Captured Fringe Patterns Research Articles

3D shape measurement of complex transparent liquid surfaces using monoscopic deformed fringe transmissometry

The measurement of transparent liquid surface topography is of great importance in many fields such as hydrodynamics of water-walking insects, liquid sloshing and flotation, etc. However, techniques for measuring these specular and rheological surfaces, especially for surfaces with floating objects, have been lacking. Here we report a monoscopic deformed fringe transmissometry for measuring complex transparent liquid surfaces with theoretical sensitivity up to 0.40 µm. The basic principle is to reconstruct the 3D liquid surfaces by the mathematical relations between the in-plane fringe phase-shift and the out-of-plane shape of the liquid surface. It only needs to project a single fringe pattern from the bottom of the liquid and capture a single deformed image from above. A sub-pixel sampling moiré method is proposed to process the captured fringe pattern to obtain the precise phase component and the displacement field, which is then used to reconstruct the 3D shape of the liquid surface. The discontinuous fringes in the pattern also settled by covering an extracted mask to the displacement filed. The verification test of floating 3D printed characters indicates that this new technique is robust in measuring the 3D shape of complex liquid surface. The developed technique provides a versatile tool to explore interfacial phenomena such as wetting, floating, surface wave, etc.

Exponential Taylor Series Method to eliminate the gamma distortion in phase shifting profilometry

Phase shifting profilometry (PSP) is widely applied in three-dimensional (3D) shape measurement because of its high precision and simple hardware configuration. However, the gamma effect of commercial digital projector usually causes nonlinear periodical phase error. In this paper, we propose an approach named as Exponential Taylor Series Method (ETSM) to eliminate the gamma distortion. The ETSM sets the input fringe pattern intensity as an exponential Taylor series function. Along with the decoding algorithm proposed, the nonlinear phase error caused by the gamma effect can be eliminated theoretically. Simulations and experiments are conducted to verify the ETSM’s performances by comparing with the conventional method of PSP and another method based on exponent. The results show that the ETSM greatly reduces the phase error caused by gamma distortion without any additional work or pre-process. The ETSM also performs better in resisting the noise due to its quasi-sinusoidal intensity distribution of the captured fringe patterns.

Real-time three-dimensional shape measurement based on color binary fringe projection

A real-time three-dimensional (3-D) shape measurement based on single-shot color binary fringe (CBF) projection is proposed. In the traditional 3-D shape measurement methods based on binary fringe projection, the duty cycle of the binary fringe is always set to 1/2, so as to approximate the sinusoidal fringe by defocusing projection. In the proposed method, the binary fringe with a duty cycle of 1/3 is introduced. It is found that although the duty cycle is not 1/2, a nearly unbroken sinusoidal fringe pattern can be extracted from the captured fringe pattern by a filtering operation in the spatial frequency domain. In order to realize real-time 3-D shape measurement, a composite CBF was designed, in which three monochromatic binary fringes share the same duty cycle of 1/3 but misaligned 1/3 periods one by one are encoded in red (R), green (G), and blue (B) channels. When this composite CBF is projected onto the measured object, only one color-deformed pattern (CDP) needs to be captured and three monochromatic sinusoidal deformed patterns with a phase-shifting of 2π / 3 one another can be extracted from the single-shot captured CDP. So the 3-D shape of the measured object can be reconstructed with three-step phase measuring profilometry. The experimental results show the feasibility and validity of the proposed method. It can either effectively avoid the color overlapping in traditional color sinusoidal fringe or avoid the grayscale nonlinearity of sinusoidal fringe caused by the effect of gamma.

An improved 2+1 phase-shifting algorithm

An improved 2+1 phase-shifting algorithm based on the flat image fitting calibration is proposed. The 2+1 phase-shifting algorithm proposed by Zhang uses one uniform gray image whose gray is the same as the mean gray of the phase-shifting sinusoidal grating patterns instead of one of these grating patterns in traditional three-step phase measuring profilometry to reduce measuring error caused by motion. It maintains that the light intensity distribution of captured flat image is equal to the dc component of the captured phase-shifting deformed patterns. But extensive experimental results show that there is a certain inconsistency between the light intensity distribution of the captured flat image and the dc component of the deformed patterns. It will bring some nonnegligible errors. So a calibration procedure is introduced before measuring to obtain a calibration coefficient between the captured flat image and the dc component of the captured fringe pattern by the least squares fit. While measuring, the veritable dc component of the captured deformed pattern is extracted by just multiplying the calibration coefficient with captured flat image. Thus, the above nonnegligible errors can be effectively improved. Furthermore, the proposed algorithm is more flexible because its shifting-phase is no longer fixed to be π/2 but optional. The experimental results show the effectiveness and practicability of the improved 2+1 phase-shifting algorithm.

Full-field phase error analysis and compensation for nonsinusoidal waveforms in phase shifting profilometry with projector defocusing

Phase shifting profilometry (PSP) using binary fringe patterns with projector defocusing is widely used for high-speed three-dimensional (3D) shape measurement. In this technique, the projector projects binary fringe patterns instead of the sinusoidal fringe patterns to perform 3D shape measurement. However, inappropriate defocusing degree of the projector causes the captured fringe patterns to be nonsinusoidal waveforms or remaining binary structure, which results in phase error and therefore measurement error. In this paper, a novel method was proposed not only to measure the phase error directly but also to extract the full-field phase error distribution in a large depth range of projector defocusing. Then a mathematical phase error function in terms of the depth z was established. Finally, with help of the detected phase error distribution, the phase error function was calibrated and used to compensate the phase error at arbitrary depth ranges within the calibration volume. The experimental results demonstrated that the proposed method can detect full-field phase error distribution successfully, and improve the phase accuracy significantly.

Generic exponential fringe model for alleviating phase error in phase measuring profilometry

Phase measuring profilometry (PMP) is susceptible to phase error caused by gamma distortion that leads to the captured fringe patterns deviating from ideal sinusoidal waveforms. Existing phase-error compensation methods are generally complex and require significant computational resources for implementation. This paper proposes a generic exponential fringe model expressed as an exponential function of the generated fringe patterns. Based on this model, a straightforward gamma correction method is presented to alleviate phase error without the need for any estimation of nonlinear coefficients or complex calibration. Experimental results demonstrate that the proposed method improves the quality of measurements by suppressing phase error.

Noise reduction in modulation measurement profilometry based on the wavelet transform method

Modulation measurement profilometry can achieve the reconstruction of a tested object with complex or step-like surface. The technology applying coaxial and coimage plane optical systems in which the projection axis coincides with the observation axis can effectively avoid the defects of shadow, shade, and phase discontinuity. All of these weaknesses are largely present in optical three-dimensional sensing based on trigonometric measurement principle. However, in the practical application, the captured fringe patterns are always inevitably influenced by noise, which will directly affect the measurement accuracy. As a well-known and remarkable time-frequency analysis tool, the wavelet transform (WT) method has a strong noise immunity and is suitable to analyze nonstationary signals. Comparisons between the WT method and the Fourier transform method are made not only by computer simulations appearing robust within various types of noise but also the practical experiment. It shows that the WT method is helpful in the enhancement of measurement accuracy. Additionally, experiment further verifies its strong antinoise ability.

Dual-frequency fringe projection for 3D shape measurement based on correction of gamma nonlinearity

In the fringe projection three-dimensional (3D) shape measurement, the gamma nonlinearity of the digital projector leads to measurement errors, a novel method based on the dual-frequency fringe projection using correction of gamma nonlinearity is proposed. In the proposed method, the pre-coded gamma values of computer generated fringe patterns in digital projector can guarantee the sinusoidal waveforms of the captured fringe pattern, and alleviate the measurement errors caused by the gamma nonlinearity. After the correction of gamma nonlinearity, the shape measurement results by dual-frequency grating projection are analyzed in detail. By combining the advantages of both low frequency and high frequency gratings, the object surface profile is obtained accurately. The simulation and experiment results are presented to validate the proposed method.

Adaptive projection intensity adjustment for avoiding saturation in three-dimensional shape measurement

Phase-based fringe projection methods have been commonly used for three-dimensional (3D) measurements. However, image saturation results in incorrect intensities in captured fringe pattern images, leading to phase and measurement errors. Existing solutions are complex. This paper proposes an adaptive projection intensity adjustment method to avoid image saturation and maintain good fringe modulation in measuring objects with a high range of surface reflectivities. The adapted fringe patterns are created using only one prior step of fringe-pattern projection and image capture. First, a set of phase-shifted fringe patterns with maximum projection intensity value of 255 and a uniform gray level pattern are projected onto the surface of an object. The patterns are reflected from and deformed by the object surface and captured by a digital camera. The best projection intensities corresponding to each saturated-pixel clusters are determined by fitting a polynomial function to transform captured intensities to projected intensities. Subsequently, the adapted fringe patterns are constructed using the best projection intensities at projector pixel coordinate. Finally, the adapted fringe patterns are projected for phase recovery and 3D shape calculation. The experimental results demonstrate that the proposed method achieves high measurement accuracy even for objects with a high range of surface reflectivities.

High-precision measurement of low reflectivity specular object based on phase measuring deflectometry

Phase measuring deflectometry (PMD) is a robust, noncoherent technique for the characterization of specular surface. For measuring high specular reflectivity surface, PMD can deliver micron radian range local gradient. However, when the measured surface has low specular reflectivity, the accuracy of the measured gradient is low since the captured fringe pattern shows low signal to noise ratio. The phase error characteristics in PMD system when testing low reflectivity surfaces are analyzed. The analysis illustrates that the random phase error increases rapidly while the nonlinear error drops slowly with the decreasing of the tested surface reflectivity. In order to attain high precision measurement of low reflectivity specular surface, a robust error reduction method based on wavelet de-noising is proposed to reduce the phase error. This error reduction method is compared with several other normally used methods in both simulation and experiment work. The method based on the wavelet de-noising shows better performance when measuring the low reflectivity specular surface.

Specular reflection compensation in homography fringe projection profilometry

Fringe projection profilometry (FPP) is widely used in the industrial 3D measurement. When measuring an object using FPP, inclined projection will cause problems of periodic inconsistency and light-intensity inhomogeneity to fringe patterns. This paper presents an improved homography fringe generation method to correct the fringes. In this paper, a homography transform is used not only for the coordinate transformation between the projection plane and the reference plane, but also for the modulation of the period and light intensity of the fringe. Based on this method, the captured fringe patterns have good periodicity and homogenized intensity, meanwhile, the impact of high reflection is evidently weakened in the recovery of 3D surface shape with specular component, thus effectively reducing the phase errors. Experiments show that the 3D rail shape is well recovered and the error is apparently decreased based on the proposed method.

Using a single-cube beam-splitter as a fringe pattern generator within a structured-light projection system for surface metrology

We show how a sinusoidal fringe pattern can be obtained by using a single cube beam splitter based on the Gates’ interferometer configuration. When an expanded and collimated laser beam hits the binding edge of a nonpolarizing cube beam splitter parallel to the splitter coating, it generates interference fringes at the exit due to the internal reflections and refractions of the laser beam passing through the cube. Unlike common digital projection systems, the proposed optical arrangement generates a perfectly sinusoidal and continuous fringe pattern, minimizing the problems associated with the discretization of a synthetic digital signal. The fast Fourier transform and phase-shifting techniques are used to demodulate the captured fringe patterns. Experimental results are presented for the three-dimensional shape reconstruction of the relief of a coin and of a spherical indentation on a piece of aluminum with a maximum height of about 150 μm. In addition, we evaluate the accuracy and resolution of the proposed measuring device: shape reconstruction accuracy is about 1.4% and axial resolution is 0.15 μm. Due to its simple and compact setup, the proposed system is particularly suited to be miniaturized.

Anti-spoof touchless 3D fingerprint recognition system using single shot fringe projection and biospeckle analysis

Fingerprint is a unique, un-alterable and easily collected biometric of a human being. Although it is a 3D biological characteristic, traditional methods are designed to provide only a 2D image. This touch based mapping of 3D shape to 2D image losses information and leads to nonlinear distortions. Moreover, as only topographic details are captured, conventional systems are potentially vulnerable to spoofing materials (e.g. artificial fingers, dead fingers, false prints, etc.). In this work, we demonstrate an anti-spoof touchless 3D fingerprint detection system using a combination of single shot fringe projection and biospeckle analysis. For fingerprint detection using fringe projection, light from a low power LED source illuminates a finger through a sinusoidal grating. The fringe pattern modulated because of features on the fingertip is captured using a CCD camera. Fourier transform method based frequency filtering is used for the reconstruction of 3D fingerprint from the captured fringe pattern. In the next step, for spoof detection using biospeckle analysis a visuo-numeric algorithm based on modified structural function and non-normalized histogram is proposed. High activity biospeckle patterns are generated because of interaction of collimated laser light with internal fluid flow of the real finger sample. This activity reduces abruptly in case of layered fake prints, and is almost absent in dead or fake fingers. Furthermore, the proposed setup is fast, low-cost, involves non-mechanical scanning and is highly stable.

3-D surface profilometry based on modulation measurement by applying wavelet transform method

A new analysis of 3-D surface profilometry based on modulation measurement technique by the application of Wavelet Transform method is proposed. As a tool excelling for its multi-resolution and localization in the time and frequency domains, Wavelet Transform method with good localized time-frequency analysis ability and effective de-noizing capacity can extract the modulation distribution more accurately than Fourier Transform method. Especially for the analysis of complex object, more details of the measured object can be well remained. In this paper, the theoretical derivation of Wavelet Transform method that obtains the modulation values from a captured fringe pattern is given. Both computer simulation and elementary experiment are used to show the validity of the proposed method by making a comparison with the results of Fourier Transform method. The results show that the Wavelet Transform method has a better performance than the Fourier Transform method in modulation values retrieval.

Adaptive digital fringe projection technique for high dynamic range three-dimensional shape measurement.

It is a challenge for any optical method to measure objects with a large range of reflectivity variation across the surface. Image saturation results in incorrect intensities in captured fringe pattern images, leading to phase and measurement errors. This paper presents a new adaptive digital fringe projection technique which avoids image saturation and has a high signal to noise ratio (SNR) in the three-dimensional (3-D) shape measurement of objects that has a large range of reflectivity variation across the surface. Compared to previous high dynamic range 3-D scan methods using many exposures and fringe pattern projections, which consumes a lot of time, the proposed technique uses only two preliminary steps of fringe pattern projection and image capture to generate the adapted fringe patterns, by adaptively adjusting the pixel-wise intensity of the projected fringe patterns based on the saturated pixels in the captured images of the surface being measured. For the bright regions due to high surface reflectivity and high illumination by the ambient light and surfaces interreflections, the projected intensity is reduced just to be low enough to avoid image saturation. Simultaneously, the maximum intensity of 255 is used for those dark regions with low surface reflectivity to maintain high SNR. Our experiments demonstrate that the proposed technique can achieve higher 3-D measurement accuracy across a surface with a large range of reflectivity variation.

Pixel-to-pixel correspondence alignment method of a 2CCD camera by using absolute phase map

An alignment method of a 2CCD camera to build pixel-to-pixel correspondence between the infrared (IR) CCD sensor and the visible CCD sensor by using the absolute phase data is presented. Vertical and horizontal sinusoidal fringe patterns are generated by software and displayed on a liquid crystal display screen. The displayed fringe patterns are captured simultaneously by the IR sensor and the visible sensor of the 2CCD camera. The absolute phase values of each pixel at IR and visible channels are calculated from the captured fringe pattern images by using Fourier transform and the optimum three-fringe number selection method. The accurate pixel corresponding relationship between the two sensors can be determined along the vertical and the horizontal directions by comparing the obtained absolute phase data in IR and visible channels. Experimental results show the high accuracy, effectiveness, and validity of the proposed 2CCD alignment method. By using the continuous absolute phase information, this method can determine the pixel-to-pixel correspondence with high resolution.

Simple quasi-common path point diffraction interferometer with adjustable fringe contrast and carrier frequency

This paper presents a simple quasi-common path point diffraction interferometer (PDI) that allows fringe contrast and fringe spatial frequency to be adjusted conveniently. The novel aspect of this PDI is the use of a polarizer with pinhole as the PDI mask - and then, with a compact circular optical setup, reference and measurement waves are obtained. Furthermore, a linear tilting modulation is added into two interfering waves and adjusted easily by tilting the polarizing beam splitter - and hence the Fourier transform method can be perfectly applied to extract the wavefront phase from the captured fringe pattern, with the highest fringe contrast and suitable carrier frequency. Detailed theoretical analysis and experimental results are presented.

Modulation measuring profilometry with auto-synchronous phase shifting and vertical scanning.

To determine the shape of a complex object with vertical measurement mode and higher accuracy, a novel modulation measuring profilometry realizing auto-synchronous phase shifting and vertical scanning is proposed. Coaxial optical system for projection and observation instead of triangulation system is adopted to avoid shadow and occlusion. In the projecting system, sinusoidal grating is perpendicular to optical axis. For moving the grating along a direction at a certain angle to optical axis, 1D precision translation platform is applied to achieve purposes of both phase-shifting and vertical scanning. A series of fringe patterns with different modulation variations are captured by a CCD camera while scanning. The profile of the tested object can be reconstructed by the relationship between the height values and the modulation distributions. Unlike the previous method based on Fourier transform for 2D fringe pattern, the modulation maps are calculated from the intensity curve formed by the points with definite pixel coordinates in the captured fringe patterns. The paper gives the principle of the proposed method, the set-up of measurement system and the method for system calibration. Computer simulation and experiment results proved its feasibility.

Phase error elimination considering gamma nonlinearity, system vibration, and noise for fringe projection profilometry

Fringe projection profilometry (FPP) using a digital video projector is widely used for three-dimensional shape measurement. However, the gamma nonlinearity, system vibration, and noise cause the captured fringe patterns to be nonsinusoidal waveforms and have a grayscale deflection from their true value. This leads to an additional phase measurement error for a general phase-shifting algorithm. Based on the theoretical analysis, we propose a method to eliminate the phase error considering two factors. In this method, four-step phase-shifting is done four times with an initial phase offset of 22.5 deg and the average of these four phase maps precisely results in the real phase. As a result, phase error caused by gamma nonlinearity can be effectively suppressed. In addition, every image in phase shifting is replaced by the average of 20 fringe images continuously captured at the same state to avoid the phase error caused by system vibration and noise. Experimental results show that this method is effective in eliminating the phase error in practical phase-shifting FPP. In general, more than 90% of the phase error can be reduced.

Shrinkage during holographic recording in photopolymer films determined by holographic interferometry

Shrinkage of photopolymer materials is an important factor for their use in holographic data storage and for fabrication of holographic optical elements. Dimensional change in the holographic element leads to a requirement for compensation in the reading angle and/or wavelength. Normally, shrinkage is studied at the end of the polymerization process and no information about the dynamics is obtained. The aim of this study was to use holographic interferometry to measure the shrinkage that occurs during holographic recording of transmission diffraction gratings in acrylamide photopolymer layers. Shrinkage in photopolymer layers can be measured over the whole recorded area by real-time capture of holographic interferograms at regular intervals during holographic recording using a complimentary metal-oxide-semiconductor camera. The optical path length change, and hence the shrinkage, are determined from the captured fringe patterns. Through analysis of the real-time shrinkage curves, it is possible to distinguish two processes that determine the value of shrinkage in the photopolymer layer. These processes are ascribed to monomer polymerization and crosslinking of polymer chains. The dependence of shrinkage of the layers on the conditions of recording such as recording intensity, single or double beam exposure, and the physical properties of the layers, such as thickness, were studied. Higher shrinkage was observed with recordings at lower intensities and in thinner layers. Increased shrinkage was also observed in the case of single beam polymerization in comparison to the case of double beam holographic exposure.