1. Introduction

Fringe pattern processing has been an interesting topic in optical metrology and interferometry; owing to its relevance nowadays, it is a widely studied discipline. Digital fringe pattern processing is used in optical measurement techniques such as optical testing [1, 2], electronic

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

speckle pattern interferometry (ESPI), holographic interferometry, and moiré interferometry or profilometry [3–5]. They are quite popular for non-contact measurements in engineering and have been applied for measuring various physical quantities like displacement, strain, surface profile, refractive index, etc. In optical methods of measurement, the phase, which is related to the measured physical quantity, is encoded in an intensity distribution represented in an image which is, in general, the result of the interference phenomena. This phenomenon is used in classical interferometry, in holographic interferometry, and in electronic speckle pattern interferometry to convert the phase of a wave of interest into an intensity distribution. As the physical quantity to be measured is codified as the phase of a fringe pattern image, the main task of fringe pattern processing is to recover such phase.

is that they require the previous estimation of the so-called fringe orientation which, as it uses the computation of the image gradient, could be an inaccurate procedure in the presence of noise and low modulation of fringes. This is not the case for the Fourier-based methods [21, 22]; however, as in the case of the Windowed Fourier transform technique, several parameters have to be adjusted depending on the particular image and it may require a long

The 2D Continuous Wavelet Transform: Applications in Fringe Pattern Processing for Optical Measurement…

http://dx.doi.org/10.5772/intechopen.74813

175

In the field of phase recovery from fringe images, there have been a lot of researches along the last decades. For the case of phase-shifting algorithms, outstanding summaries of them can be found in [2, 26]. For the case of spatial and frequency domain methods from a single pattern image, two of the most popular techniques are the well-known Fourier Transform method reported by Takeda et al. [27] and the Synchronous detection method [28]. Other methods that use the regularization theory were also proposed [29, 30]. However, although these methods are efficient and easy to implement, they are limited to be used in fringe images with frequency carrier, which just in few experimental situations these kinds of images can be obtained. In most cases, experimental conditions in optical measurement techniques yields fringe images without a dominant frequency (i.e., closed fringes) which becomes the phase recovery problem difficult, therefore more complicated algorithms must be used. One of the first proposals for phase demodulation from single closed fringe images was reported by Kreis using a Fourier based approach [31]. In the last decade of the twentieth century, it was a boom in the research of closed fringe images, specially using the regularization theory. The Regularized phasetracking technique was reported by Servín et al. [32]. Marroquín et al. reported the regularized adaptive quadrature filters [33] and the regularization method that uses the local orientation of fringes [34]. At the beginning of this century, Larkin et al. proposed the spiral-phase quadrature transform [35] and Servín et al. reported the General n-dimensional quadrature transform [36]. Also, we proposed the orientational vector-field-regularized estimator to demodulate

As will be shown, closed fringe and wrapped-phase images have certain characteristics that make them to be treated in a special manner. First, it is common that this kind of images present structures with high anisotropy at the same time that many frequencies are dispersed over the entire image. For these reasons, in most situations, the use of linear-translation-spatial (LTI) filters, which are spatially invariant and independent of image content, do not give proper results. Furthermore, owing that the Fourier transform is a global operation, this technique is

It is widely known that the wavelet transform is a powerful tool that provides local, sparse, and decorrelated multiresolution analysis of signals. In the last years, 2D wavelets have been used for image analysis as a proper alternative to the weakness of LTI filters and linear transforms as the Fourier one. In particular, it has been shown that 1-D and 2D continuous wavelet transform (CWT) using Gabor atoms is a natural choice for proper analyses of fringe images. This kind of analysis has been used for fringe pattern denoising and fringe pattern demodulation showing several advantages, for example in laser plasma interferometry [38], in shadow moiré [39–41], in profilometry [42–44], in speckle interferometry [45], in digital holography

not always suitable for accurately model the local characteristics of closed fringe images.

[46], and other optical measurement techniques [47–55].

processing time.

closed fringe images [37].

The methods for phase recovery from fringe patterns can be classified mainly in three categories [2, 6]: (a) Phase-stepping or phase-shifting methods which require a series of fringe images to recover the phase information. (b) Spatial domain methods which can compute the phase from a single fringe pattern in the spatial domain. (c) Frequency domain methods which uses some kind of transformation to the frequency domain to compute the phase. In this category, the Fourier and Wavelet transforms are the most common mathematical tools to carry out the task.

Apart from the phase recovery, there are other important steps in fringe pattern processing. For example, many times the fringe patterns are corrupted by noise, such as the case of the electronic speckle pattern interferometry. Then, fringe image enhancement by means of lowpass filtering is usually required. Owing that most algorithms to retrieve the phase from a fringe pattern give the phase wrapped in the interval ½ Þ �π; π , other important step is the wellknown phase-unwrapping process [6, 7]. In the field of fringe image enhancement, such as fringe image denoising or phase denoising, there has been a wide research activity in the last years. Researchers have realized that improving the quality of fringe images and wrapped phase fields is of main relevance for a successful phase recovery or phase unwrapping. However, enhancing fringe images or wrapped phase fields has resulted to be a task that must be realized in a special manner, so that ordinary techniques for image enhancement are not always adequate. Owing that frequencies of fringes and noise usually overlap and normally cannot be properly separated, common filters for image processing have blurring effects on fringe features, especially for patterns with high density fringes. For these cases, the use of anisotropic filters is a better way for removing noise without the harmful blurring effects.

In the fields of fringe pattern denoising and wrapped phase map denoising, there have been many proposals to realize these tasks. Some of the first contributions in this field were mainly based on convolution filters using different kinds of anisotropic filtering masks [8–12]. Other set of the main contributions in the last years is based on the variational calculus approach by solving partial differential equations [13–18], and by means of the regularization theory [19, 20]. The use of the Fourier transform for fringe or phase map denoising has also been proposed in [21, 22] (Localized Fourier transform filter and windowed Fourier transform, respectively). There have been other proposals that used different methodologies such as coherence enhancing diffusion [23], image decomposition [24], and multivariate empirical mode decomposition [25]. The great disadvantage of already reported methods for fringe and phase map denoising is that they require the previous estimation of the so-called fringe orientation which, as it uses the computation of the image gradient, could be an inaccurate procedure in the presence of noise and low modulation of fringes. This is not the case for the Fourier-based methods [21, 22]; however, as in the case of the Windowed Fourier transform technique, several parameters have to be adjusted depending on the particular image and it may require a long processing time.

speckle pattern interferometry (ESPI), holographic interferometry, and moiré interferometry or profilometry [3–5]. They are quite popular for non-contact measurements in engineering and have been applied for measuring various physical quantities like displacement, strain, surface profile, refractive index, etc. In optical methods of measurement, the phase, which is related to the measured physical quantity, is encoded in an intensity distribution represented in an image which is, in general, the result of the interference phenomena. This phenomenon is used in classical interferometry, in holographic interferometry, and in electronic speckle pattern interferometry to convert the phase of a wave of interest into an intensity distribution. As the physical quantity to be measured is codified as the phase of a fringe pattern image, the main

The methods for phase recovery from fringe patterns can be classified mainly in three categories [2, 6]: (a) Phase-stepping or phase-shifting methods which require a series of fringe images to recover the phase information. (b) Spatial domain methods which can compute the phase from a single fringe pattern in the spatial domain. (c) Frequency domain methods which uses some kind of transformation to the frequency domain to compute the phase. In this category, the Fourier and Wavelet transforms are the most common mathematical tools to carry out the

Apart from the phase recovery, there are other important steps in fringe pattern processing. For example, many times the fringe patterns are corrupted by noise, such as the case of the electronic speckle pattern interferometry. Then, fringe image enhancement by means of lowpass filtering is usually required. Owing that most algorithms to retrieve the phase from a fringe pattern give the phase wrapped in the interval ½ Þ �π; π , other important step is the wellknown phase-unwrapping process [6, 7]. In the field of fringe image enhancement, such as fringe image denoising or phase denoising, there has been a wide research activity in the last years. Researchers have realized that improving the quality of fringe images and wrapped phase fields is of main relevance for a successful phase recovery or phase unwrapping. However, enhancing fringe images or wrapped phase fields has resulted to be a task that must be realized in a special manner, so that ordinary techniques for image enhancement are not always adequate. Owing that frequencies of fringes and noise usually overlap and normally cannot be properly separated, common filters for image processing have blurring effects on fringe features, especially for patterns with high density fringes. For these cases, the use of anisotropic filters is a better way for removing noise without the harmful blurring effects.

In the fields of fringe pattern denoising and wrapped phase map denoising, there have been many proposals to realize these tasks. Some of the first contributions in this field were mainly based on convolution filters using different kinds of anisotropic filtering masks [8–12]. Other set of the main contributions in the last years is based on the variational calculus approach by solving partial differential equations [13–18], and by means of the regularization theory [19, 20]. The use of the Fourier transform for fringe or phase map denoising has also been proposed in [21, 22] (Localized Fourier transform filter and windowed Fourier transform, respectively). There have been other proposals that used different methodologies such as coherence enhancing diffusion [23], image decomposition [24], and multivariate empirical mode decomposition [25]. The great disadvantage of already reported methods for fringe and phase map denoising

task of fringe pattern processing is to recover such phase.

task.

174 Wavelet Theory and Its Applications

In the field of phase recovery from fringe images, there have been a lot of researches along the last decades. For the case of phase-shifting algorithms, outstanding summaries of them can be found in [2, 26]. For the case of spatial and frequency domain methods from a single pattern image, two of the most popular techniques are the well-known Fourier Transform method reported by Takeda et al. [27] and the Synchronous detection method [28]. Other methods that use the regularization theory were also proposed [29, 30]. However, although these methods are efficient and easy to implement, they are limited to be used in fringe images with frequency carrier, which just in few experimental situations these kinds of images can be obtained. In most cases, experimental conditions in optical measurement techniques yields fringe images without a dominant frequency (i.e., closed fringes) which becomes the phase recovery problem difficult, therefore more complicated algorithms must be used. One of the first proposals for phase demodulation from single closed fringe images was reported by Kreis using a Fourier based approach [31]. In the last decade of the twentieth century, it was a boom in the research of closed fringe images, specially using the regularization theory. The Regularized phasetracking technique was reported by Servín et al. [32]. Marroquín et al. reported the regularized adaptive quadrature filters [33] and the regularization method that uses the local orientation of fringes [34]. At the beginning of this century, Larkin et al. proposed the spiral-phase quadrature transform [35] and Servín et al. reported the General n-dimensional quadrature transform [36]. Also, we proposed the orientational vector-field-regularized estimator to demodulate closed fringe images [37].

As will be shown, closed fringe and wrapped-phase images have certain characteristics that make them to be treated in a special manner. First, it is common that this kind of images present structures with high anisotropy at the same time that many frequencies are dispersed over the entire image. For these reasons, in most situations, the use of linear-translation-spatial (LTI) filters, which are spatially invariant and independent of image content, do not give proper results. Furthermore, owing that the Fourier transform is a global operation, this technique is not always suitable for accurately model the local characteristics of closed fringe images.

It is widely known that the wavelet transform is a powerful tool that provides local, sparse, and decorrelated multiresolution analysis of signals. In the last years, 2D wavelets have been used for image analysis as a proper alternative to the weakness of LTI filters and linear transforms as the Fourier one. In particular, it has been shown that 1-D and 2D continuous wavelet transform (CWT) using Gabor atoms is a natural choice for proper analyses of fringe images. This kind of analysis has been used for fringe pattern denoising and fringe pattern demodulation showing several advantages, for example in laser plasma interferometry [38], in shadow moiré [39–41], in profilometry [42–44], in speckle interferometry [45], in digital holography [46], and other optical measurement techniques [47–55].

In this chapter, the theoretical basis of fringe pattern image formation and processing is described. Also, in general, the theory and advantages of the 2D continuous wavelet transform (CWT) for fringe pattern processing is described. We also explain some of the main applications in fringe pattern processing, such as phase recovery and wrapped phase map denoising, showing some examples of applications in different optical measurement techniques.
