An improved windowed Fourier transform for fringe demodulation

A modified algorithm of windowed Fourier transform (WFT) for phase retrieval from electronic speckle-shearing fringe patterns with carriers is proposed. The algorithm is based on the introduction of a fast Fourier transform (FFT) in WFT to reduce computation time for fringe demodulation. Since boundary effects in FFT will influence the accuracy of phase retrieval, the Gerchberg method is employed to extrapolate the fringe pattern at the boundaries to reduce boundary effects. A theoretical analysis of the algorithm is presented. Both simulated and experimental results show that the proposed method has reduced the computation time significantly compared with the convolution method of WFT without sacrificing measurement accuracy.     

A simple phase unwrapping approach based on filtering by windowed Fourier transform

Phase unwrapping is an interesting yet challenging problem in optical interferometry. In this paper, we limit our interest to unwrapping noisy phase maps as it is a common but difficult task. Our aim is to propose a simple solution to phase unwrapping. We first remove the noise by a novel windowed Fourier transform approach and then use sequential line scanning method for phase unwrapping. This simple approach is verified to be very effective.     

A simple phase unwrapping approach based on filtering by windowed Fourier transform: The phase near edges

A simple phase unwrapping approach based on windowed Fourier filtering (WFF) was proposed recently. In this paper, the effects of the WFF on edges are further investigated, which show that the phase near an edge remains unchanged and no special consideration is needed for the edges. This useful property will be theoretically proven and numerically verified.     

Applications of windowed Fourier fringe analysis in optical measurement: A review

The applications of windowed Fourier fringe analysis in the past decade are reviewed. Because fringe patterns from different optical measurement systems are similar, the reviewed applications are classified according to the functions of the windowed Fourier transform being used in fringe pattern analysis: denosing exponential phase fields, demodulating carrier fringe patterns, getting phase derivatives, and utilizing local properties. From these applications, the windowed Fourier transform is shown to be effective and versatile for fringe pattern analysis.     

A simple phase unwrapping approach based on filtering by windowed Fourier transform: A note on the threshold selection

A simple phase unwrapping approach based on windowed Fourier filtering was proposed recently [K. Qian et al. A simple phase unwrapping approach based on filtering by windowed Fourier transform. Opt Laser Technol 2005;37:458–62]. The windowed Fourier filtering algorithm is an essential ingredient that suppresses the noise effectively and makes the phase unwrapping trivial. This paper adds a note on the threshold selection in the windowed Fourier filtering algorithm. A large interval can be selected as the threshold to obtain almost optimal filtering results. Once the selected threshold is suitable, it is almost optimal. This makes the threshold selection in the windowed Fourier filtering algorithm extremely easy.     

Windowed Fourier assisted two-dimensional Hilbert transform for fringes phase extraction

In this article, two-dimensional windowed Fourier assisted Hilbert transform for fringes phase extraction has been proposed. The mathematical derivations of the filtered fringes image and its Hilbert transformations are presented using windowed Fourier transform. The proposed method has been verified by calculating the wrapped and unwrapped phase of the experimental moire fringes. In comparison to the conventional fringes phase extraction methods, the proposed method is found to be effective.     

Orthogonal projection technique for resolution enhancement of the Fourier transform fringe analysis method

The spatial resolution of the phase map in the Fourier transform fringe analysis method is determined by the size of the filter’s window in frequency domain. This article reports a straightforward technique to improve the method’s resolution by a factor of nearly two. The technique requires capturing a second image with a fringe pattern orthogonal to the first one, therefore using the information from both patterns to eliminate the central component in frequency space. The resulting spectrum supports double sized filter windows for removal of the carrier frequency without leaking into adjacent orders. The overall spatial resolution of the method is thus increased. In the following, the Fourier fringe analysis method is briefly reviewed, the new technique is described and analyzed and the experimental results are shown and discussed.     

Two-dimensional cell parameters measurement of nematic liquid crystal using optical interferometry and Fourier transform fringe analysis technique

We report a method for the measurement of nematic liquid crystal (NLC) cell parameters i.e., switching voltage, birefringence, retardation, dielectric anisotropy, average tilt angle and change in refractive index with applied DC voltage to LC material. The proposed method is based on optical interferometry and Fourier transform fringe analysis technique, in which we obtain 2-dimensional (2D) phase map of the interferograms as a function of applied voltage. Mach-Zehnder interferometer (MZI) was used for the study of cell parameters and interferograms were recorded at different applied DC voltages to NLC cell using CCD camera. From the phase map, 2D-refractive index distribution of the LC cell with applied voltage was reconstructed. Analytical equations are derived based on optical interferometry and then solved to obtain cell parameters. The present method is fast and can give 2D-cell parameters from only two quick interferograms.     

Parallel computing for fringe pattern processing: A multicore CPU approach in MATLAB environment

In the process of measurements such as optical interferometry and fringe projection, an important stage is fringe pattern analysis. Many advanced fringe analysis algorithms have been proposed including regularized phase tracking (RPT), partial differential equation based methods, wavelet transform, Wigner–Ville distribution, and windowed Fourier transform. However, most of those algorithms are computationally expensive. MATLAB^® is a general algorithm development environment with powerful image processing and other supporting toolboxes. It is also commonly used in photomechanical data analysis. With rapid development of multicore CPU technique, using multicore computer and MATLAB^® is an intuitive and simple way to speed up the algorithms for fringe pattern analysis. The paper introduces two acceleration approaches for fringe pattern processing. The first approach is task parallelism using multicore computer and MATLAB^® parallel computing toolbox. Since some algorithms are embarrassing problems, our first approach makes use of this characteristic to parallelize these algorithms. For this approach, parallelized windowed Fourier filtering (WFF) algorithm serves as an example to show how parallel computing toolbox accelerates the algorithm. Second, data parallelism using multicore computer and MATLAB^® parallel computing toolbox is proposed. A high level parallel wrapping structure is designed, which can be used for speeding up any local processing algorithms. WFF, windowed Fourier ridges (WFR), and median filter are used as examples to illustrate the speedup. At last, the results show that the parallel versions of former sequential algorithm with simple modifications achieve the speedup up to 6.6 times.     

Conversion from phase map to coordinate: Comparison among spatial carrier, Fourier transform, and phase shifting methods

For full-field phase measurement methods, many algorithms have been developed to extract a phase map from fringe image(s). Both phase wrapping and unwrapping algorithms have been extensively investigated by many researchers, but few papers can be found on how to calculate the coordinates of surface points from a phase map. This paper focuses on algorithms that show how a phase map can be used to calculate coordinates. Details are given for single image methods such as Fourier transform, spatial carrier methods, and multiple image methods like traditional phase-shifting methods. Algorithms that can be used to convert a phase map to coordinates and some issues related to these conversion algorithms are discussed. An artifact is measured using these phase measurement methods. The results show that using the correct algorithm to convert a phase map to coordinates is a key to obtaining accurate measurement results.     

Comment on the paper: H.-E. Hwang, P. Han, Theoretical analysis for surface tilt and translation detection based on speckle photography in the Fresnel domain, Opt. Commun. 282 (2009) 351-354

In a recent paper [H.-E. Hwang, P. Han, Opt. Commun. 282 (2009) 351] a speckle based metrology system is proposed which it is claimed provides significant advantages over existing systems. In this paper, we show that the discussion presented in [H.-E. Hwang, P. Han, Opt. Commun. 282 (2009) 351] is deficient, and that several of the statements made are incorrect and/or misleading.