Indirect Fourier transform
In a Fourier transform (FT), the Fourier transformed function is obtained from by:
where is defined as . can be obtained from by inverse FT:
and are inverse variables, e.g. frequency and time.
Obtaining directly requires that is well known from to , vice versa. In real experimental data this is rarely the case due to noise and limited measured range, say is known from to . Performing a FT on in the limited range may lead to systematic errors and overfitting.
An indirect Fourier transform (IFT) is a solution to this problem.
Indirect Fourier transformation in small angle scattering
In small-angle scattering on single molecules, an intensity is measured and is a function of the magnitude of the scattering vector , where is the scattered angle, and is the wavelength of the incoming and scattered beam (elastic scattering) . has units 1/length. is related to the so-called pair distance distribution function via. Fourier Transformation. is a (scattering weighted) histogram of distances between pairs of atoms in the molecule. In one dimensions ( and are scalars), and are related by:
where is the angle between and , and is the number density of molecules in the measured sample. The sample is orientational averaged (denoted by ), and the Debye equation  can thus be exploited to simplify the relations by
In 1977 Glatter proposed an IFT method to obtain form , and three years later, Moore introduced an alternative method. Others have later introduced alternative and automathised methods for IFT, and automatised the process 
The Glatter method of IFT
This is an brief outline of the method introduced by Otto Glatter. For simplicity, we use in the following.
In indirect fourier transformation, a guess on the largest distance in the particle is given, and an initial distance distribution function is expressed as a sum of cubic spline functions evenly distributed on the interval (0,):
where are scalar coefficients. The relation between the scattering intensity and the is:
Inserting the expression for pi(r) (1) into (2) and using that the transformation from to 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 . The coefficients are chosen to minimise the of the fit, given by:
where is the number of datapoints and is the standard deviations on data point . The fitting problem is ill posed and a very oscillating function would give the lowest despite being physically unrealistic. Therefore, a smoothness function is introduced:
The larger the oscillations, the higher . Instead of minimizing , the Lagrangian is minimized, where the Lagrange multiplier is denoted the smoothness parameter. The method is indirect in the sense that the FT is done in several steps: .
- P. Scardi, S. J. L. Billinge, R. Neder and A. Cervellino (2016). "Celebrating 100 years of the Debye scattering equation". Acta Crystallogr A. 72: 589–590. doi:10.1107/S2053273316015680.
- O. Glatter (1977). "A new method for the evaluation of small-angle scattering data". Journal of Applied Crystallography. 10: 415–421. doi:10.1107/s0021889877013879.
- P.B. Moore (1980). "Small-angle scattering. Information content and error analysis". Journal of Applied Crystallography. 13: 168–175. doi:10.1107/s002188988001179x.
- S. Hansen, J.S. Pedersen (1991). "A Comparison of Three Different Methods for Analysing Small-Angle Scattering Data". Journal of Applied Crystallography. 24: 541–548. doi:10.1107/s0021889890013322.
- B. Vestergaard and S. Hansen (2006). "Application of Bayesian analysis to indirect Fourier transformation in small-angle scattering". Journal of Applied Crystallography. 39: 797–804. doi:10.1107/S0021889806035291.
- Petoukhov M. V. and Franke D. and Shkumatov A. V. and Tria G. and Kikhney A. G. and Gajda M. and Gorba C. and Mertens H. D. T. and Konarev P. V. and Svergun D. I. (2012). "New developments in the ATSAS program package for small-angle scattering data analysis". Journal of Applied Crystallography. 45: 342–350. doi:10.1107/S0021889812007662. PMC .