The Fourier method of reconstruction can be applied in two ways: either
directly or by using filters. In the direct Fourier method, the Fourier trans-
forms of individual acquisition projections are taken in polar coordinates
in the frequency domain, which are then used to calculate the values in rec-
tangular coordinates. Inverse Fourier transforms of these profiles are taken
to compute the image. The method is not a true backprojection and is rarely
used in reconstruction of images in nuclear medicine because of the time-
consuming computation.
A more convenient method of reconstruction is the filtered backprojec-
tion (FBP) using the Fourier technique. In this method, filters are used to
eliminate the blurring function l/rthat arises from simple backprojection of
the projection data. These filters are analogous to tone controls or equa-
lizers in radios or CD players that act as filters to vary the amplitudes of
different frequencies, bass for low-frequency amplitudes and treble for
high-frequency amplitudes. In image reconstruction, filters do the same
thing, modulating the amplitudes of different frequencies, preserving the
broad structures (the image) represented by low frequencies and removing
the fine structures (noise) represented by high frequencies.
The Fourier method of filtering the projection data is based on the initial
transformation of these data from the spatial domain to the frequency
domain, which is symbolically expressed as
F′(x,y) =Ff(x,y) (12.2)where F′(x,y) is the Fourier transform of f(x,y) and Fdenotes the Fourier
transformation. Next a Fourier filter,H() is applied in the frequency
domain; that is,
F′() =H() · F() (12.3)where F′() is the filtered projection in the frequency domain, which is
obtained as the multiplication product of H() and F(). Finally, the inverse
Fourier transformation is performed to obtain the filtered projections,
which are then backprojected. The results obtained by the Fourier method
are identical to those obtained by the convolution method. Although the
Fourier method appears to be somewhat cumbersome and difficult to
understand, the use of modern computers has made it much easier and
faster to compute the reconstruction of images than the convolution
method.
Types of Filters
A number of Fourier filters have been designed and used in the recon-
struction of images in nuclear medicine. All of them are characterized by a
maximum frequency, called the Nyquist frequency, that gives an upper limit
to the number of frequencies necessary to describe the sine or cosine curves
162 12. Single Photon Emission Computed Tomography