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By analyzing Newton’s rings, often encountered in interferometry, the parameters of spherical surfaces such as the rings’ center and the curvature radius can be estimated. First, the classical convolutional neural networks, visual geometry group (VGG) network and U-Net, are applied to parameter estimation of Newton’s rings. After these models are trained, the rings’ center and curvature radius can be obtained simultaneously. Compared with previous analysis methods of Newton’s rings, it is shown that the proposed method has higher precision, better immunity to noise, and lower time consumption. For a Newton’s rings pattern of 640 × 480 pixels comprising − 5 d B Gaussian noise or 60% salt-and-pepper noise, the parameters can be estimated by the VGG model in 0.01 s, the error of the rings’ center is less than one pixel, and the error of curvature radius is lower than 0.5%.

Newton’s rings are often encountered in interferometry, and in analyzing them, we can estimate the physical parameters, such as curvature radius and the rings’ center. The fractional Fourier transform (FRFT) is capable of estimating these physical parameters from the rings despite noise and obstacles, but there is still a small deviation between the estimated coordinates of the rings' center and the actual values. The least-squares fitting method is popularly used for its accuracy but it is easily affected by the initial values. Nevertheless, with the estimated results from the FRFT, it is easy to meet the requirements of initial values. In this paper, the proposed method combines the advantages of the fractional Fourier transform (FRFT) with the least-squares fitting method in analyzing Newton’s rings fringe patterns. Its performance is assessed by analyzing simulated and actual Newton’s rings images. The experimental results show that the proposed method is capable of estimating the parameters in the presence of noise and obstacles. Under the same conditions, the estimation results are better than those obtained with the original FRFT-based method, especially for the rings’ center. Some applications are shown to illustrate that the improved FRFT-based method is an important technique for interferometric measurements.

Newton’s rings are the fringe patterns of quadratic phase, the curvature radius of optical components can be obtained from the coefficients of quadratic phase. Usually, the coordinate transformation method has been used to the curvature radius, however, the first step of the algorithm is to find the center of the circular fringes. In recent years, deep learning, especially the deep convolutional neural networks (CNNs), has achieved remarkable successes in object detection task. In this work, an new approach based on the Faster region-based convolutional neural network (Faster R-CNN) is proposed to estimate the rings’ center. Once the rings’ center has been detected, the squared distance from each pixel to the rings’ center is calculated, the two-dimensional pattern is transformed into a one-dimensional signal by coordinate transformation, fast Fourier transform of the spectrum reveals the periodicity of the one-dimensional fringe profile, thus enabling the calculation of the unknown surface curvature radius. The effectiveness of this method is demonstrated by the simulation and actual images.

An improved FRFT-based method for estimating parameters from Newton’s rings is proposed that can estimate parameters in about 1.3 seconds when processing a 1920×1080 pixel image of Newton’s rings.

Our study investigated an object-detection method based on the faster region-based convolutional neural network (faster R-CNN). The method was designed to determine the center of either a concentric circle or concentric ellipse. Specifically, the central spot of the image (as the object region) can be marked by the bounding box when the circular or elliptical image is used as input data for the faster R-CNN model. The center point of the bounding box can then be calculated according to the coordinates of the upper left and lower right corners, that is, the center position of the concentric circle or concentric ellipse. It is important to determine the center coordinates when taking optical measurements, as the curvature radius of optical components can thus be obtained. The effectiveness of this method is demonstrated through simulation images. Furthermore, we can obtain the center coordinates of the actual Newton’s rings image using the above method; according to the coordinate transformation method, the curvature radius can be estimated based on the center.

By analyzing Newton's rings, often encountered in interferometry, using fractional Fourier transform (FRFT), we can estimate the physical parameters, such as the curvature radius of the spherical surface and the rings' center. However, parameter estimation from large images using FRFT consumes considerable time. We introduce an improved method that resamples the image before applying FRFT. Because Newton's rings are sparse in the FRFT domain, this method reduces the computational time without decreasing the accuracy. Experimental results show that compared to traditional FRFT-based algorithms, this method can estimate parameters in about 1.3 s when processing 1920×1080 pixel images.