6.1. Introduction
Chapter 5 shows how the different mammography digital detectors were characterised. To be able to adapt the image quality of acquired images it is necessary to be able to calculate the noise power spectra (NPS) at any dose and beam quality for each detector of interest. The method employed here is to model the NPS at one reference beam quality and then to adapt the model for other beam qualities (section 6.5). This chapter develops a model for the dependence of the NPS on beam quality and shows the methodology for creating a real noise image from the calculated noise power spectra (NPS). The detective quantum efficiency (DQE) is shown in chapter 5 at one beam quality for each detector, using the noise model the DQE was calculated for a range of beam qualities and doses.
6.2. Linearisation of the image
The noise model is based on the NPS and so the linearization method used for the image is important. The linearization of images is discussed in section 5.2.3. The images are linearised to absorbed energy per unit area in the detector.
6.3. Model of noise
63.1. Noise coefficients
A key requirement for the adaption of the image quality of an image is knowledge of the magnitude and colour o f the noise associated with each detector. It is assumed that there are three sources that contribute to the noise in an image: electronic, quantum and structure noise. Each has a different dependence on dose and spatial frequency (Evans et al 2002, Mackenzie and Honey 2007, Workman and Cowen 1993) and it is convenient to express the total NPS (IP) as a sum of three terms which show explicitly the dependence on absorbed energy per unit area {E^ and spatial frequency:
W(u,v) = co^(u,v) +
CD
{u,v)+ (Dfu,v)
(6.1)In this expression, cOe, coq and cOs (with units of mm^) are referred to as electronic, quantum and structure noise coefficients respectively at a reference absorbed energy per unit area (Eo). It is not essential that the three terms exactly quantify the electronic, quantum and structure noise sources, only that the NPS can be fitted by a quadratic of the form shown. Higher order terms were tried but do not improve the fit to the NPS. The value of Eg was set to 1 keV mm'^.
The electronic, quantum and structure noise coefficients were estimated by fitting equa tion 6.1 using a least squares method for each spatial frequency to the NPS data obtained at the different dose levels. The results o f the noise separation for three systems have been published in Mackenzie et al (2011) and Mackenzie et al (2012).
6.3.2. Noise coefficients: global f it
Originally the fits to equation 6.1 were made separately for each spatial frequency. The resultant noise coefficients cOe, cOg and cOs were quite noisy due to the limited number of measurements made. An improved fitting process was subsequently developed using a global fit to all data and the assumption that the variation of the coefficients with spatial frequency was smooth, with the possible exception of spikes in the structure noise coeffi cient. A measure o f the goodness of fit was required to undertake the global fitting and a reduced chi squared function (Zr ) was chosen, which is given by:
=
S S Z w(wa)
u V E. <j{u,v,Ef)
N - p - \
, , ( « N v f r '
u V
where IF'" is the measured NPS, fV-^is the fit of the NPS at Ea, G is the standard deviation of
the measured NPS, N is the number o f data points used in the calculation, p is the number of fitting parameters, w(w,v) is a weighting function, and is the number o f weighting points.
The standard deviation was estimated from the calculated uncertainty of the NPS (Dobbins, 111 et al 2006) and the variability of the output between exposures (estimated as 1%). The standard deviation will be smaller for the dose levels where multiple images were acquired, and so these will influence the fitting more than those with only one image. It was necessary to weight the z l function in equation 6.2 towards the low frequency for two reasons. Firstly, the result is naturally weighted towards the high spatial frequency as the number of points increases with the square of the spatial frequency up to the Nyquist frequency. Secondly, it is important to obtain an accurate fit o f the low spatial frequencies, as this will have a greater influence on the conversion than the high spatial frequencies. The spatial frequencies which have a spike in the structural noise coefficient were not used in the fitting and the spikes were replaced in the structural noise at the end o f fitting. Various func tional forms were tested for the noise coefficients in equation 6.1 before deciding that the functions in table 6.1 produced the best fit. The parameters {e„, q„, and s„) for each were iteratively adjusted to minimize the reduced chi squared function using the Nelder-Mead algorithm (Nelder and Mead 1965). The process was repeated many times with different starting points to obtain the best minimum available.
The noise fitting was applied to the measured NPS of six detectors described in the pre vious chapter (section 5.2.1). The best fitting functions found for each detector is shown in table 6.1. As the detectors were characterized with systems with different anodes, the coeffi cients were measured using reference beam qualities o f 28 kV, molybdenum (Mo) anode/Mo filter (Mo/Mo), 45 mm PMMA for ASEh, CRc and CSI detectors and 29kV, tungsten (W) anode/ rhodium (Rh) filter (W/Rh), 45 mm PMMA for ASEs, NIPa and NIPc detectors.
Table 6.2 shows the chi squared function calculated for the raw fit as described in section 6.3.1. The global fit for each of the detectors improves the fit compared to the raw and fitted noise coefficients for most o f the detectors. Only the NIPc produces a slightly poorer fit.