ABSTRACT
Chapter 7 indicated that the statistics of MR imaging in its data domain propagate to its image domain through image reconstruction. Following this insight, an investigation into the statistics of MR imaging is conducted for the images generated using typical MR data acquisition schemes and basic image reconstruction methods: rectilinear k-space sampling/Fourier transform (FT), and radial sampling/projection reconstruction (PR). This approach is performed at three levels of the image: a single pixel, any two pixels, and a group of pixels (i.e., an image region). In MR image analysis and other applications, pixel intensity is always as-
sumed to have a Gaussian distribution. Because an MR image is complex valued, its magnitude image is widely utilized, and its phase image is used in some cases, Gaussianity should be elaborated in a more detailed fashion. This chapter shows that (1) pixel intensity of the complex-valued MR image has a complex Gaussian distribution, (2) its real and imaginary parts are Gaussian distributed and independent, and (3) its magnitude and phase components have non-Gaussian distributions but can be approximated by independent Gaussians when the signal-to-noise ratio (SNR) of the image is moderate or large. Characterizing spatial relationships of pixel intensities in MR imaging is an
important issue for MR image analysis/processing and other applications. Although theoretical and experimental studies indicate that the pixel intensities of an MR image are correlated, the explicit statements and/or the analytic formulae on the correlation have not been given. This chapter shows that (1) pixel intensities of an MR image are statistically correlated, (2) the degree of the correlation decreases as the distance between pixels increases, and (3) pixel intensities become statistically independent when the distance between pixels approaches infinity. These properties are summarized as spatially asymptotic independence (SAI). This chapter also gives a quantitative measure of the correlations between pixel intensities, that is, the correlation coefficient of the pixel intensities of an MR image decreases exponentially with the distance between pixels. This property is referred to as the Exponential correlation coefficient (ECC).
