International audienceIn this study, we propose a method to estimate the polychromatic X-ray spectrum of a microtomograph by measuring transmissions through a series of phantoms with known composition and thickness. An initially lognormal spectrum is iteratively optimized in order to obtain the best fit for all measurements. The validity of the estimated X-ray spectrum is verified based on an independent phantom

Keskes, N.

Moonen, Peter

Sénéchal, P.

English

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HAL Id: hal-02175597
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Submitted on 5 Jul 2019
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X-ray spectrum estimation from transmission
measurements: preliminary results
P. Sénéchal, Peter Moonen, N. Keskes
To cite this version:
P. Sénéchal, Peter Moonen, N. Keskes. X-ray spectrum estimation from transmission measurements:
preliminary results. 3rd International Conference on Tomography of Materials and Structures (ICTMS
2017), Jun 2017, Lund, Sweden. ￿hal-02175597￿
 
X-RAY SPECTRUM ESTIMATION FROM TRANSMISSION 
MEASUREMENTS: PRELIMINARY RESULTS 
P. Sénéchal*1, P. Moonen†1,2 & N. Keskes‡3 
1Univ Pau & Pays Adour, CNRS, DMEX-IPRA, UMS 3360, 64000, Pau, France 
2Univ Pau & Pays Adour, CNRS, TOTAL, LFCR-IPRA, UMR 5150, 64000, Pau, France 
3TOTAL, Pau, France 
Keywords: X-ray tomography, X-ray spectrum, dual energy. 
Summary: In this study, we propose a method to estimate the polychromatic X-ray spectrum of a microtomograph by 
measuring transmissions through a series of phantoms with known composition and thickness. An initially lognormal 
spectrum is iteratively optimized in order to obtain the best fit for all measurements. The validity of the estimated X-ray 
spectrum is verified based on an independent phantom.  
1. INTRODUCTION 
Knowledge of the spectral properties of the X-ray source and detector is important for reducing different artefacts 
as beam hardening but also for developing dual energy analysis. The X-ray source in a lab-based tomography system 
is polychromatic with a source spectrum 𝑆(𝐸). This spectrum is attenuated depending on the properties of the 
medium through which the X-rays travel. The well-known Beer-Lambert law permits expressing the transmittance 
𝑇 of a tomography system as:  
𝑇 = ∫ 𝑆(𝐸)𝐷(𝐸)𝑒𝑥𝑝 [− ∫𝜇
𝐿
(𝐸, 𝑙)𝑑𝑙] 𝑑(𝐸)
𝐸
 
where 𝐷(𝐸) is the energy-dependent detector response, 𝜇(𝐸, 𝑙) is the energy-dependent X-ray attenuation map at 
each position 𝑙 along the beam path with path length 𝐿 and 𝐸 is the photon energy. Source spectrum and detector 
response can be combined in a single system response spectrum 𝑊(𝐸) = 𝑆(𝐸)𝐷(𝐸). 
Methods of X-ray spectrum estimation can be found in the literature (see e.g. [1]). In this study, we explore the 
possibility to determine the X-ray spectrum (𝑊) by measuring the transmission through a specially-designed 
phantom with known composition and thickness in combination with a robust numerical method.  
2. MATERIALS AND METHODS 
Tomograph: A Zeiss Xradia Versa 510 X-ray microscope was used. This instrument is equipped with a 10W X-ray 
source with a tuneable energy level between 30 kVp and 160 kVp, The detector is a 4MP 16bit deep-cooled CCD 
detector which is capable of acquiring radiographs with 20482 pixels at a depth of 16 bits per pixel. The Versa 510 
permits to enlarge the sample image by geometric magnification and then, a scintillator converts X -rays to visible 
light, which is then optically magnified via X-ray optics. A 0.4x lens was used in the current study.  
Phantom: The phantom is a three-dimensional staircase, machined out of 99.9% pure aluminium. The thickness of 
each step varies from 0,1 to 1,6 cm every 0,1 cm.  
Experimental protocol: The phantom was placed between source and detector. For a given acceleration tension 
and power, one raw radiograph is recorded for each thickness in absence of a filter. Image corrections were not 
applied (i.e. no flat field nor dark field). Measurements were repeated at 80, 100 and 120 kV and at a power of 7, 9, 
10 W, respectively. The exposure time 𝑡 was taken constant for each measurement series and equal to the maximum 
time permitted to avoid saturation of the detector for the smallest thickness. For each radiograph, the transmission 
𝑇 is obtained by averaging the grey scale values of the 8x8 pixels located at the centre of the radiograph. The 
reference transmission 𝑇0 corresponds to the transmission obtained with the exposure time 𝑡0 without phantom (𝑡0 <
                                                          
* e-mail: pascale.senechal@univ-pau.fr 
† e-mail: peter.moonen@univ-pau.fr 
‡ e-mail: noomane.keskes@total.com 
𝑡). Each transmission 𝑇 is normalized by 𝑇0 and by the ratio𝑡0/𝑡. 
Spectrum determination: Equation (1) is discretized and converted into a linear system of equations 𝑅𝑖 = 𝐴𝑖𝑗𝑊𝑗 
where 𝑅𝑖 are the 𝑛 = 16 measured transmissions after normalisation, 𝑊𝑗 are the 𝑛 samples of the spectrum and  
𝐴𝑖𝑗 = 𝑒𝑥𝑝 [− ∫ 𝜇
𝐿𝑗
(𝐸𝑖 , 𝑙)𝑑𝑙] 
describes the attenuation of the phantom in a discretized fashion. The attenuation coefficients 𝜇 are obtained from 
the database of the National Institute of Standards and Technology (NIST) [2]. The matrix 𝐴 is ill-conditioned, 
rendering a direct solution of the system of equations difficult. Therefore an iterative method was applied, starting 
from an initial approximation of the spectrum. We employed a lognormal distribution, multiplied by a linear 
function. The former yields a reasonable form of the Brehmsstrahlspectrum, while the latter ensures that the 
frequency falls to zero when the photon energy equals the power 𝐸𝑚𝑎𝑥 of the generator:  
𝑊𝑖𝑛𝑖 = (1 −
𝐸
𝐸𝑚𝑎𝑥  
) ln 𝒩(𝑚, 𝜎2) 
with 𝑚 = 1
3
𝐸𝑚𝑎𝑥 the location and 𝜎
2 the scale parameters of the distribution. This initial spectrum was iteratively  
refined in order to obtain the best fit for all the measured transmissions𝑇𝑖 . 
Validation: The validity of the discretized spectrum 𝑊𝑗 was verified by the estimation of the thickness of a second 
known phantom, constructed out of 99.9% pure copper. 
3. FIRST RESULTS 
Figure 1a shows the map of 𝜇(𝐸) for the atomic element numbers from 1 to 29 and for the acceleration tension from 
0 to 160 keV, taken from the NIST database. The measured curves of normalized transmissions 𝑅 are presented on 
figure 1b for different acceleration tensions, and figure 1c depicts the corresponding X-ray spectra. The spectrum 
does not exhibit spectral lines corresponding to the X-ray source target, as the applied sampling (16 samples over 
the full photon energy range) is too low to detect them. Validation tests on other phantoms are ongoing. Additional  
tests are in progress to improve the estimation of the X-ray spectrum. 
References 
[1] E.Y. Sidky, L Yu, X. Pan, Y. Zou, and M. Vannier. A robust method of x-ray source spectrum estimation from 
transmission measurements: Demonstrated on computer simulated, scatter-free transmission data. Journal of applied 
Physics. 97, 2005. 
[2] https://www.nist.gov/pml/x-ray-mass-attenuation-coefficients. 
 
 
Figure 1: (a) map of μ(E) for atomic element number from 1 to 29 and for the acceleration tension from 0 to 
160 keV, (b)  curves of ratio T/T0 normalized by t0/t related to the phantom thickness for acceleration t ension 
of 80, 100 and 120 keV, (c) the corresponding estimation of X ray spectrums obtained with each curve of (b).  
Acknowledgement 
UMS 3360 DMEX acknowledges the inkind contribution of Total, more precisely the availability of the Zeiss Xradia  
Versa 510 X-ray microscope used in this study. 


X-ray spectrum estimation from transmission measurements: preliminary results

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X-ray spectrum estimation from transmission measurements: preliminary results

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