RESUMO
In cryo-electron microscopy (cryo-EM), a microscope generates a top view of a sample of randomly oriented copies of a molecule. The problem of single particle reconstruction (SPR) from cryo-EM is to use the resulting set of noisy two-dimensional projection images taken at unknown directions to reconstruct the three-dimensional (3D) structure of the molecule. In some situations, the molecule under examination exhibits structural variability, which poses a fundamental challenge in SPR. The heterogeneity problem is the task of mapping the space of conformational states of a molecule. It has been previously suggested that the leading eigenvectors of the covariance matrix of the 3D molecules can be used to solve the heterogeneity problem. Estimating the covariance matrix is challenging, since only projections of the molecules are observed, but not the molecules themselves. In this paper, we formulate a general problem of covariance estimation from noisy projections of samples. This problem has intimate connections with matrix completion problems and high-dimensional principal component analysis. We propose an estimator and prove its consistency. When there are finitely many heterogeneity classes, the spectrum of the estimated covariance matrix reveals the number of classes. The estimator can be found as the solution to a certain linear system. In the cryo-EM case, the linear operator to be inverted, which we term the projection covariance transform, is an important object in covariance estimation for tomographic problems involving structural variation. Inverting it involves applying a filter akin to the ramp filter in tomography. We design a basis in which this linear operator is sparse and thus can be tractably inverted despite its large size. We demonstrate via numerical experiments on synthetic datasets the robustness of our algorithm to high levels of noise.
RESUMO
In many practical applications, it is desirable to solve the interior problem of tomography without requiring knowledge of the attenuation function fa on an open set within the region of interest (ROI). It was proved recently that the interior problem has a unique solution if fa is assumed to be piecewise polynomial on the ROI. In this paper, we tackle the related question of stability. It is well-known that lambda tomography allows one to stably recover the locations and values of the jumps of fa inside the ROI from only the local data. Hence, we consider here only the case of a polynomial, rather than piecewise polynomial, fa on the ROI. Assuming that the degree of the polynomial is known, along with some other fairly mild assumptions on fa , we prove a stability estimate for the interior problem. Additionally, we prove the following general uniqueness result. If there is an open set U on which fa is the restriction of a real-analytic function, then fa is uniquely determined by only the line integrals through U. It turns out that two known uniqueness theorems are corollaries of this result.
