Results

The results of a simple trial for sample shifting section are now reported. The one-dimensional image range was normalized (i.e., the image existed for ). A sample space-variant ``impulse response'' used as a test on the system was

is not a true impulse response, since it is not normalized. But provides a smooth function which possesses a clearly defined transformation, . Then , allowing the minimization to concern itself with finding only the sample positions, and not the multipliers and . The image plane was uniformly sampled at 11 points, 10 points inside the image and one point outside the right boundary of the image (, , , and ). was then minimized for two cases: when and (greatly restricting the range of the PSF used for calculating ), and again for and (which maximizes the range of the PSF). For the greatly restricted trial (), the algorithm came upon a number of local minima but was eventually was able to find a global minimum. For the unrestricted trial (), the global minimum was found directly. Both trials were run for 100 iterations.

Fig. 1 shows examples of the impulse responses of Eq. 9 for different object positions, the uniformly spaced sample positions used as an initial estimate of the sample positions, and the sample positions which resulted from both the greatly and lightly restricted applications of the algorithm.

Table 1 lists the sample positions and , which should be constant for ideal sampling in this case. In the unrestricted trial, the ideal sample locations were found after 100 iterations with negligible error, as expected. But, even for the greatly restricted trial, the sample locations were were found to a good approximation. The advantage of the restriction was a substantially decreased computing time per iteration. The difficulty with the restricted trial was the abundance of local minima which needed to be negotiated.