運用窗函數及稀疏傅立葉轉換於干涉條紋解相之研發

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

李晨宇(Chen-Yu Lee)

指導教授 ： 李世光

國立臺灣大學/工學院/工程科學及海洋工程學研究所/碩士(2016年)

https://doi.org/10.6342/NTU201602347

摘要

本論文的研究領域屬於三維表面形貌量測技術，其應用範圍相當廣泛，例如半導體產業的元件封裝、光電產業的鏡面形狀與粗糙度量測、通訊產業的微機電裝置檢測、以及生醫產業的細胞外觀輪廓觀察等等。此技術目前最重視的就在檢測精準度和速度，希望透過精準度的提高和檢測分析時間的縮短達到產線生產成本降低的目標。因此，本研究也分成兩大部分：一是表面形貌量測技術之精準度的提升，二是介紹新的訊號分析技術-稀疏傅立葉轉換(Sparse Fourier Transform, SFT)，並探討SFT在現有干涉條紋圖分析法方面的可適性。
本研究的表面形貌量測技術是針對一張干涉條紋圖進行傅立葉轉換，並透過提取其頻譜中的相位資訊重建出物體表面的形貌。雖然這種屬於全域性轉換的傅立葉轉換技術能降低雜訊對頻譜的影響，但此方法仍面臨應用過程中因為像素相互影響造成的頻譜混疊或濾波器設計的不完善等因素而造成的量測精準度不足問題。
為解決此問題，本研究採取了窗函數傅立葉轉換法（Windowed Fourier Transform, WFT）能針對頻譜局部分析的特色。首先，用一個移動窗函數將整個條紋圖切割成許多塊局部條紋，並以快速傅立葉轉換(Fast Fourier Transform, FFT)取得每塊局部條紋圖的頻譜；然後透過所有區塊頻譜的疊加，就可獲得完整條紋的頻譜分布。比起前述FT全域頻譜技術，此方法所採用的局部頻譜分析擁有更簡單的結構，也更不容易發生頻譜混疊現象。此外，藉由適當的閥域值設定，也能有效地降低雜訊影響。
本論文利用麥克森光學干涉實驗數據以及在MATLAB平台上進行的表面形貌條紋投影模擬，驗證了WFT在形貌重建精準度上的效能提升。值得注意的是，此WFT技術的九成運算時間都用於窗函數進行FFT頻譜提取；但窗函數頻域卻只存在少數非零數值，且只出現於特定座標位置上。因此，本論文的第二部分研究重點就放在如何有效率地處理此種稀疏分布的窗函數頻域，也就是2012年才提出的SFT技術。研究中將針對稀疏傅立葉轉換進行系統化理論說明，介紹其演算法的理論架構及其演算法重建訊號的誤差準則，以及MIT Hassanieh等人最新提出的重建訊號頻譜方法。此外，也搭配一些MATLAB演算模擬範例描述了統整演算法所可能面臨的關鍵問題與解決技術，其中包括頻譜隨機重新排列、平坦窗函數濾波器的設計和降取樣FFT。此技術顯示以SFT取代窗函數FFT運算將在條紋分析應用方面大幅縮短干涉圖案解相的整體運算時間。

關鍵字

干涉條紋分析 ； 傅立葉轉換法 ； 窗函數傅立葉轉換法 ； 稀疏傅立葉轉換法 ； 頻譜隨機重排 ； 平坦窗函數

並列摘要

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.

並列關鍵字

fringe pattern analysis ； Fourier transform method ； windowed Fourier transform ； sparse Fourier transform ； Spectrum permutation ； flat window Function

參考文獻

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