Partially Observed Tomographic Reconstruction Alignment Using Matrix Norm Minimization

In this project, we propose a new algorithm that recovers the rigid motion parameters and the underlying density function from a set of noisy measurements of three dimensional (3D) density functions. We measure the Radon transform of each 3D function which are randomly rotated and translated version of a target function that we try to recover. Radon transform is a set of parallel line integrals of a density function taken at various projection angles as shown in the 2D example in Figure 1. We can recover a 3D density function from the associated Radon transform using filtered back-projection method. (See Figure 1 B and C.) These are called reconstructions or tomograms. In the field of structural biology, aligning 3D tomographic reconstructions is a common practice to improve the resolution of the density function of a molecular structure of bacterial cells and viruses [MS09]. The Radon transform of frozen cells and viruses are measured by transmission electron microscope, and this particular imaging modality is called cryo-Electron tomography. One major complication of using a transmission electron microscope to acquire Radon transforms is that we cannot cover the entire angular range of the 3D function due to the sample holder geometry as shown in Figure 1 A. These missing angular components are called missing wedge. According to the projection-slice theorem [Bra56], pictorially described in Figure 2, each Radon integral value corresponds to a frequency component of the function,