Optical phase extraction algorithm based on the continuous wavelet and the Hilbert transforms

ODAY, optical techniques have been employed in many sciences and engineering applications to compute several physical magnitudes which are codified as the phase of a periodic intensity profiles, so the development of more sophisticated phase extraction algorithms is continuously needed [1,2]. Two classical methods for phase extraction are phase-shifting technique [3] and Fourier transform technique [4]. Phaseshifting technique processes the fringe patterns pixel by pixel. Each pixel is processed separately and does not influence the others. However, three, four or more images are needed. On the contrary, Fourier transform technique processes the whole frame of a fringe pattern at the same time, but it requires a spatial carrier and the pixels will influence each other. Thus a compromise between the pixelwise processing and global processing is necessary. One of the solutions is to process the fringe patterns locally. For this, different wavelet algorithms are conceived to extract the phase distribution of the fringe patterns. The wavelet concept and its applications, is becoming a useful tool in various studies for analyzing localized variations and particularly to analyze non-stationary or transient signals [5]. The aim of this paper is the application of the continuous wavelet analysis to extract the optical phase distribution from a single recorded fringe pattern without high frequency spatial carrier. The method applied requires a second phase shifted fringe pattern which will be generated numerically via the Hilbert Transform [6]. The use of a single image can lead to the phase distribution of dynamic processes. It seems suitable where only one fringe pattern can be taken and it is also applicable on the fringe patterns without spatial carrier [7]. The paper first presents an introduction of the continuous wavelet transform. In section 3, we expose the wavelet phase extraction method. In sections 4 and 5, we present the mathematical description of the corrected phase shift via Hilbert Transform. Finally, results of numerical simulations are presented in section 6.

The continuous wavelet transform (CWT) is a powerful tool to obtain a space-frequency description of a signal. Unlike the Fourier transform that uses an infinitely oscillating terms

, the wavelet analysis technique use a set of a specially designed pulse functions, called “wavelets”, to analyze the local information of the signal [8]. A wavelet is an oscillating function (x), centered at x=0 and decay to zero such

is the Fourier transform of ψ(x), then condition (1) is equivalent to the requirement that

A family of the analyzing wavelets is generated from this “mother wavelet” ψ(x) by translations and dilations and it can be expressed as

Where s≠0 is the scale parameter related to the frequency concept, and R is the shift parameter related to position.

We note that the wavelets with small values of s have narrow spatial support and consequently rapid oscillations, making them well adapted to selecting high-frequency components of a signal. The converse is true for wavelets with large values of s. Many different types of mother wavelets are available for the phase evaluation applications, but in our case the study reveals that the Complex Morlet wavelet gives the best results. It is defined as

where fc is the wavelet center frequency. Fig. 1 shows the real and the imaginary part of the Complex Morlet wavelet.

where * denotes the complex conjugation. The continuous wavelet transform can be expressed, using the Parseval identity, as

Where f ˆ and ˆ and k are respectively the Fourier transform of the signal, the Fourier transform of the mother wavelet and the angular frequency. If the inverse wavelet transform exist, the original signal can be reconstructed by

where

This reconstruction of the signal is possible when  C has a finite value.

There are many techniques for extracting phase distributions from two-dimensional fringe patterns. The fringe patterns, derived from two-beam interferometers, can be mathematically formulated by the sinusoidal dependence of the intensity on the spatial coordinates (x,y) of the image plane:

Where I0 is the bias intensity, V the visibility or fringe contrast and ِ  the optical phase.

The one-dimensional continuous wavelet transform of the fringe pattern’s row x (in the y direction) is given by

Exploiting the wavelet localization property and assuming a slow variation of the intensity bias and the visibility, the wavelet transform becomes

and the Parseval identity leads to

Finally, the wavelet transform becomes

The two terms in the previous equation in general overlap and have to be separated in order to retrieve the phase. The method commonly used is to add a carrier frequency m to the signal satisfying [9,10]:

In this study we use an alternative method based on the use of a second /2 phase shifted fringe pattern expressed by

Then using ( 14) and (17) we get

of the row x is a matrix whi

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