Limited Angle Problem

Section 2.2.1 discussed the FST and the angular sampling requirements for unique re- construction. If angular samples from a parallel geometry are acquired over 180◦, then the frequency space of an object has been completely sampled and a unique reconstruc- tion exists. If there are missing angles, the corresponding spatial frequencies simply have not been measured. The exact values cannot be recovered from the measured data alone. Coping with this missing information is the limited angle problem. This is not to say that having complete information is always necessary. Projection radio- graphy is an extreme case of missing information, yet it retains high clinical utility. Limited angle X-ray tomography can be a valid diagnostic technique on its own. This method is traditionally called tomosynthesis [30]. Tomosynthesis is entering the clinic as a proposed replacement for mammography [59, 21] and chest radiography [71], where it offers the ability provide improved diagnostic sensitivity and specificity by approx- imately separating tissues that would be superposed in a single projection image at a lower dose than CT. I have avoided this term in favor of limited angle tomography

which implies a superset of tomosynthesis that includes other limited angle problems such as 4D CBCT and does not carry any historical baggage of analog tomosynthesis [30], where analog tomosynthesis is an imaging modality where an X-ray source and film cassette are moved to selectively sharpen a single plane in the 3D object while blurring others.

digital phantom derived from a clinical CT. The images were reconstructed with 10 iterations of the simultaneous algebraic reconstruction technique (SART) algorithm [1] and one projection taken per degree of angular coverage. The angular coverage was taken about the AP direction. Image quality decreases gracefully as angular sampling decreases, resulting in decreasing resolution in the AP direction. This is known as

artifact spread. This loss of resolution is supported by the FST since in acquiring a cone beam projection in the AP an approximate plane in the coronal plane of the frequency domain has been measured. This gives rise to a preferred viewing plane that is orthogonal to the central imaging direction. The coronal plane remains more useful for human viewing since the frequency components that give rise to it are better sampled. In current implementations, the imaging directions are usually chosen such that the preferred viewing plane is either the coronal or sagittal because these are anatomically familiar to interpreters. An axial preferred viewing plane is an infeasible geometry for human subjects, since projections would need to be acquired from the superior- inferior (SI) direction. However, it may actually be desirable to have the preferred viewing plane oblique to these planes, due to either the spread of high contrast objects obscuring low contrast objects of interest or due to the specific deformation space of these objects. The downside of this approach is that even though such images may be better interpretable to computers, they may become much more difficult for humans to interpret due to artifact spread from oblique structures and the desire for humans to interpret medical images in the traditional orthogonal planes due to symmetries in the human body. This problem is more a lack of human familiarity, rather than a methodological weakness, and could be overcome with practice. For the time being, humans make the final decisions, and their ability to judge correctly is paramount.

Figure 2.12 shows the response of a limited angle geometry and reconstruction to a small sphere, illustrating the problems of limited angle geometries in isolation. The

Source 180◦

135◦ 90◦

45◦ 20◦

Figure 2.10: Limited angle reconstructions simulated from a high quality clinical FBCT scan of the Rando anthropomorphic torso phantom. Images were reconstructed from projections over the specified angular coverage with one projection per degree of cov- erage and 10 iterations of SART. Image quality gracefully decreases with increasing artifact spread with a decrease in angular coverage. Even though the axial images with small angular coverage may be difficult to interpret, the coronal images are in the pre- ferred viewing plane and are easily interpretable. Since this is a cone beam acquisition, 180◦ is not complete angular sampling, analogous to the fan beam case.

Figure 2.11: Limited angle reconstructions analogous to those in figure 2.10, but sim- ulated with the NST geometry. The line artifacts are the result of truncation where the artifacts indicate the edges of the intersection of the rays from a particular view and the reconstruction domain. The NST geometry is fixed, but angular sampling and image quality could be improved by rotating the linear accelerator gantry.

image is reconstructed from a 20◦arc centered about the vertical axis of the image, using the 10 iterations of SART. Artifact spread in the image is obvious. The sphere is spread out in the direction of imaging but remains well localized in the horizontal direction because Fourier space is poorly sampled in the vertical direction but well sampled in the horizontal one. The image also shows a decrease in intensity. This is because the total intensity in the image is conserved under projection and reconstruction, but that intensity is spread out over a larger area. This means that the reconstructed intensities are not comparable to their true values. The intensity that CT, at least approximately, measures is attentuation. Attenuation is a real, quantifiable property of matter interrogated by X-rays. Knowing the calibrated spatial attenuation distribution function of a subject has applications in radiation oncology with application to dose calculation for treatment planning and adaptive therapy and is also important in image registration applications. Because of this phenomena, limited angle images have limited value for dose calculation. It also makes performing registrations between limited angle images and completely sampled images difficult.