Wrapped Phase Map Research Articles

Unsupervised-clustering-driven noise-residue filter for phase images

We analyze the characteristics of noise-induced phase inconsistencies, or residues, in a wrapped phase map. Because residues are the potential source of phase-error propagation, it is essential to filter them before two-dimensional phase unwrapping. We propose an unsupervised-clustering-driven noise-residue filter, and apply it as a preprocessing procedure of phase unwrapping. The filter is based on the fact that most residues are present in the form of adjacency caused by noisy wrapped phases. These noisy phases differ from the other correct ones numerically within the local k(1) x k(2) window containing the adjacent residues, and it is possible to group the correct and noisy wrapped phases into different clusters. The window size is determined adaptively according to the local noise level. The proposed procedure avoids constructing branch cuts, and converts path-following unwrapping to path independence, which improves the operating speed of phase unwrapping significantly. The tests performed on simulated and real projected fringe patterns confirm the validity of our approach in residue reduction, fringe preservation, and rapidity.

Unwrapping techniques for speckle phase maps: study and comparison of the performance of principal methods

Holographic interferometry (HI) techniques are powerful tools for experimental mechanics which can be used for displacement fields assessment on a loaded object surface. Used with no contact and high resolution HI can be combined with image processing methods for data analysis. In these techniques the speckle should be fully resolved by the image detector to record an interferogram. This leads to noisy recordings which makes it's post-processing much difficult. Each measurement with HI ends normally in a phase map computation representing the object displacement. Phase calculation algorithms are based on spatial or temporal phase modulation and lead to wrapped phase maps. To obtain the continuous phase distribution several solutions have been implemented. This paper describes the main filtering and unwrapping algorithms available and compares it's performance to solve discontinuities in noisy recordings.

Phase-retrieval techniques in fringe-projection profilometry

This paper discusses several phase-retrieval techniques commonly used in fringe-projection profilometry (FPP). The FPP has become a suitable and powerful method for the measurement of object surface shape, and some algorithms have been proposed to extract phase maps, such as Fourier transform profilometry (FTP). In this paper, FTP, phase-shifting and continuous wavelet transform (CWT) are briefly described and discussed. In addition, two-dimensional short-time Fourier transform (STFT) is also presented to extract wrapped phase maps. Simulations and experimental results are presented to show the performance of these phase-retrieval techniques in FPP.

Analysis of reconstructed interference fields in digital holographic interferometry using the polynomial phase transform

A noisy wrapped phase map is the end-output of commonly employed phase estimation methods in digital holographic interferometry. Hence filtering and unwrapping are necessary to obtain continuous phase distributions. This paper introduces a new approach for phase estimation in digital holographic interferometry using the polynomial phase transform. The proposed approach directly provides an accurate estimation of the unwrapped phase distribution from a noisy reconstructed interference field, thereby bypassing cumbersome and error-prone filtering and 2D phase unwrapping procedures.

Comparative study on determining the refractive index profile of polypropylene fibres using fast Fourier transform and phase-shifting interferometry

The refractive index profile measurement is desirable for fabricating and characterizing fibres. The fast Fourier transform method (FFT) and phase-shifting interferometry (PSI) were used to determine the refractive index profiles of polypropylene (PP) fibres. Interference fringe patterns produced by Mach–Zehnder interferometer have been analysed using the suggested methods. Wrapped phase maps and contour lines of the obtained fringes were produced. The methods used are relatively simple, accurate and are capable of conducting fully automatic measurement of refractive index profiles for fibres. By comparing the obtained refractive index profiles using the two methods, it is found that there is good agreement between them and also with the literature.

Determination of displacement derivative in digital holographic interferometry

Digital holography (DH) and digital shearography (DS) both play an important role in non-destructive evaluation. In this paper, a novel method based on digital holographic interferometry (DHI) and complex phasor (CP) is proposed to determine displacement derivative. An algorithm is employed to filter the imaginary and real parts of complex values without the need of direct phase manipulation. Two-dimensional short time Fourier transform (STFT) is employed subsequently to process wrapped phase maps. An experiment is conducted to demonstrate the validity of the proposed method.

Dual-wavelength image-plane digital holography for dynamic measurement

A dual-wavelength image-plane digital holography, combined with a windowed Fourier analysis, is presented for dynamic measurement of a vibrating object. In order to expand the range of the velocity measurement, the object is simultaneously illuminated by two lasers with different wavelengths. A sequence of digital holograms of a vibrating object is captured by a CCD camera and two wrapped phase maps are retrieved from each digital hologram. At each instant, a new phase distribution with a synthetic wavelength is obtained by subtracting these two wrapped phase maps. A windowed Fourier analysis is then applied spatially and temporally to retrieve the high-precision displacement and velocity of the tested object. Experimental results show the requirement on the camera capture frequency is reduced by the proposed method.

Windowed Fourier-filtered and quality-guided phase-unwrapping algorithm

We propose a windowed Fourier-filtered and quality-guided phase-unwrapping algorithm that is an extension and improvement of our previous phase-unwrapping algorithm based on windowed Fourier transform [Opt. Laser Technol.37, 458 (2005), Key Eng. Mater.326-328, 67 (2006). First, the filtered amplitude is used as a real-valued quality map, rather than a binary mask, which makes the phase-unwrapping algorithm more tolerant to low-quality regions in a wrapped-phase map, and the process is more automatic. Second, the window size selection is considered, which enables the algorithm to be adapted to tackle different phase-unwrapping problems. A large window size is useful for removing noise, building long barriers along phase discontinuities, and identifying invalid regions, while a small window size is useful for preserving local features, such as small regions and high-quality narrow channels. Eight typical examples in Ghiglia and Pritt's excellent book Two-Dimensional Phase Unwrapping: Theory, Algorithm and Software (Wiley, 1998) are used to evaluate the proposed algorithm. The proposed algorithm is able to unwrap all these examples successfully. The windowed Fourier ridges algorithm, another algorithm based on windowed Fourier transform, is also tested and found to be useful in building barriers along phase discontinuities.

Microscopic TV holography for MEMS deflection and 3-D surface profile characterization

Recent industrial demands for greater product quality in the fields of microelements and micro-electro-mechanical systems (MEMS) generate new challenges for metrology. The fast-growing MEMS industry requires a robust non-destructive quantitative measurement system for the characterization of their performance, reliability and integrity. A microscopic TV holographic system using a long working distance microscope with an extended zoom range has been developed for microelements and MEMS deformation and 3-D surface profile analysis. The system is capable of evaluating both rough and smooth surfaces. Noisy wrapped phase map is a usual problem in speckle interferometry. We have compared several phase-shifting algorithms for evaluation of speckle phase for their usefulness in generating less-noisy phase maps. The experimental results on a MEMS pressure sensor for out-of-plane deflection and 3-D surface profile analysis are presented.

Measurement of curvature and twist of a deformed object using digital holography

Measurement of curvature and twist is an important aspect in the study of object deformation. In recent years, several methods have been proposed to determine curvature and twist of a deformed object using digital shearography. Here we propose a novel method to determine the curvature and twist of a deformed object using digital holography and a complex phasor. A sine/cosine transformation method and two-dimensional short time Fourier transform are proposed subsequently to process the wrapped phase maps. It is shown that high-quality phase maps corresponding to curvature and twist can be obtained. An experiment is conducted to demonstrate the validity of the proposed method.

A modified fourier‐based phase unwrapping algorithm with an application to MRI venography

To present a single-step deterministic procedure for unwrapping MRI phase maps. Using an algorithm previously developed for optical applications, Laplacian operators were applied in the Fourier space of the MRI phase map. The original Fourier-based phase unwrapping algorithm was modified so that demodulation accomplished the required signal symmetrization in Fourier space. To evaluate the method's performance in the presence of thermal noise, a set of wrapped phase maps were simulated at different levels of noise in k-space, and the response of the algorithm at different levels of signal-to-noise ratio (SNR) was evaluated for stability. To demonstrate its utility in MRI, the algorithm was applied to the wrapped phase maps of susceptibility-weighted imaging (SWI) studies, which were then used to generate venograms. In simulated phase wrapping, the algorithm correctly reproduced the original phase for a wide range of phase gradients and noise. The procedure was fast and produced useful maps of venous structures in SWI images. A fast and stable single-step deterministic method for unwrapping MRI phase maps is available for such applications as SWI and mapping of static magnetic field inhomogeneity.

A novel high-speed 3D profilometry for complex surface measuring

Phase unwrapping is an important process in Fourier Transform Profilometry (FTP). Algorithm based on twice different precision measurements is one of the efficient solutions. But it requires at least two wrapped phase maps, which make impossible to apply in dynamic and real time 3D measurement. To overcome the obstacle, we present a novel method, in which FTP and colour fringe projections are combined to acquire two wrapped phase maps by a single snapshot of object. Thus algorithm based on twice different precision measurements can be accomplished with one exposure and dynamic 3D contouring can be implemented.

Absolute Fringe Order Determination in Digital Photoelasticity

An improvement of the already proposed phase unwrapping algorithm for directly processing the triangular-typed wrapped phase map of the isochromatic parameter, photoelastically generated by an arccosine function based on the well-known technique of phase-shifting, is presented. In phase unwrapping algorithm, at any pixel, a three-dimensional plane and its normal vector are generated using three fractional (relative) fringe order values. Its generated plane is then adjusted using the orthogonal projection associated with the reference plane. Such point is unwrapped on the basis of regularization. The quality guide map is used to guide phase unwrapping and to mask out the conflictive regions such that they are lastly processed. The circular disk under compression demonstrated the performance of the improved algorithm. Results showed the accuracy improvement in such conflictive regions with reasonable agreement to theory.

Kinematic and deformation parameter measurement by spatiotemporal analysis of an interferogram sequence

In recent years, optical interferometry has been applied to the whole-field, noncontact measurement of vibrating or continuously deforming objects. In many cases, a high resolution measurement of kinematic (displacement, velocity, and acceleration, etc.) and deformation parameters (strain, curvature, and twist, etc.) can give useful information on the dynamic response of the objects concerned. Different signal processing algorithms are applied to two types of interferogram sequences, which were captured by a high-speed camera using different interferometric setups: (1) a speckle or fringe pattern sequence with a temporal carrier and (2) a wrapped phase map sequence. These algorithms include Fourier transform, windowed Fourier transform, wavelet transform, and even a combination of two of these techniques. We will compare these algorithms using the example of a 1D temporal evaluation of interferogram sequences and extend these algorithms to 2D and 3D processing, so that accurate kinematic and deformation parameters of moving objects can be evaluated with different types of optical interferometry.

Measurement of microstrains across loaded resin–dentin interfaces using microscopic moiré interferometry

Little is known about the mechanical behavior of resin–dentin interfaces during loading. The presence of relatively compliant hybrid and adhesive layers between stiffer dentin and resin composite should deform more during compressive loading. Objective The objective of this study was to measure changes in microstrain across bonded dentin interfaces in real time using a recently developed microscope moiré interferometer. Method This system used a miniature moiré interferometer, using two CCD cameras for simultaneous recording of longitudinal and transverse deformation fields, a piezotransducer for fringe shifting and use of a microscope objective with magnification up to 600×. Small beams (1 mm × 2 mm × 6 mm) of moist resin-bonded dentin covered with cross-lined diffraction grating replica were placed in a miniature compression tester to allow controlled loading from 2 to 37 N while imaging the interference fringe patterns. Results Resin-dentin interfaces created by bonding dentin with Single Bond/Z100, under compressive loading, exhibited comparatively large strains across the adhesive–hybrid interface. When the wrapped phase maps were unwrapped to permit conversion of fringe order to displacements, the 2-D profiles of strain fields revealed non-uniform strains across the adhesive interface. In the adhesive/hybrid layer zone, the negative strain was larger (i.e. −6000 μɛ) than that seen in the adjacent resin composite or underlying mineralized dentin. The changes were elastic because they disappeared when the load was removed. Significance Microscopic moiré interferometry can be very useful in revealing real-time strain across bonded interfaces under load.

Digital speckle interferometry for assessment of surface roughness

In this work, the principle of interferometry is used to assess the surface roughness of the machined surfaces. Interferometry produces an interference fringe pattern when two or more light waves interact with each other. It is one of the important tool for precision optical metrology and testing. Well-known advantages of the phase shifting interferometry include high measurement accuracy, rapid measurement, good result even with low contrast fringes and that the polarity of the wave front can be determined. In fringe projection techniques, a known optical fringe pattern is projected onto the surface of interest. The fringe pattern on the surface is perturbed in accordance with the profile of the test surface, thereby enabling direct derivation of surface profile. In this work, an attempt has been made to assess the surface roughness using a speckle fringe analysis method of five frame phase shift algorithm for machined surface (ground surface). As these fringes are too noisy, advanced filtering technique has been used so as to reduce noise and to get improved wrapped phase map from the phase shifted fringes. A phase unwrapping software has been developed using discrete cosine transform (DCT) to generate the three-dimensional (3-D) profiles. Finally, it is compared with R a values measured using a mechanical stylus instrument, showing good agreement.

Surface strain measurement using multi-component shearography with coherent fibre-optic imaging bundles

Shearography is a full-field interferometric speckle technique used to determine displacement derivatives. Measurement of surface strain is possible using shearography if six components of displacement gradient are calculated. This can be achieved using shearography instrumentation that incorporates at least three measurement channels combined with two orthogonal shear directions. This paper presents a laser shearography instrument that utilizes coherent imaging fibre bundles to port four spatially multiplexed speckle images to a single CCD camera via a shearing Michelson interferometer. The four images are spatially multiplexed onto the sensor of a CCD camera. Wrapped phase maps are derived from the recorded speckle interferograms using temporal phase stepping. The unwrapped phase maps are combined with the measurement channel sensitivity vectors using a matrix operation to determine the required displacement derivatives. Results from an out-of-plane displacement of a flat aluminium plate are presented and compared with a computational model. Results from a second test object that show in-plane and out-of-plane strain components are also shown.

Comparative analysis on some filters for wrapped phase maps

Some effective filtering methods for wrapped phase maps, a regularized phase-tracking method (RPT) without the regularization term, a multiple-parameter least-square method (MPLS), a windowed Fourier ridges method (WFR), an autocorrelation function method (ACF), and a sine/cosine average filter (SCAF), are analyzed in order to establish their transversal relationship. The analysis shows that principles of the RPT, MPLS, WFR, and ACF are equivalent and the SCAF also leads to the WFR by some extension, which elegantly unifies all these methods for filtering unwrapped phase maps.

Fast and robust three-dimensional best path phase unwrapping algorithm

What we believe to be a novel three-dimensional (3D) phase unwrapping algorithm is proposed to unwrap 3D wrapped-phase volumes. It depends on a quality map to unwrap the most reliable voxels first and the least reliable voxels last. The technique follows a discrete unwrapping path to perform the unwrapping process. The performance of this technique was tested on both simulated and real wrapped-phase maps. And it is found to be robust and fast compared with other 3D phase unwrapping algorithms.

Phase map retrieval in digital holography: avoiding the undersampling effect by a lateral shear approach

In digital holography (DH) the numerical reconstruction of the whole wavefront allows one to extract the wrapped phase map mod, 2 pi. It can occur that the reconstructed wrapped phase map in the image plane is undersampled because of the limited pixel size in that plane. In such a case the phase distribution cannot be retrieved correctly by the usual unwrapping procedures. We show that the use of the digital lateral-shearing interferometry approach in DH provides the correct reconstruction of the phase map in the image plane, even in extreme cases where the phase profile changes very rapidly. We demonstrate the effectiveness of the method in a particular case where the profile of a highly curved silicon microelectromechanical system membrane has to be reconstructed.