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

Abstract

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...

The vector projection of normalized interferogram differences and its application in phase-shift estimation

“…The above process is based on the Gram-Schmidt orthogonalization, which can be used to retrieve phase directly as well [13]. Finally, we can further obtain the measured phase with estimated phase shifts by [11] u j ¼ arctan S 1;j cos D À S 2;j S 1;j sin D :…”

“…However, because of the requirement of filtering process in advance to eliminate the backgrounds of these two interferograms, obtained accuracy of two-step algorithms will decrease due to the bad performance of filtering. Furthermore, this kind of algorithm will be invalid when the phase shift is close to π [9][10][11][12][13][14].…”

A derivative method of phase retrieval based on two interferograms with an unknown phase shift

“…Recently, some two-step phase retrieval algorithms with an unknown phase shift have been reported [11][12][13][14][15][16][17][18][19]. For example, based on the statistical nature that the phase distribution of the object wave is random, Cai et al [11,12] derived some different kinds of algorithms to compute the phase shift and then retrieve the information related to the object wave.…”

“…In Ref. [14], Vargas proposed a different phase retrieval method based on the Gram-Schmidt orthonormalization algorithm. However, in above these two methods, the background term is required to be eliminated in advance, which is closely related to the accuracy of phase reconstruction.…”

Phase shifts extraction algorithm based on Gram–Schmidt orthonormalization of two vectors

“…Vargas has applied the GS process to retrieve the phase map (Vargas et al 2012). In this paper, an algorithm for phase shifts extraction with two interferograms based on the Gram-Schmidt orthonormalization of two vectors (TGS) is proposed.…”