Three-Dimensional Variance of Weighted Back-Projection Reconstruction

This dissertation addresses the statistical aspects of 3-D reconstruction in studying biological macromolecules from electron microscopy. The primary concentration is on the development and application of a 3-D variance estimation algorithm in the weighted back-projection reconstruction from a random conical tilt series. The purpose of the studies is to detect 3-D particle conformational changes, to assess the structural differences of two related reconstructions and to address the significance of local features in a reconstruction. Other topics include the 3-D spectral signal-to-noise ratio (SSNR) estimation algorithm, the estimation of 3-D signal-to-noise ratio (SNR) and 3-D resolution, and the statistics of quantum and photographic noises combined. Using the "two steps" strategy: defining the 3 -D variance from a "gedankenexperiment" and estimating it by comparing neighbor projections, a 3-D variance estimate (of each reconstruction voxel) is established, which maximally exploits the noise information hidden in the projections. Then the statistical properties of the 3-D variance estimate, the contributions of different projection noise sources, the effect of filtration, projection inhomogeneity and number of projections, and the statistical assessments from using 3-D variance is systematically analyzed. A practical application protocol of 3-D variance is subsequently proposed. In the two applications to four projection classes of the 50S ribosomal subunit depleted of L7/L12 proteins and to a hemocyanin-Fab complex, the 3-D locations of structurally flexible regions of the particles are demonstrated from the 3-D variance maps; the structural differences between the classes are assessed by using the statistical t-test; and the 3-D antibody binding site is revealed. Also, the concept of resolution is re-addressed, various 3-D resolution measures are compared, and a 3-D SSNR estimation algorithm is developed. Then, the 3-D SNR and 3-D SSNR of a weighted back-projection reconstruction are expressed as functions of projection sampling geometry, object size and shape, number of projections, and spectral projection signal and noise levels. Finally, the probability density function of the combination of quantum and photographic noises is derived and its possible application for characterizing the emulsion noise is discussed.