Physics of Medical Scans
Back-Projection Reconstruction
Another common method of image reconstruction in CT scanning is called back-projection (BP). The method derives its name from the technique of “smearing back” detected attenuation data all along their corresponding ray paths. Doing this at many different angles yield a somewhat blurred reconstruction of the detected image. This is visualised in Figure 1 where BP is done on an imaging phantom (a test device) containing three objects. Simple back-projection provides a logical approach to image reconstruction however lacks the sophistication by itself required for medical imaging due to low resolution [1].
Figure 1: The process of simple back-projection on three objects. (A) shows the projections of the objects from three different angles. (B) shows how the detection of each object is smeared along the direction of the X-rays. (C) shows how a number of these 'smearings' can approximately reconstruct the original image in (A). (Source: Principles of CT and CT Technology)
The low resolution of simple back-projection reconstruction is because the technique involves smearing bright or dark pixels all around the image whether they belong in those areas or not. To correct this, it is the standard in BP-type scanners to use filtered back-projection (FBP) instead: in other words, using a high-pass filter before initialising the back-projections [2]. The filter creates ‘negative pixels’ that mostly cancel out the smearing effects caused by the point-spread functions evident in both Figure 1 and 2(A). Figure 2 shows the noticeable difference between the resolutions of the simple BP and FBP.
A
B
Figure 2: The difference in projections between simple back-projection and filtered back-projection and their final image using many views. (A) shows three projections of a single point source, each one possessing a curved point spread function. The image from many views ends up with low resolution. (B) shows a filtered projection with the top of the curve now being flat to create a uniform signal in the white circle. There also are negative regions in the projections in (B) that help to minimise the 'blurring' that is present in simple BP.(Source: The Scientist and Engineer’s Guide to Digital Signal Processing)
A drawback to filtered back-projection is that the noise in scan data gets amplified from the use of high-pass filters. A way around this is to use a weaker filter based on its reconstruction kernel. This kernel is a parameter that varies the extent to which the data is filtered; a measure of the filter’s ‘softness’. Different reconstruction kernels are appropriate in different scanning contexts. For example, in the case where the area of interest has inherently high contrast, say in the area between the lungs and air, a strong kernel can be used as the difference in attenuation is far larger than the noise from the filter. The opposite is true for areas of low attenuation contrast or have large features, like in brain CT scans, a softer filter is more appropriate.
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Figure 3: an animation of the reconstructed image found using back-projection of data gathered as the CT goes around a patient's chest. Notice that the back-projection only requires 180 degrees in this case: as the rays travel parallel to each other, doing another half-revolution gives data that is a mirror of the first half-revolution. (Source: Adam Leon Kesner)
References:
[1] CT Physics: CT Reconstruction and Helical CT, http://xrayphysics.com/ctsim.html [Accessed: 26/02/2019]
[2] Brink, J.A. et al. (1994). "Helical CT: principles and technical considerations", RSNA Radiographics.