ABSTRACT
Chapters 6 and 8 describe statistical properties of X-ray CT imaging and MR imaging at three levels of the image: a single pixel, any two pixels, and a group of pixels (i.e., an image region). When a probabilistic distribution of any pixel intensity with respect to all other pixel intensities in the image is viewed as a stochastic model for the image, then this model can be thought of as the statistical property of an image at its image level. In this way, statistical properties at the three bottom levels of X-ray CT and MR images described in Chapters 6 and 8 can be integrated into those at this top level to build stochastic models. Thus, this chapter is a continuation of Chapters 6 and 8. Chapters 2 and 3 show that X-ray CT imaging and MR imaging are based
on different physical phenomena and their imaging principles are very different. Chapters 5 and 7 show that data acquired in X-ray CT and MR imaging processes represent different physical quantities and their statistical properties are also different. However, Chapters 6 and 8 show that X-ray CT imaging and MR imaging have the very similar statistical properties. For example, in these two types of images, the intensity of a single pixel has a Gaussian distribution; intensities of any two pixels are spatially asymptotically independent; intensities of a group of pixels (i.e., an image region) form a stationary and ergodic random process. These common statistical properties suggest that these two imaging modalities may have some fundamental and intrinsic links. One possible reason for X-ray CT imaging and MR imaging having very
similar statistical properties may be the fact that they both belong to nondiffraction computed tomographic imaging, which is briefly discussed in Chapter 4. In nondiffraction CT imaging, the interaction model and the external measurements (e.g., projections) are characterized by the straight line integrals of some indexes of the medium and the image reconstruction is based on the Fourier slice theorem. The convolution reconstruction method (FBP) for X-ray CT and the projection reconstruction method (PR) for MRI have shown this common feature. The common statistical properties at the three bottom levels of X-ray CT
and MR images also suggest that we can create unified stochastic models for both X-ray CT and MR images. Based on our view of a stochastic image
model (i.e., a probabilistic distribution of any pixel intensity with respect to all other pixel intensities in the image), these stochastic image models, in fact, are the probability density functions (pdfs) of the image. This chapter shows two stochastic image models. The first model can be treated as a special case, that is, a simple version of the second model. The use of stochastic models depends on the application and especially on
the image quality. Chapters 5 and 7 indicate that the imaging SNR is one of the fundamental measures of image quality. Chapters 6 and 8 show that the statistical properties of X-ray CT and MR images are related to the image SNR. This chapter will show that SNR plays an important role in the model selection and the model order reduction.
