1989)