穿透式電子顯微鏡以提供高解析度、材料內部型態、晶體原子結構、及部分成分的訊息，成為現在奈米科技材料分析的時代新利器。電子利用通過電磁透鏡系統及光欄的組合，來控制成像、繞射與成分分析的訊號，然而這也產生了一些無可避免的像差，使得影像因此模糊而降低了分辨率，或無法完全反映樣品的真正訊息。除了像差的問題，穿透式電子顯微鏡在紀錄影像的同時，不幸的有一半的訊息也同時遺失了。在穿透式電子顯微鏡中，紀錄下來的影像則是波的強度(intensity)，為 與它共軛複數 的乘績，正比於振幅的平方。這便是著名的相位問題(phase problem)。這一半的遺失訊息裡，影像空間與繞射空間中的相位分別和原子的內位能(inner potential)與原子之間的相互排列位置(relative position)有關。換言之，失去了相位，材料裡的一些必要的訊息也不見了，這不僅僅是發生在穿透式電子顯微鏡，包括x光繞射、中子繞射等等亦是如此。科學家因此在這方面做了相當多的努力，試圖將失去的相位回復(phase retrieval)。
在穿透式電鏡中，倒空間的相位回復可以幫助我們的得到實空間的無像差影像，本論文中，利用過取樣方法(oversampling)以氧化鎂的實驗為例，尤其繞射圖形將之影像回復。包括實驗值與模擬值的比較接在此論文中加以討論。進一歩而言，結合diffractive imaging 以及electron tomography，如果有數個不同方向的被回復的投影結果，三維影像也可以被重構，這個方法被稱為diffraction tomography，在本論文中使用模擬的27個單位晶胞，由不同方向的繞射圖形，成功回復實空間三維的無像差影像。優點則是可以避開直接在實空間取像而產生的像差問題。

Transmission electron microscopy (TEM) is a modern instrument for analyzing materials, including the imaging, structure and element analysis. However, as the diffraction patterns and images are recorded, the phase of the electronic waves is lost that the half information is unknown at the same time. Besides, as the electronic waves pass through the magnetic lens system, the unavoidable lens aberrations blur the images that it is not actually reflect the specimen when the images are recorded.
The phase in reciprocal space reveals the relative position of the atoms. From the phase retrieval the recovered image will not contain the aberrations because the lens aberrations do not affect the intensity of diffraction beam. To be brief, the goal in the thesis is to solve the phase problem and lens aberrations in TEM at the same time.
The oversampling method is still in the early stage in the phase retrieval for TEM, although the some simulations and experimental cases for x-ray are discussed in recent years. In the thesis we take MgO for example because its simple structure and light mass. Some limitation for overampling method will be discussed, and after overcome the difficulties of limitations, the experimental data will be shown and discussed. To do the comparison, not only the experimental but the simulation case will be discussed. After recovering the images of two-dimensions, it is further to do the three-dimensional reconstruction. That is, in the thesis we will discuss from several diffraction patterns in different directions, the diffraction tomography can be done and the 3-D image can be recovered in the simulation case.

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