3.1. Spatial filtering

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

Figure 3. Michelson interferometer.

4 Optical Interferometry

Figure 2. Young's experiment, where a double slit produces two wave fronts that interfere on a screen.

The mathematical entity applied in spatial filtering is the convolution operation, also known as windowing [2], written as follows:

$$f(\mathbf{x}, y) \* \mathbf{g}(\mathbf{x}, y) = \sum\_{m = -\frac{M + 1}{2}}^{\frac{M + 1}{2}} \sum\_{n = -\frac{N + 1}{2}}^{\frac{N + 1}{2}} f(\mathbf{x} - m, y - n) \mathbf{g}(m, n), \tag{8}$$

where f, g, (x, y), (m, n) and M +N are the image to be filtered, the convolution mask, the original image coordinates, the coordinates where the convolution is performed and the size of the convolution mask, respectively. The kind of convolution filter is determined by the chosen convolution filter (averaging, Gaussian, quadratic, triangular, trigonometric, etc.). These convolution functions are presented in a mathematical form as follows: [3]

$$\mathbf{g}(m,n) = \frac{1}{\sum\_{m=1}^{M} \sum\_{n=1}^{N} \omega\_{m,n}} \begin{bmatrix} \omega\_{1,1} & \cdots & \omega\_{1,N} \\ \vdots & \ddots & \vdots \\ \omega\_{M,1} & \cdots & \omega\_{M,N} \end{bmatrix},\tag{9}$$

where

$$\omega\_{m,n} = \begin{cases} \begin{array}{c} A e^{(-B[m^2+n^2])}, \\ A - B(m^2+n^2), \\ \end{array} & \text{Gaussian} \\ \begin{array}{c} A + \frac{A}{4}[\cos(Bm) + \cos(Bn)], \\ A, & \text{average} \end{array} \end{cases} \tag{10}$$

Here A, B and ω are the amplitude, the width factor and the weight of the spatial filter function, respectively. The study about the magnitude spectrums of Eq. (10) (low-pass masks) was reported in Ref. [2], where the Gaussian and quadratic masks delivered the best results. Low frequencies were conserved by these filters, while the high frequencies were attenuated. However, the filtering results may vary depending on the interferogram under process, as can be seen in Figure 4, where the results of the filter on a ronchigram (a particular kind of interferogram) can be seen in Figure 4a. The four kinds of masks presented in Eq. (10) were employed to generate the filtered fringe patterns shown in Figure 4b. The parameters used by the filters were A = 1 and B = 0.1, with mask sizes of 3 +3, 5 +5 and 7 +7 pixels.

Figure 4. Spatial low-pass filters: (a) original ronchigram, (b) convolution image filtering with Gaussian, quadratic, trigonometric and average masks, as well as 3 +3, 5 +5 and 7 +7 convolution mask sizes.

#### 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 [2]

$$F(\mu, \upsilon) = \mathcal{F}\{f(\mathbf{x}, y)\} = \sum\_{x=1}^{U} \sum\_{y=1}^{V} f(\mathbf{x}, y) e^{-2\pi i \left(\frac{\mu}{U} + \frac{\upsilon}{V}\right)},\tag{11}$$

and

convolution filter (averaging, Gaussian, quadratic, triangular, trigonometric, etc.). These con-

2 4

ω1, <sup>1</sup> ⋯ ω1,<sup>N</sup> ⋮⋱ ⋮ ωM, <sup>1</sup> ⋯ ωM,<sup>N</sup> 3

Gaussian quadratic trigonometric average

5 and 7

7 convolution mask sizes.

+

7 pixels.

:

5, (9)

(10)

<sup>n</sup>¼<sup>1</sup>ωm,<sup>n</sup>

Aeð−B½m2þn2�Þ, <sup>A</sup>−Bðm<sup>2</sup> <sup>þ</sup> <sup>n</sup><sup>2</sup>Þ,

<sup>4</sup> <sup>½</sup> cos <sup>ð</sup>BmÞ þ cosðBnÞ�, A,

Here A, B and ω are the amplitude, the width factor and the weight of the spatial filter function, respectively. The study about the magnitude spectrums of Eq. (10) (low-pass masks) was reported in Ref. [2], where the Gaussian and quadratic masks delivered the best results. Low frequencies were conserved by these filters, while the high frequencies were attenuated. However, the filtering results may vary depending on the interferogram under process, as can be seen in Figure 4, where the results of the filter on a ronchigram (a particular kind of interferogram) can be seen in Figure 4a. The four kinds of masks presented in Eq. (10) were employed to generate the filtered fringe patterns shown in Figure 4b. The parameters used by

Figure 4. Spatial low-pass filters: (a) original ronchigram, (b) convolution image filtering with Gaussian, quadratic,

+3, 5 +5 and 7 + +3, 5 +

volution functions are presented in a mathematical form as follows: [3]

<sup>g</sup>ðm, <sup>n</sup>Þ ¼ <sup>1</sup> ∑<sup>M</sup> <sup>m</sup>¼<sup>1</sup>∑<sup>N</sup>

> A 2 þ A

8 >>><

>>>:

the filters were A = 1 and B = 0.1, with mask sizes of 3

trigonometric and average masks, as well as 3

ωm,<sup>n</sup> ¼

where

6 Optical Interferometry

$$f(\mathbf{x}, y) = \mathcal{F}^{-1}\{F(\mathbf{u}, \mathbf{v})\} = \sum\_{\mu=1}^{U} \sum\_{v=1}^{V} F(\mathbf{u}, \mathbf{v}) e^{2\pi i \left(\frac{\mathbf{w}}{U} + \frac{\mathbf{v}}{V}\right)} \tag{12}$$

where (u, v), U +V, F and F <sup>−</sup><sup>1</sup> are the frequency coordinates, the image size in pixels, the Fourier transform and the inverse Fourier transform operators, respectively. A significant

Figure 5. Frequency filtering: (a) Fourier transform of the original ronchigram, (b) low-pass filtering with different radius of binary circle mask, (c) band-pass filtering with two size of binary ring mask, and (d) band-stop filtering with two size of binary ring mask.

attribute obtained from the Fourier transform is that it gives the frequencies content of the image. Due to this property, frequency filter design is a very straight forward task for image processing. Low frequencies are located into the Fourier domain around the central coordinates; as frequencies gradually increase, they spread out from the center in a radial form. This characteristic is ideal for frequency filtering (low-pass, high-pass, band-pass and band-stop) [3]. The frequency filtering development consists in the multiplication between the Fourier transform with some kind convolution function. The kind of convolution mask will determine the class of filtering to be performed. The following is a summary of the most usual filtering masks: a white centered circle for low-pass; a black centered circle for high-pass; a white centered ring for band-pass and a black centered ring for band-stop.

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.
