Gustav Zickert | Searcharxiv

arXiv2021

Formal uniqueness in Ewald sphere corrected single particle analysis

In single particle analysis (SPA), the task is to recover the scattering potential of a macromolecular structure from cryo-electron microscope images of many copies of the structure in unknown orientations. The idealized, noise-free SPA inverse problem has been shown to be uniquely solvable - up to hand - when the forward model is based on the ray transform. More accurate forward models take the non-zero curvature of the Ewald sphere into account. We analyze an Ewald sphere corrected forward model for SPA and use the diffraction slice theorem to prove that the corresponding inverse problem is uniquely solvable, including the hand of the structure.

math.FA↗

arXiv2020

Gaussian mixture model decomposition of multivariate signals

We propose a greedy variational method for decomposing a non-negative multivariate signal as a weighted sum of Gaussians, which, borrowing the terminology from statistics, we refer to as a Gaussian mixture model. Notably, our method has the following features: (1) It accepts multivariate signals, i.e. sampled multivariate functions, histograms, time series, images, etc. as input. (2) The method can handle general (i.e. ellipsoidal) Gaussians. (3) No prior assumption on the number of mixture components is needed. To the best of our knowledge, no previous method for Gaussian mixture model decomposition simultaneously enjoys all these features. We also prove an upper bound, which cannot be improved by a global constant, for the distance from any mode of a Gaussian mixture model to the set of corresponding means. For mixtures of spherical Gaussians with common variance $σ^2$, the bound takes the simple form $\sqrt{n}σ$. We evaluate our method on one- and two-dimensional signals. Finally, we discuss the relation between clustering and signal decomposition, and compare our method to the baseline expectation maximization algorithm.

stat.ML↗