7. Discussion

Figure 10. The pole reduced data set (using DFT).

20 Fourier Transforms - High-tech Application and Current Trends

Figure 11. The pole reduced data set (using IRLS-FT).

We presented a new algorithm for the 2D Fourier transform. Our purpose was to increase the noise rejection capacity of the Fourier transform. To do this, we applied the tools of inverse problem theory. In order to discretize the continuous function of the complex spectrum, series expansion was used. It was shown, that the Jacobian matrix of the inverse problem can be written as the inverse FT of the basis functions used in the discretization. Because of this reason Hermite functions were chosen as they are eigenfunctions of the Fourier transformation. This selection gave the possibility of very quick computation of the Jacobian even in 2D problems.

The unknown parameters (series expansion coefficients) are determined by solving an overdetermined inverse problem. For having a robust 2D FT method Cauchy-Steiner weights were applied in a robust iteratively reweighted least squares algorithm. In order to characterize the accuracy and the noise rejection capacity of the new Fourier Transform method we made numerical test using synthetic data sets containing random noise of Cauchy distribution and the characteristic distance between spectra calculated by means of noisy data as well as noisefree ones was calculated. It was shown that compared to the traditional DFT the characteristic distances were reduced by a factor of 6–7 so the noise reduction capability of the new inversion-based Fourier transform method (for abbreviation we used IRLS-FT) was clearly demonstrated.

Fourier transformation is widely used in science and techniques, so the new robust 2D Fourier transform method seems to be applicable on various fields of data processing dealing with noisy data sets, especially those containing outliers. As an example, we presented its application in reduction to pole, which is a frequently used operation in the interpretation of geomagnetic data sets. By our experience, the new method shows sufficient noise rejection capability compared to the traditional reduction to pole algorithm using the well-known DFT.
