Capturing data from dynamic processes through cross-sectional measurements is seen in many fields, such as computational biology. Trajectory inference deals with the challenge of reconstructing continuous processes from such observations. In this work, we propose methods for B-spline approximation and interpolation of point clouds through consecutive averaging that is intrinsic to the Wasserstein space. Combining subdivision schemes with optimal transport-based geodesic, our methods carry out trajectory inference at a chosen level of precision and smoothness, and can automatically handle scenarios where particles undergo division over time. We prove linear convergence rates and rigorously evaluate our method on cell data characterized by bifurcations, merges, and trajectory splitting scenarios like \emph{supercells}, comparing its performance against state-of-the-art trajectory inference and interpolation methods. The results not only underscore the effectiveness of our method in inferring trajectories but also highlight the benefit of performing interpolation and approximation that respect the inherent geometric properties of the data.

Image processing problems in general, and in particular in the field of single-particle cryo-electron microscopy, often require considering images up to their rotations and translations. Such problems were tackled successfully when considering images up to rotations only, using quantities which are invariant to the action of rotations on images. Extending these methods to cases where translations are involved is more complicated. Here we present a computationally feasible and theoretically sound approximate invariant to the action of rotations and translations on images. It allows one to approximately reduce image processing problems to similar problems over the sphere, a compact domain acted on by the group of three-dimensional rotations, a compact group. We show that this invariant is induced by a family of mappings deforming, and thereby compactifying, the group structure of rotations and translations of the plane, i.e., the group of rigid motions, into the group of three-dimensional rotations. Furthermore, we demonstrate its viability in two image processing tasks: multireference alignment and classification. To our knowledge, this is the first instance of a quantity that is either exactly or approximately invariant to rotations and translations of images that both rests on a sound theoretical foundation and is applicable in practice.

We study the dihedral multi-reference alignment problem of estimating the orbit of a signal from multiple noisy observations of the signal, acted on by random elements of the dihedral group. We show that if the group elements are drawn from a generic distribution, the orbit of a generic signal is uniquely determined from the second moment of the observations. This implies that the optimal estimation rate in the high noise regime is proportional to the square of the variance of the noise. This is the first result of this type for multi-reference alignment over a non-abelian group with a non-uniform distribution of group elements. Based on tools from invariant theory and algebraic geometry, we also delineate conditions for unique orbit recovery for multi-reference alignment models over finite groups (namely, when the dihedral group is replaced by a general finite group) when the group elements are drawn from a generic distribution. Finally, we design and study numerically three computational frameworks for estimating the signal based on group synchronization, expectation-maximization, and the method of moments.

We target the problem of estimating the center of mass of objects in noisy two-dimensional images. We assume that the noise dominates the image, and thus many standard approaches are vulnerable to estimation errors, e.g., the direct computation of the center of mass and the geometric median which is a robust alternative to the center of mass. In this paper, we define a novel surrogate function to the center of mass. We present a mathematical and numerical analysis of our method and show that it outperforms existing methods for estimating the center of mass of an object in various realistic scenarios. As a case study, we apply our centering method to data from single-particle cryo-electron microscopy (cryo-EM), where the goal is to reconstruct the three-dimensional structure of macromolecules. We show how to apply our approach for a better translational alignment of molecule images picked from experimental data. In this way, we facilitate the succeeding steps of reconstruction and streamline the entire cryo-EM pipeline, saving computational time and supporting resolution enhancement.

In the multireference alignment model, a signal is observed by the action of a random circular translation and the addition of Gaussian noise. The goal is to recover the signal’s orbit by accessing multiple independent observations. Of particular interest is the sample complexity, i.e., the number of observations/samples needed in terms of the signal-to-noise ratio (SNR) (the signal energy divided by the noise variance) in order to drive the mean-square error to zero. Previous work showed that if the translations are drawn from the uniform distribution, then, in the low SNR regime, the sample complexity of the problem scales as$\omega (1/ \mathrm {SNR}^{3})$. In this paper, using a generalization of the Chapman–Robbins bound for orbits and expansions of the$\chi ^{2}$divergence at low SNR, we show that in the same regime the sample complexity for any aperiodic translation distribution scales as$\omega (1/ \mathrm {SNR}^{2})$. This rate is achieved by a simple spectral algorithm. We propose two additional algorithms based on non-convex optimization and expectation–maximization. We also draw a connection between the multireference alignment problem and the spiked covariance model.

We present a method for improving a non-local means (NLM) operator by computing its low-rank approximation. The low-rank operator is constructed by applying a filter to the spectrum of the original NLM operator. This results in an operator, which is less sensitive to noise while preserving important properties of the original operator. The method is efficiently implemented based on Chebyshev polynomials and is demonstrated on the application of natural images denoising. For this application, we provide a comparison of our method with other denoising methods.