ABSTRACT
As shown in Chapter 2, X-ray CT images are reconstructed from projections, while the projections are formulated from photon measurements. In X-ray CT imaging, photon measurements and projections are called CT imaging data, abbreviated as CT data.∗
Similar to any type of realistic data, each type of CT data consists of its signal and noise components. In photon measurements, the instrumental and environmental noise form the noise component of photon measurements, which is random. The emitted and detected photons (numbers) form the signal component of the photon measurements, which is also random, due to the intrinsic variations in the emission and detection processes. The parallel projection is a translation-rotation mode (Sections 2.3 and 2.4).
Within one view, the photon measurements and the projections are acquired and formulated sequentially in time from one projection location to another. The divergent projection is a rotation mode (Sections 2.3 and 2.4). Within one view, the photon measurements and the projections are acquired and formulated simultaneously for all projections. Over all views, both parallel and divergent projections are collected sequentially in time from one view location to another. Thus, projections are spatio-temporal in nature. Because the time interval of CT data collection, particularly in the dievergent projection, is very short, the time argument in CT data is excluded in the process of image reconstruction. This chapter describes the statistics of both signal and noise components of
each type of CT data, and is focused on their second-order statistics. Based on the physical principles of X-ray CT described in Chapter 2 and according to CT data acquisition procedures, the statistical description of X-ray CT imaging is progresses in the following order: photon measurement (emission → detection → emission and detection) =⇒ projection. This chapter also provides signal processing paradigms for the convolution
image reconstruction method for the parallel and divergent projections. Then,
it gives a statistical interpretation to CT image reconstruction. CT image reconstruction can be viewed as a transform from a set of random variables (projections) to another set of random variables (pixel intensities). These new random variables form a spatial random process, also known as a random field. Statistics of CT data in the imaging domain propagate to the statistics in the image domain through image reconstruction. Discussions in this chapter are confined to the monochromatic X-ray, the
basic parallel and divergent projections, and the convolution image reconstruction method [2, 2-4, 7-9, 18, 37, 53].
