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Projection-Slice Theorem

Module by: Jason Ryan

Summary: This module describes the projection-slice theorem and its application to spotlight-mode SAR.

Brief Review of Computer-Aided Tomography

Computer-Aided Tomography, or CAT (as in CAT scan) is a technique for remote 2-D and 3-D imaging. By moving a sensor around a target, one can collect sufficient 1-dimensional data to reconstruct the original multidimensional image. This process utilizes an amazing relationship called the Projection-Slice Theorem, which states that each piece of projection data at some angle is the same as the Fourier transform of the multidimensional object at that angle. Using a range of data from a range of angles, one can, given sufficient computation resources, reconstruct the actual image by taking the inverse transform. The Projection-Slice Theorem has found a range of applications in remote sensing, the most famous of which is the 3-D imaging of humans, popularly known as the CAT scan. The focus of this project, Spotlight-Mode Synthetic Aperture Radar, uses the Projection Slice Theorem in a way quite similar to CAT scan technology, except the way radar projections are generated by the image is slightly different from the way CAT scans use X-rays.

Projection-Slice Theorem

Let g(x,y) represent the radar reflection of our image. The two-dimensional Fourier transform of g is defined as

Figure 1
Figure 1 (graphics1.png)

And

Figure 2
Figure 2 (graphics2.png)

We can model the reflection behavior of the incident radar by considering the following overhead diagram

Figure 3
Figure 3 (graphics3.png)

The smooth line outlines our image g(x,y), and the horizontal and vertical axes x,y are overlaid. The radar is incident upon the target along the axis of the path u at an angle theta. For a target which is far away, the radar wave front is approximately flat, and so this means that a reflected beam which has traveled a certain unique distance to and from the sensor comes from a straight path across the image, perpendicular to u. This path is in the direction of v and can be represented by a line integral in the direction of v at position u0. The formula for this is given by

Figure 4
Figure 4 (graphics4.png)

The 1-D Fourier transform of p(u) is given by

Figure 5
Figure 5 (graphics5.png)

And then, through applying the equation for p(u) and simplifying, we are left with

Figure 6
Figure 6 (graphics6.png)

This is the Projection Slice Theorem! What this states is that the Fourier transform of a projection taken at an angle theta is equal to the 2-D Fourier transform of the image at that same angle theta. To reconstruct the original image, one must merely take the inverse Fourier transform in two dimensions of a set of data P(U). This is not as easy as it sounds for reasons discussed later. Notice how the Fourier transform of the image G does not have the usual form, G(X,Y). It is instead expressed in polar form, and the variable theta lets us know that we have only a slice of the transform for each P(U).

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