Barth M, Bryan R K, Hegerl R, Baumeister W
Max-Planck-Institut for Biochemistry, Martinsried, F.R.G.
Scanning Microsc Suppl. 1988;2:277-84.
The range of tilt angles for which projected images of two-dimensionally periodic specimens can be obtained in electron microscopy is limited both by technical aspects, such as goniometer design, and by the more fundamental limitation of object thickness. The lack of a full set of projections causes a missing cone in the reciprocal space data for the object, which will give an anisotropic resolution in a three-dimensional reconstruction and may cause the quality to be impaired by spurious features. The problem is governed by a linear operator which maps the three-dimensional object onto the set of projections. The eigenvalue spectrum of this operator is determined by the range of tilt angles and the spatial extent of the object. If the object is spatially restricted, the eigenvalues are all positive, and it is in principle possible to retrieve experimentally unavailable structure data from those that are measured. However, with restricted angle data, some of the eigenvalues are extremely small, so the problem is 'ill-conditioned' or sensitive to small perturbations in the data, such as noise, and it is necessary to regularize the solution. We applied two methods of band-limited extrapolation and inference on electron microscope data. Alternating projections onto convex sets regularized by a regularization parameter and a least squares estimation regularized by the Shannon entropy functional yield similar results if a close object extent constraint is available. The criterion of maximum entropy, however, allows a relaxation of this constraint.
在电子显微镜中能够获取二维周期性标本投影图像的倾斜角范围，受到诸如测角仪设计等技术方面的限制，同时也受到标本厚度这一更为根本的限制。缺少一整套投影会导致物体的倒易空间数据中出现缺失锥，这将在三维重建中产生各向异性分辨率，并可能导致质量因虚假特征而受损。该问题由一个将三维物体映射到投影集上的线性算子所控制。此算子的特征值谱由倾斜角范围和物体的空间范围决定。如果物体在空间上受到限制，特征值均为正，原则上有可能从已测量的数据中检索出实验上无法获取的结构数据。然而，对于角度受限的数据，一些特征值极小，所以该问题是“病态的”，即对数据中的小扰动（如噪声）敏感，因此有必要对解进行正则化。我们对电子显微镜数据应用了两种带限外推和推断方法。如果有接近的物体范围约束，通过正则化参数进行正则化的交替投影到凸集上以及通过香农熵泛函进行正则化的最小二乘估计会产生相似的结果。然而，最大熵准则允许放宽这一约束。
