Statistics of Medical Imaging

9

Stochastic Image Models

9.1 Introduction

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 distr ibutio n of

any pixel intensity with respect to all other pixel intensities in the image is

viewed as a stochastic model for the ima ge, then this model can be tho ught of

as the statistical property of an image at its image level. In this way, statistical

properties a t 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 princ iple s are very differ-

ent. 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, Chapter s 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 distri-

bution; intensities of any two pixels are spatially asymptotically independent;

intensities of a group of pixels (i.e., an image regio n) form a stationary and er-

godic r andom 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 sta tistical properties may be the fact that they both belong to non-

diffraction computed tomographic imaging, which is brie fly disc ussed in Chap-

ter 4. In nondiffraction CT imaging, the interaction model and the external

measurements (e.g., projections) are characterized by the straight line inte-

grals of some indexes of the medium and the ima ge reconstruction is based

on the Fourier slice theorem. T he convolution reconstructio n 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 v iew of a stochastic image

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