3. Fringe pattern processing

#### 3.1. Phase-shifting methods for phase recovery

One of the most popular methods for phase recovery is the well-known phase-shifting. This method requires a set of phase-shifted fringe patterns which are experimentally obtained in different ways depending on the optical measurement technique. For example, in interferometry the phase shifting is realized by moving some mirrors in the optical interferometer. The set of N phase-shifted fringe patterns is defined as

$$I\_n(\mathbf{x}, y) = a(\mathbf{x}, y) + b(\mathbf{x}, y) \cos \left[\phi(\mathbf{x}, y) + a\_n\right] \quad n = 1, 2, \dots, N. \tag{5}$$

The pointwise solution for ϕð Þ x; y from the non-linear system of equations is obtained by using the last-squares approach (see [2] for details):

$$\mathcal{W}\{\phi(x,y)\} = \tan^{-1}\left(-\frac{\sum\_{n=1}^{N} I\_n \sin\left(\alpha\_n\right)}{\sum\_{n=1}^{N} I\_n \cos\left(\alpha\_n\right)}\right) \in [-\pi,\pi),\tag{6}$$

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

#### 3.2. Phase recovery from single fringe patterns with carrier

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

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).

One of the most popular methods for phase recovery is the well-known phase-shifting. This method requires a set of phase-shifted fringe patterns which are experimentally obtained in different ways depending on the optical measurement technique. For example, in interferometry

shown in Figure 3.

178 Wavelet Theory and Its Applications

3. Fringe pattern processing

3.1. Phase-shifting methods for phase recovery

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

$$\max\{\|\nabla\phi\|\|\} < \|\mu\_0\|.\tag{7}$$

The analytic signal g xð Þ ; y to recover the phase ϕð Þ x; y can be computed with the Fourier transform method [27], which can expressed as

$$\log(\mathbf{x}, y) = \mathcal{F}^{-1}\{H(\mathbf{u}, \mathbf{v})\mathcal{F}\{I(\mathbf{x}, y)\}\} = e^{i2\pi \left[\boldsymbol{\mu}\_0 \mathbf{x} + \phi(\mathbf{x}, y)\right]},\tag{8}$$

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 phase is computed with

$$\mathcal{W}\{\phi(\mathbf{x},y)\} = \tan^{-1}\left(\frac{\text{Real}\{\mathbf{g}(\mathbf{x},y)e^{-i2\pi u\_0}\}}{\text{Imag}\{\mathbf{g}(\mathbf{x},y)e^{-i2\pi u\_0}\}}\right) \in [-\pi,\pi). \tag{9}$$

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

$$g(\mathbf{x}, y) = h(\mathbf{x}, y) \* \left[ I(\mathbf{x}, y) e^{i2\pi u\_0} \right] = e^{i2\pi \phi(\mathbf{x}, y)},\tag{10}$$

where <sup>∗</sup> represents the convolution operator and h xð Þ ; <sup>y</sup> a low-pass convolution filter in the spatial domain. The wrapped phase can be computed with

$$\mathcal{W}\{\phi(\mathbf{x},y)\} = \tan^{-1}\left(\frac{\text{Real}\{\mathbf{g}(\mathbf{x},y)\}}{\text{Imag}\{\mathbf{g}(\mathbf{x},y)\}}\right) \in [-\pi,\pi). \tag{11}$$

Δψð Þ¼ x; y ½ � ψð Þ� x; y ψð Þ x � 1; y ;ψð Þ� x; y ψð Þ x; y � 1 , (17)

<sup>Δ</sup>ϕð Þ¼ <sup>x</sup>; <sup>y</sup> <sup>ϕ</sup>ð Þ� <sup>x</sup>; <sup>y</sup> <sup>ϕ</sup>ð Þ <sup>x</sup> � <sup>1</sup>; <sup>y</sup> ; <sup>ϕ</sup>ð Þ� <sup>x</sup>; <sup>y</sup> <sup>ϕ</sup>ð Þ <sup>x</sup>; <sup>y</sup> � <sup>1</sup> � �: (18)

kΔϕk < π, ∀ ð Þ x; y : (19)

h i, (20)

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

181

(22)

where ð Þ x � 1; y and ð Þ x; y � 1 are contiguous horizontal and vertical sites, respectively. In a

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

It can be deduced that the problem of the recovery of ϕ from ψ can be properly solved if the sampling theorem is reached, that is, if the distance between two fringes is more than two pixels (the phase difference between two fringes is 2π). In phase terms, the sampling theorem

Δϕ ¼ Wf g Δψ ¼ ψx;ψ<sup>y</sup>

Note that Wf g Δψ (the wrapped phase differences) can be obtained from the observed wrapped phase field ψ. Then, the unwrapped phase ϕ can be achieved by two-dimensional

A simple way to compute the unwrapped phase ϕ from the wrapped one ψ is by means of

where L is the set of valid pixels in the image. Unfortunately, in most cases noise is present, therefore, inequality (19) is not always satisfied and the integration does not provide proper results. Therefore, denoising wrapped phase maps is a fundamental step before the phase

4. The 2D continuous wavelet transform for processing fringe patterns

It is clear that the phase demodulation of fringe images with carrier may be easily realized. Owing that, in this case, the fringe image may represent a quasi-stationary signal along the direction of the frequency carrier, the use of classical linear operators such as the Fourier

ψ<sup>x</sup> ¼ Wf g ψð Þ� x; y ψð Þ x � 1; y and ψ<sup>x</sup> ¼ Wf g ψð Þ� x; y ψð Þ x; y � 1 : (21)

<sup>ψ</sup>xð Þ� <sup>x</sup>; <sup>y</sup> <sup>ϕ</sup>ð Þ� <sup>x</sup>; <sup>y</sup> <sup>ϕ</sup>ð Þ <sup>x</sup> � <sup>1</sup>; <sup>y</sup> � � � � <sup>2</sup> <sup>þ</sup> <sup>ψ</sup>yð Þ� <sup>x</sup>; <sup>y</sup> <sup>ϕ</sup>ð Þ� <sup>x</sup>; <sup>y</sup> <sup>ϕ</sup>ð Þ <sup>x</sup>; <sup>y</sup> � <sup>1</sup> � � h i<sup>2</sup> � �,

similar manner, we can also define the unwrapped phase-difference field:

is reached if the phase difference between two pixels is less than π or, in general

If this condition is satisfied, the following relation can be established:

where

U ϕ

� � <sup>¼</sup> <sup>X</sup>

integration of the vector field Wf g Δψ .

minimizing the cost function

ð Þ x; y ∈ L

unwrapping process.

#### 3.3. Phase recovery from single fringe patterns without carrier

As described in [34–37], for the case in which u<sup>0</sup> ¼ 0, the previous computation of the fringe direction is necessary to compute the analytic function g xð Þ ; y , for example, using the quadrature transform [36]:

$$\text{Imag}\{\mathbf{g}(\mathbf{x},\boldsymbol{y})\} = \sin\left[\phi(\mathbf{x},\boldsymbol{y})\right] = \mathbf{n}\_{\phi}(\mathbf{x},\boldsymbol{y}) \cdot \frac{\nabla I\_{n}(\mathbf{x},\boldsymbol{y})}{\|\nabla \phi(\mathbf{x},\boldsymbol{y})\|}\tag{12}$$

where Inð Þ¼ <sup>x</sup>; <sup>y</sup> cos <sup>ϕ</sup>ð Þ <sup>x</sup>; <sup>y</sup> � � <sup>¼</sup> Realf g g xð Þ ; <sup>y</sup> is a normalized version of I xð Þ ; <sup>y</sup> , and <sup>n</sup><sup>ϕ</sup> is the unit vector normal to the corresponding isophase contour, which points to the direction of ∇ϕð Þ x; y . It is well known that the computation of n<sup>ϕ</sup> is by far the most difficult problem to compute the phase using this method.

Also, the modulo-2π fringe orientation angle αð Þ x; y can be used to compute the quadrature fringe pattern by means of the spiral-phase signum function S uð Þ ; v in the Fourier domain [35]:

$$\text{Imag}\{\mathbf{g}(\mathbf{x},y)\} = \sin\left[\phi(\mathbf{x},y)\right] = -\text{ie}^{-i\mathbf{a}(\mathbf{x},y)}\mathcal{F}^{-1}\{\mathcal{S}(\mathbf{u},\mathbf{v})\mathcal{F}\{I\_n(\mathbf{x},y)\}\},\tag{13}$$

where

$$S(\mu, v) = \frac{\mu + iv}{\sqrt{\mu^2 + v^2}},\tag{14}$$

and <sup>i</sup> <sup>¼</sup> ffiffiffiffiffiffi �<sup>1</sup> <sup>p</sup> . However, the most difficult problem in this method is the computation of <sup>α</sup>ð Þ <sup>x</sup>; <sup>y</sup> . It can be deduced that Eqs. (12) and (13) are closely related because

$$\alpha(\mathbf{x}, y) = \text{angle}\{\mathbf{n}\_{\phi}(\mathbf{x}, y)\} \in (0, 2\pi]. \tag{15}$$

#### 3.4. Wrapped phase maps denoising

The unwrapping process can be, in many cases, a difficult task due to phase inconsistencies or noise. In order to understand the phase unwrapping problem of noisy phase maps, we define the wrapped and the unwrapped phase as ψð Þ x; y and ϕð Þ x; y respectively. As it is known that ψð Þ x; y ∈½ Þ �π; π , the following relation can be established:

$$
\psi(\mathbf{x}, y) = \phi(\mathbf{x}, y) + 2\pi k(\mathbf{x}, y),
\tag{16}
$$

where k xð Þ ; y is a field of integers such that ψð Þ x; y ∈½ Þ �π; π . The wrapped phasedifference vector field Δψð Þ x; y which can be computed from the wrapped phase map, is defined as

The 2D Continuous Wavelet Transform: Applications in Fringe Pattern Processing for Optical Measurement… http://dx.doi.org/10.5772/intechopen.74813 181

$$
\Delta\psi(\mathbf{x},y) = [\psi(\mathbf{x},y) - \psi(\mathbf{x}-1,y), \psi(\mathbf{x},y) - \psi(\mathbf{x},y-1)].\tag{17}
$$

where ð Þ x � 1; y and ð Þ x; y � 1 are contiguous horizontal and vertical sites, respectively. In a similar manner, we can also define the unwrapped phase-difference field:

$$
\Delta\phi(\mathbf{x},y) = \left[\phi(\mathbf{x},y) - \phi(\mathbf{x}-1,y), \phi(\mathbf{x},y) - \phi(\mathbf{x},y-1)\right].\tag{18}
$$

It can be deduced that the problem of the recovery of ϕ from ψ can be properly solved if the sampling theorem is reached, that is, if the distance between two fringes is more than two pixels (the phase difference between two fringes is 2π). In phase terms, the sampling theorem is reached if the phase difference between two pixels is less than π or, in general

$$\|\Delta\phi\| < \pi, \quad \forall \quad (\mathbf{x}, y). \tag{19}$$

If this condition is satisfied, the following relation can be established:

$$
\Delta\phi = \mathcal{W}\{\Delta\psi\} = \left[\psi\_x, \psi\_y\right]\_{\prime} \tag{20}
$$

where

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

Imagf g g xð Þ ; <sup>y</sup> <sup>¼</sup> sin <sup>ϕ</sup>ð Þ <sup>x</sup>; <sup>y</sup> � � <sup>¼</sup> <sup>n</sup>ϕð Þ� <sup>x</sup>; <sup>y</sup>

Imagf g¼ g xð Þ ; <sup>y</sup> sin <sup>ϕ</sup>ð Þ <sup>x</sup>; <sup>y</sup> � � ¼ �ie�iαð Þ <sup>x</sup>;<sup>y</sup> <sup>F</sup> �<sup>1</sup>

It can be deduced that Eqs. (12) and (13) are closely related because

ψð Þ x; y ∈½ Þ �π; π , the following relation can be established:

S uð Þ¼ ; v

3.3. Phase recovery from single fringe patterns without carrier

ture transform [36]:

180 Wavelet Theory and Its Applications

[35]:

where

and <sup>i</sup> <sup>¼</sup> ffiffiffiffiffiffi

defined as

compute the phase using this method.

3.4. Wrapped phase maps denoising

Imagf g g xð Þ ; y � �

As described in [34–37], for the case in which u<sup>0</sup> ¼ 0, the previous computation of the fringe direction is necessary to compute the analytic function g xð Þ ; y , for example, using the quadra-

where Inð Þ¼ <sup>x</sup>; <sup>y</sup> cos <sup>ϕ</sup>ð Þ <sup>x</sup>; <sup>y</sup> � � <sup>¼</sup> Realf g g xð Þ ; <sup>y</sup> is a normalized version of I xð Þ ; <sup>y</sup> , and <sup>n</sup><sup>ϕ</sup> is the unit vector normal to the corresponding isophase contour, which points to the direction of ∇ϕð Þ x; y . It is well known that the computation of n<sup>ϕ</sup> is by far the most difficult problem to

Also, the modulo-2π fringe orientation angle αð Þ x; y can be used to compute the quadrature fringe pattern by means of the spiral-phase signum function S uð Þ ; v in the Fourier domain

<sup>u</sup> <sup>þ</sup> iv ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

�<sup>1</sup> <sup>p</sup> . However, the most difficult problem in this method is the computation of <sup>α</sup>ð Þ <sup>x</sup>; <sup>y</sup> .

The unwrapping process can be, in many cases, a difficult task due to phase inconsistencies or noise. In order to understand the phase unwrapping problem of noisy phase maps, we define the wrapped and the unwrapped phase as ψð Þ x; y and ϕð Þ x; y respectively. As it is known that

where k xð Þ ; y is a field of integers such that ψð Þ x; y ∈½ Þ �π; π . The wrapped phasedifference vector field Δψð Þ x; y which can be computed from the wrapped phase map, is

∈½ Þ �π; π : (11)

<sup>k</sup>∇ϕð Þk <sup>x</sup>; <sup>y</sup> , (12)

f g S uð Þ ; v Ff g Inð Þ x; y , (13)

<sup>u</sup><sup>2</sup> <sup>þ</sup> <sup>v</sup><sup>2</sup> <sup>p</sup> , (14)

<sup>α</sup>ð Þ¼ <sup>x</sup>; <sup>y</sup> angle <sup>n</sup>ϕð Þ <sup>x</sup>; <sup>y</sup> � �<sup>∈</sup> ð � <sup>0</sup>; <sup>2</sup><sup>π</sup> : (15)

ψð Þ¼ x; y ϕð Þþ x; y 2πk xð Þ ; y , (16)

∇Inð Þ x; y

$$
\psi\_x = W\{\psi(\mathbf{x}, y) - \psi(\mathbf{x} - \mathbf{1}, y)\} \quad \text{and} \quad \psi\_x = W\{\psi(\mathbf{x}, y) - \psi(\mathbf{x}, y - 1)\}. \tag{21}
$$

Note that Wf g Δψ (the wrapped phase differences) can be obtained from the observed wrapped phase field ψ. Then, the unwrapped phase ϕ can be achieved by two-dimensional integration of the vector field Wf g Δψ .

A simple way to compute the unwrapped phase ϕ from the wrapped one ψ is by means of minimizing the cost function

$$\mathcal{U}\{\phi\} = \sum\_{(\mathbf{x},\mathbf{y})\in L} \left\{ \left[ \psi\_{\mathbf{x}}(\mathbf{x},\mathbf{y}) - \left( \phi(\mathbf{x},\mathbf{y}) - \phi(\mathbf{x}-\mathbf{1},\mathbf{y}) \right) \right]^2 + \left[ \psi\_{y}(\mathbf{x},\mathbf{y}) - \left( \phi(\mathbf{x},\mathbf{y}) - \phi(\mathbf{x},\mathbf{y}-1) \right) \right]^2 \right\},\tag{22}$$

where L is the set of valid pixels in the image. Unfortunately, in most cases noise is present, therefore, inequality (19) is not always satisfied and the integration does not provide proper results. Therefore, denoising wrapped phase maps is a fundamental step before the phase unwrapping process.

#### 4. The 2D continuous wavelet transform for processing fringe patterns

It is clear that the phase demodulation of fringe images with carrier may be easily realized. Owing that, in this case, the fringe image may represent a quasi-stationary signal along the direction of the frequency carrier, the use of classical linear operators such as the Fourier transform may be adequate. It works well mainly for few components in the frequency domain (i.e., for narrow spectrums); however, this is not the case for many signals in the real world. This dependence is a serious weakness mainly in two aspects: the degree of automation and the accuracy of the method specially when fringes produce spread spectrums due to localized variations or phase transients. Additionally, in the case of closed fringes there may be a wide range of frequencies in all directions. Then, evidently standard Fourier analysis is inadequate for treating with this kind of images because it represents signals with a linear superposition of sine waves with "infinite" extension. For this reason, an image with closed fringes should be represented with localized components characterizing the frequency, shifting, and orientation. A powerful mathematical tool for signal description that has been developed in the last decades is the wavelet analysis. Fortunately, for our purposes, a key characteristic of this type of analysis is the finely detailed description of frequency or phase of signals. In consequence, it can have a good performance especially with fringes that produce spread spectrums. Additionally, one of the main advantages using wavelets compared with standard techniques is its high capability to deal with noise. In particular, the 2D continuous wavelet transform have recently been proposed for the processing of interferometric images. Advantages of denoising and demodulation of interferograms using the 2D CWT has been discussed in [44–55].

Considering an interferometric image (an interferogram or a wrapped-phase field) G rð Þ, where <sup>r</sup> <sup>¼</sup> ð Þ <sup>x</sup>; <sup>y</sup> <sup>∈</sup> <sup>R</sup>2, its 2D CWT decomposition can be defined as

$$\mathcal{G}\_{\mathcal{W}}(\mathbf{s}, \boldsymbol{\theta}, \boldsymbol{\eta}) = \mathcal{W}\{\mathbf{G}(\mathbf{r})\} = \int\_{\mathbb{R}^2} \mathcal{G}(\mathbf{r}) \boldsymbol{\varphi}\_{s, \boldsymbol{\theta}, \boldsymbol{\eta}}^\*(\mathbf{r}) d\mathbf{r}.\tag{23}$$

φs,θ,ηð Þ¼ r exp �π

Figure 5. Frequency localization of the 2D wavelets in the Fourier domain (<sup>f</sup> <sup>¼</sup> <sup>ν</sup>

because noise and fringes are mixed in the Fourier domain.

and positive frequencies are mixed in the Fourier domain.

4.1. Phase recovery with the 2D CWT

<sup>∥</sup><sup>r</sup> � <sup>s</sup>∥<sup>2</sup> η 

where Θ ¼ ð Þ cos θ; sin θ , ð Þ� represents the dot product, and ν∈ R is the frequency variable. Figure 5 shows that the 2D CWT is performed along different directions and frequencies.

Owing that fringe pattern images with closed fringes generally contain elements with high anisotropy and sparse frequency components, the phase recovery is a complex procedure. Compounding the problem, the presence of noise makes the process even more complicated

Also, it has been shown that a single fringe pattern without carrier frequency, is not easy to deal with. Owing to ambiguities in the image formation process, a main drawback analyzing them is that several solutions of the phase function can satisfy the original observed image. Therefore, it is necessary to restrict the solution space of ϕ in Eq. (4). Fortunately, as in most practical cases the phase to be recovered is continuous, the algorithm to process the fringe pattern usually seeks for a continuous phase function. However, the recovery of the continuous phase function is not a simple task to carry out as occur with fringe patterns with carrier frequency. It can be observed that the phase gradient represents the local frequencies of the fringe pattern in the x and y directions; however, the sign of ∇ϕ is ambiguous because negative

The following is a general description of the phase recovery method using the 2D CWT. First, it is necessary to consider a normalized version of the fringe pattern. The normalization

� exp i2π

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

ν

η).

<sup>η</sup> ð Þ ð Þ� <sup>r</sup> � <sup>s</sup> <sup>Θ</sup>

, (24)

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

183

In Eq. (23), φ represents the 2D mother wavelet and <sup>∗</sup> indicates the complex conjugated. The variable <sup>s</sup><sup>∈</sup> <sup>R</sup><sup>2</sup> represents the shift, <sup>θ</sup>∈½ Þ <sup>0</sup>; <sup>2</sup><sup>π</sup> the rotation angle, and <sup>η</sup> the scaling factor. It has been shown that a proper mother wavelet for processing interferometric images is the 2D Gabor wavelet (see Figure 4). The mathematical representation of this kind of wavelet can be defined as

Figure 4. Example of a 2D Gabor wavelet. (a) Real part and (b) imaginary part.

The 2D Continuous Wavelet Transform: Applications in Fringe Pattern Processing for Optical Measurement… http://dx.doi.org/10.5772/intechopen.74813 183

Figure 5. Frequency localization of the 2D wavelets in the Fourier domain (<sup>f</sup> <sup>¼</sup> <sup>ν</sup> η).

$$\varphi\_{s,\theta,\eta}(r) = \exp\left[-\pi \frac{\|r-s\|^2}{\eta}\right] \times \exp\left[i2\pi \frac{\nu}{\eta}((r-s)\cdot\Theta)\right] \tag{24}$$

where Θ ¼ ð Þ cos θ; sin θ , ð Þ� represents the dot product, and ν∈ R is the frequency variable.

Figure 5 shows that the 2D CWT is performed along different directions and frequencies.

#### 4.1. Phase recovery with the 2D CWT

transform may be adequate. It works well mainly for few components in the frequency domain (i.e., for narrow spectrums); however, this is not the case for many signals in the real world. This dependence is a serious weakness mainly in two aspects: the degree of automation and the accuracy of the method specially when fringes produce spread spectrums due to localized variations or phase transients. Additionally, in the case of closed fringes there may be a wide range of frequencies in all directions. Then, evidently standard Fourier analysis is inadequate for treating with this kind of images because it represents signals with a linear superposition of sine waves with "infinite" extension. For this reason, an image with closed fringes should be represented with localized components characterizing the frequency, shifting, and orientation. A powerful mathematical tool for signal description that has been developed in the last decades is the wavelet analysis. Fortunately, for our purposes, a key characteristic of this type of analysis is the finely detailed description of frequency or phase of signals. In consequence, it can have a good performance especially with fringes that produce spread spectrums. Additionally, one of the main advantages using wavelets compared with standard techniques is its high capability to deal with noise. In particular, the 2D continuous wavelet transform have recently been proposed for the processing of interferometric images. Advantages of denoising and demodulation of interferograms using the 2D

Considering an interferometric image (an interferogram or a wrapped-phase field) G rð Þ, where

In Eq. (23), φ represents the 2D mother wavelet and <sup>∗</sup> indicates the complex conjugated. The variable <sup>s</sup><sup>∈</sup> <sup>R</sup><sup>2</sup> represents the shift, <sup>θ</sup>∈½ Þ <sup>0</sup>; <sup>2</sup><sup>π</sup> the rotation angle, and <sup>η</sup> the scaling factor. It has been shown that a proper mother wavelet for processing interferometric images is the 2D Gabor wavelet (see Figure 4). The mathematical representation of this kind of wavelet can be

ð R2

G rð Þφ<sup>∗</sup>

s,θ,ηð Þr dr: (23)

CWT has been discussed in [44–55].

182 Wavelet Theory and Its Applications

defined as

<sup>r</sup> <sup>¼</sup> ð Þ <sup>x</sup>; <sup>y</sup> <sup>∈</sup> <sup>R</sup>2, its 2D CWT decomposition can be defined as

Figure 4. Example of a 2D Gabor wavelet. (a) Real part and (b) imaginary part.

GWð Þ¼ s; θ; η Wf g G rð Þ ¼

Owing that fringe pattern images with closed fringes generally contain elements with high anisotropy and sparse frequency components, the phase recovery is a complex procedure. Compounding the problem, the presence of noise makes the process even more complicated because noise and fringes are mixed in the Fourier domain.

Also, it has been shown that a single fringe pattern without carrier frequency, is not easy to deal with. Owing to ambiguities in the image formation process, a main drawback analyzing them is that several solutions of the phase function can satisfy the original observed image. Therefore, it is necessary to restrict the solution space of ϕ in Eq. (4). Fortunately, as in most practical cases the phase to be recovered is continuous, the algorithm to process the fringe pattern usually seeks for a continuous phase function. However, the recovery of the continuous phase function is not a simple task to carry out as occur with fringe patterns with carrier frequency. It can be observed that the phase gradient represents the local frequencies of the fringe pattern in the x and y directions; however, the sign of ∇ϕ is ambiguous because negative and positive frequencies are mixed in the Fourier domain.

The following is a general description of the phase recovery method using the 2D CWT. First, it is necessary to consider a normalized version of the fringe pattern. The normalization procedure can be carried out using the method proposed in [56]. Consider we represent the normalized fringe pattern in complex form:

$$G(r) = \cos\left[\phi(r)\right] = \frac{\exp\left[i\phi(r)\right]}{2} + \frac{\exp\left[-i\phi(r)\right]}{2}.\tag{25}$$

In this case, νθ is the two-dimensional frequency variable. Note that for a fixed s, Wf g G rð Þ

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

To detect the analytic function and consequently compute the phase ϕð Þs at a given pixel s (i.e.,

that the sign of the phase gradient cannot be determined from the image intensity, there exists a sign ambiguity of the phase in the θ � η map. In Figure 6, it can be observed that in this situation, there are two maximum in each θ � η map. Also, it can be deduced that the magnitude of the coefficients map is periodic with respect to θ with period π. To solve the problem of sign ambiguity, Ma et al. [48] proposed a phase determination rule according to the phase distribution continuity. Also, Villa et al. [55] proposed a sliding 2D CWT method that assumes that the phase is continuous and smoothly varying, in this way, the ridge detection is realized assuming that the coefficient maps are similar in adjacent pixels, reducing

Once detected the ridge Wf g G rð Þ ridge that represents a 2D function, the wrapped phase can be

Figures 7 and 8 show examples of fringe pattern phase recovery using the 2D CWT method reported in [55]. It is important to remark that this method is highly robust against noise.

Figure 6. (a) Example of noisy simulated fringe pattern. The square indicates a region around a pixel s where the phase is estimated. (b) Magnitude of the θ � η map at the pixel s, codified in gray levels. Horizontal direction represents the rotation angle while the vertical direction represents the scale. The two white regions represent the two terms in Eq. (30).

0 @ Real <sup>W</sup>f g G rð Þ ridge n o

1

Imag <sup>W</sup>f g G rð Þ ridge n o

<sup>η</sup> ; θ � �. It 185

<sup>2</sup><sup>π</sup> . Owing

<sup>2</sup><sup>π</sup> or νθ ¼ � <sup>∇</sup>ϕð Þ<sup>s</sup>

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

A: (31)

represents two Gaussian filters in the Fourier domain localized at polar coordinates <sup>ν</sup>

the ridge detection), we can choice one of two possibilities: at νθ <sup>¼</sup> <sup>∇</sup>ϕð Þ<sup>s</sup>

<sup>W</sup> <sup>ϕ</sup>ð Þ<sup>r</sup> � � <sup>¼</sup> tan �<sup>1</sup>

the processing time too.

computed with

can also be visualized as an orientation and frequency decomposition of the fringe pattern.

In this particular case, the 2D CWT of G rð Þ is

$$\begin{split} \mathcal{W}\{\mathbf{G}(r)\} &= \int\_{\mathbb{R}^2} \frac{\exp\left[i\phi(r)\right]}{2} \exp\left[-\pi \frac{\left\|r-s\right\|^2}{\eta}\right] \times \exp\left[-i2\pi \frac{\nu}{\eta}((r-s)\cdot\Theta)\right] d\mathbf{x} \\ &+ \int\_{\mathbb{R}^2} \frac{\exp\left[-i\phi(r)\right]}{2} \exp\left[-\pi \frac{\left\|r-s\right\|^2}{\eta}\right] \times \exp\left[-i2\pi \frac{\nu}{\eta}((r-s)\cdot\Theta)\right] d\mathbf{x}. \end{split} \tag{26}$$

Note that Wf g G rð Þ represents a four-dimensional function depending on x, y, η, and θ. The process to recover the phase ϕð Þr using the 2D CWT consists on realizing the well-known ridge detection. To understand the phase recovery from the ridge detection, first it is necessary to know the meaning of Eq. (26). To do so, let <sup>~</sup><sup>r</sup> <sup>¼</sup> <sup>r</sup> � <sup>s</sup> and νθ <sup>¼</sup> <sup>ν</sup> <sup>η</sup> ð Þ cos <sup>θ</sup>; sin <sup>θ</sup> , where νθ <sup>∈</sup> <sup>R</sup><sup>2</sup> . Using Taylor's expansion we know that

$$
\phi(\tilde{r} + \mathbf{s}) \approx \phi(\mathbf{s}) + \nabla\phi(\mathbf{s}) \cdot \tilde{r}. \tag{27}
$$

Then, we can now rewrite Eq. (26) as

$$\mathcal{W}\{\mathbf{G}(r)\} \approx \frac{\exp\left[i\phi(\mathbf{s})\right]}{2} \int\_{\mathbb{R}^2} \exp\left[i\big(\nabla\phi(\mathbf{s})\cdot\mathring{r}\)\right] \times \exp\left[-\pi\frac{\|\mathring{r}\|^2}{\eta}\right] \exp\left[-i2\pi(\mathring{r}\cdot\nu\_\theta)\right] d\mathring{r}$$

$$+\frac{\exp\left[-i\phi(\mathbf{s})\right]}{2} \int\_{\mathbb{R}^2} \exp\left[-i\big(\nabla\phi(\mathbf{s})\cdot\mathring{r}\big)\right] \times \exp\left[-\pi\frac{\|\mathring{r}\|^2}{\eta}\right] \exp\left[-i2\pi(\mathring{r}\cdot\nu\_\theta)\right] d\mathring{r},\tag{28}$$

or, which is the same

$$\mathcal{W}\{G(r)\} \approx \frac{\exp\left[i\phi(s)\right]}{2} \mathcal{F}\left\{\exp\left[i\left(\nabla\phi(s)\cdot\ddot{r}\right)\right] \times \exp\left[-\pi\frac{\|\ddot{r}\|^2}{\eta}\right]\right\}$$

$$+\frac{\exp\left[-i\phi(s)\right]}{2} \mathcal{F}\left\{\exp\left[-i\left(\nabla\phi(s)\cdot\ddot{r}\right)\right] \times \exp\left[-\pi\frac{\|\ddot{r}\|^2}{\eta}\right]\right\}.\tag{29}$$

The two terms in (29) contains Fourier transforms of complex periodic functions of frequencies ∇ϕð Þs =2π and �∇ϕð Þs =2π. Then, applying the Fourier's similarity and modulation theorems this last equation can be finally written as

$$\begin{split} \mathcal{W}\{\mathbf{G}(r)\} & \approx \eta \frac{\exp\left[i\phi(s)\right]}{2} \exp\left[-\eta\pi\left\|\begin{matrix} \nu\_{\theta} - \frac{\nabla\phi(s)}{2\pi} \right\|^{2} \right] \\ + \eta \frac{\exp\left[-i\phi(s)\right]}{2} \exp\left[-\eta\pi\left\|\begin{matrix} \nu\_{\theta} + \frac{\nabla\phi(s)}{2\pi} \end{matrix} \right\|^{2} \right]. \end{split} \tag{30}$$

In this case, νθ is the two-dimensional frequency variable. Note that for a fixed s, Wf g G rð Þ represents two Gaussian filters in the Fourier domain localized at polar coordinates <sup>ν</sup> <sup>η</sup> ; θ � �. It can also be visualized as an orientation and frequency decomposition of the fringe pattern.

procedure can be carried out using the method proposed in [56]. Consider we represent the

<sup>∥</sup><sup>r</sup> � <sup>s</sup>∥<sup>2</sup> η � �

Note that Wf g G rð Þ represents a four-dimensional function depending on x, y, η, and θ. The process to recover the phase ϕð Þr using the 2D CWT consists on realizing the well-known ridge detection. To understand the phase recovery from the ridge detection, first it is necessary to

exp <sup>i</sup> <sup>∇</sup>ϕð Þ� <sup>s</sup> <sup>~</sup><sup>r</sup> � � � � � exp �<sup>π</sup>

exp �<sup>i</sup> <sup>∇</sup>ϕð Þ� <sup>s</sup> <sup>~</sup><sup>r</sup> � � � � � exp �<sup>π</sup>

<sup>2</sup> <sup>F</sup> exp <sup>i</sup> <sup>∇</sup>ϕð Þ� <sup>s</sup> <sup>~</sup><sup>r</sup> � � � � � exp �<sup>π</sup>

<sup>2</sup> <sup>F</sup> exp �<sup>i</sup> <sup>∇</sup>ϕð Þ� <sup>s</sup> <sup>~</sup><sup>r</sup> � � � � � exp �<sup>π</sup>

<sup>2</sup> exp �ηπ νθ � <sup>∇</sup>ϕð Þ<sup>s</sup>

<sup>2</sup> exp �ηπ νθ <sup>þ</sup>

� � � �

<sup>2</sup> " #

� � � �

The two terms in (29) contains Fourier transforms of complex periodic functions of frequencies ∇ϕð Þs =2π and �∇ϕð Þs =2π. Then, applying the Fourier's similarity and modulation theorems

exp <sup>i</sup>ϕð Þ<sup>s</sup> � �

exp �iϕð Þ<sup>s</sup> � �

� � � �

� � � �

2π

<sup>2</sup> " #

� � � �

∇ϕð Þs 2π

� � � �

:

<sup>∥</sup><sup>r</sup> � <sup>s</sup>∥<sup>2</sup> η � �

<sup>2</sup> <sup>þ</sup> exp �iϕð Þ<sup>r</sup> � �

� exp �i2π

� exp �i2π

<sup>ϕ</sup> <sup>~</sup><sup>r</sup> <sup>þ</sup> <sup>s</sup><sup>Þ</sup> <sup>≈</sup> <sup>ϕ</sup>ð Þþ <sup>s</sup> <sup>∇</sup>ϕð Þ� <sup>s</sup> <sup>~</sup>r: � (27)

∥~r∥<sup>2</sup> η � �

∥~r∥<sup>2</sup> η � � ν

ν

<sup>η</sup> ð Þ ð Þ� <sup>r</sup> � <sup>s</sup> <sup>Θ</sup> � �

exp ½�i2πð � ~r � νθÞ d~r

∥~r∥<sup>2</sup> η

> ∥~r∥<sup>2</sup> η

<sup>η</sup> ð Þ ð Þ� <sup>r</sup> � <sup>s</sup> <sup>Θ</sup> � �

<sup>η</sup> ð Þ cos <sup>θ</sup>; sin <sup>θ</sup> , where νθ <sup>∈</sup> <sup>R</sup><sup>2</sup>

exp ½�i2πð � ~r � νθÞ d~r, (28)

: (29)

(30)

<sup>2</sup> : (25)

dx

dx:

(26)

.

G rð Þ¼ cos <sup>ϕ</sup>ð Þ<sup>r</sup> � � <sup>¼</sup> exp <sup>i</sup>ϕð Þ<sup>r</sup> � �

<sup>2</sup> exp �<sup>π</sup>

<sup>2</sup> exp �<sup>π</sup>

normalized fringe pattern in complex form:

In this particular case, the 2D CWT of G rð Þ is

R2

þ Ð R2

Using Taylor's expansion we know that

Then, we can now rewrite Eq. (26) as

2

<sup>þ</sup> exp �iϕð Þ<sup>s</sup> � � 2

this last equation can be finally written as

<sup>W</sup>f g G rð Þ <sup>≈</sup> exp <sup>i</sup>ϕð Þ<sup>s</sup> � �

ð R2

> ð R2

<sup>þ</sup> exp �iϕð Þ<sup>s</sup> � �

Wf g G rð Þ ≈ η

þ η

<sup>W</sup>f g G rð Þ <sup>≈</sup> exp <sup>i</sup>ϕð Þ<sup>s</sup> � �

or, which is the same

exp <sup>i</sup>ϕð Þ<sup>r</sup> � �

exp �iϕð Þ<sup>r</sup> � �

know the meaning of Eq. (26). To do so, let <sup>~</sup><sup>r</sup> <sup>¼</sup> <sup>r</sup> � <sup>s</sup> and νθ <sup>¼</sup> <sup>ν</sup>

<sup>W</sup>f g G rð Þ <sup>¼</sup> <sup>Ð</sup>

184 Wavelet Theory and Its Applications

To detect the analytic function and consequently compute the phase ϕð Þs at a given pixel s (i.e., the ridge detection), we can choice one of two possibilities: at νθ <sup>¼</sup> <sup>∇</sup>ϕð Þ<sup>s</sup> <sup>2</sup><sup>π</sup> or νθ ¼ � <sup>∇</sup>ϕð Þ<sup>s</sup> <sup>2</sup><sup>π</sup> . Owing that the sign of the phase gradient cannot be determined from the image intensity, there exists a sign ambiguity of the phase in the θ � η map. In Figure 6, it can be observed that in this situation, there are two maximum in each θ � η map. Also, it can be deduced that the magnitude of the coefficients map is periodic with respect to θ with period π. To solve the problem of sign ambiguity, Ma et al. [48] proposed a phase determination rule according to the phase distribution continuity. Also, Villa et al. [55] proposed a sliding 2D CWT method that assumes that the phase is continuous and smoothly varying, in this way, the ridge detection is realized assuming that the coefficient maps are similar in adjacent pixels, reducing the processing time too.

Once detected the ridge Wf g G rð Þ ridge that represents a 2D function, the wrapped phase can be computed with

$$\mathcal{W}\{\phi(r)\} = \tan^{-1}\left(\frac{\text{Real}\left\{\mathcal{W}\{G(r)\}\_{r \text{id} \notin \text{e}}\right\}}{\text{Imag}\left\{\mathcal{W}\{G(r)\}\_{r \text{id} \notin \text{e}}\right\}}\right). \tag{31}$$

Figures 7 and 8 show examples of fringe pattern phase recovery using the 2D CWT method reported in [55]. It is important to remark that this method is highly robust against noise.

Figure 6. (a) Example of noisy simulated fringe pattern. The square indicates a region around a pixel s where the phase is estimated. (b) Magnitude of the θ � η map at the pixel s, codified in gray levels. Horizontal direction represents the rotation angle while the vertical direction represents the scale. The two white regions represent the two terms in Eq. (30).

4.2. The 2D CWT for wrapped phase maps denoising

must be realized computing the following complex function:

denoised phase map. Again, substituting (32) in (23), we now obtain

exp ½ � iψð Þr exp �π

Following the same reasoning to obtain Eq. (30), for this case, we obtain:

simpler and the filtered wrapped phase map ψfð Þr can be computed with

where <sup>i</sup> <sup>¼</sup> ffiffiffiffiffiffi

Wf g G rð Þ ¼

is only one maximum: at νθ <sup>¼</sup> <sup>∇</sup>ψð Þ<sup>s</sup>

direction represents the scale.

ð R2

Other of the most relevant tasks in fringe pattern processing is the wrapped phase maps denoising. Owing that the phase unwrapping is a key step in fringe pattern processing for optical measurement techniques, the previous denoising of the wrapped phase is crucial for a proper measurement. Several optical measurement techniques, such as the electronic speckle pattern interferometry, use different phase recovery methods, inherently produces highly noisy wrapped phase maps. In these situations, the phase map denoising is a crucial pre-process for a successful phase unwrapping. Considering the problem of denoising wrapped phase maps, the drawback is that owing to 2π phase jumps of the wrapped phase ψ, direct application of any kind of filter is not always a proper procedure to solve it. For example, the application of a simple mean filter may smear out the phase jumps. In order to avoid this drawback, the wrapped phase filtering

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

�<sup>1</sup> <sup>p</sup> . As both imaginary and real parts are continuous functions, we can properly

apply a filter over G rð Þ, and the argument of the filtered complex signal will contain the

<sup>∥</sup><sup>r</sup> � <sup>s</sup>∥<sup>2</sup> η � �

<sup>W</sup>f g G rð Þ <sup>≈</sup> <sup>η</sup>exp ½ � <sup>i</sup>ψð Þ<sup>s</sup> exp �ηπ νθ � <sup>∇</sup>ψð Þ<sup>s</sup>

The difference of this equation with the result shown in Eq. (30) is that at each θ � η map, there

Figure 9. (a) Zoom of a small square region in a noisy wrapped phase map (around some pixel s). (b) Magnitude of the θ � η map at the pixel s, codified in gray levels. Horizontal direction represents the rotation angle while the vertical

G rð Þ¼ exp ½ � iψð Þr , (32)

ν

� � � �

2π

<sup>2</sup><sup>π</sup> (see Figure 9). Thus, in this case, the ridge detection is

<sup>η</sup> ð Þ ð Þ� <sup>r</sup> � <sup>s</sup> <sup>Θ</sup> � �dx: (33)

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

187

: (34)

� exp �i2π

<sup>2</sup> " #

� � � �

Figure 7. Example of the 2D CWT method applied to phase recovery. (a) Synthetic noisy fringe pattern. (b) Recovered phase.

Figure 8. Example of the 2D CWT method applied to phase recovery. (a) Experimentally obtained moiré fringe pattern. (b) Recovered phase.

A big advantage of using the 2D CWT method to compute the phase from fringe patterns without carrier is that the sign ambiguity of ∇ϕ can be easily solved, for example, with the method reported in [55]. The key idea of the method is the assumption that the phase ϕ is smooth; in other words, the fringe frequency and fringe orientation are very similar in neighbor pixels, hence the ridge detection at each θ η map is simplified registering the previous computation of neighbor pixels.

#### 4.2. The 2D CWT for wrapped phase maps denoising

Other of the most relevant tasks in fringe pattern processing is the wrapped phase maps denoising. Owing that the phase unwrapping is a key step in fringe pattern processing for optical measurement techniques, the previous denoising of the wrapped phase is crucial for a proper measurement. Several optical measurement techniques, such as the electronic speckle pattern interferometry, use different phase recovery methods, inherently produces highly noisy wrapped phase maps. In these situations, the phase map denoising is a crucial pre-process for a successful phase unwrapping. Considering the problem of denoising wrapped phase maps, the drawback is that owing to 2π phase jumps of the wrapped phase ψ, direct application of any kind of filter is not always a proper procedure to solve it. For example, the application of a simple mean filter may smear out the phase jumps. In order to avoid this drawback, the wrapped phase filtering must be realized computing the following complex function:

$$G(r) = \exp\left[i\psi(r)\right].\tag{32}$$

where <sup>i</sup> <sup>¼</sup> ffiffiffiffiffiffi �<sup>1</sup> <sup>p</sup> . As both imaginary and real parts are continuous functions, we can properly apply a filter over G rð Þ, and the argument of the filtered complex signal will contain the denoised phase map. Again, substituting (32) in (23), we now obtain

$$\mathcal{W}\{\mathbf{G}(r)\} = \int\_{\mathbb{R}^2} \exp\left[i\psi(r)\right] \exp\left[-\pi\frac{\left\|r-s\right\|^2}{\eta}\right] \times \exp\left[-i2\pi\frac{\nu}{\eta}((r-s)\cdot\Theta)\right] d\mathbf{x}.\tag{33}$$

Following the same reasoning to obtain Eq. (30), for this case, we obtain:

A big advantage of using the 2D CWT method to compute the phase from fringe patterns without carrier is that the sign ambiguity of ∇ϕ can be easily solved, for example, with the method reported in [55]. The key idea of the method is the assumption that the phase ϕ is smooth; in other words, the fringe frequency and fringe orientation are very similar in neighbor pixels, hence the ridge detection at each θ η map is simplified registering the previous

Figure 8. Example of the 2D CWT method applied to phase recovery. (a) Experimentally obtained moiré fringe pattern.

Figure 7. Example of the 2D CWT method applied to phase recovery. (a) Synthetic noisy fringe pattern. (b) Recovered

computation of neighbor pixels.

(b) Recovered phase.

phase.

186 Wavelet Theory and Its Applications

$$\mathcal{W}\{G(r)\} \approx \eta \exp\left[i\psi(s)\right] \exp\left[-\eta\pi\left\|\begin{array}{c} \nu\_{\theta} - \frac{\nabla\psi(s)}{2\pi} \right\|^{2}\right].\tag{34}$$

The difference of this equation with the result shown in Eq. (30) is that at each θ � η map, there is only one maximum: at νθ <sup>¼</sup> <sup>∇</sup>ψð Þ<sup>s</sup> <sup>2</sup><sup>π</sup> (see Figure 9). Thus, in this case, the ridge detection is simpler and the filtered wrapped phase map ψfð Þr can be computed with

Figure 9. (a) Zoom of a small square region in a noisy wrapped phase map (around some pixel s). (b) Magnitude of the θ � η map at the pixel s, codified in gray levels. Horizontal direction represents the rotation angle while the vertical direction represents the scale.

$$\psi\_f(r) = \tan^{-1} \left( \frac{\text{Real} \{ \mathcal{W} \{ G(r) \} \_{r \text{idge}} \}}{\text{Imag} \{ \mathcal{W} \{ G(r) \} \_{r \text{idge}} \}} \right) \in [ -\pi, \pi). \tag{35}$$

removed. A comparison of the performance of this method compared with the windowed Fourier transform method [22] and the localized Fourier transform method [21] is shown in Table 1. In this case, the normalized-mean-square-error (NMSE) was used as the metric applied over a synthetic noisy phase map ψ (Figure 10). Although the performance against noise of the WFT is better that the 2D CWT method, this last is much simpler to implement, as discussed in [53].

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

2D-CWT WFT LFT 0.0692 0.0521 0.0747

Table 1. Performance comparison of the 2D CWT, WFT, and LFT methods, using the NMSE.

NMSE <sup>¼</sup> <sup>∥</sup><sup>ψ</sup> � <sup>ψ</sup><sup>f</sup> <sup>∥</sup><sup>2</sup>

It can be obviously deduced that often fringe patterns contain elements with high anisotropy, sparse frequency components, and noise, which makes the processing of this kind of images by means of classical LTI methods inadequate. Several authors have shown that the use of multiresolution analysis by means of the 2D CWT for processing fringe patterns has resulted a proper and interesting alternative for this task. The 2D CWT methods present some attractive advantages compared with other commonly used techniques. (1) The use of the Gabor mother wavelet for processing this kind of images is a natural choice to model them, as can be obviously deduced analyzing the physical theory of fringe image formation. (2) In most classical methods for processing fringe images, the previous estimation of the fringe direction or orientation is a must, especially for fringe patterns without a fringe carrier frequency. Owing that the multiresolution analysis using the 2D CWT methods models the image by means of the angle θ, fringe direction or orientation is inherently computed through the ridge detection. (3) As the 2D CWTmethods models the interferograms by means of scale and orientation, all spurious information and noise contributing in the θ � η map is efficiently removed through the ridge detection, resulting a powerful tool to remove the noise.

\*, Ismael de la Rosa<sup>1</sup>

, Gustavo Rodríguez<sup>1</sup>

, Daniel Alaniz<sup>1</sup> and Efrén González<sup>1</sup>

1 Unidad Académica de Ingeniería Eléctrica, Universidad Autónoma de Zacatecas, Zacatecas,

2 Departamento de Electrónica, Universidad de Guadalajara, Guadalajara, Jalisco, México 3 Unidad Académica de Física, Universidad Autónoma de Zacatecas, Zacatecas, México

, Jorge Luis Flores<sup>2</sup>

,

5. Conclusions

Author details

Rumen Ivanov<sup>3</sup>

México

José de Jesús Villa Hernández<sup>1</sup>

, Guillermo García<sup>2</sup>

\*Address all correspondence to: jvillah@uaz.edu.mx

<sup>∥</sup>ψ∥<sup>2</sup> : (36)

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

189

Figures 10 and 11 are examples of the results applying the 2D CWT in wrapped phase map denoising. Note the outstanding performance removing the structures due to the gratings in the experimentally obtained wrapped phase map with moire deflectometry (Figure 11).

The key step in the 2D CWT method for phase map denoising is the ridge detection. In this way, all the coefficients in the θ � η map contributed by the noise and spurious information are

Figure 10. (a) Simulated noisy wrapped phase map. (b) Filtered wrapped phase map.

Figure 11. (a) Experimentally obtained moiré noisy wrapped phase map. (b) Filtered wrapped phase map.


Table 1. Performance comparison of the 2D CWT, WFT, and LFT methods, using the NMSE.

removed. A comparison of the performance of this method compared with the windowed Fourier transform method [22] and the localized Fourier transform method [21] is shown in Table 1. In this case, the normalized-mean-square-error (NMSE) was used as the metric applied over a synthetic noisy phase map ψ (Figure 10). Although the performance against noise of the WFT is better that the 2D CWT method, this last is much simpler to implement, as discussed in [53].

$$\text{NMSE} = \frac{\left\|\psi - \psi\_f\right\|^2}{\left\|\psi\right\|^2}. \tag{36}$$

#### 5. Conclusions

<sup>ψ</sup>fð Þ¼ <sup>r</sup> tan �<sup>1</sup>

188 Wavelet Theory and Its Applications

Figure 10. (a) Simulated noisy wrapped phase map. (b) Filtered wrapped phase map.

Figure 11. (a) Experimentally obtained moiré noisy wrapped phase map. (b) Filtered wrapped phase map.

0 @ Real Wf g G rð Þ ridge n o

1

A ∈½ Þ �π; π : (35)

Imag Wf g G rð Þ ridge n o

Figures 10 and 11 are examples of the results applying the 2D CWT in wrapped phase map denoising. Note the outstanding performance removing the structures due to the gratings in the experimentally obtained wrapped phase map with moire deflectometry (Figure 11).

The key step in the 2D CWT method for phase map denoising is the ridge detection. In this way, all the coefficients in the θ � η map contributed by the noise and spurious information are

> It can be obviously deduced that often fringe patterns contain elements with high anisotropy, sparse frequency components, and noise, which makes the processing of this kind of images by means of classical LTI methods inadequate. Several authors have shown that the use of multiresolution analysis by means of the 2D CWT for processing fringe patterns has resulted a proper and interesting alternative for this task. The 2D CWT methods present some attractive advantages compared with other commonly used techniques. (1) The use of the Gabor mother wavelet for processing this kind of images is a natural choice to model them, as can be obviously deduced analyzing the physical theory of fringe image formation. (2) In most classical methods for processing fringe images, the previous estimation of the fringe direction or orientation is a must, especially for fringe patterns without a fringe carrier frequency. Owing that the multiresolution analysis using the 2D CWT methods models the image by means of the angle θ, fringe direction or orientation is inherently computed through the ridge detection. (3) As the 2D CWTmethods models the interferograms by means of scale and orientation, all spurious information and noise contributing in the θ � η map is efficiently removed through the ridge detection, resulting a powerful tool to remove the noise.

## Author details

José de Jesús Villa Hernández<sup>1</sup> \*, Ismael de la Rosa<sup>1</sup> , Gustavo Rodríguez<sup>1</sup> , Jorge Luis Flores<sup>2</sup> , Rumen Ivanov<sup>3</sup> , Guillermo García<sup>2</sup> , Daniel Alaniz<sup>1</sup> and Efrén González<sup>1</sup>

\*Address all correspondence to: jvillah@uaz.edu.mx

1 Unidad Académica de Ingeniería Eléctrica, Universidad Autónoma de Zacatecas, Zacatecas, México

2 Departamento de Electrónica, Universidad de Guadalajara, Guadalajara, Jalisco, México

3 Unidad Académica de Física, Universidad Autónoma de Zacatecas, Zacatecas, México
