Conclusion

The concepts of shift-variance and approximate shift-invariance have been presented in this paper. A metric which measures the degree of shift-variance was found and used to develop an algorithm which moved the position of image samples so that the resultant image was equivalently filtered by an approximately space-invariant system. A simple example was shown, where a transformation that made the system exactly shift-invariant was found.

The work done here is just a first step - a great deal more needs to be done. and need to be included in the minimization procedure. The minimization of needs to be improved with a quicker algorithm (e.g., conjugate gradient). This would also likely include a better way to assure the continued order of the sample positions (likely by an additional constraint term to the error metric). There is a question about the types of systems on which this technique will work well. One example of trouble is an impulse response which changes topologically with object position. For example, if an impulse response has no bumps in it (is monotonically increasing until its maximum, and monotonically decreases thereafter) in one position, but has bumps in it in another position, no amount of stretching or contracting will make them identical. Also, until this technique is tried on real images, it remains unknown how this kind of restoration affects image quality although, for the examples of exact transformations in Robbins and Huang (1972) and Sawchuck (1975), the results appear promising). The propagation of noise through the system, concentrating on the statistics of the noise in the warped image plane, needs to considered (Robbins and Huang just assumed that the noise in the warped image plane was additive and stationary). It is unknown, for example, if an MMSE solution in the warped image plane relates to an MMSE solution in the real image plane. The effects of performing the resampling from interpolated uniformly sampled images, rather than using perfect sample values at new positions, needs to be considered. Extending this procedure to two-dimensional signals appears to be straightforward, but may cause further constraints upon the types of systems for which this method will work well.