繞射斷層掃描於有限角度之三維影像重建

Abstract

近年來，在繞射斷層掃描的領域中，有限角度量測的影像重建是一項重要的研究，實際測量時，因系統架構形成的照射角度限制，往往會造成重建影像的失真，在我們的實驗架構中，照射角度的範圍亦受到限制，此研究發展一種新的方法結合凸集合投影法(projection onto convex sets, POCS)與快速迭代收縮閾值法(fast iterative shrinkage thresholding algorithm, FISTA)進行有限角度的影像重建，並比較三種迭代方法的重建結果，包括：限制條件迭代反傅利葉法(constrained iterative Fourier inversion, CIFI)、凸集合投影-最速下降法(projection onto convex sets-steepest descent, POCS-SD)、凸集合投影-快速迭代收縮閾值法(POCS-FISTA)，其中POCS-SD與POCS-FISTA應用全變差最小化法(total variation-minimization algorithm)，全變差最小化法在影像處理的領域中是一種邊緣保留的方法，此方法的優點為去除影像雜訊並保留影像邊緣，根據數值模擬的結果顯示，在未添加雜訊於散射波場的條件之下，此三種迭代方法的重建結果差異不大，當添加雜訊於散射波場時，使用POCS-FISTA的重建結果最接近理想值，POCS-SD居次，其中POCS-FISTA與POCS-SD皆能有效抑制雜訊的影響，而限制條件反傅利葉法對於雜訊的抑制功能不佳，除此之外，在實驗方面亦成功重建真實物體的折射率分布影像，且比較不同方法的重建成果與數值模擬的結果相符。

Image reconstruction from limited-angle data is an important issue in diffraction tomography (DT). The limitation of angular coverage usually occurs due to the physical constraints in measurement systems. Insufficient information will deteriorate the quality of reconstructed images. In our experimental setup, the angular range of the data scanning is limited. Therefore, in this research we developed a new reconstruction approach which consists of POCS and FISTA to resolve the limited-angle problems in DT. Besides, we compared the reconstructed results of three iterative algorithms, including the constrained iterative Fourier inversion method, projection onto convex sets-steepest descent (POCS-SD) and projection onto convex sets-fast iterative shrinkage-thresholding algorithm (POCS-FISTA). POCS-SD and POCS-FISTA utilize the total variation (TV)-minimization technique which is a kind of edge-preserving technique. According to the results of numerical simulation, the performance among these three iterative methods had little difference from noiseless limited-angle data. When Gaussian noise was present in the scattered field, the reconstructed results by POCS-FISTA were closest to the ideal values. Furthermore, both of POCS-FISTA and POCS-SD performed well on de-noising. On the contrary, the constrained iterative Fourier inversion method performed poorly about noise suppression. Finally, we have also successfully reconstructed the refractive index distribution of objects according to the experimental results. Moreover, the comparison of reconstructed results by different methods was consistent with the results of numerical simulation.