Alexander Katsevich

THEORETICALLY EXACT FILTERED BACKPROJECTION-TYPE INVERSION ALGORITHM FOR SPIRAL CT ∗

Proposed is a theoretically exact formula for inversion of data obtained by a spiral computed tomography (CT) scan with a two-dimensional detector array. The detector array is supposed to be of limited extent in the axial direction. The main property of the formula is that it can be implemented in a truly filtered backprojection fashion. First, one performs shift-invariant filtering of a derivative of the cone beam projections, and, second, the result is backprojected in order to form an image. Another property is that the formula solves the so-called long object problem. Limitations of the algorithm are discussed. Results of numerical experiments are presented.

Analysis of an exact inversion algorithm for spiral cone-beam CT

In this paper we continue studying a theoretically exact filtered backprojection inversion formula for cone beam spiral CT proposed earlier by the author. Our results show that if the phantom f is constant along the axial direction, the formula is equivalent to the 2D Radon transform inversion. Also, the inversion formula remains exact as spiral pitch goes to zero and in the limit becomes again the 2D Radon transform inversion formula. Finally, we show that according to the formula the processed cone beam projections should be backprojected using both the inverse distance squared law and the inverse distance law.

An improved exact filtered backprojection algorithm for spiral computed tomography

Proposed is a theoretically exact formula for inversion of data obtained by a spiral computed tomography scan with a two-dimensional detector array. The detector array is supposed to be of limited extent in the axial direction. The main property of the formula is that it can be implemented in a truly filtered backprojection fashion. First, one performs shift-invariant filtering of a derivative of the cone beam projections, and, second, the result is back-projected in order to form an image. Compared with an earlier reconstruction algorithm proposed by the author, the new one is two times faster, requires a smaller detector array, and does not impose restrictions on how big the patient is inside the gantry. Results of numerical experiments are presented.

A GENERAL SCHEME FOR CONSTRUCTING INVERSION ALGORITHMS FOR CONE BEAM CT

Given a rather general weight function n0, we derive a new cone beam transform inversion formula. The derivation is explicitly based on Grangeat’s formula (1990) and the classical 3D Radon transform inversion. The new formula is theoretically exact and is represented by a 2D integral. We show that if the source trajectory C is complete in the sense of Tuy (1983) (and satisfies two other very mild assumptions), then substituting the simplest weight n0 ≡ 1 gives a convolution-based FBP algorithm. However, this easy choice is not always optimal from the point of view of practical applications. The weight n0 ≡ 1 works well for closed trajectories, but the resulting algorithm does not solve the long object problem if C is not closed. In the latter case one has to use the flexibility in choosing n0 and find the weight that gives an inversion formula with the desired properties. We show how this can be done for spiral CT. It turns out that the two inversion algorithms for spiral CT proposed earlier by the author are particular cases of the new formula. For general trajectories the choice of weight should be done on a case-by-case basis.

Image reconstruction for the circle-and-arc trajectory

Proposed is an exact shift-invariant filtered backprojection algorithm for the circle-and-arc trajectory. The algorithm has several important features. First, it allows for the circle to be incomplete. Second, axial truncation of the cone beam data is allowed. Third, the length of the arc is determined only by the region of interest and is independent of the size of the entire object. The algorithm is quite flexible and can be used for even more general trajectories that consist of several circular segments and arcs. The algorithm applies also in the case when the circle (or, circles) is complete. A numerical experiment with the clock phantom demonstrated good image quality.

Cone beam local tomography

Methods, systems and processes for providing efficient image reconstruction using local cone beam tomography which provide a reduced level of artifacts without suppressing the strength of the useful features; and in a dynamic case provide reconstruction of objects that are undergoing a change during the scan. An embodiment provides a method of reconstructing an image from cone beam data provided by at least one detector. The method includes collecting CB projection data of an object, storing the CB projection data in a memory; and reconstructing the image from the local CB projection data. In the reconstructing step, a combination of derivatives of the CB projection data that will result in suppressing the artifacts are found. The combination of derivatives includes collecting cone beam data that represents a collection of integrals that represent the object.

Exact filtered backprojection reconstruction for dynamic pitch helical cone beam computed tomography

We present an exact filtered backprojection reconstruction formula for helical cone beam computed tomography in which the pitch of the helix varies with time. We prove that the resulting algorithm, which is functionally identical to the constant pitch case, provides exact reconstruction provided that the projection of the helix onto the detector forms convex boundaries and that PI lines are unique. Furthermore, we demonstrate that both of these conditions are satisfied provided the sum of the translational velocity and the derivative of the translational acceleration does not change sign. As a special case, we show that gantry tilt can also be handled by our dynamic pitch formula. Simulation results demonstrate the resulting algorithm.

On two versions of a 3π algorithm for spiral CT

A 3π algorithm is obtained in which all the derivatives are confined to a detector array. Distance weighting of backprojection coefficients of the algorithm is studied. A numerical experiment indicates that avoiding differentiation along the source trajectory improves spatial resolution. Another numerical experiment shows that the terms depending on the non-standard distance weighting 1/|x − y(s)| can no longer be ignored.

Improved cone beam local tomography

In this paper, we study cone beam local tomography (LT). A new LT function g is proposed. It is shown that g contains non-local artefacts, but their strength is an order of magnitude smaller than those of the previously known LT function gΛ of Louis A K and Maass P (1993 IEEE Trans. Med. Imaging 12 764–69). We also investigate LT reconstruction in the dynamic case, i.e. when the object f being scanned is undergoing some changes during the scan. Properties of g are studied, the notion of visible singularities is suitably generalized, and a relationship between the wave fronts of f and g is established. It is shown that the changes in f do not cause any smearing of the singularities of g. Numerical implementation of the new LT function is discussed, and results of numerical experiments 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.