2. Digital fringe patterns

#### 2.1. Elements of digital fringe image processing systems

Often, a digital fringe image processing system is represented by a sequence of devices, which typically starts with an imaging system that observes the target, a digitizer system which samples and quantizes the analog information acquired by the imaging system, a digital storage device, a digital computer that process the information, and finally, a displaying system to visualize the acquired and processed information (Figure 1).

A typical imaging system is composed by an objective lens to form images in a photosensitive plane which is commonly a CCD (charge couple devices) array.

#### 2.2. Fringe image formation

Fringe pattern images are present in several kinds of optical tests for the measurement of different physical quantities. Such tests are examples for the quality measurement of optical devices using optical interferometry, photoelasticity for stress analysis, or electronic speckle pattern interferometry (ESPI) for the measurement of mechanical properties of materials. The interference phenomena are usually used in many optical methods of measurement. We now describe a classical way to form a fringe pattern image using the two-wave interference.

E xð Þ¼ ; y A1ð Þ x; y e

ð Þ x; y

<sup>1</sup>ð Þþ <sup>x</sup>; <sup>y</sup> <sup>A</sup><sup>2</sup>

For simplicity, Eq. (2) is usually written in a general form as:

The irradiance at a given plane perpendicular to z-axis is then represented as

flat wavefront), respectively, and <sup>k</sup> <sup>¼</sup> <sup>2</sup><sup>π</sup>

reference wavefront.

I xð Þ¼ ; <sup>y</sup> E xð Þ ; <sup>y</sup> <sup>E</sup><sup>∗</sup>

removed and Eq. (3) is simplified:

<sup>¼</sup> <sup>A</sup><sup>2</sup>

ikW xð Þ ; <sup>y</sup> <sup>þ</sup> <sup>A</sup>2ð Þ <sup>x</sup>; <sup>y</sup> <sup>e</sup>

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

<sup>λ</sup> , being λ the wavelength.

<sup>2</sup>ð Þþ <sup>x</sup>; <sup>y</sup> <sup>2</sup>A1ð Þ <sup>x</sup>; <sup>y</sup> <sup>A</sup>2ð Þ <sup>x</sup>; <sup>y</sup> cos ½ � kx sin <sup>θ</sup> <sup>þ</sup> kW xð Þ ; <sup>y</sup> : (2)

I xð Þ¼ ; <sup>y</sup> a xð Þþ ; <sup>y</sup> b xð Þ ; <sup>y</sup> cos <sup>u</sup>0<sup>x</sup> <sup>þ</sup> <sup>ϕ</sup>ð Þ <sup>x</sup>; <sup>y</sup> , (3)

I xð Þ¼ ; <sup>y</sup> a xð Þþ ; <sup>y</sup> b xð Þ ; <sup>y</sup> cos <sup>ϕ</sup>ð Þ <sup>x</sup>; <sup>y</sup> : (4)

where A<sup>1</sup> and A<sup>2</sup> are the amplitudes of the wavefront under test and the reference wavefront (a

Figure 2. Interference of two wavefronts. Solid line represents the wavefront under test and dashed line represents the

where a xð Þ ; y and b xð Þ ; y are commonly called the background illumination and the amplitude modulation, respectively. The term u<sup>0</sup> ¼ k sin θ is the fringe carrier frequency and ϕð Þ¼ x; y kW xð Þ ; y is the phase to be recovered from the fringe pattern image. It must be noted that if the reference wavefront is perpendicular to z-axis (i.e., θ ¼ 0), the fringe carrier frequency is

ikx sin <sup>θ</sup>, (1)

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

177

Two-wave interference can be generated by means of several types of interferometers, and the interferograms or fringe patterns are produced by superimposing two wavefronts. An interferometer can accurately measure deformations of the wavefront of the order of the wavelength. Considering two mutually coherent monochromatic waves, as depicted in Figure 2, W xð Þ ; y represents the wavefront shape under study (i.e., the wave that contains the information of the physical quantity to be measured). The sum of their complex amplitudes can be represented as

Figure 1. Typical sequence in a digital fringe pattern image processing system.

The 2D Continuous Wavelet Transform: Applications in Fringe Pattern Processing for Optical Measurement… http://dx.doi.org/10.5772/intechopen.74813 177

Figure 2. Interference of two wavefronts. Solid line represents the wavefront under test and dashed line represents the reference wavefront.

$$E(\mathbf{x}, y) = A\_1(\mathbf{x}, y)e^{ik\mathcal{W}(\mathbf{x}, y)} + A\_2(\mathbf{x}, y)e^{ik\mathbf{x}\sin\theta},\tag{1}$$

where A<sup>1</sup> and A<sup>2</sup> are the amplitudes of the wavefront under test and the reference wavefront (a flat wavefront), respectively, and <sup>k</sup> <sup>¼</sup> <sup>2</sup><sup>π</sup> <sup>λ</sup> , being λ the wavelength.

The irradiance at a given plane perpendicular to z-axis is then represented as

$$\begin{aligned} I(\mathbf{x}, y) &= E(\mathbf{x}, y)E^\*(\mathbf{x}, y) \\ &= A\_1^2(\mathbf{x}, y) + A\_2^2(\mathbf{x}, y) + 2A\_1(\mathbf{x}, y)A\_2(\mathbf{x}, y)\cos\left[k\mathbf{x}\sin\theta + kW(\mathbf{x}, y)\right]. \end{aligned} \tag{2}$$

For simplicity, Eq. (2) is usually written in a general form as:

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

Often, a digital fringe image processing system is represented by a sequence of devices, which typically starts with an imaging system that observes the target, a digitizer system which samples and quantizes the analog information acquired by the imaging system, a digital storage device, a digital computer that process the information, and finally, a displaying

A typical imaging system is composed by an objective lens to form images in a photosensitive

Fringe pattern images are present in several kinds of optical tests for the measurement of different physical quantities. Such tests are examples for the quality measurement of optical devices using optical interferometry, photoelasticity for stress analysis, or electronic speckle pattern interferometry (ESPI) for the measurement of mechanical properties of materials. The interference phenomena are usually used in many optical methods of measurement. We now describe a classical way to form a fringe pattern image using the two-wave interference.

Two-wave interference can be generated by means of several types of interferometers, and the interferograms or fringe patterns are produced by superimposing two wavefronts. An interferometer can accurately measure deformations of the wavefront of the order of the wavelength. Considering two mutually coherent monochromatic waves, as depicted in Figure 2, W xð Þ ; y represents the wavefront shape under study (i.e., the wave that contains the information of the physical quantity to be measured). The sum of their complex amplitudes can be

examples of applications in different optical measurement techniques.

system to visualize the acquired and processed information (Figure 1).

plane which is commonly a CCD (charge couple devices) array.

Figure 1. Typical sequence in a digital fringe pattern image processing system.

2.1. Elements of digital fringe image processing systems

2. Digital fringe patterns

176 Wavelet Theory and Its Applications

2.2. Fringe image formation

represented as

$$I(\mathbf{x}, y) = a(\mathbf{x}, y) + b(\mathbf{x}, y) \cos \left[ u\_0 \mathbf{x} + \phi(\mathbf{x}, y) \right],\tag{3}$$

where a xð Þ ; y and b xð Þ ; y are commonly called the background illumination and the amplitude modulation, respectively. The term u<sup>0</sup> ¼ k sin θ is the fringe carrier frequency and ϕð Þ¼ x; y kW xð Þ ; y is the phase to be recovered from the fringe pattern image. It must be noted that if the reference wavefront is perpendicular to z-axis (i.e., θ ¼ 0), the fringe carrier frequency is removed and Eq. (3) is simplified:

$$I(\mathbf{x}, y) = a(\mathbf{x}, y) + b(\mathbf{x}, y) \cos \left[ \phi(\mathbf{x}, y) \right]. \tag{4}$$

the phase shifting is realized by moving some mirrors in the optical interferometer. The set of N

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

The pointwise solution for ϕð Þ x; y from the non-linear system of equations is obtained by using

P<sup>N</sup>

where <sup>W</sup> is the wrapping operator such that <sup>W</sup> <sup>ϕ</sup>ð Þ <sup>x</sup>; <sup>y</sup> � �∈½ Þ �π; <sup>π</sup> . Several algorithms can be

As previously mentioned, processing fringe patterns with fringe carrier frequency may be simple to carry out. The key point in the demodulation of fringe patterns with carrier is that the total phase function u0x þ ϕð Þ x; y represents the addition of an inclined phase plane u0x plus the target phase ϕð Þ x; y . In this case, a monotonically increasing (or decreasing) phase function has to be recovered. If we analyze the Fourier spectrum of Eq. (3), for a proper separation between

The analytic signal g xð Þ ; y to recover the phase ϕð Þ x; y can be computed with the Fourier

fH uð Þ ; v Ff g I xð Þ ; y g ¼ e

where H uð Þ ; v is a filter in the Fourier domain centered at the frequency u0, u the frequency variable along x direction, and v the frequency variable along y direction. Finally, the wrapped

Other technique to compute the phase from a carrier frequency fringe pattern is the synchronous detection technique [28], which is realized in the spatial domain. Using the complex

where <sup>∗</sup> represents the convolution operator and h xð Þ ; <sup>y</sup> a low-pass convolution filter in the

� � Imag g xð Þ ; y e f g �i2πu<sup>0</sup>

i2πu<sup>0</sup> � � <sup>¼</sup> <sup>e</sup>

<sup>W</sup> <sup>ϕ</sup>ð Þ <sup>x</sup>; <sup>y</sup> � � <sup>¼</sup> tan �<sup>1</sup> Real g xð Þ ; <sup>y</sup> <sup>e</sup>�i2πu<sup>0</sup>

notation, in this case, the analytic function g xð Þ ; y can be computed with

spatial domain. The wrapped phase can be computed with

g xð Þ¼ ; y h xð Þ ; y ∗ I xð Þ ; y e

<sup>n</sup>¼<sup>1</sup> In sin ð Þ <sup>α</sup><sup>n</sup> <sup>P</sup><sup>N</sup> <sup>n</sup>¼<sup>1</sup> In cos ð Þ <sup>α</sup><sup>n</sup>

!

� � <sup>n</sup> <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, …, N: (5)

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

179

max <sup>k</sup>∇ϕ<sup>k</sup> � � <sup>&</sup>lt; <sup>k</sup>u0k: (7)

� � <sup>∈</sup>½ Þ �π; <sup>π</sup> : (9)

<sup>i</sup>2<sup>π</sup>½ � <sup>u</sup>0xþϕð Þ <sup>x</sup>; <sup>y</sup> , (8)

<sup>i</sup>2πϕð Þ <sup>x</sup>; <sup>y</sup> , (10)

∈½ Þ �π; π , (6)

Inð Þ¼ x; y a xð Þþ ; y b xð Þ ; y cos ϕð Þþ x; y α<sup>n</sup>

spectral lobes in the Fourier space, the following inequality must be complied:

<sup>W</sup> <sup>ϕ</sup>ð Þ <sup>x</sup>; <sup>y</sup> � � <sup>¼</sup> tan �<sup>1</sup> �

3.2. Phase recovery from single fringe patterns with carrier

phase-shifted fringe patterns is defined as

the last-squares approach (see [2] for details):

used that require three, four, up to eight images.

transform method [27], which can expressed as

phase is computed with

g xð Þ¼ ; <sup>y</sup> <sup>F</sup> �<sup>1</sup>

Figure 3. Examples of simulated fringe pattern images with (a) and without (b) fringe carrier frequency. The phase of modulation ϕð Þ x; y (c) is the same for both fringe images (phase shown wrapped and codified in gray levels).

Equations (3) and (4) represent the mathematical expressions of fringe pattern images with and without fringe carrier frequency, respectively. Examples of these kinds of fringe images are shown in Figure 3.
