Radon Transform - Relationship With The Fourier Transform

Relationship With The Fourier Transform

The Radon transform is closely related to the Fourier transform. For a function of one variable the Fourier transform is defined by

\hat{f}(\omega)=\int_{-\infty}^\infty f(x)e^{-2\pi ix\omega }\,dx.

and for a function of a 2-vector ,

\hat{f}(\mathbf{w})=\int\limits_{-\infty}^{\infty} \int\limits_{-\infty }^{\infty} f(\mathbf{x})e^{-2\pi i\mathbf{x}\cdot\mathbf{w}}\,dx\, dy.

For convenience define as it is only meaningful to take the Fourier transform in the variable. The Fourier slice theorem then states

\widehat{R_{\alpha}}(\sigma)=\hat{f}(\sigma\mathbf{n}(\alpha))

where

Thus the two-dimensional Fourier transform of the initial function is the one variable Fourier transform of the Radon transform of that function. More generally, one has the result valid in n dimensions

Indeed, the result follows at once by computing the two variable Fourier integral along appropriate slices:

An application of the Fourier inversion formula also gives an explicit inversion formula for the Radon transform, and thus shows that it is invertible on suitably chosen spaces of functions. However this form is not particularly useful for numerical inversion, and faster discrete inversion methods exist.

Read more about this topic:  Radon Transform

Famous quotes containing the words relationship and/or transform:

    “Artists have a double relationship towards nature: they are her master and her slave at the same time. They are her slave in so far as they must work with means of this world so as to be understood; her master in so far as they subject these means to their higher goals and make them subservient to them.”
    —Johann Wolfgang Von Goethe (1749–1832)

    “But I must needs take my petulance, contrasting it with my accustomed morning hopefulness, as a sign of the ageing of appetite, of a decay in the very capacity of enjoyment. We need some imaginative stimulus, some not impossible ideal which may shape vague hope, and transform it into effective desire, to carry us year after year, without disgust, through the routine- work which is so large a part of life.”
    —Walter Pater (1839–1894)