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Chapter 3:
Windowed Fourier Filtering for Exponential Phase Fields
The windowed Fourier ridges (WFRn) algorithm can successfully estimate parameters, including local frequencies, phase, and fringe amplitude froma noisy exponential phase field (EPF). In the WFRn, many windowed Fourier kernels gjðxÞ are generated and compared with a real EPF patch. Among these kernels, the one with the highest similarity (ridge) is singled out as the best match. The parameters from the best windowed Fourier kernel are used for our estimation. Although not the best, a number of other kernels are actually quite good (the exact definition of “good” will be defined hereafter). It is of interest to determine whether a team of good members can do a better and faster job than the single best individual. This idea will be explored and developed into a windowed Fourier filtering (WFFn) algorithm, a sister algorithm of the WFRn. Different from the WFRn, which is a parametric algorithm, the WFF is nonparametric. This chapter is organized as follows. The first four sections are devoted to the 1D WFF algorithm, including a description (Section 3.1), error analysis (Section 3.2), implementation and verification (Section 3.3), and performance for a higher-order polynomial phase (Section 3.4). The subsequent four sections are devoted to the 2D WFF with the same structure and include a description (Section 3.5), error analysis (Section 3.6), implementation and verification (Section 3.7), and performance for a higher-order polynomial phase (Section 3.8). Two real examples are given in Section 3.9, and higher dimensions are considered in Section 3.10.
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