6.3 The Fourier Diffraction Theorem

Fundamental to diffraction tomography is the Fourier Diffraction Theorem, which relates the Fourier transform of the measured forward scattered data with the Fourier transform of the object. The theorem is valid when the inhomogeneities in the object are only weakly scattering. The statement of the theorem is as follows:

When an object, o(x, y), is illuminated with a plane wave as shown in Fig. 6.2, the Fourier transform of the forward scattered field measured on line TT ? gives the values of the 2-D transform, O( ? ₁ , ? ₂ ), of the object along a semicircular arc in the frequency domain, as shown in the right half of the figure.

Fig. 6.2: The Fourier Diffraction Theorem relates the projection to the Fourier Fourier transform of a diffracted transform of the object along a semicircular arc. (From [Sla83].)

The importance of the theorem is made obvious by noting that if an object is illuminated by plane waves from many directions over 360 , the resulting circular arcs in the ( ? _{1 ,} ? ₂ )-plane will fill up the frequency domain. The function o(x, y) may then be recovered by Fourier inversion.

Before giving a short proof of the theorem, we would like to say a few words about the dimensionality of the object vis- -vis that of the wave fields. Although the theorem talks about a two-dimensional object, what is actually meant is an object that doesn t...