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研究生:林瑞琳
研究生(外文):Jui-Lin Lin
論文名稱:X 射線轉換之相位恢復問題
論文名稱(外文):X-ray transform in phase retrieval
指導教授:陳鵬文陳鵬文引用關係
指導教授(外文):Peng-Wen Chen
口試委員:凃瀞珽魏秀娟
口試日期:2022-05-12
學位類別:碩士
校院名稱:國立中興大學
系所名稱:應用數學系所
學門:數學及統計學門
學類:數學學類
論文種類:學術論文
論文出版年:2022
畢業學年度:110
語文別:中文
論文頁數:38
中文關鍵詞:X 射線轉換電腦斷層掃描相位恢復交替投影法共軛梯度法
外文關鍵詞:X-ray transformcomputed tomographyphase retrievalretrieval;alternating projection methodconjugate gradient method
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電腦斷層掃描利用了X 射線轉換讓使用者能不使用侵入性方法就能了解目標物的內部構造。由於物體的密度不同會影響X 射線衰減的程度,因此X 射線轉換利用此特性去收集各角度下目標物的傅立葉強度並進行還原。由於探測器無法收集相位資料,因此在還原目標物時需要還原相位,所以還原的過程也被稱作相位恢復。本文使用交替投影法與共軛梯度法進行還原。
Instead of invasive methods, users can observe internal structural details through computed
tomography using X-ray conversion.Since the degree of X-ray attenuation is easily
affected by the density of the object, X-ray transform collects the Fourier intensity of the target at each angle and perform the inverse transform.The detector cannot collect phase data, so it is necessary to retrieve the phase when reconstructing the target. The process of inverse transformation is also known as phase retrieval. This paper uses alternating projection method and conjugate gradient method to achieve reconstruction.
摘要 i
Abstract ii
Contents iii
List of Figures v
List of Tables vi
1 Introduction 1
2 Radon transform 3
2.1 Fourier slice theorem 4
2.2 Filtered Back Projection 5
3 X-ray transform 6
4 Phase retrieval 9
4.1 Inverse transform of M 9
4.2 Least squares problems of estimating f from h 10
4.3 Convolution 11
5 Algorithm 13
5.1 Conjugate gradient 13
5.2 Main algorithm 14
5.3 Initialization:Spectral method 15
6 Experiment 16
6.1 One axis 17
6.1.1 One axis 17
6.1.2 Initialization via a spectral method 19
6.1.3 Spectral method 21
6.1.4 Different quantities of d1 and θ 23
6.1.5 Different quantities of d1 and θ 26
6.2 Three axis 29
6.2.1 Error reduction through increased surface area 29
6.2.2 Initialization via a spectral method 32
6.2.3 Spectral method 34
7 Conclusion 36
References 37
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