Abstract:We have been reporting several new techniques of analysis and synthesis applied to Phase Shifting Interferometry (PSI). These works are based upon the Frequency Transfer Function (FTF) and how this new tool of analysis and synthesis in PSI may be applied to obtain very general results, among them; rotational invariant spectrum; complex PSI algorithms synthesis based on simpler first and second order quadrature filters; more accurate formulae for estimating the detuning error; output-power phase noise estimation. We have made our cases exposing these aspects of PSI separately. Now in the light of a better understanding provided by our past works we present and expand in a more coherent and holistic way the general theory of PSI algorithms. We are also providing herein new material not reported before. These new results are on; a well defined way to combine PSI algorithms and recursive linear PSI algorithms to obtain resonant quadrature filters.

This Letter presents an efficient, fast, and straightforward two-step demodulating method based on a Gram-Schmidt (GS) orthonormalization approach. The phase-shift value has not 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 an orthonormalized interferogram basis from the two supplied interferograms using the GS method. We have applied the proposed method to simulated and experimental interferograms, obtaining satisfactory results. A complete MATLAB software package is provided at http://goo.gl/IZKF3. © 2012 Optical Society of America OCIS codes: 100.5070, 100.2650.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. The method is based on the application of the Fourier transform demodulating approach to both interferograms. Then, the phase-step map is calculated using a direct algebraic expression. As the phase step has to be equal for all pixels, it is possible to solve the local sign ambiguity and, therefore, retrieve the phase map. This method requires filtering out the DC term, but it does not need the normalization of the fringe patterns. The main drawback of this approach is that it is very sensitive to noise. In [2] is presented a self-tuning (SF) method that first retrieves the phase step between interferograms, looking for the minimum of a merit function. Then a quadrature filter is constructed from the obtained phase step and the modulating phase is determined. This method presents good results when the phase step is close to π∕2 rad, but the accuracy decreases when the phase step moves away from this value. 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. 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. In this work, we present a novel two-step demodulation method based on the Gram-Schmidt (GS) orthonormalization approach. The method is very fast, easy to implement, does not require any minimization process, and is not computationally demanding. The method requires filtering out the DC term, but it does not require normalizing the fringe patterns. In [4][5] we have shown that a sequence of phase-shifted fringe patterns free from harmonics can be expressed as a linear combination of two orthonormal signals. Therefore, any phase-shifted interferogram sequence can be described using a twodimensional vector subspace. The orthonormal...

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...

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Phase unwrapping techniques remove the modulus 2π ambiguities of wrapped phase maps. The present work shows a first-order feedback system for phase unwrapping and smoothing. This system is a fast phase unwrapping system which also allows filtering some noise since in deed it is an Infinite Impulse Response (IIR) low-pass filter. In other words, our system is capable of low-pass filtering the wrapped phase as the unwrapping process proceeds. We demonstrate the temporal stability of this unwrapping feedback system, as well as its low-pass filtering capabilities. Our system even outperforms the most common and used unwrapping methods that we tested, such as the Flynn's method, the Goldstain's method, and the Ghiglia least-squares method (weighted or unweighted). The comparisons with these methods show that our system filters-out some noise while preserving the dynamic range of the phase-data. Its application areas may cover: optical metrology, synthetic aperture radar systems, magnetic resonance, and those imaging systems where information is obtained as a demodulated wrapped phase map.