Usage of the Fourier transform as an invertible extension to the Radon transform with application to wave front extraction

The Radon transform is used in many applications in order to detect predefined geometrical structures, for instance curves in a given two-dimensional dataset. It is a mapping between a data space and a model space; the coordinates of a point in the latter correspond to the parameters of a geometrical structure in the former. The construction of an inverse operator is usually not straightforward, it is however possible to obtain numerical approximations. In this paper, an extension to the Radon transform is proposed which simplifies inversion by using a Fourier transform along the input curve instead of a line integral. Thereby, the model space is extended by one dimension containing the Fourier coefficients, from which information about the amplitude variation along the input curve can be extracted. The method is successfully applied in order to extract wave fronts from acoustical impulse response measurements taken on a circular microphone array.