A Poisson noise model

An x-ray tube can be treated as a Poisson source and fluctuations in the number of x-rays emitted from the tube can be described by a Poisson probability density function [Barr04, p1103]. Many of the detection processes for the x-rays can be described by either binomial selection or Poisson processes and it can be shown that the cascading of these processes with the Poisson x-ray tube source results in another Poisson distribution [Barr04, p638] [Yaff94].

This simplified model of a noise source with a single Poisson distribution can be used to derive a theoretical model of the relationship between the image pixel intensities and the noise in these pixels. The noise measurement of interest is the variance of pixel in- tensities in a small region of interest within the image. If measurements are made using regions of interest in the image that have a large area compared with the blur of the de- tector, then pixel intensities can be considered independent and identically distributed, and the mean pixel intensity of the region is the mean of a collection of uncorrelated pix- els [Barr04, p1101]. This independence means that the noise can be considered individu- ally for each pixel in the region. The region of interest has to be small compared with gross variations in the x-ray flux across the detector. This is so that there is no variation across the sampling region and the expected value (E{.}) of the individual pixels in the region equates to the mean pixel intensity in the region. This requirement also means that the measured variance of the pixel intensities in the region is only due to the random noise and not the gross x-ray flux variations.

As introduced in Section 4.2.1, the output from the CR plate light detection system is mapped to pixel intensities by a logarithmic calculation of the form

Y = a log(X) + b (4.3)

where a = 1000 and b incorporates scaling of X via the relationship log(kX) = log(k) + log(X). If X and Y are treated as random variables, then the mean and standard deviation for a measurement region of interest can be calculated by applying a probability density function based on a Poisson distribution, but with a change of random variables based on Eqn. (4.3) [Barr04, p1443] [Pres07, p362].

is said to be Poisson distributed or Poissonian if for a parameter λ it obeys the probability law

p(X = x; λ) = λ

x_e−λ

x!

where x is a non-negative integer for the number of events in an interval. The parameter λ is the expected number of occurences in an interval. The expected value (mean) µ and variance σ2 of X are both equal to λ [Barr04, p1467], so the probability law for the Poisson distribution can be written

p(X = x) = µ

x_e−µ

x!

meaning that the random variable X is completely specified by its mean value µ. If µ > 10 then the discrete Poisson distribution can be approximated by a continuous normal distribution with mean µ and variance µ [Barr04, p1468]:

p(X = x) = _√1 2πµe

−(x−µ)22µ _{. (4.4)}

4.3.1.1 Mean pixel intensity of a noise region of interest

If X is a random variable for the x-rays detected in the measurement region, then the mean, P , of the pixel intensities is given by the expectation operation P = E{Y } = E{a log(X) + b}.

Using the linearity of the expectation operation, E{Y } can be simplified to E{a log(X) + b} = a E{log(X)} + b. The expectation E{log(X)} can be approximated using a Taylor series expansion and only keeping terms to a second order [Barr04, p1445]:

E{log(X)} = log(E{X}) +1₂σ_X2 d 2_log(x) dx2     x=E{X} = log(E{X}) −1₂σX2 1 ln(10) E{X}2 = log(E{X}) −1 2 1 ln(10) E{X}

4.3 Radiographic noise 83

So for E{Y } = E{a log(X) + b},

P = E{Y } = a E{log(X)} + b ≈ a(log(E{X}) −1₂ 1 ln(10) E{X}) + b ≈ a(log(E{X}) −0.2171 E{X}) + b ≈ a log(E{X}) + b. (4.5)

As the expected value of X increases, this approximation improves, although E{log(X)} remains slightly less than log(E{X}). This is supported by the result that because log(x) is a concave function, the opposite of Jensen’s inequality for the relationship between functions of random variables and expectations [Barr04, p1444] can be applied to give E{log(X)} ≤ log(E{X}) (recalling that X > 0 must apply for physical detection of an x-ray). Furthermore, the approximation of a Poisson distribution with a normal distribu- tion only applies for E{X} > 10, and for these values the error in the approximation of E{Y } is less then 2.2%. As the CR pixel intensities (Pinv) increase, the value of E{X} de-

creases (because Pinv = 4095 − E{Y }) and the measurement of the mean becomes more

inaccurate. This is an important caveat when analysing areas of high attenuation in the images, for example, thick bones.

4.3.1.2 Variance of a noise region of interest

For a sampled region of interest, the variance of the random variable Y is a measure of the image noise. A first order approximation to the variance is given by [Jahn02, p82] [Knol00, p87] Var{Y } = Var{X}     dy dx     2 x=E{X}

Using y = a log(x) + b and because for a Poisson distribution of X, Var{X} = E{X} > 0 Var{Y } = E{X}     a ln(10) E{X}     2 = a 2 ln(10)2_E{X}

Applying the approximation E{X} = 10(E{Y }−ba )(inverting Eqn. (4.5)), this equation can be

written in terms of the variable Y only

Var{Y } = a 2 ln(10)210−( E{Y }−b a ) = a 2 ln(10)210 (Pinv +b−4095_a ) (4.6)

where the mean pixel intensity substitution Pinv = 4095 − P = 4095 − E{Y } has been

applied.5

Taking the square root of the variance to give the standard deviation (σ) and taking the logarithm of both sides gives

log(σ) = log( a ln(10)) + 1 2  Pinv+ (b − 4095) a  =  log( a ln(10)) + ( b − 4095 2a )  + Pinv 2a  (4.7)

This noise model suggests that for radiographs in which the x-ray statistics can be de- scribed by a single Poisson distribution (the quantum fluctuations dominate) and the trans- formation of pixel intensities is of the form of Eqn. (4.3), the image noise is signal-dependent and should vary approximately exponentially with the mean pixel intensity (Pinv). This

model is very similar in form to the result stated by Gravel et al. of the exponential rela- tionship between noise variance and optical density in screen-film radiography [Grav04]. Plotting the logarithm of the standard deviation for a region of interest versus the mean pixel intensity should produce a straight line with a slope of_2a1 . (For the Kodak CR system a = 1000 – refer to Eqn. (4.2)).