MRC Laboratory of Molecular Biology, Francis Crick Avenue, Cambridge CB2 0QH, UK.
MRC Laboratory of Molecular Biology, Francis Crick Avenue, Cambridge CB2 0QH, UK.
Ultramicroscopy. 2018 Apr;187:26-33. doi: 10.1016/j.ultramic.2017.11.001. Epub 2018 Jan 12.
Curvature of the Ewald sphere limits the resolution at which Fourier components in an image can be approximated as corresponding to a projection of the object. Since the radius of the Ewald sphere is inversely proportional to the wavelength of the imaging electrons, this normally imposes a limit on the thickness of specimen for which images can be easily interpreted to a particular resolution. Here we present a computational method for precisely correcting for the curvature of the Ewald sphere using defocused images that delocalise the high-resolution Fourier components from the primary image. By correcting for each approximately Friedel-symmetry-related sideband separately using two distinct complex transforms that effectively move the displaced Fourier components back to where they belong in the structure, we can determine the amplitude and phase of each of the Fourier components separately. This precisely accounts for the effect of Ewald sphere curvature over a bandwidth defined by the defocus and the size of the particle being imaged. We demonstrate this processing algorithm using: 1. simulated images of a particle with only a single, high-resolution Fourier component, and 2. experimental images of gold nanoparticles embedded in ice. Processing micrographs with this algorithm will allow higher resolution imaging of thicker specimens at lower energies without any image degradation or blurring due to errors made by the assumption of a flat Ewald sphere. Although the procedure will work best on images recorded with higher defocus settings than used normally, it should still improve 3D single-particle structure determination using images recorded at any defocus and any electron energy. Finally, since the Ewald sphere curvature is in a known direction in the third dimension which is parallel to the direction of view, this algorithm automatically determines the absolute hand of the specimen without the need for pairs of images with a known tilt angle difference.
Ewald 球的曲率限制了可以近似为物体投影的图像中傅里叶分量的分辨率。由于 Ewald 球的半径与成像电子的波长成反比，因此这通常限制了可以轻松解释特定分辨率图像的样品厚度。在这里，我们提出了一种使用离焦图像精确校正 Ewald 球曲率的计算方法，该方法使高分辨率傅里叶分量从主图像中弥散。通过分别使用两个不同的复数变换来校正每个近似的 Friedel 对称相关边带，这些变换有效地将位移的傅里叶分量移回它们在结构中的所属位置，我们可以分别确定每个傅里叶分量的幅度和相位。这可以精确地解释 Ewald 球曲率在离焦和被成像粒子大小定义的带宽上的影响。我们使用以下方法展示了这种处理算法：1. 只有一个高分辨率傅里叶分量的粒子的模拟图像，以及 2. 嵌入在冰中的金纳米粒子的实验图像。使用此算法处理显微照片将允许在更低能量下对更厚的样品进行更高分辨率的成像，而不会因假设平坦的 Ewald 球而导致图像降级或模糊。尽管该过程在记录图像时的离焦设置比通常使用的离焦设置更高的情况下效果最佳，但它仍应改善使用任何离焦和任何电子能量记录的图像的 3D 单颗粒结构确定。最后，由于 Ewald 球曲率在第三维中沿与观察方向平行的已知方向，因此该算法无需具有已知倾斜角差的一对图像即可自动确定样品的绝对手性。
