The method of common lines is a well-established reconstruction technique in cryogenic electron microscopy (cryo-EM), which can be used to extract the relative orientations of an object given tomographic projection images from different directions. In this paper, we deal with an analogous problem in optical diffraction tomography. Based on the Fourier diffraction theorem, we show that rigid motions of the object, i.e., rotations and translations, can be determined by detecting common circles in the Fourier-transformed data. We introduce two methods to identify common circles. The first one is motivated by the common line approach for projection images and detects the relative orientation by parameterizing the common circles in the two images. The second one assumes a smooth motion over time and calculates the angular velocity of the rotational motion via an infinitesimal version of the common circle method. Interestingly, using the stereographic projection, both methods can be reformulated as common line methods, but these lines are, in contrast to those used in cryo-EM, not confined to pass through the origin and allow for a full reconstruction of the relative orientations. Numerical proof-of-the-concept examples demonstrate the performance of our reconstruction methods.

翻译：公共线方法是低温电子显微术（cryo-EM）中一种成熟的图像重建技术，可用于从不同方向的断层投影图像中提取物体的相对取向。本文研究光学衍射层析成像中的类似问题。基于傅里叶衍射定理，我们证明物体的刚体运动（即旋转和平移）可通过检测傅里叶变换数据中的公共圆来确定。我们提出了两种识别公共圆的方法。第一种方法受投影图像公共线方法的启发，通过参数化两张图像中的公共圆来检测相对取向。第二种方法假设运动随时间平滑变化，并利用公共圆方法的无穷小形式计算旋转运动的角速度。有趣的是，通过立体投影变换，两种方法均可重构为公共线方法，但这些线不同于低温电镜中使用的公共线——它们不局限于通过原点，且能完整重建相对取向。数值概念验证示例展示了我们重建方法的性能。