<p>Phase extraction from a single closed interferogram with a quadratic phase plays an important role in optical interferometry. Based on the energy of the interference images of this type is concentrated in a very narrow range around a point in the fractional Fourier transform (FRFT) domain under matched angle, the FRFT technique is a useful parameter estimator for the fringe pattern, but it has not been used to retrieve the desired phase from the fringe pattern. Thus, the parameter estimation approach based on FRFT is extended to the extract phase. The phase extraction can be done without using a phase unwrapping algorithm. Moreover, for ensuring both the estimation precision and speed, a coarse-to-fine searching strategy that includes a direct searching process implemented with a large step size and an iterative searching based on quasi-Newton method is presented in this paper to implement the FRFT method. The feasibility and applicability of the proposed approach are demonstrated using simulation and experimental results. The experimental results show that the proposed method is robust to noise and obstacles.</p>
A method based on the fractional Fourier ridges for accurate phase demodulation of a single interferogram with quadratic phase is presented. The interferograms being analyzed may contain circular, elliptic or astigmatic fringes. In signal processing field, such interferograms can be called 2-D chirp-type signals. Since the fractional Fourier transform (FRFT) of a chirp signal is a function under the matched angle that is determined by chirp rates of the signal, so the method can be used to match the multiple chirp rates in chirp-type signals with multiple chirp components. In this work, the FRFT of all row (column) signals are firstly calculated, and the ridge of the FRFT amplitude of each row (column) signal in FRFT domain is recorded. Repeat the above process for each angle of a searching range. Then a ridge tracking approach is employed to determine the matched angle, which can be used to calculate the coefficient of the square term of row (column) coordinates. Moreover, under the matched angle, the ridge of the FRFT amplitude of each row (column) signal all lie on a straight line. The slope and constant term of the line can be used to calculate the coefficient of the linear term of row (column) coordinates and the coefficient of cross term, respectively. The same procedures are implemented to all column (row) signals to determine the coefficients of the square and liner term of column (row) coordinates. According to the obtained coefficients, the phase of the fringe pattern can be constructed without phase unwrapping operation. Furthermore, the present procedure is also capable of analysis of interferograms with or without circularly symmetry fringe distribution instead of using complex and time consuming algorithms for recovering phase from fringe patterns with closed fringes. Finally, the method is tested in simulated and real data.
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.
The fractional calculus (FC) deals with integrals and derivatives of arbitrary (i.e., non-integer) order, and shares its origins with classical integral and differential calculus. The fractional Fourier transform (FRFT), which has been found having many applications in optics and other areas, is a generalization of the usual Fourier transform. The FC and the FRFT are two of the most interesting and useful fractional areas. In recent years, it appears many papers on the FC and FRFT, however, few of them discuss the connection of the two fractional areas. We study their relationship. The relational expression between them is deduced. The expectation of interdisciplinary cross fertilization is our motivation. For example, we can use the properties of the FC (non-locality, etc.) to solve the problem which is difficult to be solved by the FRFT in optical engineering; we can also through the physical meaning of the FRFT optical implementation to explain the physical meaning of the FC. The FC and FRFT approaches can be transposed each other in the two fractional areas. It makes that the success of the fractional methodology is unquestionable with a lot of applications, namely in nonlinear and complex system dynamics and image processing.
Newton’s rings pattern always blurs the scanned image when scanning a film using a film scanner. Such phenomenon is a kind of equal thickness interference, which is caused by the air layer between the film and the glass of the scanner. A lot of methods were proposed to prevent the interference, such as film holder, anti-Newton’s rings glass and emulsion direct imaging technology, etc. Those methods are expensive and lack of flexibility. In this paper, Newton’s rings pattern is proved to be a 2-D chirp signal. Then, the fractional Fourier transform, which can be understood as the chirp-based decomposition, is introduced to process Newton’s rings pattern. A digital filtering method in the fractional Fourier domain is proposed to reduce the Newton’s rings pattern. The effectiveness of the proposed method is verified by simulation. Compared with the traditional optical method, the proposed method is more flexible and low cost.
KEYWORDS: Quantization, Fringe analysis, Fourier transforms, Error analysis, Signal to noise ratio, Commercial off the shelf technology, Signal processing, Digital filtering, Image quality, Optical engineering
Newton’s rings fringe pattern is often encountered in optical measurement. The digital processing of the fringe pattern is widely used to enable automatic analysis and improve the accuracy and flexibility. Before digital processing, sampling and quantization are necessary, which introduce quantization errors in the fringe pattern. Quantization errors are always analyzed and suppressed in the Fourier transform (FT) domain. But Newton’s rings fringe pattern is demonstrated to be a two-dimensional chirp signal, and the traditional methods based on the FT domain are not efficient when suppressing quantization errors in such signals with large bandwidth as chirp signals. This paper proposes a method for suppressing quantization errors in the fractional Fourier transform (FRFT) domain, for chirp signals occupies little bandwidth in the FRFT domain. This method has better effect on reduction of quantization errors in the fringe pattern than traditional methods. As an example, a standard Newton’s rings fringe pattern is analyzed in the FRFT domain and then 8.5 dB of improvement in signal-to-quantization-noise ratio and about 1.4 bits of increase in accuracy are obtained compared to the case of the FT domain. Consequently, the image quality of Newton’s rings fringe pattern is improved, which is beneficial to optical metrology.