Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert–Huang prefiltering

Abstract

This paper presents a novel method to extract the phase shift and phase distribution from two interferograms simultaneously. By employing Hilbert–Huang transform based prefiltering, the background intensities and modulation amplitudes of the two interferograms are suppressed and normalized respectively. With the addition and subtraction operation of the two prefiltered interferograms, two parametric equations are achieved which can be regarded as the complex harmonic motion of the Lissajous figure. The phase of the Lissajous figure can be directly demodulated by the ellipse fitting algorithm. Apart from the advantages of other well-known two-step phase demodulation algorithms, i.e., high accuracy and efficiency of the Gram-Schmidt orthonormalization (GS) method and the less stringent requirement concerning the fringe number in the extreme value of interference (EVI) method, proposed Lissajous figure and ellipse fitting (LEF) approach has another bonus related to its robustness to the fluctuations of the fringe patterns noise, background intensity and modulation amplitude. Simulations demonstrate the outstanding performance of the proposed method, and experiments further corroborate its effectiveness.

Robust two-step phase estimation using the Simplified Lissajous Ellipse Fitting method with Gabor Filters Bank preprocessing

“…In order to compare our proposal, we performed the phase step estimation by pre-filtering the synthetic patterns with the HHT as proposed in [25] (which we call LEF-HHT) by using the implementation the Enhaced Fast Empirical Decomposition (EFEMD) proposed in [29,39]. Figure 4a depicts the Lissajous Pattern (LP) for noiseless and normalized fringe patterns with a phase shift of δ = π/3; in this case, the LP is centered at the origin and it is perfectly well-marked.…”

“…The Lissajous Ellipse Fitting (LEF) method consists on using the Lissajous figure to detect the phase step between two interferograms and estimate the phase [13,[23][24][25][26].…”

“…Thus, by solving the coefficients (vector θ) by the least square method (as proposed in [13,[23][24][25][26]), the phase step is computed as…”

“…The second step is the estimation of the phase step δ using (6). To evaluate the feasibility of our proposed methods, we calculated the phase step estimation comparing them with the preprocessed LEF algorithm proposed in [25]. The first comparison consists on calculating the phase step at different Gaussian noise levels.…”

“…[10][11][12][13][14][15][16][17][18][19][20][21][22]. Particularly, various algorithms based on the LEF method have been designed to improve the estimation of the phase step and the phase distribution in two-step intertferometry, such as: iterative processes based on the least square technique [23], the use of the Gram-Schmidt orthonormalization to transforms the ellipse into a circle [24], the application of a Hilbert-Huang Transform (HHT) pre-filtering with the LEF algorithm [25] or the computation of the Euclidean distance from the points to the ellipse [26]; just to mention some of the novel techniques.…”