Inverse Filtering

Since blurring can be implemented by a product in the frequency domain, it may be possible to deblur an image by a division in the frequency domain,

This works if the modulation transfer function (MTF), , contains no zeros, but such a situation rarely occurs in practice because MTFs of most image acquisition systems do contain zeros throughout the measured spatial frequency range. When contains no zeros but has components with very small values, limited numerical precision can create significant errors in the image estimate. Such a blur irretrievably discards some information about the structure of the true image, and there is no longer a unique solution to the deblurring problem. In practice, constraints on the form of the reconstructed image (e.g., smoothness constraints) can be used to infer appropriate values for the deleted components.

When inverse filtering is not possible, the objectives of deconvolution are modified. Given an observed, blurred image, , and an estimate of the PSF, , derive an image such that . The image is the estimate of the true image, . This operation is called pseudoinverse filtering, or approximate deconvolution, which is usually implemented using iterative methods.