Conclusion
Our study provided us with 3 significant findings. The first finding is that there is a tradeoff between image quality and computation time. We feel it is necessary, to a certain extent, to sacrifice time in order to get a better image. Our next finding is that although no perfect filter exists, a ramp filter which stops right past the original signal (to retain all of the signal but not add extra information/noise) seems to perform the best in terms of balancing image clarity and reception. Lastly, we also conclude that the same filter works the best in real world situations, those in which we added random noise to the projections. Why is this the case? Through our understanding of 2-D Fourier transforms, we were able to conclude that it is due to the fact that when the projections are mapped onto the (x,y) plane, there is a concentration of points near the origin. In order to compensate for that, we must use a ramp in order to get those outer points that would be ignored in certain cases. This is why some clarity was lost in our low pass filters.
We have learned a large amount of MATLAB and the theory behind tomography in a short period of time. With some funding and more time, it would have been beneficial to have taken some actual 3-D projections to recreate 2-D cross-sections to see some of the same principles we discovered with 2-D images. In this way, we would be able to tie our findings together with some actual real world applications. But as things are, we are able to see how it ties to these applications and we have a greater appreciation for how imperfect x-rays, CAT scans, and MRIs really are. We can also see how these imperfect techniques can always be improved yet never reach perfection.