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 Katsevich's 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 spiral's 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.