Phase Measuring Deflectometry Research Articles

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 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.

Deep learning-based Phase Measuring Deflectometry for single-shot 3D shape measurement and defect detection of specular objects.

Phase Measuring Deflectometry (PMD) and Structured-Light Modulation Analysis Technique (SMAT) perform effectively in shape and defect measurements of specular objects, but the difficulty of giving consideration to accuracy and speed has also restricted the further development and application of them. Inspired by recent successes of deep learning techniques for computational imaging, we demonstrate for the first time that deep learning techniques can be used to recover high-precision modulation distributions of specular surfaces from a single-frame fringe pattern under SMAT, enabling fast and high-quality defect detection of specular surfaces. This method can also be applied to recover higher-precision phase distributions of specular surfaces from a single-frame fringe pattern under PMD, so as to realize the 3D shape measurement. In this paper, we combine depthwise separable convolution, residual structure and U-Net to build an improved U-Net network. The experimental results prove that the method has excellent performance in the phase and modulation retrieval of specular surfaces, which almost reach the accuracy of the results obtained by ten-step phase-shifting method.

Monoscopic Phase Measuring Deflectometry Simulation and Verification

The three-dimensional (3D) shape of specular surfaces is important in aerospace, precision instrumentation, and automotive manufacturing. The phase measuring deflectometry (PMD) method is an efficient and highly accurate technique to measure specular surfaces. A novel simulation model with simulated fringe patterns for monoscopic PMD is developed in this study. Based on the pre-calibration and the ray-tracing model of the monoscopic PMD system, a comprehensive model from deformed pattern generation to shape reconstruction was constructed. Experimental results showed that this model achieved high levels of measuring accuracy in both planar and concave surfaces measurement. In planar surface measurement, the peak to valley (PV) value and root mean square (RMS) value of the reconstructed shape can reach 26.93 nm and 10.32 nm, respectively. In addition, the accuracy of the reconstructed concave surface can reach a micrometre scale. This work potentially fills critical gaps in monoscopic PMD simulation and provides a cost-effective method of PMD study.

Near optical coaxial phase measuring deflectometry for measuring structured specular surfaces.

Phase measuring deflectometry (PMD) is an important technique for the form measurement of specular surfaces. However, the existing stereo-PMD techniques have noticeable weaknesses for structured specular surfaces measurement due to the optical axis of the imaging system must have a notable intersection angle with the optical axis of the display system according to the law of reflection. This leads to the imaging sensor and the fringe display screen must be located on the opposite sides of the normal of the surface under test (SUT), which results in large system volume and measurement shadows when measuring discontinuous specular surfaces. In this paper, we propose a novel near optical coaxial PMD (NCPMD) by utilizing a plate beamsplitter. With the assistance of plate beamsplitter, the optical axis of display screen can be configured much closer to the optical axis of the imaging system which makes the system more compact and has significantly reduced volume compared with the conventional PMD configuration. Moreover, imaging sensors in the proposed configuration can perpendicularly capture the SUT, which can drastically decrease measurement shadows caused by discontinuous structures on the SUT and increases measurement efficiency. A comparison between the proposed NCPMD and the conventional PDM is studied by measuring a specular step to show the advantage of the proposed configuration in reducing measurement error caused by structure shadows. A portable NCPMD prototype with stereo imaging sensors is developed and verified through experiments. Experimental results show the portable prototype has comparable measurement accuracy with the existing PMD techniques while has obviously advanced performances for portable and embedded form measurement, such as small system volume, and light weight.

Surface Shape Distortion Online Measurement Method for Compact Laser Cavities Based on Phase Measuring Deflectometry

Conventional phase measuring deflectometry (PMD) takes up a large measurement space and is not suitable for compact online measurement, as the liquid crystal display (LCD) has to be placed in parallel with the mirror under test. In this paper, a compact online phase measuring deflectometry (COPMD) with the LCD screen set perpendicular to the mirror under test is presented for surface shape distortion real-time measurement. The configuration of the COPMD in an enclosed laser cavity is proposed, and the principle of the method is theoretically derived by using the vector-form reflection law. Based on the analysis model, the fringe modulation regulation of the LCD is revealed, and the measurement errors caused by misalignments of the components are illustrated. The validity and flexibility of the COPMD method are verified in the experiment by using a single-actuator deformable mirror as the mirror under test and the PMD method as the comparison. The proposed COPMD method remarkably expands the application range of the conventional PMD method, as it could make efficient use of compact space and is applicable for real-time measurement in enclosed laser facilities and assembled laser systems.

Phase measuring deflectometry based on calibration of the entrance pupil center of the camera lens.

A camera calibration method for phase measuring deflectometry (PMD) based on the entrance pupil center (EPC) of the camera lens is proposed. In our method, the position of the entrance pupil of the camera lens is first measured; next the absolute coordinates of the EPC are calibrated by using a reference flat and an external stop that is mounted in front of the camera lens; then the EPC as the camera coordinates is used for PMD. The feasibility of the proposed method is verified by simulation. The surface shapes of a planar optical element and a planar window glass are separately measured in our experiments, and a subwavelength accuracy level is achieved. Meanwhile, the effects of the camera lens with different aperture settings on captured images are investigated (including exposure time, image contrast, and measurement accuracy). The experimental results show that the exposure time required declines with the decrease in the f-number, and the measurement accuracy is higher than others when the f-numbers are changed from f/5.6 to f/11.

Untangling parasitic reflection in phase measuring deflectometry by multi-frequency phase-shifting.

Phase measuring deflectometry (PMD) is a technique that reconstructs the three-dimensional (3D) profiles of specular surfaces. When the object under test is transparent, its bottom surface creates a parasitic reflection that superimposes with the primary reflection created by the top surface. The superimposed reflections cause phase error in decoding of the fringe patterns and reduce the reconstruction accuracy. To accurately reconstruct the 3D profile of transparent objects, the superimposed reflections must first be untangled. In this paper, a multi-frequency phase-shifting approach is proposed to untangle the superimposed signals. Based on the principle of phase-shifting, a mathematical model is developed. The unknown phase angles in the mathematical model are solved by an optimization technique with input data obtained from fringe patterns at different spatial frequencies. A procedure is also developed to obtain the set of initial conditions for the optimization process. Both simulation and experiments were conducted to verify the proposed method. The results show that the proposed method can accurately untangle the phase angles corresponding to primary and parasitic reflections. The surface reconstruction result was compared to a reference measurement given by an interferometer, and a root-mean-square error of 32.95 nm was recorded. The accuracy achieved by the proposed method is compatible with another existing multi-frequency approach, while using roughly eight times fewer images. With the proposed method, better time efficiency can be achieved, and the computer's memory requirement can be lowered.

Surface Measurement of a Large Inflatable Reflector in Cryogenic Vacuum

The metrology of membrane structures, especially inflatable, curved, optical surfaces, remains challenging. Internal pressure, mechanical membrane properties, and circumferential boundary conditions imbue highly dynamic slopes to the final optic surface. Here, we present our method and experimental results for measuring a 1 m inflatable reflector’s shape response to dynamic perturbations in a thermal vacuum chamber. Our method uses phase-measuring deflectometry to track shape change in response to pressure change, thermal gradient, and controlled puncture. We use an initial measurement as a virtual null reference, allowing us to compare 500 mm of measurable aperture of the concave f/2, 1-meter diameter inflatable optic. We built a custom deflectometer that attaches to the TVAC window to make full use of its clear aperture, with kinematic references behind the test article for calibration. Our method produces 500 × 500 pixel resolution 3D surface maps with a repeatability of 150 nm RMS within a cryogenic vacuum environment (T = 140 K, P = 0.11 Pa).

Curved LCD based deflectometry method for specular surface measurement

Phase measuring deflectometry (PMD) technique is widely used to measure the specular surface. However, the Current PMD methods are generally based on the planar LCD for the measurement, which limits the measurable curvature of the surface under test (SUT). To increase the measurable range of curvature, this paper proposes a novel PMD method by using a curved LCD monitor. A calibration method based on the cylindrical coordinate system is introduced for the proposed system calibration. To extend the measurable range further, the curved LCD is applied in stereoscopic PMD. The partial surface of SUT in the field of view (FOV) of each camera and curved LCD are reconstructed with iterative algorithm respectively, the calibration parameters as well as the seed point are used for point cloud registration. Simulations show the advantage of a curved LCD over a planar one and the feasibility of the proposed method. The actual performance is demonstrated in the experiments for specular optics and a phone touchscreen.

Comparison of Camera Calibration and Measurement Accuracy Techniques for Phase Measuring Deflectometry

Phase measuring deflectometry (PMD) is a competitive method for specular surface measurement that offers the advantages of a high dynamic range, non-contact process, and full field measurement; furthermore, it can also achieve high accuracy. Camera calibration is a crucial step for PMD. As a result, a method based on the calibration of the entrance pupil center is introduced in this paper. Then, our proposed approach is compared with the most popular photogrammetric method based on Zhang’s technique (PM) and Huang’s modal phase measuring deflectometry (MPMD). The calibration procedures of these three methods are described, and the measurement errors introduced by the perturbations of degrees of freedom in the PMD system are analyzed using a ray tracing technique. In the experiment, a planar window glass and an optical planar element are separately measured, and the measurement results of the use of the three methods are compared. The experimental results for the optical planar element (removing the first 6 terms of the Zernike polynomial) show that our method’s measurement accuracy reached 13.71 nm RMS and 80.50 nm PV, which is comparable to accuracy values for the interferometer.

Active speckle deflectometry based on 3D digital image correlation.

A method based on 3D digital image correlation (DIC) to measure the shape of specular surface is proposed. The proposed active speckle deflectometry (ASD) utilizes a stereo-camera system to monitor the liquid crystal display (LCD), which is deliberately moved during the measurement. Another testing camera (TC) is used to capture the single-shot speckle pattern displayed on the LCD screen after reflection by the test surface. With this proposal, the movement of the LCD screen can be arbitrary as long as the TC can capture the reflection of speckle pattern. The distance as well as the direction of the movement is not required to be known. The coordinates of the point source are determined by applying the 3D DIC technique with the monitoring stereo-cameras (MSC) before and after the movement of the LCD screen, then the slope and surface shape are obtained. The measurement accuracy of this method is evaluated by measuring a flat glass with a diameter of about 80 mm, compared with the measurement results of interferometer, the shape measurement difference is 0.278um in root mean square (RMS). The shape of two wafers is also measured, and the measurement results are compared to that of the traditional phase measuring deflectometry (PMD). ASD has the advantages of fast measurement, low cost, arbitrary LCD movement, tolerance for the out-of-plane shape of the LCD screen. It provides a new method for specular surface measurement.

Infrared phase measuring deflectometry by using defocused binary fringe.

Three-dimensional surface information acquisition of specular objects plays an important role in the fields of automobile industry, aerospace, cultural relic protection, intelligent robotics, equipment manufacturing, and so on. Most of the existing specular surface measurement methods are based on focused sinusoidal fringe patterns, so there are certain requirements for the range of the depth of field (DOF) of the camera on the focus position. However, for many specular surfaces with a large gradient, the tested objects may not always be in the DOF of the camera, so sinusoidal fringe patterns are defocused to be vulnerable to the noise. In this Letter, a new infrared phase measuring deflectometry (PMD) based on defocused binary fringe is proposed that combines a binary fringe defocusing technique and direct PMD. The measurement principle and the corresponding system calibration method are described. The feasibility and measurement accuracy of fringe defocus in specular measurement are studied in principle. The experimental results on several specular objects show that the proposed method can effectively measure specular surfaces out of the DOF of the camera.

3D reconstruction of the specular surface using an iterative stereoscopic deflectometry method.

Phase measuring deflectometry (PMD) is an effective technique for three-dimensional measurement of specular surfaces. However, the ambiguity of monoscopic PMD and the time-consuming searching process of stereoscopic PMD are challenges for specular surface reconstruction. To solve it, we propose an iterative reconstruction algorithm for the stereoscopic phase measuring deflectometry system free of the time-consuming searching process for each pixel. An arbitrary seed point on the specular surface is accurately obtained via a coarse-to-fine optimization means without any other expensive and complicate auxiliaries. Then, a plane with the height of seed point is set as the initial surface form for the iteration, in which the pinhole model is used to find the linear relation to update the surface form. The converging height is the output as the final result. Simulations and experiments verify the feasibility and efficiency of our proposed method based on the stereoscopic phase measuring deflectometry system. The accuracy and robustness are comprehensively evaluated as well.

Phase measuring deflectometry for obtaining 3D shape of specular surface: a review of the state-of-the-art

Phase measuring deflectometry (PMD) is a superior technique to obtain three-dimensional (3D) shape information of specular surfaces because of its advantages of large dynamic range, noncontact operation, full-field measurement, fast acquisition, high precision, and automatic data processing. We review the recent advances on PMD. The basic principle of PMD is introduced following several PMD methods based on fringe reflection. First, a direct PMD (DPMD) method is reviewed for measuring 3D shape of specular objects having discontinuous surfaces. The DPMD method builds the direct relationship between phase and depth data, without gradient integration procedure. Second, an infrared PMD (IR-PMD) method is reviewed to measure specular objects. Because IR light is used as a light source, the IR-PMD method is insensitive to the effect of ambient light on the measured results and has high measurement accuracy. Third, a proposed method is reviewed to measure the 3D shape of partial reflective objects having discontinuous surfaces by combining fringe projection profilometry and DPMD. Then, the effects of error sources that mainly include phase error and geometric calibration error on the measurement results are analyzed, and the performance of the 3D shape measurement system is also evaluated. Finally, the future research directions of PMD are discussed.