以二維窗函數傅立葉轉換進行干涉條紋相位解析之研發

Research and Development of Two-dimensional Windowed Fourier Transform Based Phase Unwrapping Algorithms for Interference Fringe Analysis

徐冠宇(Kuan-Yu Hsu)

指導教授 ： 李世光

國立臺灣大學/工學院/應用力學研究所/碩士(2015年)

https://doi.org/10.6342/NTU.2015.01437

摘要

本論文所思考的實驗情境，主要藉由邁克森干涉儀(Michelson interferometer)一類的光學非破壞檢測方法，先取得干涉條紋後進行相位解析，以量測待測物之微小變形量。由於將此一研究情境轉換為幾近可即時進行變形檢測地基處瓶頸乃是相位解析，因此本論文研判因為CCD/CMOS等影像擷取晶片的快速發展，使得空間信號調變頻率可以大幅提升，這個趨勢也使得過往運用傅立葉轉換方法為基礎的相位解析技術有了更寬廣的提升空間。因此本論文以Mitsuo Takeda在1981年時，研發之傅立葉轉換法(Fourier transform method)為起點，利用傅立葉轉換來取得干涉條紋影像之頻譜，接著運用預先導入的空間條紋，以類似於都卜勒干涉儀(Doppler interferometer)中運用空間移頻來解決待測物移動方向的不可分辨性(Directional ambiguity)，分辨出正確的變形方向，所以僅只需要一張干涉條紋圖即可恢復其相位資訊，將過去各種如五一相移法(5,1 phase-shifting algorithm)等需要多張影像來進行相位解析方法的計算量龐大的缺點進行改進。隨後並利用Qian Kemao在2004年提出之窗函數傅立葉轉換法(Windowed Fourier Transform; WFT)的概念，來處理信雜比問題。另外，本論文將Mallat在1999年提出的窗函數傅立葉轉換數學式完整地推導，將窗函數的條件嚴謹的使用，並且從推導中可將逆傅立葉轉換時的效果移除，處理窗函數定義域(domain)問題，接著使用最小平方法(Least Squares method; L-S method)來進行相位重建(phase unwrapping)，終能順利快速擷取待測物之量測資訊。值得一提的乃是，前述這個方法在操作時，將變成使用傅立葉變換來求解一個符合Neumann邊界條件的泊松方程式(Poisson’s equation)。
由於本論文所整合開發完成的整個相位計算均奠基於與傅立葉轉換相關的運算，本論文所研究的整個相位解析計算法可運用如自1965年就被積極研究的快速傅立葉轉換(Fast Fourier Transform)、2012年發展出來的稀疏傅立葉轉換（sparse Fast Fourier transform；SFT）、還有如圖形處理器（Graphic Processing Unit，GPU）等硬體加速技術來建構今日的超高速相位解析系統。
本論文在整體的量測上，除了架設邁克森干涉儀外，在軟體分析上結合MATLAB強大的數值運算和LabVIEW特有的人機介面做後端的處理，此方法整合了實驗中的流程，讓實驗量測一氣呵成且執行順暢，希望利用以上的方法能使量測效率提升，達成實驗中即時量測的目標。為了初步驗證演算法的精確度，本論文在實驗上並且進行平面鏡及不鏽鋼薄板變形的相位分析，驗證確認當平面鏡斜率越來越大或量測不鏽鋼薄板之曲度變大時，相位重建的結果能夠成功的反應變形所造成的相位變化，終能成功驗證了本論文所提出演算法的可行性與精確性。

關鍵字

邁克森干涉儀 ； 傅立葉轉換法 ； 窗函數傅立葉轉換法 ； 最小平方法 ； 相位重建

並列摘要

The main application scenario of this thesis is using Michelson interferometer as the non-destructive testing method, which measures tiny deformation of objects by retrieving the phase of the interference fringes generated due to deformation. To retrieve the deformation induced phase, we measure the interference fringe first and then analyze the phase by a process called phase-unwrapping. Trying to take advantages of the rapid development of CCD/CMOS, which provides a platform to significantly increase the spatial modulation frequency, Fourier transform based phase-unwrapping algorithm was adopted in this thesis.
Trying to remove the bottleneck associated with phase-unwrapping, Fourier spectrum of interference fringe obtained with pre-introduced spatial carrier frequency so as to distinguish the correct direction of object deformation, was adopted. This approach first proposed by Mitsuo Takeda in 1981 is similar to the Doppler interferometer that solves the directional ambiguity by providing a frequency shift. The only difference lies on either temporal or spatial frequency was pre-introduced. It is with this pre-introduced spatial frequency shift, retrieving the phase information by using only one interference fringe (intensity map) becomes feasible. More specifically, the above-mentioned approach circumvent the disadvantages associated with phase-shifting algorithms such as the 5,1 phase-shifting algorithm, etc. that require more than one image to analyze the phase information. It is to be noted that retrieve phase map from intensity map with a single intensity map not only saves valuable computation time but also provides us with a platform for dynamic measurement as high-speed camera can be used to record the time-varying interference fringes (intensity maps) first and then compute phase map after the deformation is completed.
Furthermore, in dealing with problems related to valid or effective functional domain (domain with valid interference fringes) and regions with vastly different signal-to-noise ratios (SNR), Windowed Fourier Transform (WFT) first proposed by Qian Kemao in 2004 was also introduced in this thesis. For phase unwrapping, this study used Least-Squares method to get the information of measured object rapidly. It is to be noted that this method leads to the use of Fourier transform to solve a Poisson’s equation with Neumann boundary conditions.
As Fourier transform algorithm was used in converting the intensity map to the phase map and then perform phase-unwrapping, these algorithms developed in this thesis provides us with an opportunity to adopt the many attempts over the last 50 years in speeding up the computation time associated with Fourier transform. Some of these methods include Fast Fourier Transform (FFT), Sparse Fast Fourier Transform (SFFT) and hardware solution such as Graphic Processing Unit (GPU), etc. All of which can then be integrated to develop an ultrafast phase analysis system, which can found applications potentials ranging from in-situ real-time optical field measurement, production-need driven automatic optical inspection (AOI), etc.
This works completed throughout this research include setting the Michelson interferometer, integrating MATLAB and LabVIEW to transfer experimentally induced optical intensity map to computers for signal post-processing, etc. To verify the overall effectiveness of this system, this study analyzed the phase information by measuring the mirror and stainless steel deformation by using the Michelson interferometer set up. The results of unwrapped phase matched the object deformation, successfully validated the accuracy and the feasibility of integrating these algorithms in this thesis.

並列關鍵字

Michelson interferometer ； Fourier transform method ； windowed Fourier transform ； Least squares method ； phase unwrapping

參考文獻

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[7] (2015, June)Sparse Fast Fourier Transform, Available: http://groups.csail.mit.edu/netmit/sFFT/