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 with, relationship and/or transform:

    “When a mother quarrels with a daughter, she has a double dose of unhappiness—hers from the conflict, and empathy with her daughter’s from the conflict with her. Throughout her life a mother retains this special need to maintain a good relationship with her daughter.”
    —Terri Apter (20th century)

    “I began to expand my personal service in the church, and to search more diligently for a closer relationship with God among my different business, professional and political interests.”
    —Jimmy Carter (James Earl Carter, Jr.)

    “Government ... thought [it] could transform the country through massive national programs, but often the programs did not work. Too often they only made things worse. In our rush to accomplish great deeds quickly, we trampled on sound principles of restraint and endangered the rights of individuals.”
    —Gerald R. Ford (b. 1913)