Inversion of the broken ray transform in the case of energy-dependent attenuation.

Broken Ray transform (BRT) arises when one considers a narrow x-ray beam propagating through medium under the assumption of single scattering. Previous algorithms for inverting the BRT assumed that the medium is characterized by a single attenuation coefficient µ. However x-rays lose their energy after Compton scattering and the energy loss depends on the scattering angle. Since the attenuation coefficient depends on energy, the µ's before and after scattering are different. When there are three or more detectors one should distinguish not only between µ's that are 'seen' by x-rays before and after scattering, but also between µ's that are 'seen' by x-rays traveling towards different detectors.The main thrust of this paper is inversion of the BRT with N ⩾ 3 detectors under the assumption that the attenuation coefficient can be accurately approximated by a linear function of energy within the window of relevant energies. When the number of detectors is four or greater, we derive a family of inversion formulas. If N > 4, we find the optimal formula, which provides the best stability with respect to noise in the data. If N = 4, the family collapses into a single formula and no optimization is possible. If µ is independent of energy, N = 3 is sufficient for inversion. We also develop iterative reconstruction algorithms that can use global and local data. The results of testing the algorithms are presented.

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 FOR POLYNOMIAL REGION OF INTEREST.

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.

A note on computing the derivative at a constant direction.

The derivative at constant direction is frequently used in inversion of cone-beam data. Several algorithms for computing the derivative have been proposed in the literature. The best algorithm to date has been proposed recently by Noo et al (2007 Phys. Med. Biol. 52 5393-414). In this note we propose a new, simple and efficient formula for computing the derivative. Numerical experiments with helical CT show that our formula and the one of Noo et al provide fairly similar spatial resolution and noise stability, even though the new formula is more efficient and easier to implement.