ABSTRACT
As shown in Section 5.4, X-ray CT image reconstruction constitutes a transform from a set of random variables (projections) to another set of random variables (pixel intensities). Statistics of X-ray CT data in the imaging domain propagate to the statistics in the image domain through image reconstruction. This chapter describes the statistics of the X-ray CT image generated by using basic CT data acquisition schemes (parallel and divergent projections) and the typical image reconstruction technique (convolution method). A statistical description of X-ray CT image is given at three levels of the image: a single pixel, any two pixels, and a group of pixels (also referred to as an image region). The Gaussianity of the intensity of a single pixel in an X-ray CT image
is proved in two ways. Then this chapter gives conditions for two pixel intensities to be independent with a probability measure for the parallel and divergent projections, respectively. These conditions essentially imply that (1) the intensities of any two pixels are correlated, (2) the degree of the correlation decreases as the distance between two pixels increases, and (3) in the limiting case of the distance approaching the infinity, the intensities of two pixels become independent. These properties are summarized as spatially asymptotically independent, abbreviated as SAI. An X-ray CT image consists of piecewise contiguous regions. This fact re-
veals that each image region, that is, a group of pixels, may have some unique statistical properties. This chapter shows that, based on Gaussianity and SAI, pixel intensities in each image region are stationary in the wide sense, hence, in the strict sense; and ergodic, hence, satisfy the ergodic theorems. Gaussianity, spatially, asymptotically independent, stationarity, and ergod-
icity are described in the order of a single pixel =⇒ any two pixels =⇒ a group of pixels. These properties provide a basis for creating a stochastic image model and developing new image analysis methodologies for X-ray CT image analysis, which are given in Chapters 9, 10, and 11.
