Covariance Matrix Estimation for the Cryo-EM Heterogeneity Problem

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

Local Tomography and the Motion Estimation Problem

In this paper we study local tomography (LT) in the motion contaminated case. It is shown that microlocally, away from some critical directions, LT is equivalent to a pseudodifferential operator of order one. LT also produces nonlocal artifacts that are of the same strength as useful singularities. If motion is not accurately known, singularities inside the object f being scanned spread in different directions. A single edge can become a double edge. In such a case the image of f looks cluttered. Based on this observation we propose an algorithm for motion estimation. We propose an empiric measure of image clutter, which we call edge entropy. By minimizing edge entropy we find the motion model. The algorithm is quite flexible and is also used for solving the misalignment correction problem. The results of numerical experiments on motion estimation and misalignment correction are very encouraging

Exponential instability in the fractional Calder\'on problem

In this note we prove the exponential instability of the fractional Calder\'on problem and thus prove the optimality of the logarithmic stability estimate from \cite{RS17}. In order to infer this result, we follow the strategy introduced by Mandache in \cite{M01} for the standard Calder\'on problem. Here we exploit a close relation between the fractional Calder\'on problem and the classical Poisson operator. Moreover, using the construction of a suitable orthonormal basis, we also prove (almost) optimality of the Runge approximation result for the fractional Laplacian, which was derived in \cite{RS17}. Finally, in one dimension, we show a close relation between the fractional Calder\'on problem and the truncated Hilbert transform.Comment: 17 page

Automatic Optimization of Alignment Parameters for Tomography Datasets

As tomographic imaging is being performed at increasingly smaller scales, the stability of the scanning hardware is of great importance to the quality of the reconstructed image. Instabilities lead to perturbations in the geometrical parameters used in the acquisition of the projections. In particular for electron tomography and high-resolution X-ray tomography, small instabilities in the imaging setup can lead to severe artifacts. We present a novel alignment algorithm for recovering the true geometrical parameters \emph{after} the object has been scanned, based on measured data. Our algorithm employs an optimization algorithm that combines alignment with reconstruction. We demonstrate that problem-specific design choices made in the implementation are vital to the success of the method. The algorithm is tested in a set of simulation experiments. Our experimental results indicate that the method is capable of aligning tomography datasets with considerably higher accuracy compared to standard cross-correlation methods

Microscopic Optical Projection Tomography In Vivo

We describe a versatile optical projection tomography system for rapid three-dimensional imaging of microscopic specimens in vivo. Our tomographic setup eliminates the in xy and z strongly asymmetric resolution, resulting from optical sectioning in conventional confocal microscopy. It allows for robust, high resolution fluorescence as well as absorption imaging of live transparent invertebrate animals such as C. elegans. This system offers considerable advantages over currently available methods when imaging dynamic developmental processes and animal ageing; it permits monitoring of spatio-temporal gene expression and anatomical alterations with single-cell resolution, it utilizes both fluorescence and absorption as a source of contrast, and is easily adaptable for a range of small model organisms

Towards Omni-Tomography—Grand Fusion of Multiple Modalities for Simultaneous Interior Tomography

We recently elevated interior tomography from its origin in computed tomography (CT) to a general tomographic principle, and proved its validity for other tomographic modalities including SPECT, MRI, and others. Here we propose “omni-tomography”, a novel concept for the grand fusion of multiple tomographic modalities for simultaneous data acquisition in a region of interest (ROI). Omni-tomography can be instrumental when physiological processes under investigation are multi-dimensional, multi-scale, multi-temporal and multi-parametric. Both preclinical and clinical studies now depend on in vivo tomography, often requiring separate evaluations by different imaging modalities. Over the past decade, two approaches have been used for multimodality fusion: Software based image registration and hybrid scanners such as PET-CT, PET-MRI, and SPECT-CT among others. While there are intrinsic limitations with both approaches, the main obstacle to the seamless fusion of multiple imaging modalities has been the bulkiness of each individual imager and the conflict of their physical (especially spatial) requirements. To address this challenge, omni-tomography is now unveiled as an emerging direction for biomedical imaging and systems biomedicine

Stability of the interior problem with polynomial attenuation in the region of interest

In many practical applications, it is desirable to solve the interior problem of tomography without requiring knowledge of the attenuation function f(a) on an open set within the region of interest (ROI). It was proved recently that the interior problem has a unique solution if f(a) 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 f(a) inside the ROI from only the local data. Hence, we consider here only the case of a polynomial, rather than piecewise polynomial, f(a) on the ROI. Assuming that the degree of the polynomial is known, along with some other fairly mild assumptions on f(a), 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 f(a) is the restriction of a real-analytic function, then f(a) is uniquely determined by only the line integrals through U. It turns out that two known uniqueness theorems are corollaries of this result

Covariance Matrix Estimation for the Cryo-EM Heterogeneity Problem

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

Stability Of The Interior Problem With Polynomial Attenuation In The Region Of Interest

With the extensive use of two-photon fluorescence microscopy (2PFM) in the biomedical field, the need for development of fluorescent probes with improved two-photon fluorescence (2PF) properties has triggered extensive studies in the synthesis of new probes that undergo efficient two-photon absorption (2PA). In order to provide a more comprehensive comparison of fluorophores for 2PF bioimaging, a figure of merit (F M) was developed by normalizing the 2PA action cross-section, a commonly used parameter for characterizing bioimaging 2PF probes, by the photodecomposition quantum yield. Another important aspect of developing 2PA fluorophores is hydrophilicity. Although hydrophilic fluorophores are generally preferred in 2PFM bioimaging, hydrophobic fluorophores are typically easier to synthesize and purify, and have been used successfully in 2PFM bioimaging. The methodologies of dispersing hydrophobic fluorophores into aqueous media, such as in a DMSO/water mixture, micelles, silica nanoparticles, or forming polymer nanoparticles, are reviewed. The design and synthesis of hydrophilic 2PA fluorophores, achieved by introducing polyethylene glycol (PEG), anionic acid groups, cationic ammonium salt, and PAMAM dendrimers, is presented. Introduction of specificity to target certain biomarkers via labeling of antibodies, DNA, smallbioactive molecules, and peptides, and for the sensing of sepcific cations and pH, is also reviewed. Copyright © 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim