Sampling strategies and resolution in limited data cone-beam tomography

This paper concerns cone-beam tomography from limited data. Cone-beam tomography is a technique used to visualize the interior of a three-dimensional (3-D) object in a non-invasive way. This technique involves two steps. In the first step, measurements called cone-beam projections are taken. In the second step, digital signal processing techniques are used to process the measurements to form a 3-D image. The 3-D image represents a function f(x) which associates each voxel x in the object with the value of a physical quantity. In x-ray imaging, this quantity is the x-ray attenuation factor which roughly corresponds to the density of the object. In single photon emission computed tomography (SPECT), the physical quantity is the concentration of a radioactive tracer injected into the 3-D object (the patient). A cone-beam projection is a 2-D set of integrals of f(x) measured along lines which diverge from a given vertex point a. In xray imaging, the vertex point is an x-ray source and the projection is usually called a radiograph. The object lies between the x-ray source and an area detector as shown in figure 1. Physically, the x-rays travel along lines which diverge from a and undergo a net attenuation depending on the densities encountered in the object. The cone-beam projection is the set of intensity losses (integrals of f(x)) observed along each line. These intensity losses are measured on the area detector; they form a 2-D image (radiograph) of the 3-D object under study. In SPECT, the area detector is replaced by a gamma camera with a collimator. The gamma camera is used to count photons; the sum of photons emitted in a given direction is proportional to a line integral of f(x). The collimator restricts the photons to specific lines by an arrangement of cylindrical holes. To obtain a cone-beam projection, the holes are oriented towards a single focal point in space. The vertex point a is the collimator focal point. With ideal (noiseless) projections, the imaging capabilities of a cone-beam tomographic system are determined by the locations of the vertices relative to the object. Most works on cone-beam tomography are based on Tuy’s condition [1] which requires the vertices to be finely sampled along certain kinds of curves in space. This condition ensures accurate tomographic reconstructions but requires many (typically hundreds of) projections. If data collection is slow however, it may not be possible to acquire enough projections. Also, sampling the projections along an appropriate vertex curve is not always possible due to physical constraints. One example is imaging during interventional surgery. For such an application, only a small number of projections can be considered to ensure fast image update. Also, the physical constraints of a crowded operating room will preclude certain locations for tak object detector