Research and Development of Windowed Fourier Transform & SFT Algorithm on Phase Retrieval for Interference Fringes Analysis

This thesis is related to three-dimensional metrology system, a widely used technique that has been applied in monitoring semiconductor elements packing, shape/roughness of optical lenses/mirrors, microelectromechanical devices, panel geometry checking, and the cell morphology. The two most significant performance factors in morphology technique are precision and speed, and the corresponding improvements, i.e. enhancing the measurement accuracy and reducing the processing time can ensure the production quality and lower its costs. Consequently, there are two major aspects in this thesis. One is the accuracy improvement of metrology techniques, and the other is the introduction of a new signal analysis method called Sparse Fourier Transform (SFT) and exploring its applicability in the current fringe analysis techniques. The metrology technique used in this research starts with the Fourier transformation of a single frame of fringe pattern, and reconstruct the object morphology through the phase information obtained from its spectrum. Although the noise effect on the spectrum can be reduced by using this full-field based FT algorithms, there still exist some drawbacks on the accuracy due to either frequency aliasing caused by pixels’ interactions or the imperfect design of the filters. To solve the problem mentioned above, we take the advantage of local spectral analysis used in Windowed Fourier Transform (WFT). Firstly, the original spectrum of fringes pattern is partitioned into many small blocks by a moving window function, and the frequency spectrum of each block is obtained by Fast Fourier Transform (FFT). And then, the final spectrum is achieved by adding up the spectrum of each block. As compared to the conventional full-field FT, this local analysis based technique is simpler in its structure and less affected by aliasing. In addition, by a proper choice of cutoff frequency, this technique can still guarantee an effective noise reduction. By comparing with the experimental pattern obtained from Michelson interferometer and simulated pattern generated on the MATLAB coding, we have proved the WFT’s advantages on the accuracy of morphology reconstruction. It is worth noting that 70% of computer processing time is devoted to FFT of window function, whereas there are only a few non-zero values on the frequency spectrum of window function and exist only on specific coordinates. Hence, the major content of the 2nd part in this thesis is focusing on an efficient processing theory proposed since 2012, i.e. SFT, in dealing with these sparsely distributed spectrum. We’ll systemically introduce the theoretical aspects of SFT, its algorithm’s theoretical structure, the error guarantee of signal recovery, as well as the signal reconstruction technique newly proposed by MIT’s Hassanieh. Besides, by implementing a few MATLAB simulation examples, we also discussed some key technique problems and possible solutions such as spectrum permutation, subsampled FFT and the design of flat window function. It appears that replacing window functions’ FFT by SFT is very likely to greatly reduce the processing time of fringe pattern analysis.