Circular Fringe Pattern Research Articles

Comparative analysis of circular and linear fringe projection profilometry: from calibration to 3D reconstruction

This study compares the accuracy of circular and linear fringe projection profilometry in the aspects of system calibration and 3D reconstruction. We introduce, what we believe to be, a novel calibration method and 3D reconstruction technique using circular and radial fringe patterns. Our approach is compared with the traditional linear phase-shifting method through several 2 × 2 experimental setups. Results indicate that our 3D reconstruction method surpasses the linear phase-shifting approach in performance, although calibration efficiency does not present a superior performance. Further analysis reveals that sensitivity and estimated phase error contribute to the relative underperformance in calibration. This paper offers insights into the potentials and limitations of circular fringe projection profilometry.

A optomechanical design for Fabry-Perot Interferometer

Fabry-Perot Interferometer are particularly useful for high-resolution spectroscopy and other applications including astronomy and intra-cavity laser-line narrowing. Fabry-Perot etalon with diameters 1 mm square to 100 mm circular are commercially available and using a conglomerate of optomechanical components and lenses form a Fabry-Perot Interferometer (FPI) system. The etalon when illuminated with a known wavelength laser beam through fore-optics transmits the interference pattern onto a detector through back-optics and thus the image formed on the detector is a concentric circular interference fringe pattern. A sharp pattern with a maximum number of fringes is achieved by introducing a correct combination of the fore and back optomechanical system. Commercially available interferometer inclusive of the optics and the etalon are expensive and beyond the purchasing capacity of budget-conscious users. An alternative to these ready-made interferometer is to manufacture an indigenous optomechanical system and integrate it with the commercially available etalon that delivers a low cost, less weight, and a fairly simple FPI. This report discusses in detail one such system designed and developed in our institute laboratory. The important parameter finesse is calculated and its accuracy is found to be 85%. The developed system with further modifications, like the insertion of the field-of-view module, may serve as an alternative for the large size-based integrating sphere systems that are deployed in the field for observations.

Sub-Nanometer Acoustic Vibration Sensing Using a Tapered-Tip Optical Fiber Microcantilever.

We demonstrate a highly sensitive acoustic vibration sensor based on a tapered-tip optical fiber acting as a microcantilever. The tapered-tip fiber has a unique output profile that exhibits a circular fringe pattern, whose distribution is highly sensitive to the vibration of the fiber tip. A piezo transducer is used for the acoustic excitation of the fiber microcantilever, which results in a periodic bending of the tip and thereby a significant output power modulation. Using a multimode readout fiber connected to an electric spectrum analyzer, we measured the amplitude of these power modulations over the 10-50 kHz range and observed resonances over certain frequency ranges. Two types of tapered-tip fibers were fabricated with diameter values of 1.5 µm and 1.8 µm and their frequency responses were compared with a non-tapered fiber tip. Thanks to the resonance effect as well as the sensitive fringe pattern of the tapered-tip fibers, the limit of detection and the sensitivity of the fiber sensor were obtained as 0.1 nm and 15.7 V/nm, respectively, which were significantly better than the values obtained with the non-tapered fiber tip (i.e., 1.1 nm and 0.12 V/nm, respectively). The sensor is highly sensitive, easy to fabricate, low-cost, and can detect sub-nanometer displacements, which makes it a promising tool for vibration sensing, particularly in the photoacoustic sensing of greenhouse gases.

Tapered tip optical fibers for measuring ultra-small refractive index changes with record high sensitivity.

Here we demonstrate an inexpensive, simple, and ultra-sensitive refractive index sensor based on a tapered tip optical fiber combined with a straightforward image analysis method. The output profile of this fiber exhibits circular fringe patterns whose intensity distribution dramatically changes even with ultra-small refractive index variations in the surrounding medium. The sensitivity of the fiber sensor is measured using different concentrations of saline solutions with a transmission setup consisting of a single wavelength light source, a cuvette, an objective lens, and a camera. By analyzing the areal changes in the center of the fringe patterns for each saline solution, we obtain an unprecedented sensitivity value of 24,160 dB/RIU (refractive index unit), which is the highest value reported so far among intensity-modulated fiber refractometers. The resolution of the sensor is calculated to be 6.9 ×10-9. Moreover, we measure the sensitivity of the fiber tip in the backreflection mode using salt-water solutions and obtained a sensitivity value of 620 dB/RIU. This sensor is ultra-sensitive, simple, easy to fabricate, and low-cost, which makes it a promising tool for on-site measurements and point-of-care applications.

Dynamic Fizeau interferometer with assistant defocused source.

We report and demonstrate a dynamic Fizeau interferometer (DFI) capable of high-precision measurements under severe vibration. The method involves two interferometric passes: a null-interference pass (NP) and a circular carrier-frequency pass (CP). The NP adopts an on-axis Fizeau design, preserving the common path of the reference and test beams. The CP uses an assistant defocused source to generate circular fringe patterns, assisting the NP in phase extraction and sign correction. The feasibility and performance of the proposed method are demonstrated experimentally. To the best of our knowledge, this is the first proposal of a general asynchronous phase-shifting DFI that can not only demodulate null-fringe patterns and correct phase-sign ambiguity but can also maintain the common-path advantages of the Fizeau interferometer.

An exact approach for radius of curvature measurement in a calibration test sphere

In this paper we present a technique to measure the radius of curvature of a test sphere based on the relation between acquired images of a circular cosine fringe pattern and size of virtual image formed on the calibration test surface. Radius of curvature is calculated with the exact equation proposed, using the parameters of the optical setup. Fringe pattern evaluation was performed by locating extrema indices. The mathematical formulation as well as the experimental setup and results are presented. After applying a linear fit algorithm to the data as a method of compensation, obtained results show an error within the tolerance established by the ISO 10343 specifications.

Calibrating large-FOV stereo digital image correlation system using phase targets and epipolar geometry

We propose an easy-to-implement and accurate calibration method for large-scale stereo-digital image correlation (stereo-DIC). First, the intrinsic parameters of each camera are separately calibrated using a regular-sized phase target. Using phase-shifted circular fringe patterns that feature the advantage of being insensitive to the defocus, the phase target can be placed in the close range of each camera. Then, scale-free extrinsic parameters are computed from the epipolar geometry, which can be easily retrieved from DIC registration of homologous point pairs in the stereo images of a test specimen surface. These intrinsic and scale-free extrinsic parameters are used as the initial guess for further optimizing the calibration results. Further, by measuring objects with known physical size, the scale information of the stereo-DIC system can be determined. The metrological performance of the proposed method is evaluated by large-scale shape reconstruction and deformation measurement tests. The obtained results demonstrate that the proposed method is a viable solution for the large-scale 3D deformation measurement tasks.

Single-shot circular fringe projection for the profiling of objects having surface discontinuities.

Fringe projection profilometry (FPP) is a widely used non-contact optical method for 3D profiling of objects. The commonly used linear fringe pattern in FPP has periodic intensity variations along the lateral direction. As a result, the linear fringe pattern used in FPP cannot uniquely represent the lateral shift induced by the objects having surface discontinuities. Thus, unambiguous surface profiling of objects, especially with surface discontinuities, using a single linear fringe image having a single fringe frequency, is unfeasible. This paper proposes using a radially symmetric circular fringe pattern as the structured light pattern for accurate unambiguous surface profiling of sudden height-discontinuous objects. To the best of our knowledge, this is the only method that can reconstruct discontinuous height profiles with the help of a single fringe image having a single frequency. The performance of the proposed algorithm is evaluated on several synthetic and real objects having smooth variations and discontinuities. Compared to the well-known fringe projection methods, the results depict that for a tolerable range of error, the proposed method can be applied for the reconstruction of objects with 4 times higher dynamic range and even at much lower fringe frequencies.

Two point-source interference: a study of the formation of concentric circular fringes

In this short paper, the hyperbola-based analysis of wave interference recently developed by Thomas is employed to study the pattern of concentric circular fringes captured on a distant screen, when its plane is oriented orthogonal to the line joining two coherent point-sources. Newton’s rings are another instance of concentric circular fringes, generated with an extended source and a plano-convex lens-glass plate combination. The underlying geometry of the latter scenario, as described in the literature, allows for the estimation of two important physical parameters viz. the wavelength of light and the refractive index of a liquid medium. It is demonstrated here that the geometrical arrangement of the former scenario can in principle, be utilized to reach the same ends as well. Additionally, the hallmark distinguishing features of both types of circular fringe patterns are qualitatively and quantitatively elucidated.

Separation of Two-Dimensional Mixed Circular Fringe Patterns Based on Spectral Projection Property in Fractional Fourier Transform Domain

In this paper, a method for automatically separating the mixed circular fringe patterns based on the fractional Fourier transform (FrFT) analysis is proposed. Considering the mixed two-dimensional (2-D) Gaussian-based circular fringe patterns, detected by using an image sensor, we propose a method that can efficiently determine the number and parameters of each separated fringe patterns by using the FrFT due to the observed higher sparsity in the frequency domain than that in the spatial domain. First, we review the theory of FrFT and the properties of the 2-D circular fringe patterns. By searching the spectral intensities of the various fractional orders in the FrFT projected along both the frequency axes, the proposed method can automatically determine the total fringe number, the central position, binary phase, and the maximum fringe width of each 2-D circular fringe pattern. In the experimental results, both the computer-simulated and optically mixed fringe patterns are used to verify the proposed method. In addition, the additive Gaussian noise effects on the proposed method are investigated. The proposed method can still successfully separate the mixed fringe pattern when the signal-to-noise ratio is higher than 7 dB.

Development of prototype of electronic speckle interferometry based spirometer

The paper reports the design, construction, and calibration of the prototype of a spirometer based on electronic speckle interferometry (ESPI). The conventional ESPI setup is modified by incorporating a DNM (Diaphragm-Nozzle-Mouthpiece) module comprising a metallic diaphragm, regulated airflow channel, and a mouthpiece. The exhaled air after a deep breathe is channelled to the DNM module where the diaphragm gets deformed. From the circular fringe pattern obtained by subtracting the speckled images before and after deformation of the metallic diaphragm, the radius of curvature (R) due to deformation is calculated using the principle of Newton's rings. The value of R and peak expiratory flow rate (PEFR) from the standard spirometer reading are correlated. From the 640 observations spread over the range 100 - 500 L/min in the standard spirometer, an empirical relation is set in terms of R from the scatter plot. The ESPI spirometer (ESPIS) is validated by determining the value of R corresponding to a particular PEFR from the empirical relation and also from the standard spirometer. The PEFR calculated from ESPIS matches well with the standard spirometer reading, which suggests that the system designed and constructed can be used for biomedical applications for assessing lungs’ efficiency.

Real-time 3-D shape measurement based on radial spatial carrier phase shifting from circular fringe pattern

In this paper, we propose a method to realize real-time three-dimensional (3-D) measurement with a radial spatial carrier phase shifting algorithm. The circular fringe patterns are continuously projected onto the moving object; the distorted fringe patterns are recorded by CCD. In the 3-D reconstruction process, three patterns consisting of one recorded image and two artificial phase shifting fringe patterns are used to implement phase shifting, where the two artificial patterns are formed by numerically shifting the recorded image along the radial direction. Compared with traditional carrier phase shifting method, the proposed method possesses lower phase error and higher accuracy. The simulations and two experiments are carried out to testify the proposed method. From the experimental results of the static object, we find that the standard deviation of the proposed method is better than that of the traditional one. The dynamic experiment of a moving lampshade shows that the time resolution of our method is the same as the CCD. As a real-time quantitative dynamic measurement method, it can be applied in various fields.

Circular fringe projection technique for out-of-plane deformation measurements

Fringe projection is a widely used technique for measurement of out-of-plane deformation of diffusely reflecting objects. Among the various fringe patterns employed for projection, usage of linear fringes is most common. However, the phase function of linear fringes has 2π periodicity, thereby leading to ambiguity in determining whole-field out-of-plane deformation of dynamic objects. This paper presents a new method of measurement of whole-field out-of-plane deformation of targets by projecting a circular fringe pattern. New technique of fringe analysis is introduced to estimate the underlying phase distribution of the fringe pattern from a single image. In this technique, the fringe pattern is coordinate transformed into polar coordinates, and then Fourier fringe analysis technique is employed to obtain its phase. Subsequent to unwrapping, the phase map is re-transformed to Cartesian coordinates. The phase map obtained after coordinate transformation is converted to out-of-plane deformation map using camera calibration parameters. Simulations are carried out to establish the analysis method. Experimental validation is performed with the development of circular grating based fringe projection system.

Simulation of optical vortex measurement for out-of-plane displacement by using fringe-based optical-flow

Fringe-based optical flow is a novel method for measuring the deformation of an object with a high sensitivity. When it is used for out-of-plane displacement measurement, there would be greatly error in low fringe-frequency area. To solve the problem, a new method is proposed, in which an optical vortex beam is used as the object light and a spherical wave as the reference light in double-optical-path interferometry. The interference system and principle are introduced. Some simulations are completed. The results show that there is an unmeasurable area with the circular closed fringe patterns formed by two plane waves. But with the non-closed circular fringe patterns formed by the OV object light and the spherical wave reference light, the measured phase differences are in good agreement with the standard values. And then the measurement errors would be decreased greatly. Besides, the method by using patterns obtained by the interference of an OV light and a spherical wave have a certain degree of anti-noise performance.

Circular fringe projection profilometry.

In this Letter, a novel three-dimensional (3D) measurement method, called the circular fringe projection profilometry (CFPP), is proposed. Similar to the conventional fringe projection profilometry, CFPP also requires fringe pattern projection and capture, phase demodulation, and phase unwrapping. However, it works with a totally different mechanism. CFPP recovers the height of a point by calculating its distance to the optical center of a projector along the optical axis. This distance is calculated with the aid of the divergence angle of a projected light ray and the distance between the measured point and the optical axis. The distance between the measured point and the optical axis is detected by a camera with telecentric lenses, while the divergence angle can be calculated from the phase of a captured circular fringe pattern. The validity of CFPP is confirmed by a set of experiments.

Smartphone-based platform optical setup measuring π/256 optical phase difference in an interference process.

Utilizing the camera of a smartphone and simple laboratory optical components, we demonstrate an optical technique that measures an optical phase difference (OPD) of π/256 in an interference process. We develop a compact optical setup for viewing circular interference fringe patterns through the camera of the smartphone. By introducing OPD between the interfering beams, variation in fringe pattern is recorded using the smartphone camera. We envision that the proposed optical setup could emerge as an ultrasensitive optical tool for measurement of inclination of a given surface.

Variational method for integrating radial gradient field

We propose a variational method for integrating information obtained from circular fringe pattern. The proposed method is a suitable choice for objects with radial symmetry. First, we analyze the information contained in the fringe pattern captured by the experimental setup and then move to formulate the problem of recovering the wavefront using techniques from calculus of variations. The performance of the method is demonstrated by numerical experiments with both synthetic and real data.

A Panoramic Fringe Projection system

In this work, we present a novel Panoramic Fringe Projection system (PFP) to retrieve the three-dimensional topography of quasi-cylindrical objects along their full length and around the entire circumference. The proposed procedure uses a 45° concave conical mirror to project a circular sinusoidal fringe pattern onto a specimen placed coaxially to the mirror, and at the same time, to image the modulated fringe pattern diffused from the object surface. In order to obtain the required sensitivity, an axicon is used to create a divergent fringe pattern with constant pitch. By processing the phase map, information on the radius over the full 360° surface of the sample is obtained. To verify the feasibility of the PFP measurement, a target sample with known shape-discontinuities has been tested. The proposed method demonstrated to be able to retrieve the full topography of complex shaped samples for a large variety of applications in the industrial and biomedical fields.

Center positioning of circular interference fringe patterns for fine measurement

A kind of center positioning method of the circular interference fringe patterns was reported in this paper. The whole process was programmed by Matlab for processing the gray-scale images without denoising preprocessing and binary format transformation. The gradient vector field (GVF) based on a complex-valued cost function was used to extract the skeletons of circular interference fringe patterns, and the coordinate of the center was calculated by statistical method in which partial skeletons were selected to calculate the center according to the space topological relationship. Finally, subpixel accuracy and perfect anti-noise ability of the proposed method are demonstrated by the simulation and experimental fringe patterns, respectively. Thus the technique can be used in precision measurement and fine interferometry.

单幅圆条纹相位展开光刻对准研究

The phase unwrapping method based on the polar coordinate to get the alignment offset was proposed for the wrapped phase generated after the single closed interference fringe phase extraction in alignment of nanolithography.The phase distribution of the fringe pattern occurs in the relative movement between two circular gratings was transformed from rectangular coordinates system to polar coordinates system firstly;then the relationship between the alignment offset and the phase parameter was analysis in polar coordinates system;the initial phase amplitude and phase delay in different radial radius were got to acquire the alignment offset finally.The feasibility of the method was verified by numerical simulation and experiment and the comparisons of the tradition least squares method and path tracking method are given.The results show that the wrapped phase can be unwrapped for the alignment with high precision in tens of nanometers through the method with good adaptability.