The easiest way to reduce the noise in the images is via averaging. If the noise in the images is additive with zero mean, averaging over a number of K consecutive measurements of x reduces the standard deviation of x by a factor of 1/√K, since the uncorrelated noise contributions in the
images will level out. Even if only a single measurement is performed, in some experimental situations averaging over a symmetry axis of the sample or a certain constant region-of-interest can yield a higher SNR for a given application. In the following, the average over a number of
K measurements of a variable x will be denoted x.
In DPC imaging, the assumption of additive noise with zero mean does only apply to the raw data and the attenuation-contrast channel, i.e. the calculation of the mean of the phase- stepping oscillation, due to the linearity of the reconstruction algorithm. In contrast, the calculation of the amplitude and phase of the phase-stepping oscillation is non-linear and shows a transition in its statistical behavior, from Gaussian in high SNR for both parameters, to a uniform distribution for the differential phase-contrast and a Rayleigh distribution for the amplitude. Therefore, the noise properties of these two contrast channels cannot be described by an additive noise with zero mean. For this reason, the arithmetic average over a number of measurements will converge towards the expectation values of the noise distributions, not however towards the parameters ϕ and a1 of the phase-stepping oscillation that are desired.
From a more abstract and analytical point of view, when performing averaging, we are actually interested in an adequate estimator for the parametersϕ and a1of the underlying distributions
of an ensemble of reconstructed data. While arithmetic averaging is an appropriate estimator only for the mean of the phase-stepping oscillation, for the estimation of the phaseϕ and the amplitude a1from a number of noisy measurements, more elaborate means are necessary.
That the arithmetic average is an unsuitable estimator for the phase can be seen by averaging ϕ1=π−ε andϕ2=−π+ε, with ε> 0. The arithmetic mean isϕ = 0, whereas the desired
result would be ϕ =−π. This problem can be solved by choosing an estimator adapted to the circular character of the underlying distribution. The distribution of the phase ϕ is a wrapped normal distribution after Eq. (5.13). Instead of describing the statistical moments of the wrapped distribution by the random variable x, the properties of the distribution can be described more directly by a circular variable z = eix. The expectation value of the circular variable z is given by [Fisher93]
E(z) = exp
(
iϕ−σϕ2/2
)
. (5.53)
The expectation value of the original angular variable x can therefore be calculated by taking the complex argument of the expectation value of z
E(x) = arg (E(z)) =ϕ. (5.54)
If the standard definition of variance is applied to the measured data, the variance of ϕ will also be affected by the transition from a Gaussian distribution to a uniform distribution and converge towards a value of V ( ˆϕ) =π2/3 as described by Eq. (5.19). Therefore, the concept
0.01 0.1 −1 −0.5 0 0.5 1 noise level σ differential phase
a) Differential phase estimation
raw estimator complex estimator phase average 0.01 0.1 100 102 104 noise level σ amplitude b) Amplitude estimation average ref estimtor ref average Al estimator Al high SNR
Fig. 38: Different estimators for the (a) the differential phase and (b) for the amplitude. (a) shows that
while averaging over the reconstructed phases leads to a decrease of the mean phase, estimation by averaging over the raw and complex data respectively leads to a correct estimation of the phase, only with different noise properties. In (b), the amplitude estimator is shown relative to the mean estimator, for both the reference ROI and the Al bar.
of circular variance was developed in directional statistics. It is equal to the variance of z and can be calculated by V (x) = ln ( 1 |E(z)|2 ) =σϕ2. (5.55)
If these considerations are taken into account, we see that the average of the phase of the phase-stepping oscillation should be calculated by identifying z = c1 and thus performing the
averaging over the complex first Fourier coefficient. The average complex Fourier coefficient
c1is known as mean resultant vector in directional statistics
c1= a1· exp(i ·ϕ) . (5.56)
The average phaseϕ can be derived from the resultant vector by
ϕ= arg (c1) , (5.57)
while the circular variance ofϕ is
V (ϕ) = ln (
1/|c1|2
)
. (5.58)
The mean of an ensemble of phase measurements has to be calculated by the average over the complex Fourier coefficients. In other words: arg(c1) is a biased estimator for the average
phaseϕ and ln(1/|c1|2) is a biased estimator for the circular variance V (ϕ). The average over
the reconstructed phases will tend towards zero for low SNR and can thus be excluded as an estimator for∆ϕ.
For the reconstruction of the differential phase shift ∆ϕ caused by the sample, an additional possibility has to be taken into account: averaging over the raw data Sn and averaging over the complex Fourier coefficients. While both methods lead to the correct reconstruction of the parameterϕ of the phase distribution, there are differences in the variance of the estimation. This is due to the fact that
arg ( c1,s c1,r ) 6= arg (( c1,s c1,r )) . (5.59)
As the step from raw data to Fourier coefficient is linear, the right hand side is obtained by averaging over the raw data, while the left hand side can only be obtained by averaging the
Analysis of signal and noise properties 75
Fourier coefficients in the complex plane. The estimation of the oscillation phase from the ROI of the PMMA wedge by the use of the three averaging methods can be seen in Fig. 38 (a). While the arithmetic average decreases towards zero for increasing noise level, the raw and complex estimators remain constant. Nevertheless, there are differences in their noise properties. Similarly, the average a1= 2|c1| is an unbiased estimator for E(a1). Nevertheless, in a low
SNR measurement, it is not the expectation value of the amplitude that is of interest, but the parameter a1 of the Rician distribution as presented in Eq. (5.21). While for high SNR,
the expectation value of the Rician distribution is approximately equal to a1 as described by
Eq. (5.28), for low SNR, the expectation value of the amplitude rises with the noise level, as described by Eq. (5.25). Thus, while the mean reconstructed amplitude is easily obtainable by Eq. (5.58), the actual oscillation amplitude and thus the parameter that carries information considering the sample remains unknown. The simplest estimator for the parameter a1 of the
Rician distribution can be found by solving Eq. (5.30) for the amplitude
a1=
√
V (x) + E(x)2− 4 · a2
0· ˜σ2. (5.60)
While E(x) can be estimated by averaging over the measured data and V (x) by the squared standard deviation of the data, ˜σ can be extracted from the attenuation-contrast channel. The estimation of the amplitudes for the reference and sample measurements from the ROI of the Al bar as presented in section 5.1.7 can be seen in Fig. 38 (b). The calculations were performed by the use of the arithmetic average on the one hand and by Eq. (5.60) on the other.
While the estimation leads to better results than the estimation using solely averaging, for low SNR, the estimator still leads to deviations. Following the argumentation in Ref. [Sijbers99], this is due the biased nature of this estimator. The problem of estimating the parameters a1and
V (a1) from a given number of amplitude data has been widely studied in the literature and will
not be followed further in this work. The interested reader is referred to three different methods: the methods of moments [Talukdar91], the method of maximum likelihood [Bonny96] and the method of least squares [Sijbers99].