MEASURES OF IMAGE QUALITY
4.3. CONTRAST 1. Definition
4.5.3. Noise power spectra
the correlation of noise can be determined either in the spatial domain using autocorrelation (as we have seen in the previous section) or in the spatial frequency domain using the noise power spectrum (nPs), also known as the Wiener spectrum, after n. Wiener, who pioneered its use. there are a number of requirements that must be met for the nPs of an imaging system to be tractable.
these include: linearity, shift invariance, ergodicity and wide sense stationarity.
in the case of digital devices, the latter requirement is replaced by wide sense cyclostationarity. if the above criteria are met, then the nPs completely describes the noise properties of an imaging system. in point of fact, it is impossible to meet all of these criteria exactly; for example, all practical detectors have finite size and thus are not strictly stationary. however, in spite of these limitations, it is generally possible to calculate the local nPs.
by definition, the nPs is the ensemble average of the square of the ft of the spatial density fluctuations:
2 i( ) 2
, 1 1
( , ) lim g( , )e d d
2 2
|
X Y ux vy|
X Y X Y
u v x y x y
X Y
+ +
− +
→∞ − −
=
∫ ∫
∆ πW (4.41)
the nPs and the autocovariance function form an ft pair. this can be seen by taking the ft of eq. (4.36) and applying the convolution theorem.
the nPs of a discrete random process, such as when measured with a digital X ray detector, is:
2 i( ) 2
, ,
( , ) lim x y
|
( , )e|
x y ux vy
N N x y
x y
a
u v a x y
N N
− +
= →∞
∑
∆ πW g (4.42)
Nx and Ny are the number of dels in x and y, and ax and ay are the pitch of each del in x and y. equation (4.42) requires that we perform the summation over all space. in practice, this is impossible as we are dealing with detectors of limited extent. by restricting the calculation to a finite region, it is possible to determine the fourier content of the fluctuations in that specific region. We call this simple calculation a sample spectrum. it represents one possible instantiation of the noise seen by the imaging system, and we denote this by W:
2 i( , ) 2
( , ) x y
|
m n, ( m, n)e uxm vyn|
x y
a a
u v x y
N N
=
∑
∆ −π
W g (4.43)
an estimate of the true nPs is created by averaging the sample spectra from M realizations of the noise:
1
( , ) 1 M i( , )
u v i u v
M =
=
∑
W W (4.44)
ideally, the average should be done by calculating sample spectra from multiple images over the same region of the detector. however, by assuming stationarity and ergodicity, we can take averages over multiple regions of the detector, significantly reducing the number of images that we need to acquire.
now, the estimate of the nPs, W, has an accuracy that is determined by the number of samples used to make the estimate. assuming gaussian statistics, at frequency (u, v), the error in the estimate W( , )x y will have a standard error given by:
( , ) c u v
MW (4.45)
where c = 2 for u = 0 or v = 0, and c = 1 otherwise. the values of c arise from the circulant nature of the ft.
typically, 64 × 64 pixel regions are sufficiently large to calculate the nPs.
approximately 1000 such regions are needed for good 2-D spectral estimates.
Remembering that the autocorrelation function and the nPs are ft pairs, it follows from Parseval’s theorem that:
0 0 ,
(0, 0) 1 ( , )
x y u v
x y N N u v
=
∑
K W (4.46)
this provides a useful and rapid method of verifying a nPs calculation.
there are many uses of the nPs. it is most commonly used in characterizing imaging device performance. in particular, the nPs is exceptionally valuable in investigating sources of detector noise. for example, poor grounding often causes line frequency noise (typically 50 or 60 hz) or its harmonics to be present in the image. nPs facilitates the identification of this noise; in such applications, it is common to calculate normalized noise power spectra, since the absolute noise power is less important than the relative noise power. as we shall see in section 4.6, absolute calculations of the nPs are an integral part of detective quantum efficiency (DQe) and noise equivalent quanta (neQ) measurements, and the nPs is required to calculate the snR in the application of signal detection theory.
unlike the Mtf, there is no way to measure the ‘presampling nPs’. as a result, high frequency quantum noise (frequencies higher than those supported by the sampling grid) will be aliased to lower frequencies, in the same way that high frequency signals are aliased to lower frequencies. Radiation detectors with high spatial resolution, such as a-se photoconductors, will naturally alias high frequency noise. Radiation detectors based on phosphors naturally blur both the signal and the noise prior to sampling, and thus can be designed so that both signal and noise aliasing are not present. there is no consensus as to whether noise aliasing is beneficial or detrimental. ultimately, the role of noise aliasing is determined by the imaging task, as we shall see in section 4.6.3.1.
as with the Mtf, it is sometimes preferable to display 1-D sections through the 2-D (or 3-D) noise power spectrum or autocovariance. there are two presentations that are used, the central section:
C( )u = ( , 0)u
W W (4.47)
and the integral form:
I( )u =
∑
v ( , )u vW W (4.48)
similarly, if the noise is rotationally symmetrical, it can be averaged in annular regions and presented radially. the choice of presentation depends upon the intended use. it is most common to present the central section. Regardless, the various 1-D presentations are easily related by the central slice theorem, as shown in fig. 4.11.
FIG. 4.11. Both 1-D integral and central sections of the NPS and autocovariance can be presented. The various presentations are related by integral (or discrete) transformations. Here, the relationships for rotationally symmetrical 1-D noise power spectra and autocovariance functions are shown.