Statistics of Medical Imaging

8

Statistics of MR Imaging

8.1 Introduction

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 fo r

the imag es generated using typical MR data acquisition schemes and basic

image reconstruction methods: rec tilinear k-space sampling/Fourier transform

(FT), and radial sampling/projection rec onstruction (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, a nd its pha se image is used in

some cas es, Gaussianity should be elaborated in a more detailed fashion. This

chapter shows that (1) pixel intensity of the c omplex-valued MR image has a

complex Gaussian distribution, (2) its real and imaginary parts a re Gaussian

distributed a nd independent, and (3) its magnitude and phase components

have non-Gaus sian distributions but can be approximated by indep endent

Gaussians when the signal-to-noise ratio (SNR) o f the image is moderate o r

large.

Characterizing spatial r elationships of pixel intensities in MR imaging is an

important issue for MR image a nalysis/processing and other applications. Al-

though theoretical and experimental studies indicate that the pixel intensities

of an MR image are correlated, the e xplicit sta tements 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 pix els increases, and (3 ) pixel

intensities become statistically independent when the distance between pixels

approaches infinity. These properties are summarized as spatially asymptotic

independenc e (SAI). This chapter also gives a quantitative measure of the

correlations between pixel intensities, that is, the correlation coefficient of the

pixel intensities o f an MR image decreases exponentially with the distance

between pixels. This pro perty is referred to as the Exponential correlation

coefficient (ECC).

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214 Statistics of Medical Imaging

An MR image appears piecewise contiguous. This scenario suggests that

each image region (i.e., a group of pixels) may possess some unique statistics.

This chapter proves that each image region is stationary and ergodic (hence

satisfies ergodic theo rems). Thus, an MR image is a piecewise stationary and

ergodic random field. Furthermore, the autocor relation function (acf) and the

sp ectral density function (sdf) of pixel intensities in an image region of an

MR image a re derived and expressed in analytic formulae.

Six statistical pro perties of an MR image—Gaussianity, spatially asymp-

totic independence, exponential correlation coefficient, stationarity, ergodic-

ity, autocorrelation function, and spectral density function—are described in

the order of a single pixel =⇒ any two pixels =⇒ a group of pixels. In addi-

tion to theoretical derivations and proo fs, experimental results obtaine d using

real MR images are also included. Theoretical and experimental results are in

good agree ment. These statistics provide the bas is for c reating stochastic im-

age models and developing new image analysis methodologies for MR image

analysis, which a re g iven Chapters 9, 10, and 11.

8.2 Statistics of the Intensity of a Single Pixel

This sectio n analyzes the statistics of the intensity of a s ingle pixel in an MR

image. It first derives probability density functions (pdfs) of (1) the complex-

valued pixel intensity, (2) its real and imaginary parts, and (3) its magnitude

and phase components, and the associated statistical parameters. Then it

proves and interprets the Gaussianity of the pixel intensity of an MR image.

For the c onvenience of description, all proofs and derivations of these s tatistics

are given in Appendix 8A.

In general, x(i, j) and x

i,j

are used to represent a pixe l intensity at the

location (i, j) in a 2-D image. When the pixel location (i, j) is not required,

it is convenient to change notations slightly by suppressing the location index

(i, j), that is, s imply to use x. Let x be a pixel intensity of the complex-valued

MR image, x

R

and x

I

be its real and imaginary parts, and x

s

and x

n

be its

(underlying) signal and noise components. Similar to Eq. (7.49), we have

x = x

R

+ ix

I

= (x

s

R

+ x

n

R

) + i(x

s

I

+ x

n

I

)

or (8.1)

x = x

s

+ x

n

= (x

s

R

+ ix

s

I

) + (x

n

R

+ ix

n

I

),

where x

s

R

and x

n

R

are the signal and noise components of the real part x

R

of the pixel intensity x, and x

s

I

and x

n

I

are the signal and nois e components

of the imaginary part x

I

of the pixel intensity x. x

R

and x

I

are real-valued

quantities, x

s

and x

n

are complex-valued quantities.