Conclusions

Back-projection

We implemented (in MATLAB) the filtered back-projection algorithm for reconstruction of an image from its projections obtained using parallel beams. Two different implementations of the filter were created: convolution based and FFT based. The convolution back-projection method performed better than the FFT based approach in terms of the squared error between the original and reconstructed images. The error in the reconstructed image was mostly concentrated along the edges in the image. This results from the use of 1-D interpolation used during back-projection.

We also experimentally evaluated the number of projections and number of samples in each projection needed to avoid aliasing. In our case, we got the best 250x250 image using 400 projection angles and 500 samples in each projection. This result was obtained with a detector length of one. A length one detector provided an aperture that always contained the target, but did not enclose the entire unit square. Since convolutional filtering was used, the sampling density was improved compared to the 1.5 length detector array. Surprisingly, relatively good results can also be obtained with a high number of detectors using only 250 projection angles. This is important when considering radiation exposure.

With all of our techniques combined (i.e., projections = 400, detector length = 1, number of detectors = 500, and the use of zeroneg.m and median filtering), the lowest squared error obtained was 9.61 (total squared error over the whole image).

Image Enhancement

There is a lot we can gather from our attempts at image enhancement. Table 5.1 tells us quite a bit about our reconstruction process and ways to improve on it. To begin with, we notice that the squared error for the image with 250 projection angles and 250 detectors is not terribly worse than the image with 400 projection angles and 500 detectors, after thresholding. This implies that most of the error in the Fourier transform reconstruction method lies outside the body cavity. Therefore, a simple thresholding filter can greatly reduce the patients exposure to x-rays as well as reduce the imaging costs. Second, we see that Wiener filtering and median filtering offered only mild improvements to our squared error measurements. Mesh plots of the error tell us that the reduction in error is occurring in what should be the smoother regions of the image. Therefore, since most of the differences between the reconstructed and original images lie in the edges, we are only reducing a small amount of the error when we operate on the smoother portions of the image.

The localization of error to the image edges led to our attempt at edge sharpening through the Haar wavelet transform. While the metric of squared error increased through this process, we saw from a representative image in Figure 5.7 that our edges were indeed sharpened. The resulting edges, however, were artificially located over a reduced area. For instance, where there was blurring in the heart, the blur was removed and the new edge was located on the inside portion of the blur. This accounts for the higher squared error, because the blurred sections were closer to the original image than an image with the blur removed. Decreasing this squared error through edge thinning is difficult because wherever we have a blurred edge, it is impossible to tell where the new edge should be placed (i.e., on the inside, middle, or outside of the blurred section). There is also the possibility that blur removal may remove desired image components. We operated on smooth, circular edges, but how would bumps be handled? Could we potentially remove evidence of a tumor on the edge of a tissue? Further research in this area should be explored.

Finally, we see that we were not able to gain significant improvements with post-reconstruction filters in our images corrupted by shifts or expansions. The best way to restore these images (post-reconstruction) would be to have accurate models of degradation effects so that Wiener filtering will be more accurate. However, these models would be space varying and generally difficult to come by. We would have to know when the components shifted or expanded to select an accurate degradation model. A better way to reduce error was presented in Case IV, where the projections themselves were shifted to account for shifts in the body cavity. This had a much better effect on error reduction than any post-reconstruction filters. Detection of shifts, however, is easier than detection of organ expansions and contractions. It is possible to measure a patient's heart beat and respirations through monitoring when recording the projections, and thus account for things like lung or heart expansion. In this manner we can process the projections directly and achieve significant reductions in reconstruction artifacts.

The theory has been outlined and the code provided so that others can expand on this work in the future.