Jiehua Zhu

Minimum detection windows, PI-line existence and uniqueness for helical cone-beam scanning of variable pitch

The goal of this paper is to study Cone-beam CT scanning along a helix of variable pitch. First the rationale and applications in medical imaging of variable pitch CT reconstruction are explained. Then formulas for the minimum detection window are derived. The main part of the paper proves a necessary and sufficient condition for the existence and uniqueness of PI-lines inside this variable pitch helix. These results are necessary steps toward an exact reconstruction algorithm for helix scanning of variable pitch, generalizing Katsevichs formula on constant pitch exact reconstruction. It is shown through an example that, when the derivative of the pitch function is not convex, or when the pitch function passes a inflection point and begins to slow down, PI-lines may be not unique near the rim of the helix cylinder. The conclusion is that the restriction on the pitch function is weaker, if the object is placed well within the helix cylinder and far from its rim, in order to preserve the uniqueness of PI-lines. If the object is near the rim, the restriction condition on the allowable pitch functions becomes stronger.

Geometric studies on variable radius spiral cone-beam scanning

The goal is to perform geometric studies on cone-beam CT scanning along a three-dimensional (3D) spiral of variable radius. First, the background for variable radius spiral cone-beam scanning is given in the context of electron-beam CT/micro-CT. Then, necessary and sufficient conditions are proved for existence and uniqueness of PI lines inside the variable radius 3D spiral. These results are necessary steps toward exact cone-beam reconstruction from a 3D spiral scan of variable radius, adapting Katsevichs formula for the standard helical cone-beam scanning. It is shown in the paper that when the longitudinally projected planar spiral is not always convex toward the origin, the PI line may not be unique in the envelope defined by the tangents of the spiral. This situation can be avoided by using planar spirals whose curvatures are always positive. Using such a spiral, a longitudinally homogeneous region inside the corresponding 3D spiral is constructed in which any point is passed by one and only one PI line, provided the angle omega between planar spirals tangent and radius is bounded by [omega - 90 degrees] < or = < epsilon for some positive epsilon < or = 32.48 degrees. If the radius varies monotonically, this region is larger and one may allow epsilon < or = 51.85 degrees. Examples for 3D spirals based on logarithmic and Archimedean spirals are given. The corresponding generalized Tam-Danielsson detection windows are also formulated.

Analysis on the strip-based projection model for discrete tomography

Discrete tomography deals with image reconstruction of an object with finitely many gray levels (such as two). Different approaches are used to model the raw detector reading. The most popular models are line projection with a lattice of points and strip projection with a lattice of pixels/cells. The line-based projection model fits some applications but involves a major approximation since the x-ray beams of finite widths are simplified as line integrals. The strip-based projection model formulates projection equations according to the fractional areas of the intersection of each strip-shaped beam and the rectangular grid of an image to be reconstructed, so is more realistic in some applications. In this paper, we characterize the strip-based projection model and establish an equivalence between the system matrices generated by the strip-based and line-based projection models.

Computed tomography simulation with superquadrics

Accurate and efficient simulation of an x-ray transform for representative structures plays an important role in research and development of x-ray CT, for the evaluation and improvement of CT image reconstruction algorithms, in particular. Superquadrics are a family of three-dimensional objects, which can be used to model a variety of anatomical structures. In this paper, we propose an algorithm for the computation of x-ray transforms for superellipsoids and tori with a monochromatic x-ray. Their usefulness is demonstrated by projection and reconstruction of a superquadric-based thorax phantom. Our work indicates that superquadric modeling provides a more realistic visualization than quadratic modeling, and a faster computation than spline methods.

Numerical studies on Feldkamp-type and Katsevich-type algorithms for cone-beam scanning along nonstandard spirals

In this paper, we perform numerical studies on Feldkamp-type and Katsevich-type algorithms for cone-beam reconstruction with a nonstandard spiral locus to develop an electron-beam micro-CT scanner. Numerical results are obtained using both the approximate and exact algorithms in terms of image quality. It is observed that the two algorithms produce similar quality if the cone angle is not large and/or there is no sharp density change along the z-direction. The Katsevich-type algorithm is generally preferred due to its nature of exactness.

System Generated by Strip-Based Projections in Discrete Tomography

The projection equations in the research of discrete tomography are obtained by either counting the number of points that each line passes or computing the fractional areas of the intersection of each strip and the grid. In this work, a system of linear equations for strip-based projections with rational slopes is obtained. The linear dependency number of these equations is derived.

A pointwise limit theorem for filtered backprojection in computed tomography.

Computed tomography (CT) is one of the most important areas in the modern science and technology. The most popular approach for image reconstruction is filtered backprojection. It is essential to understand the limit behavior of the filtered backprojection algorithms. The classic results on the limit of image reconstruction are typically done in the norm sense. In this paper, we use the method of limited bandwidth to handle filtered backprojection-based image reconstruction when the spectrum of an underlying image is not absolutely integrable. Our main contribution is, assuming the method of limited bandwidth, to prove a pointwise limit theorem for a class of functions practically relevant and quite general. Further work is underway to extend the theory and explore its practical applications.