Digital image processing techniques are needed in order to recover the object information encoded in fringe patterns generated in a determined interferometric setup. Main fringe analysis techniques are reviewed in order to give the reader the most fundamental insights for the interpretation of interferograms. Phase shifted, open fringe, lateral shear and other types of interferograms make use of specific procedures to correctly retrieving the searched phase. Here, algorithms are described and tested in numerical simulations and real data.
- fringe analysis
- phase recovery
- phase unwrapping
Interferometric techniques are widely used for measuring a wide range of physical variables including refraction index, deformations, temperature gradients, etc. Typically, an interferometer is used to generate one or several interferometric fringe patterns that contain the information of the physical variable that is being measured. Those images must be interpreted in order to recover the parameters that are encoded in the fringe patterns generated by the interferometric setup. Thus, fringe analysis methods deal with the problem of a three-dimensional reconstruction (the object information) from a two-dimensional intensity patterns (interferograms) acquired by a CCD camera. Digital interferometry became extensively used since the development of lasers and CCD devices. In those years, however, the resulting interferograms had to be interpreted visually, and only qualitative results were often achieved. Visual interpretation of an interferogram with only straight or circular fringes is not difficult, but things become more complicated for a fringe pattern that combines several regions with circular, straight and crossed fringes of varying density. Rapidly, it was recognized the need of automatic methods for fringe analysis. The first great advance arises with the development of the phase-shifting techniques. With those procedures, a set of interferograms is acquired with a phase shift among them. The phase shifts are usually introduced by a piezoelectric transducer moving the reference mirror in such way that the phase difference between two consecutive interferograms is a constant term. With phase-shifting techniques, it is possible to isolate the sine and cosine of the phase allowing the calculation of the wrapped phase distribution and consequently the continuous phase with an unwrapping algorithm. Another great success came with the method proposed by Takeda (also referred as the Fourier method) performing a band-pass filtering in the Fourier domain. The method of Takeda works only with interferograms that contain open fringes (patterns that consist in nearly straight fringes). In order to generate such interferograms, the reference beam (e.g., in a two arm interferometer) is tilted introducing a large carrier function to the phase. The Fourier transform of these interferograms is composed of three lobules, one at the center that corresponds to the background term and two lobules located symmetrically respect to the origin. One of this lobules and the one that is located at the origin are filtered out. The remaining spectrum is transformed back to the spatial domain from which the so-called wrapped phase can be calculated. A final step is to apply a phase unwrapping technique to recover the continuous phase. Interferometric measurements and fringe analysis techniques are a growing and fast-changing field of research. Through this chapter, we will review the most known procedures.
2. Interferogram acquisition
The wave nature of light can be studied theoretically by a homogeneous partial differential equation of second order, which satisfies the superposition principle:
If two waves of the same frequency are superimposed on a point in space, they excite oscillations in the same direction:
In the preceding equations and the subsequent ones in this section, we will drop the spatial dependence for displaying purposes. The amplitude of the resulting oscillation at that point is determined by the equation:
If the phase difference,
So, we can conclude that the intensity observed in the superposed point by noncoherent waves equals the sum of the intensities, which create each separately. However, if the difference
The interference of two or more electromagnetic waves can be usually achieved in two ways: by division of the wave front and by division of the amplitude. A mechanism used for the division of the wave front is, for example, the Young’s experiment or double slit that is showed in Figure 2 .
A mechanism used for dividing the amplitude of the wave is, for example, the Michelson interferometer that is shown in Figure 3 .
Both mechanisms produce a pattern of light and dark intensities in the plane of interference fringes. The image resulting from the interference is known as interferogram. When the interferogram is captured by a recording medium, that is, a photographic film or a CCD camera, the process commonly involves some optical system, which introduces imperfections with respect to the ideal image. Such imperfections are known as aberrations. Aberrations can be classified as chromatic and monochromatic. Chromatic aberrations are present to illuminate the object with white light or polychromatic light, that is, light with different wavelengths. These aberrations are the only ones that can be predicted by the theory of the first order, which states that an optical system consisting of lenses has different focal lengths for different wavelengths. These variations are related to the change of refractive index with respect to wavelength causing that both the position and the image size are different for each wavelength. Monochromatic aberrations occur when the object is illuminated with monochromatic light (i.e., light of a single wavelength), and the reflected or transmitted light is registered by a recording medium. This type of aberration causes that the captured image of a punctual object is no longer a point, but a blurred point. Monochromatic aberrations can be calculated roughly in the third-order theory, using the first two terms of the expansion in power series of the
3. Interferogram filtering
In order to enhance the interferogram images and reduce the noisy caused by external factors at the interference phenomenon in an interferometer (interference produced by two or more controlled wave fronts), low-pass techniques are used as preprocessing filtering step. Interferogram smoothing and denoising are the principal purposes for the application of low-pass filters in interferometry. Filtering techniques can be grouped in spatial or frequency domains, where the spatial filtering is directly applied to the interferogram image, pixel to pixel, while the frequency filtering is usually performed in the Fourier domain. Band-pass and band-stop are some filtering techniques that can be applied in the Fourier domain, and these filters are used to attenuate some frequencies of specific noise.
3.1. Spatial filtering
The mathematical entity applied in spatial filtering is the convolution operation, also known as windowing , written as follows:
3.2. Frequency filtering
Frequency filtering is usually performed in the Fourier domain. The Fourier transform represents the change from spatial to frequency domain. Eq. (11) and Eq. (12) represent a pair of discrete Fourier transforms in two dimensions 
In Figure 5 , it is showed the masks described above along with the results of the filtering process. Low-pass filtering masks and results are presented in Figure 5b , band-pass filters and filtered images are seen in Figure 5c and finally, band-stop filters and filtering results can be appreciated in Figure 5d . The kind of filter or the size of geometrical mask depends directly in the image and in its noise content. There is no ideal filter; the kind of applied filter to process an image is dependent of the characteristics that are pretended to enhance or eliminate.
4. Phase-shifting interferometry
Interference fringe patterns obtained by means of interferometric techniques can be evaluated by using digital image processing techniques for the estimation of phase map distributions. The intensity of an image in an interference fringe pattern, according with Eq. (7), can be represented by Eq. (13)
Since there are three unknown terms in the representation of the interference intensity equation, then the measurement of at least three interferograms at known phase shifts is needed to determine the relative phase difference. However, one of the simplest modes to determinate the phase considers the use of four interferograms equally spaced by
Rearranging the simultaneous equation system from the above formulas, in which the spatial dependence (
The phase difference estimated from the four equations is determined in the range between −
, a set of four phase-shifted interferograms with phase shifts of π/2 is shown. A first interferogram with
In applied phase-shifting interferometry, there are concerns about the presence of errors that may affect the accurate phase extraction from phase-shifted interferograms. A typical systematical source of error introduced PZT arises when there is miscalibration of the phase-shifting actuator, causing detuning in the phase extraction process. The inclusion of phase-shifting algorithms with more than three or four interferograms can be implemented in order to reduce this systematic error .
5. The Fourier method
The method of Takeda or the Fourier method was developed in 1982 . Unlike phase-shifting methods, see Ref. , a single interferogram with open fringes can be analyzed. This is useful when the object under study changes dynamically or environmental disturbances (vibrations or air turbulence) do not allow the use of phase-shifting methods unless special, and often, very expensive hardware is used to acquire several images simultaneously. However, in general, the accuracy and the dynamical range of the phase that can be measured are reduced. The Fourier method makes use of the fast Fourier transform technique to separate, in the frequency domain, the background and phase terms of the interferogram. Employing complex notation, an interferogram can be written as follows:
The separation of the terms in the Fourier spectrum due to the introduction of a linear carrier can be observed in
. An open-fringe interferogram and its Fourier spectrum can be seen in
. It can be noted that the three terms of Eq. (21) are clearly separated. The central peak corresponds to the delta function δ(
In order to recover the phase, we need to isolate one of the lateral lobules of the Fourier domain. To this end, we employ a band-pass filter. The filtered spectrum is then transformed back to the spatial domain to obtain
The band-pass filter has the following form:
Transforming back to spatial domain we obtain:
The wrapped phase is found by:
The final step is to apply an unwrapping method to obtain the continuous phase related with the object under study. This last procedures are shown in
where the wrapped phase
The Fourier method is not the unique procedure for phase retrieval from one interferogram with open fringes. Besides the Fourier approach there are other procedures in the spatial domain including phase locked loop  and spatial carrier phase-shifting methods [9, 10].
6. Phase unwrapping
Phase unwrapping is a common step to finally find a continuous phase for several fringe analysis techniques such as phase shifting, Fourier, the phase synchronous and others methods that use the arctangent function of the ratio of the sine and cosine of the phase to obtain a wrapped phase. In its simplest form, phase unwrapping consists in adding or subtracting
The described procedure works well only for wrapped phases with no inconsistencies and low noise levels, however, delivers wrong results when dealing with noisy wrapped phases or those obtained from interferograms with broken or unconnected fringes. A more consistent approach is achieved with the least squares phase unwrapping method . The least square technique integrates the discretized laplacian of the phase. To this end, the laplacian of the phase is calculated as follows:
In the last equations, we have used pixel subscript notation in order to limit the extension of the equations. The pupil function
Solving for the phase in the above equation, we obtain that the unwrapping problem under the least square approach consist in the resolution of a linear system of equations, as:
An iterative technique that solves the above system of linear equations is the overrelaxation method in which the following equation is iterated until convergence:
In the last equations,
If desired, the constant term in the retrieved phase may be corrected easily in the following form:
The wrapped phase seen in Figure 13a was constructed as follows:
In the above equations
Finally, results on a noisy wrapped phase are presented. Random noise with uniform distribution in the range of
7. Phase recovery from lateral shearing interferograms
Lateral shearing interferometry is a very important field in experimental optical measurements, in which, the test beam interferes with a laterally displaced version of itself instead of a reference beam. The resulting fringe patterns are thus related with the object wavefront derivative in a given direction. This is very useful when the object information of interest is related with the derivative as in strain analysis or when the dynamical range of the object wave front is too high that cannot be measured with direct interferometry. Let us consider a laterally shear interferogram with a beam displacement in the
In Eq. (45)
Let us considered that we have retrieved the phase differences
Once the phase differences
In the above equations we have changed the (
Eq. (51) represents a linear system of equations; however, it is ill posed because there are more unknowns than equations due to the effects of the sheared pupils. Nevertheless, a regularization term may be aggregated to overcome this problem [13–15]. The regularization term is in the form of discrete Laplacians of the phase among adjacent pixels. The following equation that incorporates the regularization term is iterated until convergence:
Here, α is a parameter that controls the effects of the regularization term. The phase reconstruction is seen in Figure 18 . A two-dimensional view of the retrieved phase is observed in Figure 18a , the same phase but in a three-dimensional perspective is shown in Figure 18b and the phase error (the actual phase minus the reconstructed one) can be appreciated in Figure 18c . This reconstruction was achieved after 800 iterations with a parameter α = 0.1 obtaining a maximum error of about 0.0004 radians.
The actual phase was constructed as follows:
Digital processing techniques applied to interferometric measurements allow to obtain the phase from fringe patterns. The fringe analysis methods described here can be used to recover the phase that is associated with the physical variable under study. Under controlled conditions, phase-shifting techniques are the most used methods to retrieve the wrapped phase. If experimental conditions suffer from vibrations, air turbulences or the object changes dynamically, among other factors, then a Fourier method may be preferable to analyze an open-fringe interferogram. Those procedures deliver a wrapped phase. Then, an unwrapping algorithm is needed to reconstruct a continuous phase related with the object being studied. The aim of this chapter is to present, to the reader, the fundamentals of principal fringe analysis techniques. Numerical simulations are provided, in such way that the reader can reproduce them by its own. The extension of the chapter is insufficient to introduce many important techniques. However, the methods presented here were described as clearly and briefly as we could. We hope that the reader finds this information useful in the interpretation of interferograms obtained in the study of some object or phenomena by using an interferometric setup.
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