A New Algorithm for Automatic Double Bright Fringe of Multiple-Beam Fizeau Fringe Skeletonization Using Fourier Transform Method of Fringe Pattern Analysis

The interferogram of multiple-beam Fizeau fringe technique plays an important role to investigate the optical properties of fiber because this interferogram provides us with useful information which can used to determine the dispersion curve of the fiber sample. A common problem in any interferogram analysis is the accuracy in locating fringe centers (fringe skeleton). There are a lot of computer-aided algorithms, which depend on the interferogram types, used to fringe skeleton extraction of various digital interferogram. In this paper, as far as I know, a novel algorithm for fringe skeleton extraction of double bright fringe of multiple-beam Fizeau fringe is presented. The proposed algorithm based on using the different order of Fourier transform and the derivative-sign binary image. Also the proposed algorithm has been successfully tested by using a computer simulation fringe and an experimental pattern. The results are compared with the original interferogram and shown a good agreement.


Introduction
Nowadays, the intensive progress in development of laser sources and optical elements has challenged scientists to use the interferometry as an important diagnostic tool for accurate and gross field measurements of a various phenomena.These challenges have resulted in an accurate measurement in almost all engineering domains.For example in chemical engineering, materials science and physical fields, refractive index measurement is frequently needed.
There are many non-invasive interferometric techniques for measuring the refractive index profile and other related optical properties of fibers [1][2][3][4].All of these techniques have their own advantages and disadvantages.The main advantages of these interferometric techniques, with respect to well established techniques, are that the sensitivity, accuracy, and non-contact characteristics.Furthermore, the most important of interferometric techniques over other techniques is that they are full-field, i.e. information about all points in the field can be recorded and observed simultaneously.
All of interferometric techniques based on the transformation of the phase differences of the wave fields, which is in turn related to a variety of physical properties, into observable intensity variation.In other words, the direct result of an interferometric technique is generally fringe pattern or interferogram.This interferogram provides us with rich quantitative information regarding sample material behavior.
In the past, the interferogram must be photographed and enlarged to a suitable magnification then the required data are obtained from the magnified image [5][6][7][8].The manual quantitative analysis process is time consuming and very tedious, furthermore cannot make a full use of the interferogram's data.For this reason, automatic image processing of these interferograms is an active topic in recent literature [9][10][11].
Recently, the trend has been toward the development of image processing algorithms capable of digitizing the image of the interferogram by using CCD camera.Each point is referred to as a pixel or picture element.The camera digitizes each pixel to a value between 0 (black) and 255 (bright) depends on the brightness of the pixel being digitized.The digital interferogram contains several millions of pixels.How to correctly interpret and make a full use of the digital interferogram data by eye is a big difficult task.Over the last thirty years or so, due to the greet ability of the image processing techniques, there are a lot of computer-aided processing algorithms, which depend on the image types, that can be used to process various digital interferometric fringe patterns to enable semiautomatic or fully automatic analysis of the fringe pattern [12][13][14][15][16][17][18].
Because of the location of the geometric fringe centers or fringe skeleton are very convenient for fringe analysis.One of the basic steps in fringe analysis is to extract fringe center lines or the fringe skeleton.A number of methods for extracting the skeleton have been published and developed and many aspects of these techniques have been deeply investigated [19][20][21][22][23][24].
In short, although a variety of methods for extracting the fringe skeleton exist, these methods can be classified into two main groups.In the first group, the fringe pattern is first threshold to provide a binary gray level image.Then horizontal or vertical scan lines are used to determine the midpoints of each dark and bright region [25].
In the second group, the intensity variation within a fringe is used in devising algorithms for fringe skeleton determination by using two methods.The first method is a thinning algorithm which erodes the fringes from all sides [26][27][28][29].The disadvantage of this algorithm is that it usually requires numerous iteration in order to peel the fringe pattern.The second method consists of differential filter techniques [24,30], this algorithm is very simple because it requires one step to find the center line of the fringe.
It is evident from the above paragraphs that the algorithm for determining the fringe center lines (skeletons) is a valuable task in the automatic fringe analysis technique and is still open.
Multiple-beam Fizeau interferometer is a well known technique frequently used to investigate the optical and structural properties of fiber [31].The power of multiplebeam Fizeau interferometer changed dramatically with the introduction of automated method of detection and analysis of its fringe patterns [32,33].But, in general, it is well known that most algorithms for fringe analysis are tailored to a specific type of interferogram and thus may require substantial modifications when applied to other types of interferogram.
In General, the multiple-beam Fizeau fringes characteristic by sharp bright fringe on a dark background 31 , as shown in Figure 1(a).But in some cases the light source not gives only one wavelength but it gives a light having two wavelengths simultaneously.For example the sodium lamp gives two wavelengths 5890 Å and 5896 Å.In this paper all type of multiple-beam Fizeau fringe are fully automated for determination of the contour line using different order of Fourier transform technique.

Description of the Algorithm
The principle problem in accurately measuring the refractive index using multiple-beam Fizeau fringe system, is how to determine the contour line of the fringe pattern for; 1) interference fringe shift [3] or 2) the area enclosed under the fringe shift [34,35], and interfringe spacing.The proposed algorithm based on determine the spectrum of the Fourier transform and the derivative-sign binary image of the multiple-beam Fizeau interferogram.This algorithm is summarized as follows.The intensity data contained in such a fringe pattern can be written as where , where r(x, y) is a fringe constant factor that depends on non-uniform illumination and reflection from the object surface, A n is a Fourier coefficient, n is the order of the Fourier series, f o is the carrier frequency, and (x, y) is the phase of the object.The algorithm consists of four simple steps: Step 1 Applying Fourier transform (FT) algorithm, we compute the one dimension (1-D) Fourier transform of Equation (1) for the variable x only, with y being fixed.The Fourier transform of Equation ( 1) is given by where G(f, y) and Q n (f, y) are the 1-D Fourier spectra of g(x, y) and q n (x, y), respectively.It is clear from Equation (2) that the Fourier spectrum consists of many peaks.These peaks is generally separated by carrier frequency f o .Figure 2 shown the Fourier spectrum of the interferograms shown in Figure 1, where 0, 1, 2, 3 orders Fourier component are shown.
Step 2 we select the suitable order component (this order depend on the interferogram as we will explain after) and the unwanted orders are filtered out.

  
, and using the inverse Fourier transform, if we compute the inverse Fourier transform to Equation ( 3) with respect to f, the resulting function is a complex function.
We select only the real part.
Step 3 we determine the derivative signs binary image [33,36] to the real part.In the binary image, if the deriva-  tive signs of the real part are positive, the points are setsto be a bright value.On the other hand, if the derivative signs of the real part are negative, the points are sets to be a dark value.
Step 4 After constructing the derivative sign binary image, it is easy to extract the fringe center lines using the following two steps: First, we select the positive peaks of the derivative sign and cancel the negative peaks by applying the threshold process over all the derivative sign binary image.Second examine each point neighborhood and quantify the slope then the result is the fringe center lines over the image field.

Results
To verify this algorithm, first the algorithm was tested by using a computer simulated of multiple-beam Fizeau fringe in case of the refractive index of the immersion liquid is mismatching with the refractive index of the clad of the optical fiber, which shown in Figure 1(a).shows the extracting fringe contour lines in a dark background.Figure 4 is obtained by applying the suggested algorithm and using the first order Fourier transform, in other words, in the step 2 of the algorithm, the first peak is selected and the other peaks are filtered out.rithm with the first order transform.From Figure 5, it is clear that the algorithm with the first order transform cannot obtain the contour line of the double bright fringe on a dark background.Figure 6 is the same as in Figure 5 but Figure 6(a) obtained by using the third order Fourier transform, in other words, in the step 2 of the algorithm, the third peak is selected and the other peaks are filtered out.It is clear that the algorithm successes to determine the contour lines not only of the bright fringe but also the dark fringe.In fact the wanted results are get out from the bright fringes only in case of the multiple-beam Fizeau technique in transmission, So if we want the contour lines of the bright fringe only we should be select the first, second, and third peaks together as shown in Figure 6(b).
The previous test was done with free noise interferogram.In most case of practical application, various noises can be introduced into interferogram by various intrinsic or extrinsic adverse experimental conditions.So it is instructive to test the algorithm with noise interferogram.Without of using any filtering process, Figure 7 and Figure 8 show the result of testing in case of four different random noise ratio added to the original interferogram (Figure 1) by using MatLab7 program.
The results clearly illustrate the capability of the algorithm to construct the contour lines for the bright and dark fringes simultaneously and the effect of random noise is very small.

Conclusions
In  original interferogram.The importance of the demonstrated algorithm lies in the following benefits: 1) Simple to automate, the interferogram can be analyzed without interruption or user intervention.
2) Fast, within few seconds, the full-field fringe pattern of the interferogram which contain at least million pixels can be analyze.
3) The algorithm performs better under certain circumstance such as noise and low contrast.

Figure 1 (
b), characteristics by double bright fringe on a dark background.

Figure 1 .
Figure 1.Simulated Microinterferogram of multiple-beamFizeau technique in transmission in case of optical fiber as a object.

Figure 2 .
Figure 2. Two dimensional plot of the Fourier transform intensity of Figure 1, respectively.

Figure 3 ,Figure 3 .
Figure 3. (a) Superimposed the one line intensity distribution (column 100) of (a) with the derivative data of its derivative-sign binary, (b) is magnified part of (a) for more clearly illustrate.

Figure 4 (
a) shows the contour lines plotted inside the original pattern of the interferogram, while Figure 4(b)

Figure 5 Figure 4 .Figure 5 .
Figure 4. Extracted fringe skeleton of Figure 1(a), the first order Fourier transform, (a) Shown on the background of the original interferogram; (b) Shown on a dark background.

Figure 9 (Figure 6 .
Figure 6.The same as in Figure 5 but (a) The third peak of the Fourier spectrum is selected; (b) The first, second and third peaks are selected together.

Figure 10 .
Figure 10.Experimental multiple-beam Fizeau interferogram in case of using optical fiber as a object; (a) Extracted fringe skeleton, shown on the background of the original interferogram; (b) Extracted fringe skeleton, shown on the dark background.

Figure 11 .
Figure 11.Extracted fringe skeleton, shown on the background of the original interferogram, of experimental of double bright fringe of multiple-beam Fizeau interferogram; (a) In case of using irregular textile fiber as a object; (b) In case of using single mode optical fiber as a object.

A
New Algorithm for Automatic Double Bright Fringe of Multiple-Beam Fizeau Fringe Skeletonization Using Fourier Transform Method of Fringe Pattern Analysis 413