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We demonstrate a method for improving the accuracy of phase retrieval based on the Transport of Intensity equation by using intensity measurements at multiple planes to estimate and remove the artifacts due to higher order axial derivatives. We suggest two similar methods of higher order correction, and demonstrate their ability for accurate phase retrieval well beyond the 'linear' range of defocus that TIE imaging traditionally requires. Computation is fast and efficient, and sensitivity to noise is reduced by using many images.
Waller et al. (Thu,) studied this question.