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Fringe centering technique, also known as “fringe skeletonizing”, was developed for the traditional manual fringe analysis, in which the fringe spacings are used to calculate displacements or strains. Before the phase measurement techniques became available, the fringe centering technique was the only processing tool available for the automatic analysis of interferograms. This class of techniques remains a vital element in the repertoire of fringe analysis methods. The fringe centering method is the only viable automatic fringe analysis technique for interferograms if only photographic records of the interferograms are available or the experimental event is dynamic such as impact testing. The flow chart of the fringe centering method is shown below.

1. Fringe centerline detection
 
In this method, only the integer fringe orders along the fringe centerlines are sought. The process of fringe centerline detection is to find points representing the fringe centerlines by eliminating all other parts of the fringes. There are two different methods for extracting the fringe centerlines from the fringe patterns. The first one involves binarizing the fringe patterns and then skeletonizing the binary fringes; the other method detects the local maxima or minima of fringe intensities in a gray-scale. The peak detection methods are more sensitive to noise than their binary counterparts, but they offer the prospect of higher resolution detection of the fringe centers.
 
1.1  Binarization
In the binarization process, the gray levels above or below a threshold value are truncated to the maximum or zero intensity, respectively, to convert the image into a binary intensity image. The mean intensity of the fringe pattern can be chosen as the binarization threshold. When the image has an uneven background, the image can be segmented into small blocks and binarizing operation is applied to each block. After binarization, the geometric centerline of the black fringe is regarded as the fringe centerline. This assumption can give rise to a large error if the fringe intensity is asymmetric. Because of this disadvantage, the fringe binary method is not widely used.

1.2  Peak detection
Peak detection means finding the local maxima or minima of the gray-scale images. In this method, the whole image is subjected to a peak detection matrix and the gray level of the pixels in the matrix is reduced to zero value if a peak does not exist and to logical ‘1’ if a peak is found. Among many fringe centerline detection methods, a 5´5 window pixel peak detection scheme is one of the easiest and most effective methods.
 

The fringe peak detection uses two-dimensional peak detection, locally performed within a 5´5 pixel matrix, as shown in the figure. With respect to the four directions shown in the figure, the peak conditions are defined as follows:
For the X-direction,
\[ P_{0,0}+P_{0,-1}+P_{0,1}>P_{-2,0}+P_{-2,-1}+P_{-2,1} \]
\[ P_{0,0}+P_{0,-1}+P_{0,1}>P_{2,0}+P_{2,-1}+P_{2,1} \]
For the Y-direction,
\[ P_{0,0}+P_{-1,0}+P_{1,0}>P_{0,-2}+P_{-1,-2}+P_{1,-2} \]
\[ P_{0,0}+P_{-1,0}+P_{1,0}>P_{0,2}+P_{-1,2}+P_{1,2} \]
For the XY-direction,
\[ P_{0,0}+P_{-1,-1}+P_{1,1}>P_{-2,2}+P_{-2,1}+P_{-1,2} \]
\[ P_{0,0}+P_{-1,-1}+P_{1,1}>P_{-2,2}+P_{2,-1}+P_{1,-2} \]
For the -XY-direction,
\[ P_{0,0}+P_{-1,1}+P_{1,-1}>P_{-2,-2}+P_{-2,-1}+P_{-1,-2} \]
\[ P_{0,0}+P_{-1,1}+P_{1,-1}>P_{2,2}+P_{2,1}+P_{1,2} \]
 
When the peak conditions are satisfied for any of two or more directions, the object point is recognized as a point on a fringe skeleton.
  
1.3 Fringe thinning
In many cases, the fringe obtained from the image binary or the peak detection schemes is usually wider than the width of one pixel.  Binary image thinning algorithms are required to further reduce the fringe width and thus to obtain a true fringe centerline.  Among many proposed algorithms, Rosenfeld thinning algorithm and Hilditch thinning algorithm are employed in this study.

1.4 Fringe centerline improvement
Fringe patterns uniquely define the displacement fields.  Because of the noise, the fringe centerlines obtained from above steps inevitably have undesired defects, such as broken fringe centerlines, cross-connected fringe centerlines and fringe centerlines with short branches.  These defects must be eliminated before next processing.  A series of corresponding algorithms to cope with these fringe centerline defects are utilized in this dissertation; these algorithms include automatic broken-line connection, automatic cross-connection line seperation, short fringe branch elimination, manual fringe connection and elimination, and so on.
Figure below shows an example of fringe centerline detection using binarization and peak detection methods. (a) is the original experiment horizontal field fringe pattern of a diametrical compression circular disc with a hole in the center; (b) is the fringe pattern after low-pass filtering of (a); (c) is the image after binarizing; (d) is the fringe centerline after thinning and (e) is the image after improving fringe centerlines; (f) is the overlap of original fringe pattern (a) and the fringe centerline pattern (e) where the color of the centerlines were changed to be white.  It is evident that the fringe centerlines are well recognized.  Similarly, (g) through (j) are the results of using fringe peak detection method.
   
  

2. Semi-automatic fringe orders assignment
Fringe centerlines represent the contours of equal displacements; the fringe centerline itself does not contain information of fringe orders.  Therefore, fringe order assignment is necessary for the whole-field fractional fringe order calculation.
The adjacent fringe orders differ by -1 or 1 except in zones of local maxima and minima.  Based on this feature, a semi-automatic process to assign fringe orders has been developed to achieve fringe ordering.  This semi-automatic process requires a simple human-computer interaction and the increasing or decreasing directions of some fringe orders should be known. The later requirement can be provided through a judgment based on the mechanical behavior of the testing specimen or the moving orientation of the fringes when small phase change is added into the experiment system.
 

Figure above shows an example of the process. Suppose the order of a fringe located at L1 is 0 and the direction from L1 to H1 is known (increasing or decreasing).  Then the orders of the fringes spanned from L1 to H1 can be determined.  Now, fringe orders at the fringes that contain H1’ and H2 are known; according to this information, fringe orders from H2 to L2 can also be determined automatically.  The same procedure can be used for other segments except the circular fringe located in the center whose fringe order determination requires a human judgment.  The fringe order at L1’ is compared with the fringe order at L1 to double check whether the fringe orders are assigned correctly.  Finally, a constant fringe order can be added to all the fringes to reflect the real fringe orders.
 

3. Fractional fringe order calculation: fringe order interpolation
In the fringe centering method, only the integral fringe orders along the fringe centerlines are determined.  Interpolation is required to obtain fractional fringe orders at every pixel.  The following sections describe the interpolation methods used in this study.

3.1 1-D interpolation
The most widely used 1-D interpolation algorithm is cubic spline interpolation, and it has been successfully applied to fringe order interpolations.  Cubic spline interpolation requires the boundary conditions to be known, which is the most critical limitation of this algorithm since the boundaries are often regions of interest.  For this reason, a segment-by-segment curve fitting interpolation algorithm is proposed to avoid the requirements of boundary conditions.  The proposed algorithm is based on a continuous differential of the first derivative (2nd order differential of displacement). The algorithm is illustrated in the figure below

3.2 Limitation of 1-D interpolation
The main disadvantage of 1-D interpolation is that interpolations along x-direction and y-direction do not yield the exactly same results.  This implies that 1-D interpolation is not sufficient to describe the full-field experimental parameters.  The error can be expected when the derivatives are calculated along one direction using data obtained from interpolation along another direction (e.g., calculation of strain x in the x-direction using the U-field displacement data that were obtained from the interpolation along y-direction).

3.3 New approach: Improved 1-D interpolation
Instead of using the fixed Cartesian coordinates, the directions normal to the fringe orientations can be used to improve the efficiency of 1-D interpolation.  For example, the interpolation along the virtual-lines in the figure below should yield better results for the points along those lines simply because fringe orders change more rapidly along them. Once the fractional fringe orders are obtained at the points of the virtual lines, the virtual-lines can be regarded as regular fringe centerlines. These lines are called as “assistant fringes”; although they are not real fringe centerlines, they are useful for the fringe order interpolation. 
 

            

3.4 Proposed 2-D interpolation approach: tile-by-tile interpolation
Since fringe pattern is a 2-D image, a 2-D fringe order interpolation seems more attractive.  For simple and uniform fringe patterns, a global 2-D polynomial interpolation can be used.  However, for the real engineering problems, the 2-D interpolation should be based on tile-by-tile processing.  To use a proper number of fringe centerlines, the size of the tile should be adjusted according to the fringe density.  After the 2-D polynomial interpolation within every tile is conducted, the displacement data at the edge of adjacent tile are compared and smoothly connected.  Figure below shows the result of the fringes in the figure shown above obtained by using the 2-D interpolation. 

Currently, 2-D interpolation is still in somewhat immature state, more development efforts are need to complete the 2-D interpolation.
 

 

4 Fringe order gradient or strain calculation
Fringe order gradient or strain can be calculated using the interpolation function, namely differentiating the polynomial function used in interpolation. Figure below shows an U-field example of a tensile specimen with a hole.  The theoretical stress (also strain) concentration factor at the top point of the hole is 3.0, and the fringe centering method gives 3.08.