Fourier Based Reconstruction

The Radon transform defines a mathematical mapping that relates a two-dimensional object, f (x, y), to its one-dimensional projections, p(r, 0), measured at different angles around the object [4,30]:

and lTo0 represents a straight line that has a perpendicular distance r from the origin and is at an angle 0 with respect to the x-axis. It can be shown that an object can be uniquely reconstructed if its projections at various angles are known [4,30]. Here, p(r, 0) is also referred to as line integral. It can also be shown that the Fourier transform of a one-dimensional projection at a given angle describes a line in the two-dimensional Fourier transform of f (x, y) at the same angle. This is known as the central slice theorem, which relates the Fourier transform of the object and the Fourier transform of the object's Radon transform or projection. The original object can be reconstructed by taking the inverse Fourier transform of the two-dimensional signal which contains superimposed one-dimensional Fourier transform of the projections at different angles, and this is the so-called Fourier reconstruction method. A great deal of interpolation is required to fill the Fourier space evenly in order to avoid artifacts in the reconstructed images. Yet in practice, an equivalent but computationally less demanding approach to the Fourier reconstruction method is used which determines f (x, y) in terms of p(r, 0) as:

where f(r) is a filter function that is convolved with the projection function in the spatial domain. Ramachandran and Lakshminarayanan [31] showed that exact reconstruction of f (x, y) can be achieved if the filter function f (r) in equation (2.8) is chosen as

0 otherwise where } represents the Fourier transform of f(r) and ma is the highest frequency component in f (x, y). The filter function f (r) in the spatial domain can be expressed as:

This method of reconstruction is referred to as the filtered-backprojection, or the convolution-backprojection in the spatial domain. The implementation of FBP involves four major steps:

1. Take the one-dimensional Fourier transform for each projection.

2. Multiply the resultant transformation by the frequency filter.

3. Compute the inverse Fourier transform of the filtered projection.

4. Back-project the data for each projection angle.

However, the side effect of the ramp filtering using equation (2.9) is that high-frequency components in the image that tend to be dominated by statistical noise are amplified [32]. The detectability of lesion or tumor is therefore severely hampered by this noise amplification during reconstruction by FBP, particularly when the scan duration is short or the number of counts recorded is low. To obtain better image quality, it is desirable to attenuate the high-frequency components by using some window functions, such as the Shepp-Logan or the Hann windows, which modify the shape of the ramp filter at higher frequencies [33]. Unfortunately, the attenuation of higher frequencies in filtering process will degrade the spatial resolution of the reconstructed images, and we will briefly discuss it in Section 2.13.

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