Phase Measuring Deflectometry Research Articles

Batch Specular Plane Flatness Measurements Based on Phase Measuring Deflectometry.

Flatness is a critical parameter in the manufacturing industry, directly impacting the fit and overall product performance. As the efficiency of manufacturing continues to advance, there is an increasing demand for more accurate and efficient measurement techniques. Existing methods often struggle to strike a balance between precision and efficiency. In response, this article introduces a novel approach that is capable of achieving high-precision and rapid measurements concerning multiple surfaces. By enhancing the traditional phase measuring deflectometry (PMD) method, employing a matching technique based on polar lines and normal vector constraints to address discrete surface measurement challenges, and implementing a plane pre-positioning method to tackle low efficiency in binocular matching and solving, we successfully performed swift and synchronized measurements for a large batch of specular surfaces and obtained the three-dimensional surface profile of each measured surface. Through experimental validation, the method proposed in this paper can perform the batch measurement of specular planes while maintaining high measurement accuracy.

Parasitic Reflection Eliminating for Planar Elements Based on Multi-Frequency Phase-Shifting in Phase Measuring Deflectometry.

Phase measuring deflectometry (PMD) stands as an extremely important technique for specular surface measurement. However, the parasitic reflection from the rear surface poses a challenge for PMD. To solve this problem, this paper proposes an effective method based on multi-frequency and phase-shifting to search for the correct phase. Firstly, the relationship between the phase error and fringe frequency is adequately investigated. Subsequently, an auxiliary function is established to find the special frequency at which the phase error is zero theoretically and the unwrapped phase is the phase of the top surface exactly. Then, the shape of the top surface can be reconstructed correctly. A standard plane element with a thickness of 40 mm and a flat glass with 19 mm were measured. The experimental results verify the feasibility of the proposed method. Considering the result of the interferometer as a reference, the RMSE of the error map is up to 20 nm for the standard plane element. The experimental results demonstrate that the proposed method can successfully untangle the superposed reflections and reliably reconstruct the top surface of the object under test.

DYnet++: A Deep Learning Based Single-Shot Phase-Measuring Deflectometry for the 3-D Measurement of Complex Free-Form Surfaces

DYnet++: A Deep Learning Based Single-Shot Phase-Measuring Deflectometry for the 3-D Measurement of Complex Free-Form Surfaces

Laser-tracker-based reference measurement for geometric calibration of phase-measuring deflectometry with active display registration

Abstract. Phase-measuring deflectometry (PMD) with active display registration (ADR) is a ray-optics-based technique for the shape measurement of specular surfaces. To obtain quantitative results, the relative position of the cameras of the PMD–ADR setup needs to be determined by geometric calibration. Geometric calibration can be performed by inserting a planar mirror into the setup that brings all camera fields of view to overlap on an active pattern display. The mirror is tilted to multiple positions and each time the cameras capture the displayed images, which yields sufficient data to obtain the relative camera positions and the positions of the mirror. In this article, we give a more detailed description of PMD–ADR and its calibration. We also implement a laser-tracker-based reference method to measure the mirror positions and use its result to expose systematic errors in the geometric calibration.

Multi-view stitching phase measuring deflectometry for freeform specular surface metrology.

Phase measuring deflectometry (PMD) offers notable advantages for precision inspection of specular elements. Nevertheless, if confronts challenges when measuring freeform specular surfaces due to the dispersion of reflection rays from surfaces with high local slopes. Here, we propose a multi-view stitching PMD. It utilizes distinct sensors combining with a screen to capture the appearance of each region. After precisely calibrating the entire system to correct the absolute depth of each region, the appearances of all regions are precisely stitched together, reconstructing the comprehensive appearance of the surface. Through experimental setup, we measured the 3D morphology of a spherical lens with a curvature radius of 155.04 mm and a peak-to-valley (PV) value of 2.9 mm, which yielded a measurement accuracy of 5.3 µm (relative error: 0.18 %). Furthermore, we successfully measured the appearance of a curved mobile phone screen with local slopes ranging from -46.1° to 51.3°, and freeform acrylic sheet with local slopes ranging from -6.7° to 7.7° and a PV value of 5.3 mm.

A deep learning-based approach to solve the height-slope ambiguity in phase measuring deflectometry

Phase measuring deflectometry (PMD) shows excellent performance in measuring the shape of specular surfaces. However, the existing height-slope ambiguity affects the measurement of PMD. In this paper, we propose a deep learning-based method to solve the ambiguity and reconstruct the specular surface. A neural network was built to reconstruct the surface. A set of simulation surfaces were generated using two-dimensional Legendre polynomials to train and test the network. A spherical-shaped surface was reconstructed in a simulation to evaluate the validity of the proposed method and compare our method to the state-of-the-art method. Experimental results demonstrate a 35.88% improvement in this method’s accuracy. Our method provides a great improvement in measurement accuracy.

High-precision dynamic three-dimensional shape measurement of specular surfaces based on deep learning.

In order to solve the difficulty of traditional phase measuring deflectometry (PMD) in considering precision and speed, an orthogonal encoding PMD method based on deep learning is presented in this paper. We demonstrate for, what we believe to be, the first time that deep learning techniques can be combined with dynamic-PMD and can be used to reconstruct high-precision 3D shapes of specular surfaces from single-frame distorted orthogonal fringe patterns, enabling high-quality dynamic measurement of specular objects. The experimental results prove that the phase and shape information measured by the proposed method has high accuracy, almost reaching the results obtained by the ten-step phase-shifting method. And the proposed method also has excellent performance in dynamic experiments, which is of great significance to the development of optical measurement and fabrication areas.

Aided Imaging Phase Measuring Deflectometry Based on Concave Focusing Mirror

With the rapid development of aerospace, high-speed train, and automotive industries, the demand for the measurement of high-precision specular components is increasing. The acquisition of high-precision three-dimensional (3D) data is conducive to improving the performance of and extending the service life of these components. However, the existing 3D measurement methods of specular surfaces are affected by the inherent limitation of the depth of field (DOF) of camera lenses. Based on the principle of paraxial reflection imaging of a concave mirror, this paper introduces a concave mirror into a phase measuring deflectometry (PMD) system and proposes an aided imaging PMD (AIPMD) based on a concave focusing mirror. The proposed system realizes the clear imaging of the encoded patterns and the surface under test in the DOF of the camera lens, simultaneously. Meanwhile, the iterative coefficient specular reconstruction algorithm is studied based on this system. The feasibility and accuracy of the proposed method are verified by simulations and experiments.

Measurement uncertainty of phase measuring deflectometry.

Phase measuring deflectometry (PMD) is a proven optical measurement method for measuring the shapes of objects. This method is suitable for measuring the shape of an object with an optically smooth (mirror-like) surface. The measured object is used as a mirror through which the camera observes a defined geometric pattern. We derive the theoretical limit of measurement uncertainty using the Cramér-Rao inequality. It shows that the measurement uncertainty is expressed in the form of an uncertainty product. The factors of the product are the angular uncertainty and lateral resolution. The magnitude of the uncertainty product depends on the mean wavelength of the light used and the number of photons detected. The calculated measurement uncertainty is compared with the measurement uncertainty of other deflectometry methods.

Specular Surface Shape Measurement with Orthogonal Dual-Frequency Fourier Transform Deflectometry

Three-dimensional (3D) shape measurement for specular surfaces is becoming increasingly important in various applications. A novel orthogonal dual-frequency fringe is proposed in the specular surface shape measurement to overcome the phase jumping and discontinuities in spatial phase unwrapping. The fringe recalibrated high-accuracy phase information from its high-frequency fringe component with low-ambiguity phase information from its low-frequency fringe component. An improved Fourier transform deflectometry method based on the orthogonal dual-frequency fringe is proposed to measure 3D specular surface shapes. Simulation results showed that the orthogonal dual-frequency Fourier transform deflectometry (ODD) method could precisely reconstruct flat surfaces with an error of 2.16 nm rms, and concave surfaces with an error of 1.86 μm rms. Experimental results showed that the reconstructed shapes of both the flat mirror and the concave mirror measured by the ODD measurement system were highly comparable to those obtained by the phase-measuring deflectometry (PMD) method. This new fringe provides a distinctive approach to structured pattern construction and reduces the phase unwrapping ambiguities in specular surface shape measurement. The ODD method can achieve accurate 3D shape measurement for specular surfaces by sampling only one fringe, providing a possible basis for future real-time measurement of specular surfaces.

Screen-monitored stitching deflectometry based on binocular stereo vision

The measurement accuracy of the current phase measuring deflectometry (PMD) methods for in-situ measurement of the surface shape of large-aperture optics, which are placed at different inclination angles, is affected by the varied pose and deformed shape of screen due to manufacturing, external force and temperature variation, etc. This paper proposes a novel PMD method, the screen-monitored stitching deflectometry (SMSD), by combining binocular stereo vision with stitching deflectometry. And a four-camera PMD measurement system is constructed accordingly. The coordinates of the point source are determined using binocular stereo vision technology; based on the acquired coordinates, the stereo-iterative reconstruction and stitching algorithm are used to achieve the surface shape measurement. Simulations and experiments are accomplished to validate the advantages of the proposed method in reducing measurement errors caused by the varied pose and deformed shape of screen.

A Wavefront Integration Algorithm Based on Radial Basis Function for Off-Axis PMD

Sampled points on a measured surface are not evenly spaced on rectangular grids but, rather, have a more general quadrilateral geometry due to the perspective effect of the off-axis configuration, which is extensively utilized in phase measuring deflectometry (PMD). The current procedures for slope-to-wavefront integration results in reconstruction errors, which impact PMD to achieve a higher measurement accuracy. Due to the advantageous properties of the radial basis function (RBF) for irregular sampling and incomplete data, the quadrilateral geometry acquired by a CCD is first realized in this paper using the ray tracing method, which is used to obtain the actual measurement area and sampling interval in off-axis PMD instead of the area defined by the mathematical formula in the past when RBF was used. The RBF integration method is researched in this paper. The simulation and experiment show that RBF is a robust and higher accuracy algorithm for off-axis PMD. We further implement the RBF integration method to measure a crystal transparent element where an off-axis configuration is required.

Wavefront measurement of a multilens optical system based on phase measuring deflectometry

Usually, a multilens optical system is composed of multiple undetectable sublenses. Wavefront of a multilens optical system cannot be measured when classical transmitted phase measuring deflectometry (PMD) is used. In this study, a wavefront measuring method for an optical system with multiple optics is presented based on PMD. A paraxial plane is used to represent the test multilens optical system. We introduce the calibration strategy and mathematical deduction of gradient equations. Systematic errors are suppressed with an N-rotation test. Simulations have been performed to demonstrate our method. The results showing the use of our method in multilens optical systems, such as the collimator and single-lens reflex camera lenses show that the measurement accuracy is comparable with those of interferometric tests.

A Novel Phase Measuring Deflectometry Method Using a Curved Screen

Phase measuring deflectometry (PMD) is widely used to measure the surface of specular objects. The planar screen used in traditional PMD systems have the drawbacks of a small field of view of measurement and a large virtual image depth range. This problem can be effectively solved by using the curved screen. As the arc structure of the curved screen is different to the planar structure required in Zhang’s method, the existing methods mainly rely on the stereo vision of two cameras to obtain the screen position. This paper presents a novel PMD based on a curved screen and single camera. A method of calibrating a curved screen by constructing a virtual planar screen is presented. The method only needs one camera to calibrate the curved screen directly with high accuracy. In addition, the change from planar screen to curved screen makes the traditional phase-gradient-height relation invalid. Based on the comprehensive analysis of space optical path models, a novel phase-gradient-height relation is presented. The relation is a general formula. It can be used for any curved screen, including the planar screen. Simulation and quantitative comparison experiments verify the accuracy and reliability of the proposed calibration method. The measurement results of concave mirror show that the proposed method has higher accuracy and a larger field of view in measuring specular objects than traditional methods.

A General Reconstruction Framework for Deflectometric Measurement Based on Nonuniform B-Splines

Phase measuring deflectometry (PMD) is a powerful tool well suited for precision measurement of optical surfaces. However, accurate and efficient surface reconstruction from gradients is challenging for the PMD. Because most reconstruction algorithms are not well adapted to complex surface forms with local undulations, and a large number of tedious operations are demanded for calculating the ray intersections. Besides, the format of the reconstructed data is usually not compatible to the machine tools and additional format conversion is required, which increases the computational effort and lengthens the error propagation chain. Therefore, a general reconstruction framework is proposed for the deflectometric measurement based on the non-uniform B-spline (NUBS), which embeds the NUBS into the PMD framework to enhance the local representativity. The correspondences between the points on the screen, camera and workpiece can be established by the parameter mapping instead of by the ray intersections. Simulations and experiments prove the effectiveness of the proposed method, and the measurement accuracy can achieve a level of one hundred nanometers in practice.

Polynomial color compensation model with neural network solver for color-coded fringe patterns in phase-measuring deflectometry

The total image display/projection and image acquisition time in a projector-based phase-measuring deflectometry (PMD) measurement is reduced when color-coded fringe patterns are used since only a single image of the fringe pattern is sufficient to extract phase information using the phase-stepping method. This benefit is, however, offset by the difficulty in resolving various sources of color intensity errors of the fringe patterns in the camera view, such as uneven illumination, color imbalance, color crosstalk, and deviation of color intensity response of the camera and the projector from the calibrated color targets. Thus, a polynomial color compensation model is proposed in this study to compensate for color intensity errors. The model coefficients were estimated using the relationship between the captured and targeted color intensities, which are sequences of monochromatic 8-bit red, green, and blue (RGB) images for projection. The model was applied to modify the color intensities of the generated fringe patterns before projection. The modified color fringe patterns were then used for PMD measurement on spherical concave and convex mirrors. Based on the experimental results, the root-mean-squared (RMS) color intensity errors in R, G, and B channels were reduced by 33.3%, 58.3%, and 36.4%, respectively, after conducting the polynomial color compensation. The RMS phase errors in the fringe patterns were reduced by 71.9%. The RMS height errors and the radius percentage errors of the reconstructed concave and convex heightmaps were 1.87 μm (0.12%) and 1.55 μm (0.17%), respectively. The performance of the proposed polynomial color compensation model in reducing color intensity errors of the captured fringe patterns for accurate PMD measurement was demonstrated successfully.

Correction of perspective distortion and intensity errors in projected fringe patterns in phase-measuring deflectometry

The relative pose between the projector and screen in a phase-measuring deflectometry (PMD) setup can result in perspective distortion and intensity errors in the projected fringe patterns. These problems degrade the sinusoidal characteristics of the fringe patterns, thus increasing phase, slope, and height errors in the PMD measurement. To overcome these issues, inverse perspective distortion and intensity modification algorithms were developed and applied to computer-generated fringe patterns before projection. The inverse perspective distortion matrices were obtained using a single image of the projected grid distortion target. The pixel intensities in the generated fringe patterns were modified using the relationship between the generated and captured pixel intensity values in grayscale images of uniform intensity. Experiments were conducted to measure the surface topographies of spherical concave and convex mirrors using a monoscopic PMD setup. Based on the experimental results, the root-mean-squared (RMS) intensity and phase errors in the projected sinusoidal fringe patterns were reduced by about 90% after using the modified fringe patterns compared to the original fringe patterns. When comparing the measured and the theoretical heightmaps, the results in PMD measurement showed an 8% and a 14% reduction of the RMS height errors in concave and convex heightmaps, respectively. The effectiveness of the algorithms in improving the accuracy of projector-based PMD measurement was demonstrated successfully.

Error compensation for phase retrieval in deflectometry based on deep learning

Fringe patterns are widely applied in optical metrology, and phase retrieval is an essential process for decoding surface information. In the field of phase measuring deflectometry (PMD), phase errors in the phase retrieval process have more significant effects for PMD is a slope-based technique and is more sensitive to low-frequency errors. The main factors affecting the quality of the captured fringe images include the gamma effect of the liquid crystal display screen, the random noise from the charge-coupled device camera, and the random noise amplified by the defocused fringe patterns. Conventional methods compensated the phase errors of these factors separately with different methods, which are inefficient in handling the errors from coupling factors effectively. In this paper, we propose a deep neural network to compensate for the phase errors resulting from the combination of the factors. Experimental results demonstrate that the proposed network can significantly suppress the errors in phase retrieval with non-ideal fringe images. The phase errors can be reduced in both simulated and authentic data for deflectometry, which verifies the robustness and effectiveness of the proposed method.

Improved methods of binocular phase measuring deflectometry.

Phase measurement deflectometry (PMD) is an excellent method for high-precision optical surface measurement; through the simple system structure, accuracy comparable to that of interference methods can be realized. The key of PMD is to solve the ambiguity between the surface shape and normal vector. Considering all kinds of methods, the binocular PMD method is undoubtedly a method with a very simple system structure and can be easily applied to complex surfaces, such as free-form surfaces. However, this method relies on a large screen with high accuracy, which not only increases the weight of the system, but also reduces the flexibility of the system, and manufacturing errors in the large-size screen can easily become error sources. In this Letter, we have made some improvements based on the traditional binocular PMD. At first, we try to replace the large screen with two small screens to increase the flexibility and accuracy of the system. Further, we replace one small screen with a single point to simplify the system structure. Experiments show that the proposed methods not only improve the system flexibility and reduce the complexity, but can also achieve high measurement accuracy.

Front and back surface measurement of the transparent planar element based on multi-frequency fringe deflectometry.

As a highly accurate metrology, phase measuring deflectometry (PMD) can be used for in-situ surface shape measurement. However, due to the reflection off the back surface, PMD cannot measure both the front and back surfaces of the transparent planar element simultaneously. Therefore, this paper proposes a method for measuring the front and back surfaces of the transparent planar element. The phase distribution corresponding to the front and back surfaces can be firstly acquired by multi-frequency fringe deflectometry. Then, the front and back surface shapes can be obtained by inverse ray-tracing and nonlinear optimization. Numerical simulation and experiment verify the proposed method. The surface shape of window glass with a thickness of 10 mm is measured in the experiment. The surface shape error is around 50 nm in the root mean square with a diameter of 51 mm.