Two-step interferometry by a regularized optical flow algorithm

Abstract

A two-step phase-shifting method, that can demodulate open-and closed-fringed patterns without local sign ambiguity is presented. The proposed method only requires a constant phase-shift between the two interferograms. This phase-shift does not need to be known and can take any value inside the range (0, 2π), excluding the singular case where it corresponds to π. The proposed method is based on determining first the fringe direction map by a regularized optical flow algorithm. After that, we apply the spiral phase transform (SPT) to one of the fringe patterns and we determine its quadrature signal using the previously determined direction. . As a general rule, the use of few interferograms simplifies the computation process and reduces the processing speed in PSI. Moreover, it increases the robustness against uncontrolled mechanical vibrations, air turbulence, or temperature changes that are typical problems of interferometric setups. In the case of high-frequency open-fringe interferograms, there are well-known solutions as the Fourier transform method [3] but in the case of closed-fringes, we need at least-two frames to solve the local sign ambiguity if we do not have additional a priori information. In the past, there have been reported works about phase reconstruction with only two frames [4,5]. [4] presents the standard technique for obtaining the phase map from two phase-shifted interferograms. This method is based on applying the Fourier transform demodulation approach to both interferograms. After that, the phasestep value is calculated for every pixel using a direct algebraic expression. Because the phase-step has to be constant along the full image, it is possible to solve the local sign ambiguity and then retrieve the phase map. In [4], the authors recommend using the method when the phase-step is inside the range ½π=3; 2π=3 ðradÞ. Additionally, this method is very sensitive to noise. In [5], a twostep method has been recently presented. This method consists of a self-tuning (ST) approach that first retrieves the phase-step between interferograms looking for the minimum of a merit function. Then a quadrature filter is constructed and the phase is determined. This method produces good results when the phase-step is close to π=2 ðradÞ, but the accuracy decreases when the phasestep moves away from this value. Additionally, the method requires that the interferograms were previously normalized.In this Letter, we propose a two-step phase-shifting method that is capable of retrieving the modulating phase in the whole (0, 2π) range in a robust and fast way. Our method, as any two-step method, cannot retrieve the phase when the phase-shift is exactly π ðradÞ. In this singular case there is not enough information. The method first obtains the fringe direction using a regularized optical flow method. After that, the sign ambiguity problem is solved and the modulating phase can be obtained from one interferogram using the spiral phase transform (SPT).The optical flow approach is a standard method used in compute...

Shack-Hartmann spot dislocation map determination using an optical flow method

“…Note that optical flow approaches have been successfully applied previously in optical metrology and specifically in phase demodulation processes to determine a regularized phase direction map [10,11]. In this work, we have used an advanced optical flow approach based on a pyramidal implementation of the well-known Lucas-Kanade (LK) optical flow algorithm with iterative refinement [12].…”

Two-step demodulation based on the Gram–Schmidt orthonormalization method

“…Additionally, the method requires the interferograms to be previously normalized. In [3] is presented a recent demodulating two-step method based on a regularized optical-flow (OF) method. The method is robust against additive noise and different values of the phase step.…”

“…In the past there have been reported works about phase reconstruction with only two frames [1][2][3]. In [1] is presented the standard and most used technique for obtaining the modulating phase map from two phase-shifted interferograms.…”

“…Additionally, this approach does not require normalizing the fringe patterns but it requires subtracting the DC term. The main drawback of [3] resides on the computational requirements necessary to perform the OF analysis, which make this demodulating method costly from a processing and computational point of view.…”

“…We have compared the results obtained by the proposed GS method with the results obtained by the standard Kreis [1], ST [2], and OF [3] methods. The utility of this method has special interest in the case of closed-fringe interferograms, as two frames are the minimum number of interferograms to reconstruct the phase without local sign ambiguity.…”