### Principle of polar coordinate transform

The task of polar coordinate transform is that an image under the Cartesian coordinate (x - y) space is transformed to another image under polar coordinate (*ϕ*- r) space. The expression of polar coordinate transform is expressed as [19]

$$ \left\{{}_{\phi = \arctan \left(y/x\right)}^{r=\sqrt{{\displaystyle {x}^2}+{\displaystyle {y}^2}}}\right.. $$

(1)

The schematic diagram of polar coordinate transform is shown in Fig. 1.

In polar coordinate space, the meaning of r describes the distance of a point (x,y) to the origin position in Cartesian coordinate space, and *ϕ* discribes the angle of vector and its range is from 0 to 359°. Due to the origin symmetric of polar coordinate transform, the transform needs to be carried out in the range of 0° to 179°.

According to Eq. (1) and Fig. 1, one point under the Cartesian coordinate space corresponds uniquely to one point under the polar coordinate space. One circle in the Cartesian coordinate space whose center coincides with the origin, will corresponds to one line along *ϕ*-axis in the polar coordinate space, and the radius of the circle corresponds to the distance of this line to the origin in polar coordinate space.

### Determination of center of circular fringes

Newton’s ring interference fringes is composed of alternating light and dark stripes, and light and dark area are clear, as shown in Fig. 2(a). Through using the Otsu method [20] the fringes image is processed with threshold segmentation, and so a binary image B(x, y) can be obtained from the fringes image f(x, y) according to the following expression,

$$ B\left(x,y\right)=\left\{{}_{0,f\left(x,y\right)\ge T}^{1,f\left(x,y\right)<T}\right. $$

(2)

Here, assumed that light fringes are regarded as the target and the radius of light fringes will be calculated, and T is threshold value. The binary image of Fig. 2(a) is shown in Fig. 2(b).

Through using the connected component labeling algorithm [18] the first order ring from the binary image can be extracted, which is shown in Fig. 2(c). Then, the circular region is filled by the morphological operation, which is shown in Fig. 2(d). Furthermore, the edges of the target region is smoothed by using the opening operation, then the gravity ordinates (xc, yc) of circular region (i.e., the region with white gray-scale pixels) can be calculated according the following equation,

$$ \left\{{}_{y_c=\frac{1}{n}{\displaystyle \sum {y}_i}}^{x_c=\frac{1}{n}{\displaystyle \sum {x}_i}}\right., $$

(3)

where, n is the number of the white pixels as shown in Fig. 2(d). Also, the center of circular fringes is marked in Fig. 2(d).

### Calculation of circular fringes radius

Based on the calculated center the circular fringes are transformed to the polar coordinate space with polar coordinates transform method introduced in Section 2. Therefore, the circular fringes can be transformed to straight fringes. The transformed result of the original image as shown in Fig. 2(a) is shown in Fig. 3(a) and expressed as *p*(*r*, *ϕ*).

To calculate the radius of each order ring and eliminate immensely the effect of random noise, the straight fringes as shown in Fig. 3(a) is implemented the horizontal gray projection according to the following equation,

$$ {\displaystyle {R}_0}(r)={\displaystyle \int p\left(r,\phi \right)d\phi .} $$

(4)

The projection curve of Fig. 3(a) is shown in Fig. 3(b). In the case, the r coordinate value corresponding to each peak position in the projection curve denotes the radius value of each order ring, and hence the method can calculate respectively the radius parameter of each order ring from circular fringes.