Two-dimensional windowed Fourier transform for fringe pattern analysis: Principles, applications and implementations
Introduction
In optical metrology, the output is usually in the form of a fringe pattern, which should be further analyzed [1], [2], [3], [4]. For example, phase retrieval from fringe patterns is often required. Two traditional techniques for phase retrieval are phase-shifting technique [1], [5] and carrier technique with Fourier transform [1], [6]. Phase-shifting technique processes the fringe patterns pixel by pixel. Each pixel is processed separately and does not influence the others. However, this technique is sensitive to noise. As an example, Fig. 1(a) shows one of four phase-shifted fringe patterns. Phase extracted using phase-shifting algorithm is shown in Fig. 1(b), which is obviously very noisy. On the contrary, carrier technique with Fourier transform processes the whole frame of a fringe pattern at the same time. It is more tolerant to noise, but pixels will influence each other. As an example, A carrier fringe pattern and its phase extracted using Fourier transform are shown in Fig. 2(a) and (b), respectively. A better result, if possible, is expected.
Thus a compromise between the pixel-wise processing and global processing is necessary. A natural solution is to process the fringe patterns locally, or block by block. A smoothing filter is a typical local processor [7]. It assumes that the intensity values in a small block around each pixel are the same and hence the average value of that block is taken as the value of pixel . Obviously it is not reasonable for a fringe pattern since its intensity undulates as a cosine function (see next paragraph). Because of this, more advanced and effective techniques, such as regularized phase tracking (RPT) [4], [8], [9], [10], [11], [12], wavelet transform [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], Wigner–Ville distribution [19], [24] and windowed Fourier transform (WFT) [19], [25], [26], were proposed. In this paper, principle of WFT will be emphasized and compared with other techniques. Then various applications of WFT and the implementation issues will be introduced. Fig. 1, Fig. 2 show the effectiveness of WFT at a first glance. Again it is emphasized that most of the work has appeared in various papers and its originality is not claimed. Instead, this paper gives an overview and more insights of our work on WFT.
Before further discussion, a brief definition and analysis of fringe patterns are given, which will be used throughout the paper. A fringe pattern can be generally expressed aswhere , , and are the recorded intensity, background intensity, fringe amplitude and phase distribution, respectively. Fringe patterns are classified into four types: (I) exponential phase fringe patterns, (II) wrapped phase fringe patterns, (III) carrier fringe patterns and (IV) closed fringe patterns. Exponential phase fringe patterns, which are analytic signals, are fundamental for fringe processing [27] and are basic patterns considered in this paper. They can be obtained from, say, phase shifting technique. For example, given four phase-shifted fringe patterns as the combination of giveswhere . The phase can be obtained by taking the angle of , which is usually very noisy. Sometimes wrapped phase maps are given. A typical example is phase unwrapping. This type of fringe patterns can be written aswhere denotes a wrapped phase map. It can be easily converted to fI by simply multiplying it with j and taking its exponential value. In this paper, fII is always converted to fI before any processing. The third type, carrier fringe patterns, can be written aswhere and are carrier frequencies along x and y directions, respectively. They are usually constants. Since , fIII consists of a background field and two conjugated fringe patterns of fI. These three items are separable in Fourier domain provided that the carrier frequencies are high enough. Thus, it can also be converted to fI. However, this conversion is unnecessary as the separation is realized automatically in the WFT algorithms. The fourth type, closed fringe patterns, can be written as
It also consists of a background field and two conjugated fringe patterns of fI, but they are not separable. Thus converting fIV to fI is generally not easy. Other cues, such as a fringe follower [12] or fringe orientation [28], are required. Note that though the expressions of Eqs. (1), (5) are the same, the former refers to general fringe patterns while the latter refers to fIV.
Local frequencies (or instantaneous frequencies) are used to express phase derivatives as
Phase distribution in a block around a pixel can thus be approximated as a small plane,
For fIII, the carrier frequencies are usually included into the local frequencies. Note that there is slight difference in the definition of instantaneous frequencies [29]. Also note that in this paper, and are sometimes used interchangeably if no confusion is raised.
Section snippets
Principles of windowed Fourier transform
In this section, the WFT is first introduced, based on which, two algorithms, windowed Fourier filtering (WFF) and windowed Fourier ridges (WFR) are developed and discussed. To process the fringe patterns block by block, the WFT is by default two-dimensional (2-D) throughout this paper.
Phase and frequency retrieval by WFF and WFR
It can be seen from Section 2 that the phase distribution and local frequencies can be extracted. Fig. 1, Fig. 2 show two examples, which can be obtained by either WFF or WFR with very similar results. The functions of WFF and WFR are illustrated in Fig. 6, Fig. 7. Suggestion for the selection of frequency ranges is given in Table 1. Following are some notations.
- (1)
The accuracy of WFF and WFR for fI, fII and fIII was investigated in Ref. [25]. It is typically below one fiftieth of a wavelength for
Implementations of WFF and WFR
Implementation issues are addressed in this section so that the results in this paper are reproducible and the algorithms can be readily tested by the readers [65]. Define it is easy to rewrite the important equations in Section 2 equivalently as follows:
Conclusions
Demodulation of fringe patterns is necessary for many optical metrological systems. Systematic solutions using time–frequency analysis are provided. Two-dimensional windowed Fourier transform is chosen for the determination of phase and phase derivatives and it is compared with other time–frequency techniques. Two approaches are developed, one is based on the concept of filtering the fringe pattern and the other is based on the similarity measure between the fringe pattern and windowed Fourier
Acknowledgments
I would like to express my sincere gratitude to Prof. Seah Hock Soon and Prof. Anand Asundi of the Nanyang Technological University and Prof. Wu Xiaoping of the University of Science and Technology of China for their encouragement and to the reviewers for their helpful comments.
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