Multi-filter transport of intensity equation solver with equalized noise sensitivity

Abstract

Phase retrieval based on the Transport of Intensity Equation (TIE) has shown to be a powerful tool to obtain the phase of complex fields. Recently, it has been proven that the performance of TIE techniques can be improved when using unequally spaced measurement planes. In this paper, an algorithm is presented that recovers accurately the phase of a complex objects from a set of intensity measurements obtained at unequal plane separations. This technique employs multiple band-pass filters in the frequency domain of the axial derivative and uses these specific frequency bands for the calculation of the final phase. This provides highest accuracy for TIE based phase recovery giving minimal phase error for a given set of measurement planes. Moreover, because each of these band-pass filters has a distinct sensitivity to noise, a new plane selection strategy is derived that equalizes the error contribution of all frequency bands. It is shown that this new separation strategy allows controlling the final error of the retrieved phase without using a priori information of the object. This is an advantage compared to previous optimum phase retrieval techniques. In order to show the stability and robustness of this new technique, we present the numerical simulations.

Variational Phase Imaging Using the Transport-of-Intensity Equation

“…For the rest of the spectrum (denoted as "Region 2" in Figure 4), we rely on the small defocus images. We remark that similar approaches have been carried out in [20]- [23].…”

“…As τ → 0 (i.e., no regularization), one recovers a method that is in spirit of [20]- [23]. We note that the matrix inversion in (30) is welldefined, even in the absence of regularization.…”

“…In [22], multiple spectral bands are combined via the Savitzky-Golay differentiation filter. Similarly, bandpass filters that are specific to the defocus distance are used in the frequency domain for the calculation of the phase image [23]. What is common in all of these methods is that they are all linear reconstruction algorithms and that the TIE is solved directly by inverting the model.…”

“…It linearly combines high and low frequency components (together with a Tikhonov regularization) of the phase maps obtained by using small and large defocus measurements, respectively. The algorithm is compatible with the ones proposed in [20]- [23]. As described in Section III, all linear reconstructions are computed directly.…”

Absolute phase recovery in structured light illumination systems: Sinusoidal vs. intensity discrete patterns

“…Absolute phase measurements [1][2][3][4][5][6] are an essential element in science and technology. One important area is optical metrology [7][8][9] where the absolute phase measurement (or optical path difference) can directly be related to a physical quantity that commonly refers to lengths or heights in industrial applications [10].…”