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Fringe Pattern Analysis for Optical Metrology: Theory, Algorithms, and Applications The main objective of this book is to present the basic theoretical principles and practical applications for the classical interferometric techniques and the most advanced methods in the field of modern fringe pattern analysis applied to optical metrology. A major novelty of this work is the presentation of a unified theoretical framework based on the Fourier description of phase shifting interferometry using the Frequency Transfer Function (FTF) along with the theory of Stochastic Process for the straightforward analysis and synthesis of phase shifting algorithms with desired properties such as spectral response, detuning and signal-to-noise robustness, harmonic rejection, etc.

Summary An optical measurement system, which is capable of vibration measurement and response‐only experimental modal analysis for beam‐like structures, is reported by taking the concept of 2‐dimensional optical coherence vibration tomography technique. In the proposed approach, it requires only quasi‐interferogram fringe patterns as sensors and a high‐speed camera as a detector. Experimental results of beam‐like structures subjected to swept and harmonic excitations are demonstrated. The key advantage of the proposed method is the high spatial and temporal resolution with simultaneously measuring the absolute displacement of multiple points along the beam‐like structures without point‐by‐point optical scanning. Without any vibration excitation information, the quasi‐optical coherence vibration tomography technique system can capture structural modal parameters. Therefore, it is a response‐only experimental modal analysis method, making it attractive for the structural health monitoring applications in the multiple‐point measurement of real engineering structures with the absence of vibration input information.

Distortion is a key parameter affecting the imaging performance of lenses. In this study, we propose a testing method based on detector Nyquist sampling of image data to achieve high-precision measurements of the distortion distribution of lenses. The distribution patterns of distortions in horizontal and vertical directions can be obtained by analyzing the distribution patterns of Moiré fringes in images under Nyquist sampling conditions and using phase-shift algorithms. The distortion-distribution characteristics of the lens are then calculated using distortion formulas. This method is characterized by high testing accuracy and sampling resolution. The image-plane distortion distribution exhibited a consistent linear trend when the object-plane position varied within a limited spatial range. Furthermore, the proposed method achieved a magnification deviation factor repeatability accuracy of approximately ±108 nm/cm and third-order distortion-measurement accuracy of approximately ±108 nm/cm3. This method enables a high-precision distortion evaluation of conventional industrial imaging lenses.

Endoscopic fringe projection is used to perform inspections of hard-to-reach areas. In order to transfer fringe patterns from a projector to the specimens’ surface, fiber optic imaging bundles (FOIB) can be employed. To ensure maximum accessibility, a highly flexible FOIB is needed. Therefore, the number of individual fibers has to be minimized, which affects the quality of the fringe pattern. This paper presents methods and results for projecting a high frequency pattern despite a small number of fibers by adapting the FOIBs’ structure. First, the spatial structure of the FOIB is identified with regard to the projector pixels. By determining their center, it is possible to address individual fibers. It will be shown that the peak values of spots produced by individual fibers behave nonlinearly according to the modulated intensity. Furthermore, the intensity distribution within the spots changes. By recording the intensity curves, the presented algorithm is able to adapt the fringe pattern in orientation and intensity. This leads, especially for high frequency patterns, to an improved amplitude and signal-to-noise ratio.

Electronic speckle pattern interferometry (ESPI) is widely used in fields such as materials science, biomedical research, surface morphology analysis, and optical component inspection because of its high measurement accuracy, broad frequency range, and ease of measurement. Phase extraction is a critical stage in ESPI. However, conventional phase extraction methods exhibit problems such as low accuracy, slow processing speed, and poor generalization. With the continuous development of deep learning in image processing, the application of deep learning in phase extraction from electronic speckle interferometry images has become a critical topic of research. This paper reviews the principles and characteristics of ESPI and comprehensively analyzes the phase extraction processes for fringe patterns and wrapped phase maps. The application, advantages, and limitations of deep learning techniques in filtering, fringe skeleton line extraction, and phase unwrapping algorithms are discussed based on the representation of measurement results. Finally, this paper provides a perspective on future trends, such as the construction of physical models for electronic speckle interferometry, improvement and optimization of deep learning models, and quantitative evaluation of phase extraction quality, in this field.

Lens distortion can introduce deviations in visual measurement and positioning. The distortion can be minimized by optimizing the lens and selecting high-quality optical glass, but it cannot be completely eliminated. Most existing correction methods are based on accurate distortion models and stable image characteristics. However, the distortion is usually a mixture of the radial distortion and the tangential distortion of the lens group, which makes it difficult for the mathematical model to accurately fit the non-uniform distortion. This paper proposes a new model-independent lens complex distortion correction method. Taking the horizontal and vertical stripe pattern as the calibration target, the sub-pixel value distribution visualizes the image distortion, and the correction parameters are directly obtained from the pixel distribution. A quantitative evaluation method suitable for model-independent methods is proposed. The method only calculates the error based on the characteristic points of the corrected picture itself. Experiments show that this method can accurately correct distortion with only 8 pictures, with an error of 0.39 pixels, which provides a simple method for complex lens distortion correction.

In optical metrology, the output is usually in the form of a fringe pattern, from which a phase map can be generated and phase information can be converted into the desired parameters. This paper proposes an end-to-end method of fringe phase extraction based on the neural network. This method uses the U-net neural network to directly learn the correspondence between the gray level of a fringe pattern and the wrapped phase map, which is simpler than the exist deep learning methods. The results of simulation and experimental fringe patterns verify the accuracy and the robustness of this method. While it yields the same accuracy, the proposed method features easier operation and a simpler principle than the traditional phase-shifting method and has a faster speed than wavelet transform method.

Inkjet printers are key technologies in manufacturing organic light-emitting diodes and quantum dot light-emitting diode panels, but precise measurement and control of inkjet droplets remains challenging. The international standard, IEC 62899-302-1, uses shadow image-based measurement with high magnification microscopes to observe picoliter-sized droplets. However, high magnification lens results in a shallow depth of field or narrow optimal measurement area, causing the blurring image if the droplet does not pass through the optimal measurement area. To solve this, we propose using the interference image-based measurement with interference fringe patterns by inkjet droplets as a tool to measure the flight speed of droplets. The interference fringe patterns can be obtained simply passing the droplet through within the light beam path, providing approximately 1000× wider measurement area compared to the shadow image-based measurement, making it practical to use in the industry. The flight speed of droplets analyzed with the interference image-based measurement at various frequencies and amplitudes of the inkjet driving voltage were compared with the shadow image-based measurement. The interference image-based measurement showed a coefficient of variation of less than 3%, showing higher repeatability than the shadow image-based measurements.

Single-shot fringe projection profilometry (FPP) has become a more prevalently adopted technique for retrieving the absolute phase values of the objects in intelligent manufacturing, defect detection, and some other important applications. In FPP, phase unwrapping plays a decisive role as the quality of final three-dimensional reconstructions relies on how accurately the phase values have been calculated. However, noise, shadow, discontinuity, and aliasing often exist in the original wrapped phase patterns which increase the complexity and difficulty of phase unwrapping, even lead to the unwrapping failure. To deal with phase unwrapping problems, we propose a phase unwrapping method based on deep learning, called channel transformer U-net, for directly extracting absolute phase value from the wrapping phase patterns. In the proposed method, the advanced channel-wise cross-fusion transformer module is integrated into the design of deep U-Net architecture. And a new loss function by combining the Smooth L1 loss and multi-scale structural similarity function is proposed. The effectiveness of the proposed method has been verified by real dynamic FPP measurements. The experimental results demonstrate that the proposed channel transformer U-Net can obtain accurate, complete, and smooth absolute phase values in real FPP measurement.

Rapid full-field surface topography measurement for large-scale wafers remains challenging due to limitations in speed, system complexity, and scalability. This work presents a interferometric system based on thin-film interference for high-precision wafer profiling. An optical flat serves as the reference surface, forming a parallel air-gap structure with the wafer under test. A large-aperture collimated beam is introduced via an off-axis parabolic mirror to generate high-contrast interference fringes across the entire field of view. Once the wafer is fully illuminated, topographic information is directly extracted from the fringe pattern. Comparative measurements with a commercial interferometer show relative deviations below 3% in bow and warp, confirming the system’s accuracy and stability. With its simple optical layout, low cost, and robust performance, the proposed method shows strong potential for industrial applications in wafer inspection and online surface monitoring.