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Chapter 2:
Windowed Fourier Ridges for Exponential Phase Fields
Windowed fringe pattern analysis starts its journey from exponential phase fields (EPFs - the T1 fringe patterns) in this chapter. Quality guidance assumes its duty from phase unwrapping in Chapter 4. EPFs can be constructed through either phase-shifting or Fourier transform techniques. Out of the five difficulties in fringe pattern analysis, the ill-posedness problem (D1) and the sign ambiguity problem (D2) do not exist in EPFs; the order ambiguity problem (D3) exists but is discussed in Chapter 4. The noise problem (D4) is solved in this chapter, and with its solution, a clean wrapped phase map can be extracted from a noisy EPF. The discontinuity problem (D5) will be briefly discussed in Section 2.9. The phase within a small window is assumed to be a quadratic polynomial. If the coefficients of the polynomial are found, the phase is completely determined. The denoising problem thus turns into a parameter estimation problem from a noisy observation. A simple algorithm called windowed Fourier ridges (WFR) is introduced for parameter estimation. A 1D algorithm is introduced first for 1D EPFs such as temporal data. The algorithm is extended to 2D, which is most commonly used, and finally is extended to higher dimensions. This chapter is organized as follows. The first four sections are devoted to 1D EPFs, including the problem statement in Section 2.1, the WFR algorithm in Section 2.2, error analysis in Section 2.3, and implementation and verification in Section 2.4. The subsequent four sections are devoted to 2D EPFs with the same structure, including the problem statement in Section 2.5, the 2D WFR (WFR2) algorithm in Section 2.6, error analysis in Section 2.7, and implementation and verification in Section 2.8. Two real examples are given in Section 2.9, where phase discontinuity will be briefly mentioned. Higher dimensions are considered in Section 2.10.
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