Fourier spectra for nonuniform phase-shifting algorithms based on principal component analysis

We develop an error-free, nonuniform phase-stepping algorithm (nPSA) based on principal component analysis (PCA). PCA-based algorithms typically give phase-demodulation errors when applied to nonuniform phase-shifted interferograms. We present a straightforward way to correct those PCA phase-demodulation errors. We give mathematical formulas to fully analyze PCA-based nPSA (PCA-nPSA). These formulas give a) the PCA-nPSA frequency transfer function (FTF), b) its corrected Lissajous figure, c) the corrected PCA-nPSA formula, d) its harmonic robustness, and e) its signal-to-noise-ratio (SNR). We show that the PCA-nPSA can be seen as a linear quadrature filter, and as consequence, one can find its FTF. Using the FTF, we show why plain PCA often fails to demodulate nonuniform phase-shifted fringes. Previous works on PCA-nPSA (without FTF), give specific numerical/experimental fringe data to"visually demonstrate"that their new nPSA works better than competitors. This often leads to biased/favorable fringe pattern selections which"visually demonstrate"the superior performance of their new nPSA. This biasing is herein totally avoided because we provide figures-of-merit formulas based on linear systems and stochastic process theories. However, and for illustrative purposes only, we provide specific fringe data phase-demodulation, including comprehensive analysis and comparisons.

Another nonuniform phase-steps algorithm (nPSA) is the principal component analysis (PCA) of phase-shifted fringe data [33][34][35][36][37][38]; we call this the PCA-nPSA technique. This is a subspace technique because it finds two orthogonal signals from N correlated nonuniform phase-shifted fringe images. Unmodified (or plain) PCA-nPSA demodulate the phase from fringe images without the explicit knowledge about their nonlinear phase shifts. The PCA-nPSA technique has low computational cost, it is linear, it is non-iterative, and it can deal with spatially varying background illumination and fringe contrast. Therefore, it seems at first glance, that PCA-nPSA would deal with all possible situations of nonuniform/linear phaseshifted phase demodulation [33][34][35][36][37][38]..
In spite of all those good properties, the PCA-nPSA has however some disadvantages which often gives inacceptable phase demodulation errors [37][38][39][40]. The PCA-nPSA users may not be aware of the phase-demodulation errors, and may therefore reach erroneous conclusions in phase metrology engineering [37][38][39][40]. A well studied PCA-nPSA limitation occurs when less-than-one spatial fringe is present within the fringe pattern [33][34][35][36][37][38][39][40]. But the problem of "few-spatial-fringes" is in our view, a pseudo-problem. That is because one can easily introduce as many spatial fringes as desire simply by introducing a large spatial carrier (a large tilt), and the few-spatial-fringes "problem" is gone [1]. A more recent attempt, and good review, to improve the PCA-nPSA using the Lissajous figure is given in [40].
In this work we show that the PCA-nPSA can be regarded as a linear quadrature filter applied to nonuniform phase-shifted fringe data. And as any other linear filter, it is possible to find its Fourier spectrum through its frequency transfer function (FTF) [1]. Finding the FTF of the PCA-nPSA one can see the reason why plain PCA often fails to demodulate, error-free, a set of nonuniform phase-shifted data. With the FTF at hand, one can find the signal-to-noise ratio (SNR) and fringe harmonics robustness of the PCA-nPSA from first principles of stochastic process and linear systems theories [1].

Nonuniform phase-shifting fringe images
We first describe the continuous-time fringe model as, .
For all ( , ) x y L L   , and 1 i   ; see    Figure 1(a) shows 9 nonlinearly-spaced phase-shifted samples (in red), and Fig. 1(b) the Fourier spectrum of the continuous temporal fringe (in blue).

PCA-nPSA phase demodulation formula
Principal component analysis (PCA) was invented by Karl Pearson in 1901 [42] and it is a statistical procedure that uses a linear transformation that convert hundreds of correlated observations, into a subset of linearly uncorrelated signals called the principal components of the data. In phase-shifting demodulation, the PCA is used to find 2 orthogonal signals (an analytic signal) from few temporal fringe samples. The reason of using a handful (instead of hundreds) of nonuniform phase-shifted data makes that the PCA-nPSA often give erroneous phase estimation, making the PCA-nPSA (if not properly corrected) inadequate for precision optical metrology. That is why several works have been published to correct the PCA-nPSA to cope with this residual phase demodulation error [37][38][39][40][41].
We now construct the desired PCA-nPSA formula. We start by modeling N nonuniform phase-shifted samples as, The images have ( , ) x y L L   pixels. Firstly we estimate the background of the fringes as, The following step in PCA is to compute the covariance matrix, We now find the N eigenvalues  n and eigenvectors v n of matrix C as, (8) Assuming that the largest eigenvalues are 0 1 ( , ) , then the PCA-nPSA formula is given by, Where ( , ) A x y is the demodulated analytic signal, and its phase is given by arg[ ( , )] A x y . Note that the PCA as herein presented do not need to vectorize back-and-forth the fringe images [33][34][35][36]. Equation (9) is the searched non-corrected (plain) PCA-nPSA formula. Equation (9) constitutes an interesting result because however implicit in the original PCA technique [33], it has not been explicitly given as an standard phase-shifting algorithm formula (Eq. (9)).
The first work on PCA as nPSA was presented as a linear algorithm which could demodulate any set of phase-shifted fringes, almost error-free [33]. Afterwards this was found not to be exact and several attempts have been made to improve plain PCA [35,[38][39][40]. However, the PCA can be combined with AIA, obtaining a PCA-AIA algorithm which eliminate the phase-errors left by plain PCA [35]. The PCA providing the first phase estimation making the AIA to converge faster [35]. Note that the AIA estimate both, the modulating phase ( , ) x y  and the nonuniform phase-steps 0 1

Correcting the Lissajous ellipse of the PCA-nPSA analytic signal
The Lissajous figure is obtained by the following parametric plot, Where i and j are the real and imaginary unit vectors. A good advantage of PCA-nPSA is that it always give zero-rotated Lissajous ellipses    r . Zero-rotated ellipses are easily transformed into circles (with zero phase demodulation error) by first calculating the ratio, Where | | x denotes the absolute value of x. Equation (11) is very robust to noise because a single parameter is estimated from an entire image. With , we may transform the Lissajous ellipse into a circle by modifying plain PCA-nPSA (Eq. (9)) as, .
To obtain a quadrature signal, ( ) H  must comply at least, with the following conditions, The zero ( 1) If the response at ( 1) H  is not zero, then one obtains an erroneous analytic signal given by, A non-zero ( 1) H  generate a detuning-like phase-demodulation error . This is the typical phase error given by plain PCA-nPSA.

Signal-to-noise ratio gain (G SNR ) and fringe harmonic robustness
Once the FTF (Eq. (14)) is obtained, one can find the SNR and harmonics robustness of the PCA-nPSA from basic stochastic process and linear systems theories [1]. Without the FTF people usually rely on particular synthetic/experimental fringe images (sometimes favorably biased) which may lead to over-optimistic conclusions .

Signal-to-noise ratio gain G SNR of the PCA-nPSA formula
The SNR of the analytic signal (Eq. (13)) for nonuniform sampled fringes corrupted by additive white Gaussian noise (AWGN) with power density ( ) / 2 The number SNR G is the SNR of the analytic signal with respect to the SNR of the fringe data.
For example SNR G N  means that the analytic signal has N-times higher SNR than the fringe data. The number SNR G reduces substantially for highly nonuniform phase-step fringes [41].

Fringe harmonic robustness R H for N-steps PCA-nPSA
Phase-shifted harmonic-distorted fringes may be modeled by, Then the demodulated analytic signal ( , ) A x y for harmonic distorted fringes is given by, A large H R number means high fringe harmonics robustness. In contrast, a low R H means low fringe harmonics robustness.

Computer simulations
For illustrative purposes only, we now offer two simulation for plain and corrected PCA-nPSA applied to 3 and 9 nonuniform phase sampled fringe images. These examples are given to show the Fourier spectral response of plain/corrected PCA-nPSA.

Plain PCA-nPSA applied to 3 nonuniform phase-step fringe images
The PCA-nPSA has been applied to many temporal fringe samples for better results [33][34][35][36]. That is because PCA was conceived to extract uncorrelated signals from hundreds correlated statistical data. The PCA was not invented to extract an analytic signal from, let say, 5 fringe samples. However the herein corrected PCA-nPSA, can demodulate the phase from just 3 temporal samples.
Let us start with the 3 nonuniform phase-shifted fringe images shown in Fig. 2 and Fig. 3.  Then we apply the PCA to these 3 images as, Taking the largest eigenvalues The demodulated phase arg [ ( , )] A x y is given in Fig. 4 along with its phase error.
Fib. 4. Here we show plain PCA-nPSA demodulated phase, and its phase-error.
The FTF ( ) H  of this (plain) PCA-nPSA is, The FTF of plain PCA-nPSA is plotted in Fig. 5(a). We see that | ( 1) | 0 H   and the erroneous analytic signal is given by, 2 2 ( 1)  Figure 5(a) shows plain PCA's FTF, its analytic signal and its Lissajous ellipse. We then use  to correct the PCA-nPSA obtaining the Lissajous circle in Fig. 5(b). From Fig. 5 we see that G SNR reduces from 1.96 to 1.2. On the other hand, the harmonic robustness R H decreases from 1.3 to 0.66. That is, the corrected PCA-nPSA is more sensitive to noise and harmonics than plain PCA-nPSA. However the phase error of plain PCA-nPSA is intolerable (see Fig. 4).

Correction of plain PCA-nPSA applied to 9 nonuniform phase-step fringe data
We now phase demodulate 9 nonlinear phase-shifted fringe images; Fig. 9 shows 4 images,   The SNR-gain G SNR =8.11 is higher for plain PCA-nPSA, but the phase-error is intolerable. The harmonic robustness for plain PCA-nPSA is R H =4.316, while for the corrected PCA-nPSA is R H =3.381. Plain PCA-nPSA has better harmonics robustness. Figure 8 also shows the Lissajous ellipse for plain PCA and the Lissajous circle for the corrected PCA-nPSA.

Conclusion
We have presented a very simple way to correct the technique of principal component analysis (PCA) applied to phase-demodulation of nonuniform phase-shifting fringes. We can summarize the contributions of this work as,  We have presented a PCA-nPSA procedure which do not need to vectorize back and forth the nonuniform phase-sampled fringe images (Eqs. (5)- (9)).
 Applying the PCA-nPSA formula we found that the Lissajous figures of the demodulated analytic signal are always non-rotated ellipses.
 The non-rotated Lissajous ellipses are corrected to Lissajous circles using the corrected PCA-nPSA formulas given in Eq. (11) and Eq. (12).  The FTF of the PCA-nPSA is then used to estimate the SNR-gain (G SNR ) Eq. (19) from basic stochastic processes theory for fringes corrupted by AWGN [1].
 Also the FTF ( ( ) H  ) is used to estimate the harmonics robustness R H (Eq. (22)) of the plain/corrected PCA-nPSA formulas.