The under-determined case: The least-squares inverse of m for each frequency is
The over-determined case: The least-squares inverse for m for each frequency is
An interesting problem occurs when the conditioning of the matrix quickly deteriorates as frequencies increase. This effect is partly caused by the frequency truncation of the operator. A solution to this problem is to increase , which appears in equations (14) and (15), with the frequency. To do so, I set
The size of the matrix to invert makes possible the use of direct inversion as opposed to iterative processes. Figure 5 shows the result of inversion using the aliased operator. The aliasing artifacts are still strong, but some energy has focused, decreasing the width of the horizontal events caused by near-offset aperture effects. The resulting data, which appear in the right panel of Figure 5, are almost perfectly recovered. The residual in the left panel of Figure 7 demonstrates that the data fitting is nearly perfect.
Now, using the antialiasing conditions on the operator, we obtain the data in Figure 6: events are better focused in the radon domain and most of data aliasing artifacts have disappeared. We observe a loss of resolution for the non-flat events caused by the antialiasing conditions. The aliasing of the input data produces the remaining noisy events in the radon domain. Again, the data in the right panel of Figure 6 are almost completely recovered, and the residual in the right panel of Figure 7 is very small.
This result proves that the antialiasing PRT can be inverted as long as is provided. It also proves that the use of antialiasing conditions on the operator helps to mitigate data aliasing artifacts. However, as Figure 6 indicates, data aliasing effects can't be totally removed. The next section shows that sparseness conditions in the radon domain can destroy data aliasing artifacts.