Dipartimento di Fisica E. Fermi
Corso di Laurea Magistrale in Fisica
Characterization of a system for breast-CT with
synchrotron radiation
Relatore Candidato
prof. Pasquale Delogu Vittorio Di Trapani
Introduction vi
1 Computed Tomography 1
1.1 Physical background of X-ray imaging . . . 1
1.1.1 Lambert-Beer’s law . . . 1
1.1.2 The linear absorption coefficient . . . 2
1.1.3 Interactions radiation-matter in medical imaging . . 3
1.1.4 Radiography . . . 6
1.2 Computed tomography . . . 7
1.3 Radon transform . . . 9
1.4 Fourier based methods . . . 13
1.4.1 Direct Fourier reconstruction . . . 13
1.4.2 Filtered backprojection . . . 14
1.4.3 Practical implementation of the FBP . . . 16
1.5 Algebraic reconstruction algorithms . . . 18
1.5.1 Data representation . . . 19
1.5.2 The inversion problem . . . 19
1.6 Artifacts in CT reconstruction . . . 21
1.6.1 Detector based artifacts . . . 21
1.6.2 Undersampling artifacts . . . 21
1.7 The advantages of monochromatic radiation compared to conventional sources in CT . . . 22
1.8 Monochromatic X-ray sources . . . 24
1.8.1 Synchrotron radiation . . . 25
2 Metrics of linear systems 29
2.1 Introduction . . . 29
2.2 The sampling process . . . 30
2.3 Metrics for the evaluation of the spatial resolution . . . 32
2.3.1 Impulse response function . . . 32
2.3.2 PSF, LSF and ESF . . . 33
2.3.3 Modulation transfer function (MTF) . . . 34
2.3.4 Presampling transfer functions: pESF, pLSF and pMTF . . . 35
2.3.5 MTF in Computed Tomography . . . 36
2.4 Metrics for the evaluation of the quality of the digital images 37 2.4.1 Contrast . . . 38
2.4.2 SNR . . . 40
2.4.3 NPS . . . 41
2.4.4 NEQ . . . 43
3 Materials and methods 45 3.1 Overview of the chapter . . . 45
3.2 Description of the experimental apparatus . . . 45
3.2.1 Beamline . . . 45
3.2.2 Detector . . . 49
3.2.3 Geometry and acquisition procedure . . . 54
3.2.4 Artifacts of continuous rotation . . . 56
3.3 Image preprocessing . . . 57
3.3.1 Resampling from hexagonal to orthogonal matrix . . 57
3.3.2 Flat field correction . . . 61
3.3.3 Recover of the gaps between detector blocks . . . 62
3.3.4 Logarithm . . . 64
3.3.5 Re-binning . . . 64
3.3.6 Individuation of the rotation axis and of the number of projections . . . 66
3.5 Analysis of planar images . . . 68
3.5.1 pLSF and pMTF on planar images . . . 68
3.5.2 Optimization of the resampling of the images from hexagonal to orthogonal lattice . . . 74
3.5.3 NNPS on planar images . . . 75
3.5.4 NEQ on planar images . . . 76
3.6 Analysis of the CT reconstructions . . . 77
3.6.1 PSF, LSF and MTF on CT images . . . 77
3.6.2 Comparison between the measured and the theoreti-cal MTF . . . 81
3.6.3 Measure of NNPS . . . 81
3.6.4 Experimental NEQ . . . 83
4 Experimental results 84 4.1 Outline of the Chapter . . . 84
4.2 Analysis of planar images . . . 85
4.2.1 pLSF and pMTF on planar images . . . 85
4.2.2 NNPS on planar images . . . 89
4.2.3 NEQ on planar images . . . 92
4.3 Optimization of the resampling of the images from hexagonal to orthogonal lattice . . . 93
4.4 Analysis of CT reconstructions . . . 99
4.4.1 Measure of the spatial resolution with the wire phantom 99 4.4.2 PSF and MTF of CT reconstructions with V120 voxel size . . . 102
4.4.3 Comparison between measured and theoretical MTF with V60 voxel size . . . 105
4.4.4 NNPS of CT reconstructions with V60 voxel size . . . 106
4.4.5 NNPS of CT reconstructions with V120 voxel size. . . 111
4.4.6 NEQ on CT reconstructions with V60 voxel size . . . 114
5 Conclusions 119
A Appendix to chapter 1 122
A.1 Interactions radiation-matter . . . 122
A.1.1 Rayleigh scattering . . . 122
A.1.2 Compton scattering . . . 122
A.1.3 Photoelectric absorption . . . 124
A.1.4 Pair production . . . 125
A.2 CT reconstructions . . . 125
A.2.1 Central Slice Theorem . . . 125
A.2.2 Derivation of FBP . . . 127
A.2.3 BP and FBP . . . 128
A.2.4 Practical implementation of FBP . . . 128
A.3 Iterative algorithms: Kaczmarz’s method . . . 130
A.3.1 ART . . . 133
A.3.2 SIRT . . . 133
A.3.3 SART . . . 134
A.4 Synchrotron radiation . . . 137
A.4.1 Essential accelerator physics . . . 137
A.4.2 Bending magnets and insertion devices . . . 140
A.4.3 Index of quality for synchrotron radiation: brightness and emittance . . . 141
A.4.4 Radiated power by charged particles . . . 142
A.4.5 Angular distribution of synchrotron radiation . . . . 144
A.4.6 Monochromator crystals: Bragg’s law, Mosaicity and Rocking curves . . . 145
B Appendix to chapter 2 147 B.1 Sampling theorem . . . 147
B.2 MTFin CT reconstructions with FBP algorithm . . . 149
C Optimization of the resampling process 153 C.1 Correlation function . . . 154 C.2 MSE and PSNR . . . 155
Early detection of breast cancer greatly increases the chances for successful treatments reducing mortality by at least 20% [1]. To date, although mammography is the main tool for detecting breast cancer, it poses several limitations to the detectability of tumors due to the superimposition of breast glandular structures [2].
Various studies have shown that breast-computed tomography (CT) increases the sensitivity of breast cancer detectability eliminating superim-position [3]. Moreover, it has been demonstrated that breast-CT improves contrast resolution compared to mammography, leading to a better identi-fication and classiidenti-fication of breast tissues at the cost of an increased dose [4] [5].
In the past, the main disadvantages of breast-CT (using whole-body CT scanners) were low spatial resolution and high dose to the patient compared to mammography [6]. Recently, the increased interest in breast-CT among radiologists led to the implementation of dedicated breast-CT scanners able to meet the needs for high spatial and contrast resolution. In particular, a dedicated breast-CT imaging system with synchrotron radiation at the SYRMEP (Synchrotron Radiation for Medical Physics) beamline of Elettra (Trieste) was developed through the SYRMA-3D (SYnchrotron Radiation
Mammography-3D) project.
SYRMA-3D uses ‘Pixirad-8’, a direct detection photon counting device with CdTe sensor and pixel size of 60µm, which potentially provides improved performances in terms of noise, efficiency and spatial resolution.
The quality of CT images depends mainly on the source, the detector and the reconstruction software.
This work focuses on the characterization of the detection system and on the optimization of the CT reconstructions using the objective metrics for the evaluation of the spatial resolution and the noise.
The detector imaging properties have been assessed from planar images. In frequency domain (FD), the noise has been evaluated by means of the Normalized Noise Power Spectrum (NNPS). The spatial resolution has been measured with an innovative technique, which allows analysis at single pixel level. The presampling Edge Spread Function (pESF) and the presampling Line Spread Function (pLSF) have been employed as metrics in the spatial domain (SD), while the presampling Modulation Transfer Function (pMTF) has been used in FD. This analysis has shown how the imaging properties of this system can be influenced by physical limitations, specific pixel geometry and resampling process.
The choice of the algorithm and of the parameters of reconstruction for CT-images, such as the voxel size, are not univocal. For this system, the optimal algorithm for CT reconstructions has been investigated comparing the Filtered Back Projection (FBP) with different filters, the Simultaneous Algebraic Reconstruction Technique (SART) and the Simultaneous Iterative Reconstruction Technique (SIRT). CT images have been reconstructed,
using two voxel sizes (V60 = 603 µm3 and V120 = 1203 µm3), with ASTRA
toolbox (All Scales Tomographic Reconstruction Antwerp), an open source software characterized by its modularity and customization [7].
For CT reconstructions, noise and spatial resolution have been assessed through the NNPS (measured on a homogeneous phantom [8]) and the Point Spread Function (PSF), the LSF and the MTF (measured with the thin wire method [9], [10]).
Finally, the measurements of the NNPS and the MTF have been used to calculate the Noise Equivalent number of Quanta (NEQ). This quantity summarizes the informations about the spatial resolution and the noise and can be used to directly compare the overall quality of the images. In particular, NEQ has been used to directly compare the CT reconstructions
performed in different conditions. This analysis allowed to characterize the performances of the reconstruction algorithms. This provides a guide for choosing the best algorithm and voxel size to be used, in order to achieve optimal performances in terms of spatial resolution and noise.
Computed Tomography
1.1
Physical background of X-ray imaging
The concept of X-rays imaging is very simple: a homogeneous beam of X-rays interacts with an object and, passing through it, a fraction of photons is absorbed by the matter depending on its thickness and chemical composition, the remaining fraction is impressed on the support producing a gray scale image.
This image can be interpreted as the map of the absorptions of the crossed objects respect to the background.
1.1.1 Lambert-Beer’s law
The removal of the photons from the incident monochromatic beam due to the interactions with the crossed matter is a stochastic event and is described by Lambert-Beer’s law
I(x) = I0 · e−µx (1.1)
where I0 is the intensity1 of the X-rays passing through an object and µ is the attenuation coefficient. The Lambert-Beer’s Law (1.1) means that a
beam of intensity I0 is exponentially absorbed by the crossed matter (see
Figure 1.1).
1_{For intensity I we consider the flux per unit of time dt of N photons through an arbitrary unit}
area dA perpendicular to the direction of propagation: I =_{(dt)·(dA)}N .
Figure 1.1: Schematic of Lambert-Beer’s law: before the object the beam intensity is constant (I0); crossing the object the beam is attenuated exponentially I(x) = I0e−µx;
after the object, the beam is attenuated by the entire thickness ∆x and I1 = I0e−µ∆x.
1.1.2 The linear absorption coefficient µ
The parameter µ is called linear absorption coefficient. It is measured in (cm−1) and its reciprocal is the mean free path λ of the photons which pass through a medium; both parameters (µ and λ) are function of the energy of the beam E, of the density ρ and the atomic number Z of the crossed object
µ = µ(ρ, Z, E) (1.2)
The coefficient µ includes all the interactions radiation-matter2 and can be rewritten in function of the cross-sections as follow
µ = ρNA
A · σtot (1.3)
where ρ is the density of target material is ρ = _{V}A ∝ Z_{V}, NA = 6.23 · 10−23
is the Avogadro constant and A3 the atomic weight.
In (1.3) the dependence of µ from the energy E and from the atomic number Z is inside σtot. In many calculations, the coefficient µ is replaced by the mass absorption µ/ρ coefficient that is independent from the density of the crossed object and is useful to determine the thickness of a material which absorb a defined fraction of input intensity.
The global cross section σtot is the result of various interactions
radiation-2_{A brief summary of the interactions radiation-matter is in Appendix A.1.} 3_{The atomic mass A is linearly dependent on the atomic number Z}
matter and can be rewritten in explicit form as follow
σtot = σR + τ + σC + k (1.4)
where σR, τ , σC and k are respectively the cross-sections of: • Rayleigh scattering: σR ∝ Z2.5 (Eγ)2 and σR ρ ∝ Z1.5 (Eγ)2 (1.5)
• Photoelectric absorption (for E < 100 KeV )
τ ∝ Z 4 (Eγ)3 and τ ρ ∝ Z Eγ 3 (1.6) • Compton scattering σcatom = Zσc and σc ρ ∝ σc (1.7) • Pair production (Eγ ≥ 1.022M eV ) k ρ ∝ Z (1.8)
1.1.3 Interactions radiation-matter in medical imaging
The diagnostic energy window for medical imaging goes from about 20KeV (for mammography) to a maximum of 140KeV (for CT). In this range
of energies the dominant interactions are photoelectric absorption and Compton scattering while pair production doesn’t occur; in Figure 1.2 and in Figure 1.3, some examples of the various contributions of the interactions radiation-matter in the mass attenuation coefficient for lead, water and breast tissue are shown.
In principle all the interactions radiation-matter remove photons from the primary beam and are calculated as absorption with the Lambert-Beer’s law contributing to the production of the images. In common setups for
(a) (b)
Figure 1.2: Mass attenuation coefficient versus incident photon energy for lead (a) and water (b). In the diagnostic energy window of CT, E = [50keV ˘140keV ], photoelectric absorption is dominant for lead and Compton scattering is dominant for water.[11]
Figure 1.3: Mass attenuation coefficient versus incident photon energy for breast tissue. The diagnostic energy window (in the box) extent from the mean energy of mammography and maximum energy for breast CT E = [18KeV − 40KeV ]. Compiled with the data from NIST database [12].
mammography or radiography the detector is close to the object to scan and most of the secondary Compton scattered photons are detected contributing negatively to the production of the images (see Figure 1.4).
The negative effects of the detection of Compton scattering can be reduced by increasing the distance detector-object (working in condition of good geometry) or by using anti-scattering grids (which remove secondary photon, but also a fraction of primary photon). The condition of ‘good’ or ‘bad’ geometry can be analytically described by introducing a corrective factor in
Figure 1.4: Schematic of interaction radiation-matter in medical imaging: A) Back-ground photon; B) Absorbed photon; C) Scattered Compton photon detected in a wrong position contributing to the noise of the image; D) Scattered Rayleigh photon; E) Photon which doesn’t interact with the crossed object as predict by Lambert-Beer’s law.
Lambert-Beer’s law (1.1) which take in account not only primary photons, but also the possibility to detect secondary photons
I(x) = I0 · B(x, E) · e−µx (1.9)
B(x, E) ≥ 1 is called buildup factor and depends on the energy and on the geometry of acquisition.
Practically the condition of ‘good geometry’ (which means B → 1) can be reached by increasing the distance object-detector. In fact, in the simplest, but also little realistic case, in which the incident beam is parallel and scattered N photons are homogeneously diffused into a semisphere, the number n of detected scattered photons depends on the solid angle covered by the detection area A
n ≈ ΩA
Ωsemisphere
= A
2πR2 (1.10)
where R is the distance object-detector. Thus, in the limit R → ∞ the fraction n → 0 leading to B ≈ 1.
1.1.4 Radiography
Applying the Lambert-Beer’s law in monochromatic case E = E0 to a
com-plex object, the linear absorption depends only on the spatial distribution of the materials µ(E = E0, x, y, z) = µ(x, y, z). If the setup of acquisition grant a buildup factor B(y, E) = 1, the spatial distribution of the detected photon which passes the object along the direction ‘y’ is given by
I(x, L, z) = I0 · e− RL
0 µ(x,y,z)dy (1.11)
where L is the thickness of the object.
The total absorption of the X-ray beam, called projection, can be obtained from Equation 1.11 with some simple manipulation:
P (x, z) = Z L 0 µ dy = −logI I0 (1.12) In Figure 1.5 there is graphic example of radiography and of extraction of the projection in the simple case of an object with absorption coefficient µ1 including a small detail with µ2 > µ1 .
Figure 1.5: Radiography in the simple case of an object with absorption coefficient µ1
including a small detail with µ2 > µ1, the projection obtained are the results of the
integral along the direction L of the spatial distribution of µ(x, z).[13]
In polychromatic case, the (1.11) must be rewritten as follow I(x, L, z) = Z Emax 0 dE I0(E) · e− RL 0 µ(E,x,y,z) dy (1.13)
The total absorption of the X-ray beam depends both on energy and on the thickness L P (x, z) = −log I I0 = −log REmax 0 dE I0(E) · e −RL 0 µ(E,x,y,z) dy REmax 0 dE I0(E) (1.14)
Both using monochromatic and polychromatic sources, radiography doesn’t allow a measurement of the linear absorption coefficient µ(x, y, z), because even knowing the thickness L of the object, in general case µ varies along of y.
In X-ray imaging we need to maximize the contrast between two anatomical structures with different µ, reduce the dose and noise as more as possible. These three requests depend on energy, the optimal energy is that which at fixed dose both maximizes the contrast and minimizes the noise. In
practice the range of optimal energies is narrow. If the energy E = E0
is the optimal energy, all photons with energy E > E0 are less probably
absorbed reducing contrast and increasing noise, all photons with energy
E < E0 are more probably absorbed increasing dose. Thus, while the use
of monochromatic sources allow to set the optimal energy E = E0, the use
of polychromatic sources allows only to set a broad spectrum with mean energy ¯E = E0, hence, all the other components of the spectrum contribute negatively to image and dose.
1.2
Computed tomography
A classical radiography gives only 2D projections of a 3D object and in many cases cannot be useful for diagnostic instances because the information along the direction ‘y’ of the projection is averaged. Besides it can happen that some structures are superimposed on each other preventing their proper display. Sometimes these problems can be solved with more than one projection at different angles, but when these solutions are insufficient a tomographic technique is required for correct diagnosis. The computed
tomography (CT) provides detailed informations on the morphology of the organs giving a map of the attenuation coefficients in a selected plane instead of the map of the projections. CT reconstruction consists on a series of mathematical techniques to obtain the map of attenuations µ(x, y) (see Figure 1.6) of a 3D object section starting from a collection of 1D
independent projections at different angles (CT-scan).
Figure 1.6: Conceptual Tomographic slice extraction: the aim of tomographic recon-struction is to recover the distribution of the linear attenuation coefficients µ(x, y) on a section of the object.[13]
A CT scan can be performed by rotating the source-detector system or the sample. The acquisition geometries for CT are shown in Figure 1.7.
The simplest geometry is the parallel beam (Figure 1.7 (a)) in which the source is at infinite distance from the sample. This condition is well approximated by the beams produced in synchrotron facilities, for instance SYRMEP beamline provides a distance source-experimental hutch of 20m granting a well approximated parallel beam.
The reconstruction algorithms which depend on the acquisition geometry can be grouped in two families of analytic and iterative methods.
The algorithms used in this thesis refer to the parallel beam geometry and are the common filtered back projection (FBP) which belong to the family of analytic solutions and two iterative algorithms SIRT and SART that will be discussed in the following sections.
(a) Parallel beam geometry (simplest geometry applied in I and II generation CT scanners).
(b) Fan-beam geometry (scanner CT of III and IV generation).
(c) Cone-beam geometry (usually adopted in micro-CT and dental CT).
Figure 1.7: CT acquisition geometries.
1.3
Radon transform
CT aims to reconstruct the object function f (x, y) = µ(x, y) of a section in a specified plane as explained in Figure 1.6. The reconstruction starts
from the 1D projections acquired at various angles4. In the parallel beam
geometry, a CT scan is the collection of 1D projections acquired at different angles ϕ source-object (Figure 1.7 (a)).
The reformulation of the line integral for CT acquisition is a generalization of Eq. (1.12) in the case in which the source-detector system rotates around
the object5. Referring to Figure 1.8, fixing the plane x-y of the object function f(x,y), the axis x0 of the rotating system (x0-y0) can be written in x-y system
x0 = xcos(ϕ) + ysin(ϕ) (1.15)
where ϕ is the angle between the two systems. Thus the line integral can
be rewritten in function of x0 and of the acquisition angles ϕ
p(x0, ϕ) = Z ∞ −∞ f (x, y) dy0 = −logIϕ(x 0_{)} Iϕ0 (1.16) The function p(x0, ϕ) assigns to each point x0 the value of the corresponding line integral (see Figure 1.8) in other words is a collection of 1D projections of the sampled angles.
Figure 1.8: Acquisition of the n-th projection of object function f (x, y). The object function is on the plane x-y while the system beam-detector rotates with the coordinate system x0-y0. The projection at n-th angle p(x0, ϑn) is composed by the line integrals
p(x0_{i}, ϑ0) for each x0i. The sampling step is given by ∆ϕ = x0i+1−x0i and is a characteristic
of the detector employed.
Finally, rewriting the expression (1.16) by the use of the Dirac δ function
5_{CT acquisition can be attained also by rotating the object in a fixed source-detector system, but}
the mathematical description of CT reconstruction depends only on the relative positions of the two systems and is the same for the two cases.
it can be formulated the expression of the Radon transform R{f(x, y)} = p(x0_{, ϕ) =} Z ∞ −∞ f (x, y) dy0 = = Z ∞ −∞ Z ∞ −∞ f (x, y) δ(xcos(ϕ) + ysin(ϕ) − x0) dx dy (1.17)
In Figure 1.9 there is an application of the Radon transform for a simple object which shows how this operator transforms points into sinusoids, for this reason the result of Radon transform is called sinogram.
Figure 1.9: Radon transform: (left) the sample with a circular detail inserted is scanned in parallel beam geometry. CT scan leads to the sinogram (right) which isR{f(x, y)}.
To reconstruct the image of the object function f (x, y) from the sinogram, the inversion of the Radon transform is required
f (x, y) = R−1{R{f(x, y)}} (1.18)
This inversion is possible6 thanks to the central slice theorem7 which
gives the fundamentals for the Fourier based methods of reconstruction such as the Direct Fourier Reconstruction (DFR) and the Filtered Back Projection (FBP).
Referring to Figure 1.10, in mathematical language the statement of the
6_{The inversion of the Radon transform is possible only for an ideal CT scan with infinite projections}
and punctiform pixel sizes. In practice the inversion of the Radon transform is an approximation of the exact solution and f (x, y) 6=R−1{R{µ(x, y)}}.
central slice theorem8 can be formulated as follow:
If the 1D Fourier transform of the Radon transform (R{f(x, y)} = p(x0, ϕ)) is
F1D{p(x0, ϕ)} = P (ν, ϕ), (1.19)
and if the 2D Fourier transform of the object function is
F2D{f (x, y)} = F (u, υ) (1.20)
then (see Figure 1.10)
P (ν, ϕ) = F (u0, υ0)|υ0_{=0} (1.21)
Figure 1.10: Central slice theorem: the 1D Fourier transform of the projection profile p(x0, ϕ) across the direction (cosϕ, sinϕ) equals the the profile extracted from 2D Fourier transform of the object function f(x,y) at angle ϕ (along the axis u0 = ν).
1.4
Fourier based methods
1.4.1 Direct Fourier reconstruction
The central slice theorem provides a mathematical tool to obtain the 2D Fourier transform (F (u, υ)) of the object function (f (x, y)) from the acquired projections (i.e. from the Radon transform of the object). The simplest and direct method which can be thought to recover the object function is provided by the 2D inverse Fourier transform of the function F (u, υ) f (x, y) = F_{2D}−1{F (u, υ)} = Z ∞ −∞ Z ∞ −∞
F (u, υ)e2πi(ux+υy)dudυ (1.22)
This theoretical approach cannot be perfectly realized in practice because the angular sampling of the ‘real’ Radon transform is not continuous, but discrete. Another obstacle due to the sampling is that Radon transform is sampled in its natural polar system (p(ρ = x0, ϕ)), but the Fast Fourier Transform (FFT) works only in Cartesian grids. For this reason it is neces-sary the ‘regridding’ process which creates a Cartesian grid of the spectral data via an interpolation process (such as ‘nearest-neighbor’, ‘bilinear’, ‘Spline’ etc.).
Figure 1.11: Regridding of the spectral projection data: the spectral projection data lying on a polar grid must be interpolated to be re-gridded on Cartesian grid. The distances between the spectral projection data increase with frequency.
As direct consequence of the radial data arrangement illustrated in Figure 1.11, the lower frequencies are oversampled while the higher frequencies are undersampled.
The interpolation process introduces a not reversible error that increases with frequency. Because higher frequencies contain the finest details of the image, the regridding process degrades the quality of the reconstructed images.
1.4.2 Filtered backprojection9
The expression for FBP algorithm directly derives by rewriting the
expres-sion of the theoretical DFR (Eq. (1.22)) in polar coordinates10
f (x, y) = Z π 0 dϕ Z ∞ −∞ |ν| P (ν, ϕ) e2πiν·x0 dν (1.23)
As suggested by its name, FBP includes two step: 1) filtration with the ramp filter |ν|; 2) backprojection (BP) i.e. the integral of the function p(x0, ν) = F−1{P (ν, ϕ)} over the angles ϕ ∈ [ 0, π) (see Figure 1.12). In
Figure 1.12: (left) acquisition of two projection of a simple rectangular object; (right) in the reconstruction process the projections are backprojected on the image plane. [13]
FBP algorithm, the ramp filter |ν|, takes into account the fact that while in F (u, ν) the lower frequencies are oversampled, the higher frequencies
are subsampled.11
9_{A complete derivation of FBP algorithm is in Appendix A.2.2.} 10_{(u, υ) → (ν cosϕ, ν sinϕ) with ν ∈ [0, ∞), ϕ ∈ [0, 2π).}
Filtering window function for FBP
The FBP expression in (A.19) doesn’t take into account the discrete nature of the detector, in real cases the ‘ramp’ filter can be insufficient for a satisfying quality of reconstructed images. In fact, according to the sampling theorem, if ∆x is the pixel size of the detector the maximum reconstructed frequency is:
νmax =
1
2∆x (1.24)
Thus to avoid Aliasing effects which can degrade the reconstructed images
a cutoff frequency νc ≤ νmax must be imposed. Besides, the ramp filter
increases the high frequency noise. These unwanted effects can be reduced by composing the ramp filter with an additional window A(ν), a low-pass filter which can provide some amount of smoothing. Thus, the FBP formula can be rewritten more generally by including the appropriate window as follow f (x, y) = Z π 0 dϕ Z ∞ −∞ A(ν) · |ν| P (ν, ϕ) e2πiν·x0 dν (1.25)
The most common additional window filter used to reduce the high frequency noise are the following:
• Butterworth: A(ν) = _{√} 1 1+(ν/νo)2n , for n > 0 • Shepp-Logan: A(ν) = sinc ν 2νc
• Hann: A(ν) = [0.5 + 0.5 cos(πν/νc)] rect
ν 2νc
• Hamming: A(ν) = [0.54 + 0.46 cos(πν/νc)] rect
ν 2νc
The reduction of high frequency noise isn’t priceless, in fact the smoothing at higher frequencies is counterbalanced by a reduction of the spatial resolution (see Figure 1.13). In practice the choice of the filters results from a trade-off between spatial resolution and high frequency noise.
0 0.5 ν_{c} ν_{c} 0 0.5 1 A( ν ) Hamming Hann Shepp Logan Butterworth n=3 0 0.5 ν_{c} ν_{c} 0 0.5 A( ν )*| ν | |ν| Butterworth*|ν| Hann*|ν| Hamming*|ν| Shepp Logan*|ν|
Figure 1.13: (left) Common window functions used for band limitation and high frequency noise reduction; (right) Composition of ramp filter with the window functions: a lower high frequency weighting smooths the high frequency noise, but reduces also the spatial resolution reintroducing part of the blurring corrected by the unweighted ramp filter.
1.4.3 Practical implementation of the FBP
Since a real CT scan is a collection of a discrete number of projections, sinograms are constituted by a discrete collection of projections at discrete angles ϕi, thus the (1.25) must be discretized as follow
f (x, y) ≈ π nϕ nϕ X i=1 F pϕi(x cos(ϕi) + y sin(ϕi)) (1.26)
where it is assumed that the angular sampling is uniformly spaced ϕi =
i − 1 nϕ
πi, with i = 1, ..., nϕ (1.27)
F p(x, ϕ) = R_{−∞}∞ |ν|A(ν)p(ν, ϕ) dν is the filtered projection.
In real cases the sinogram is also sampled along the radial direction (fi-nite pixel size of the detection system). This usually requires the use of interpolationsin reconstructing images12.
Observations on radial and angular sampling
The practical implementation of FBP starts from a discrete collection of projections, thus it has to be determined the minimum number of pixel per
projection and the minimum angular step ∆ϕ to avoid aliasing effects. The determination of the radial sampling depends on a series of factors such as the detector width, the source size etc.
An approximated estimation of radial sampling is guided by the Nyquist
sampling theory which gives the practical rule13
nR ≥
2 · F OV dmin
(1.28)
where dmin is the minimum diameter of the object, nR is the number of
radial samples and FOV is the field of view of the object.
Once determined the radial sampling for a given FOV it can be determined
also the minimum angular sampling. In Fourier domain nR radial samples
spaced by ∆R corresponds to a spacing of ∆ν = 1/(nR∆R). Thus, referring to Figure 1.14, the minimum angular step is
∆ϕ = ∆ν
1/(2∆R)
= 2
nR
(1.29)
The minimum number of angular samples over 180◦ must be
nϕ ≥
π
∆ϕ
= π
2nR (1.30)
In practice a number of angular samples nϕ ≈ nR is sufficient because
Figure 1.14: Angular sampling: 1/(2∆R) is the maximum frequency, ∆ϕ is the angular
step and ∆ν is the frequency spacing.
the spatial resolution of the real detectors is generally lower than that
provided by the pixel size. The previous rule is not a strict rule and the number of angular samples can be further reduced for technical limitations or to fasten the FBP algorithm, but it must take into account that the angular subsampling can leads to significant Aliasing artifacts. The effects of angular subsampling is shown in Figure 1.15.
(a) original (b) 4 angles (c) 16 angles
(d) 32 angles (e) 64 angles (f) 1024 angles
Figure 1.15: Illustration of the aliasing effects on the FBP reconstructed images due to the angular undersampling.
1.5
Algebraic reconstruction algorithms
The analytic reconstruction algorithms assume that the object function is continuous. Since the discrete nature of the acquisition, approximation errors or aliasing effects due to angular undersampling are possibles (see Figure 1.15).
The algebraic reconstruction algorithms assume that the object function is discrete.
1.5.1 Data representation
The object function can be represented as a matrix where to the j-th cell is assigned a constant value f (x, y) = fj. If N is the the total number of
cells, the values fj can be organized as shown in Figure 1.16. Besides, the
rays are considered as lines of a certain width14 through the (x-y) plane.
The line integral of the i − th ray is denoted as pi. Referring to Figure 1.16
Figure 1.16: Schematic of data representation in algebraic reconstruction approach: the unknown object function f (x, y) is discretized in a matrix (N=n × n); at each cell is assigned an unknown constant value fj. The quantity wij is the fraction of cell area
covered by the ray.
the line integrals can be written in a mathematical compact form
pi = N X
j=1
wijfj, i = 1, 2, ..., M (1.31)
1.5.2 The inversion problem
In the previous relation (1.31), M is the total number of rays and wij are
the coefficient of the weighting matrix (called also matrix of the system) which take into account the fact that the rays passes generally a fractional area of cells as shown in Figure 1.16.
The relation (1.31) can be rewritten in matrix form ~
P = W ~f (1.32)
where ~P = (P0, P1, ..., PM) and ~f = (f0, f1, ..., fN).
The object function ~f can be recovered by inverting the matrix W
~
f = W−1P~ (1.33)
In practical cases, M<N and the system is under-determined, thus the weighting matrix W is predominantly composed by zeros because only a small number of cells contributes to a given line integral and the direct
inversion W−1 is not practicable.
Even in the unrealistic case in which M>N, the ‘real’ system is under-determined. In fact, the processes involved in a CT scan introduce a random noise, thus, the relation (1.32) should be written as follow:
~
P = W ~f + ~n (1.34)
In this correct formulation, the system has M independent equations, but the unknowns are N’+M, thus the system has not an exact solution. An approximated solution can be obtained with the least square method by minimizing the equation
χ2 = (|W ~f − ~P |)2 (1.35)
The relation (1.35) leads ever to a solution, but even in this case a direct solution can be computationally impractical when both N and M are large. An alternative approach to the direct problem is the use of iterative methods which can lead to a computationally accessible solution. Iterative methods such as ART (Algebraic Reconstruction Techniques), SIRT (Simultaneous Iterative Rieonstructive Technique) and SART (Simultaneous Algebraic Reconstruction Techniques) are different implementations based on the ‘methods of projections’ proposed in 1937 by Kaczmartz [14].
The complete derivation of these methods is in Appendix A.3. The advan-tages of the iterative approach include improved performances compared to FBP in cases of unequally spaced projections or low statistic (number of projections and/or photons).
1.6
Artifacts in CT reconstruction
In the ideal case, the analytic algorithms will perfectly reconstruct the input object function f (x, y). The physical and technical limitations of CT acquisition process lead to a series of approximations that can corrupt the reconstructed images. The sources of the artifacts can be identified in a series of factors as the physical processes involved in a CT scan, the patient movements, the technical limitations of the detector and of the entire acquisition system.
In following sections are described the artifacts that occurred in this work of thesis.
1.6.1 Detector based artifacts
The most common artifacts have a ring shape in CT reconstructions. Ring artifacts occur when one or more pixels are out of calibration recording erroneous values in each projection. As shown in Figure 1.17, the recon-struction algorithm interprets the inconsistencies of the sinogram as a circularly symmetric feature of the scanned object.
These artifacts can be easily individuated and corrected using an interpola-tion of the nearest pixels in projecinterpola-tions or in the sinograms.
1.6.2 Undersampling artifacts
As shown in Figure 1.15, the number of angular samplings is one of the determining factors of the quality of CT reconstruction. When the object function f (x, y) is undersampled, the reconstructed images suffers of
alias-(a) (b)
Figure 1.17: Ring artifacts are generated by the uncalibrated pixels. (a) The uncalibrated pixel introduces a rotationally invariant noise which is detectable as a straight line without physical meaning in the sinogram. (b) The inconsistencies of the sinogram leads to a semi ring artifact in case of reconstruction over 180◦ or to a complete ring in case of reconstruction over 360◦.
ing effects (Figure 1.15 (d)) which appears as stripes originating from the undersampled detail of the object function. Generally undersampling arti-facts don’t degrade the diagnostic quality of the images because the stripes do not mimic anatomic structures. When it is necessary undersampling aliasing can be reduced by increasing the number of projections (i.e the angular samples).
1.7
The advantages of monochromatic radiation
com-pared to conventional sources in CT
Usually, CT reconstruction softwares and algorithms (such as FBP) assume that the attenuation coefficient µ is only function of the spatial coordinates. In monochromatic case, the measured values of the projection p depend linearly by the path y, hence p ∝ y. As shown in Sect 1.1.4 by (1.14), in polychromatic case this assumption is untrue. This leads to inconsistencies in reconstructed images. These inconsistencies result from the fact that when an X-ray beam with a broad-band energy spectrum passes through an object, the spectrum changes along the path because the bands of the
frequency spectrum are differently attenuated, depending on the specific attenuation coefficients µ(x, y, z, E). Thus, the non-linear relation between the attenuation values, µ, and the measured values of the projection, p (see Eq. (1.14)), induces the so-called beam-hardening artifacts.
Figure 1.18: Beam hardening artifact: the projection values actually measured, p, should ideally be proportional to the path length y of the beam through the object. However, in polychromatic case p is always smaller than the ideal value. [11]
In other words, if a monochromatic beam (E = E0) is used to acquire
a CT scan, the value measured on a single CT reconstruction voxel is
equal to the exact local linear absorption coefficient µ(E0) allowing the
quantitative distinction between different materials. In polychromatic case, the values of µ reconstructed are lower compared to the true values (see Figure 1.18). Moreover the value of the voxel results from a weighted
average over the spectrum (¯µ), thus, since the main informations of a
CT image are given by the capability of distinguish different materials
with µ1 and µ2 (contrast resolution), it can happen that materials with
different elemental compositions have close voxel values ¯µ1 ≈ ¯µ2 on the CT reconstruction and poor contrast respect the monochromatic case in which µ1(E0) 6= µ2(E0).
Another advantage of monochromatic sources is that the choice of the optimal monochromatic energy maximizes the contrast resolution of CT reconstructions reducing the dose to the patient (mainly due to the low energy components of the broad spectra). However, an intense source of
monochromatic radiation is provided only by synchrotron facilities which are not fully accessible (for costs and dimensions) compared to X-rays tubes (i.e. portable polychromatic sources).
1.8
Monochromatic X-ray sources
Synchrotron radiation is a powerful tool for research in various topics of science from archeology to medicine, crystallography or biology.
A synchrotron is a complex machine, but in first analysis can be outlined in six essential components (see Figure 1.19):
1. Electron gun: a source of electrons generated by thermionic emission from a hot filament;
2. Linac: a linear accelerator to accelerate electrons to about 100M eV ; 3. Booster ring: a small circular accelerator into which electrons are injected from the linac and further accelerated to a similar energy of the electrons in the main storage ring;
4. Storage ring: the main circular accelerator which contains the electrons and keeps them on a closed circular path by the use of an array of magnets. The imperfect vacuum in storage ring requires a regular supply of electrons. The storage ring isn’t exactly a true ring, but is made of alternate arced sections containing bending magnets (used to curve electrons) and straight sections in which are inserted focusing and correcting magnets (quadrupole, sextupole magnets used to cor-rect Coulombian repulsion between charged particles and aberrations produced by focusing). Sometimes in straight sections are inserted the so called insertion devices (wigglers and undulators) with the aim of generate a more intense synchrotron radiation than that produced by the only bending magnet;
5. RF cavities: specific structures to restore the energy lost by electrons due to synchrotron radiation;
6. Beamlines: the exit way of synchrotron radiation to the experiment. Beamlines run off tangentially to the storage ring. The first portion, called ‘front end’, has many functions such as isolate the beamline vacuum from that of the storage ring, filter low-energy tails of the beam, monitor and define the position and the angular acceptance of the photon beam. Then the beam is monochromated in the optics hutch and finally reaches the experimental hutch. To ensure users’ safety, the hutches are shielded by thick concrete walls.
Figure 1.19: A schematic of the essential components of a third generation synchrotron source: (1) Electrons from a source (e-gun) are accelerated by (2) a linear accelerator (linac) into an evacuated booster ring (3) where they are further accelerated. The electrons are then injected into the storage ring (4) where are maintained in a closed path by the bending magnets in the arc sections. The energy lost by synchrotron radiation is restored by a RF cavity (5). The beamline (6) conduce synchrotron radiation to the experiment.
1.8.1 Synchrotron radiation15
The Synchrotron radiation is the electromagnetic radiation emitted by radially accelerated charged particles of small mass such as electrons with relativistic energies.
The instantaneous radiated power emitted by a relativistic charged particle
(e) in a circular accelerator with radius R derives from the Larmor equation16 in relativistic case and is
PW ≈ 2 3 e2cβ4 R2 γ 4 _{=} 2 3 e2cβ4 R2 E m0c2 4 (1.36)
where c is the speed of light, β = vparticle/c and γ = 1/ p
1 − β2_{.}
In a synchrotron, the particles reach relativistic energies (γ >> 1), in these
conditions can be demonstrated17 that the radiation is collimated (along
the direction of the particle velocity ~vparticle) with a root mean square angle of
phθ2_{i ≈} 1
γ =
m0c2
E (1.37)
This result implies that the synchrotron radiation produced by a storage ring in which E = 2GeV is strongly concentrated in in a forward-pointing
cone beam of half-angle ≈ 1/γ ≈ 2.55 · 10−4rad (see figure 1.20).
Figure 1.20: Divergence of synchrotron radiation in ultra-relativistic case (γ >> 1).
Since synchrotron radiation is strongly collimated in a cone of half-angle ≈ 1/γ, on a fixed point of observation in laboratory system the radiation is visible only when the particle’s velocity is directed towards the observer. Thus, at every turn in the storage ring, the accelerated particles illuminate the observer for a time interval
16_{J Larmor. “LXIII. On the theory of the magnetic influence on spectra; and on the radiation from}
moving ions”. In: Philosophical Magazine Series 5 44.271 (1897), pp. 503–512.
∆t0 ≈ _{cγ}R
where t0 is the particle own time and R is the radius of curvature. Thus,
in terms of time of laboratory system, the point of observation will be enlightened for a time interval
∆t ≈ hdt dt0i∆t 0 _{≈} 1 γ3 R C (1.38) where h_{dt}dt0i ≈ 1 γ2.
The spectrum of this signal, which is given by the Fourier transform of
(1.38), has appreciable components up to the frequency ωc ≈ (2π)/∆t
which is called ‘critical frequency’ (see Figure1.21) ωc ≈
2π
∆t ≈ ω0γ 3
(1.39)
Here ω0 ≈ 2πc_{R} is the angular frequency of circular motion at relativistic
energies β → 1.
(a) (b)
Figure 1.21: (a) The radiating particle illuminates the observer with a pulse of F W HM ≈ ∆t; (b) The frequency spectrum is broad, but contains relevant frequencies up to ωc ≈ (∆t)−1.[16]
The relation (1.39) shows that a relativistic particle emits a broad spectrum
up to a frequency γ3 times grater than fundamental ω0. For a storage ring
with energy E = 2GeV , γmax ≈ 3914 and considering a rough estimate of
the fundamental frequency ω0 ≈ 3 · 108s−1, the critical frequency is ωc ≈ 6 · 1010ω0 ≈ 1.8 · 1019s−1
Hence the spectrum of synchrotron radiation extends from radio-waves with
of x-rays.
By means of a Fourier analysis and some calculations18 it can be shown
that this spectrum has the following characteristics:
1. Radiation from relativistically moving charge is strongly, but non completely polarized in the plane of the orbit;
2. Low-frequency components are emitted at wider angles than average
(hθi1/2 ≈ 1/γ), high frequency components are confined to an angular
range much smaller than average.
To monochromatize the broad spectrum of the synchrotron radiation, the beamlines use a system of monochromator crystals.
Usually these crystals use the Bragg’s law19.
One of the most common systems of monochromator is the ‘double-crystal monochromator’ (DCM) which uses two crystals to select a bandwidth of wavelengths. By setting relative geometries and crystallographic orientation, the radiation emerging from the DCM can be more or less monochromatic (see Figure 1.22).
(a) Nondispersive mode of DCM. (b) Dispersive mode with different ori-entations of the crystals.
Figure 1.22: Examples of DCM configurations: (a) in non dispersive mode the incident angle of any bandwidth is the same for the two crystals; (b) in dispersive the Bragg’s condition is more stringent, the bandwidth become narrower, but the flux decreases.[17]
18_{For a complete derivation see J.D Jackson Classical Electrodynamics par 14.6 p. 481-488.} 19_{See Appendix A.4.6.}
Metrics of linear systems
2.1
Introduction
The goal of X-ray imaging is to produce the images of the object function f (x, y, z) through the use of an acquisition system. In general, the acqui-sition system can be seen as a black box (see Figure 2.1) which take an input object f (~x) and gives, as output, the image g(~x).
Figure 2.1: Schematic of images production: the system of acquisition take as input a 3D object f (~x) giving an output image g(~x).
In X-ray imaging, the object function f (~x) is the spatial distribution of the linear attenuation coefficient ~µ of the anatomical structures of the patient.
The output image depends on the actual technique: for radiography, g(~x)
is the 2D projection of a 3D object; for CT, g(~x) is the reconstruction of
the 3D distribution of the absorption coefficient.
In this conceptual form, the system can be described by a function S{}
which takes an input h(x) and produce an output S{h(x)}.
For linear shift-invariant (LSI) systems, the quantitative and unambiguous characterization of the performances can be performed trough transfer functions.
A system is linear if for two input functions h1(x) and h2(x) and for each constant a and b:
S{a · h1(x) + b · h2(x)} = a · S{h1(x)} + b · S{h2(x)} (2.1) In practice, no system has a perfectly linear response and the linear approach can be performed only after a calibration of the system or in the ranges of linearity. Thus the linear-system approach is always an approximation. Besides, thanks to the uniqueness and to the reversibility of the Fourier’s transform of a function, it’s useful to remember that the study of the properties of the systems can be performed both in spatial and in frequency domain, respectively SD and FD.
For planar imaging, the system which includes source, detection system, pre- and post- processing softwares usually approximates a LSI system. In CT, the system includes also the reconstruction software which clearly introduces non linear components. However, even in this case, the use of transfer functions allows a ranking of the performances of different imaging systems.
2.2
The sampling process
A digital detector is composed by a discrete number of detection elements called pixels and the input signals are sampled in a discrete number of array elements which constitutes the output image.
The sampling process of a signal h(x), is mathematically described, in the continuous case, by the use of the Dirac’s function as follow
h(x) =
Z ∞
−∞
The second term of (2.2) is the convolution integral1 which is generally indicated in the following notation
h(x) ∗ δ(x) (2.3)
Besides, remembering that the Dirac’s function has the property that
δ(x − x0) = 0 f or x 6= 0 ∞ f or x = x0 (2.4)
with the constrain that
Z ∞
−∞
δ(x − x0) dx = 1. (2.5)
it can be simply shown that the function δ(x) is the unitary operator of the convolution integral. Thus, for an ideal detector, the sampling process is the convolution of the input signal with the Dirac’s delta function.
h(x) = h(x) ∗ δ(x) (2.6)
A digital detector samples the input signal d(x) with discrete values dn,
each value correspond to the evaluation of the function at the n-th pixel
x = n · x0. In this notation, the expression (2.2) can be rewritten in the
discrete case with uniform spacing (x0) between the sampling elements
(pixels) ˜ d(x) = d(x) ∞ X n=−∞ δ(x − nx0) = ∞ X n=−∞ dnδ(x − nx0) (2.7)
where d(x) is called the presampling signal and the sampled signal ˜d(x)
is the sequence of δ functions scaled by the sampled value of the function (see Figure 2.2).
The sampling process can lead to sampling artifacts (aliasing). The sampling
Figure 2.2: The sampling of the function d(x) consist of sequence of δ functions scaled by dn. The sampling step is of ∆x = x0.
theory gives a sufficient condition to avoid these artifacts.2
According to Shannon’s criterion, if an image contains frequencies higher than the Nyquist frequency aliasing occurs. For a digital detector the Nyquist frequency (fN y) is defined as follow: fN y = 1/(2 · x0), where x0 is the pixel pitch in mm.
2.3
Metrics for the evaluation of the spatial resolution
2.3.1 Impulse response function
The (2.2) implies that any input function can be expressed as a super-imposition of many (or infinite) impulse functions. For an ideal system
S{δ(x − x0)} = δ(x − x0), but a real system transforms the input delta
function in a distribution with non negligible width (see Figure 2.3).
Figure 2.3: The input δ(x − x0) produces an irf (x, x0).
The response of the system to the impulse δ(x − x0) is called impulse
response function (‘irf’) and is
irf (x, x0) = S{δ(x − x0)} (2.8)
Thus, the output of a linear system results from the superimposition of many irf
S{h(x)} =
Z ∞
−∞
h(x0)irf (x − x0) dx0 (2.9)
In other words, the sampling process of an acquisition system is the convo-lution between the input function and the ‘irf’
S{h(x)} = h(x) ∗ irf (x) (2.10)
If the system performs imaging, the FWHM of the ‘irf’ gives useful infor-mations on the spatial resolution which is defined as the minimum distance between two objects for which these are distinguished as separate objects (see Figure 2.4).
Figure 2.4: Two near impulses are seen as a unique largest impulse because of the superimposition of the two irf.
In practice two objects are perceived as distinct if their reciprocal distance is greater than one FWHM of the ‘irf’.
2.3.2 PSF, LSF and ESF
Other useful descriptors of the spatial resolution related to the irf are: the point spread function (PSF), the line spread function (LSF) and the edge spread function.
system and correspond to the 2D ‘irf’.
The LSF is response of the system to a line x = x0 of delta functions
LSF (x − x0) = R∞ −∞ R∞ −∞δ(x − x0) P SF (x, y) dxdy R∞ −∞ R∞ −∞ P SF (x, y) dxdy (2.11)
For shift invariant systems the expression of the LSF (2.11) simplifies to
LSF (x − x0) = R∞ −∞psf (x, y) dy R∞ −∞ R∞ −∞ P SF (x, y) dxdy (2.12)
In other words, the LSF describes the response of the system in one direction. Finally, the ESF is defined as the integral of the LSF:
ESF (x) =
Z x
−∞
LSF (x0 − x0) dx0 (2.13)
2.3.3 Modulation transfer function (MTF)
The 2D Fourier’s transform of the PSF(x,y) is called optical transfer
function OT F (u, ν) = F {P SF (x, y)}. The accepted descriptor of the
spatial resolution in frequency domain is the modulation transfer function (MTF) which is defined as follow
M T F (u, ν) = |OT F (u, ν)| (2.14)
The MTF is usually normalized to its zero frequency and shows how well an input signal is transferred to the output at each spatial frequency. This metric is commonly adopted to evaluate and compare the spatial resolution of the systems.
The MTF can be also evaluated in 1D from the FFT of the LSF. Figure 2.5 shows the typical profiles of the MTF resulting from the LSF: the MTF decreases more rapidly as the width of the LSF increases.
(a) irf normalized at maximum. (b) MTF normalized at zero frequency.
Figure 2.5: Comparison between profiles of frequency dependent M T F (u): the MTF of a system with poorer spatial resolution (that with a larger irf) decreases faster.
2.3.4 Presampling transfer functions: pESF, pLSF and pMTF
The assessment of the LSI systems is generally performed by measuring the presampled transfer functions (pESF, pLSF, pMTF) which describe the system response up to, but not including, the stage of sampling.
In practice, the presampling transfer functions can be measured by over-sampling the matrix of pixels of the detection system. There are various methods to oversample the matrix of pixels of a detector in order to measure the presampling transfer functions. The most commonly employed is the edge3 method [18, 19, 20].
This method uses images of a sharp edge placed at a shallow angle with respect to the matrix of the pixels in order to oversample it in one direction as shown in Figure 2.6.
The extraction of the profile along a vertical (or horizontal) profile gives the oversampled ESF (i.e. the pESF) whose derivative leads to the pLSF. Finally the Fourier’s transform of the pLSF gives the pM T F (u).
These functions, include the response of the system to the blur introduced by the detection processes (as the optical blur of the indirect detectors or the charge sharing in direct detectors) and the modulations due to size and geometry of the pixels active area.
3_{An edge is the limiting region between two different materials such as tungsten/air, lead/plastic}
Figure 2.6: Schematic of pMTF measurement with the original edge method.
2.3.5 MTF in Computed Tomography
The evaluation of the MTF is also useful in CT to obtain an objective
evaluation of the quality of the system (hardware+software).4
In radiography the MTF gives the response of the system which includes
only the imaging hardware and the source. In CT, the M T FCT results from
the imaging hardware and the imaging software (such as the reconstruction algorithm, the preprocessing etc.)
M T FCT(u) = M T Fimaging hardware(u) · M T Fimaging sof tware(u) (2.15) The two contributes M T Fimaging hardware(u) and M T Fimaging sof tware(u)
can be evaluated separately and finally combined to obtain the M T FCT(u).
For FBP algorithm in parallel beam configuration, the (2.15) can be written
as follow:
M T FCT(u) = M T Fimaging hardware(u) · |sinc(π∆ξu)| 2
· |A(u)| (2.16)
Where ∆ξ is the pixel size of detection, the second term is the contribute of the interpolations used in practical CT reconstructions and A(u) is the actual FBP filter5. In practice, the M T Fimaging hardware is the pMTF measured on planar images which includes both the modulations of the source and detector.
2.4
Metrics for the evaluation of the quality of the
digital images
The input to an X-ray imaging system is always a distribution of X-ray quanta. Before interacting with the imaging system, the interaction of the input X-ray beam with the sample modifies the input distribution of quanta. The output distribution of quanta resulting from this interaction is called quantum image. A quantum image is composed by a distribution of photons with negligible spatial extent, thus it can be mathematically
described by the sample function q(~r) resulting from the superimposition
of a great number of spatially-distributed impulse functions
q(~r) = Nq
X
i=1
δ(~r − ~ri) (2.17)
where Nq is the total number of quanta.
In this notation the quantum image is a distribution, thus, the mean value of the detected quanta in a specified area (i.e. the intensity of the signal in a region of interest called ROI) is given by the expectation value of the distribution intended as the ensemble average of many realizations
¯
qROI = EROI{q(r)} (2.18)
The stochastic variations in a ROI, with a homogeneous expected value, can be interpreted as the noise of the image
σROI =
p
varROI(q(r)) (2.19)
The quantum image is the input image to the imaging system, thus, an imaging system in ideal cases reproduces the quantum image. In a digital system, the detected quanta are converted by an ADC in digital signals
Ln which are stored in an array. Then, for a digital image, the evaluation
of the mean of signal and of the noise can be reformulated in function of intensities level Ln of the values as follow:
¯ lROI = 1 N · N X n=1 ln, σROI = v u u t 1 N · N X n=1 (ln − ¯l)2 (2.20)
In the following sections the metrics for the evaluation of the image quality in terms of signal and noise such as Contrast (C), Signal to Noise Ratio (SNR), Noise Power Spectrum (NPS) and Noise Equivalent number of
Quanta (NEQ) will be described.
2.4.1 Contrast
In the simple case of an homogeneous detail in a uniform background, the most common definition of contrast is the following
C = (¯lb − ¯_{¯} lo) lb
(2.21) where ¯lb is the mean signal of the background and ¯lo is the mean signal of the detail. The greater the contrast, the greater the ability to distinguish a detail from the background.
Physical meaning of contrast for Radiography and CT
The image that radiography aims to produce is the projection integral defined in Sect. 1.1.4, while CT image aim to give the map of the absorption
coefficient of the object function. Thus, referring to figure 2.7, while for CT the signals are µ1 and µ2, for radiography the signals are
p1 =µ1 · L1
p2 =(µ2 − µ1) · L2 + µ1 · L1
(2.22)
Figure 2.7: The contrast of a sample with µ2 respect to the background with µ1 < µ2
depends on the type of the exam (a) for radiography the signals are given by the line integrals of the absorption coefficients; (b) for CT, the signals are the absorption coefficients themselves.[13]
The intrinsic differences between the output signals for radiography and CT leads to different contrast resolution:
Cproj = p2 − p1 p1 = L2 L1 µ2 − µ1 µ1 with L2 ≤ L1 (2.23)
for radiography and
CCT =
µ2 − µ1 µ1
(2.24) for CT.
Comparing the two expressions for the contrast it is clear how CT images have a better contrast resolution than radiographic images.
The ideal contrast depends on the absorption coefficients which, in turn, depend on the energy, then the input contrast C(E) to the system can be changed by selecting the energy or the spectrum in polycromatic case.
2.4.2 SNR
The stochastic nature of the quantum image is the fundamental limit of the performance of the entire imaging system. Rose in 1948, recognized that the unavoidable quantum noise can impair the detectability of signals. To describe these facts, Rose introduced the SNR function for the evaluation of the capability of detect an object in a uniform background having mean
number of quanta per unitary area ¯qb. Rose’s model defines the Contrast
as follow:
C = (¯qb − ¯q0)/¯qb (2.25)
where ¯q0 is the mean number of quanta per unit area of a homogeneous
region of the object.
Then Rose defined ‘signal’ to be incremental change in in the number of
image quanta caused by the object integrated over the area of that object6.
In these hypothesis, the signal measured in a homogeneous region of the object with area A is
∆SRose = (¯qb − ¯q0) · A (2.26)
Therefore, in Rose’s model, the noise is the standard deviation σb of the
number of quanta in an equal area of uniform background. In the simplest case of uncorrelated quanta, noise obeys to the Poisson statistic and then σb =
√ A¯qb.
In these conditions the SNR is
∆SN RRose =
A(¯qb − ¯q0) √
A¯qb
= CpA¯qb (2.27)
Rose, demonstrates empirically that an observer needs a SN RROSE > 5 to
have a probability P > 50% of distinguish a detail from background. In this model, the detectability of a detail depends both on the difference of the signals and on the area of the detail itself. Rose’s model works fine in the strictly condition of uncorrelated and Poisson distributed quanta, but in
digital images the signals can be correlated, cannot obey to Poisson statistic or both. For instance, the images produced by a photon counting detector respect Poisson statistic, but can be affected by statistical correlations due for example to multiple counts of the same event while a charge integrating detector doesn’t respect Poisson’s statistic.
Concluding, this model can be useful for a first analysis, but must be inte-grated with a Fourier based metric which takes into account the modulations introduced by the detector.
2.4.3 NPS
It can be shown that if two identical images have the same magnitude of noise σ, but different noise textures, the visual perception of details and of the contrast differ (see Figure 2.8).
(a) binary image. (b) image with σbackground =
0.51 and with a fine grain.
(c) image with σbackground =
0.51 and with a coarse grain.
Figure 2.8: Comparison between different noise textures: (a) binary image without noise; (b) image with σbackground = 0.51 and with a fine grain; (c) image with σbackground= 0.51
and with a coarse grain.
Thus, the only measure of the σ for noise evaluation of a digital images can be insufficient. A more complete descriptor of the noise of digital image is provided by the NPS.
NPS can be defined as the decomposition of the square of the noise σ2 of
an image I in frequency domain. Mathematically, NPS is the Fourier’s
transform of the autocorrelation function7 and for digital images [21] can
7_{A complete mathematical description of the autocorrelation function and the NPS is in Appendix}
be written as follow N P S(u, ν) = = lim X,Y →∞ 1 X · Y * Z X/2 −X/2 Z Y /2 −Y /2
[I(x, y) − ¯I]e−2πi(ux+νx)dydx 2E = limX,Y →∞ 1 X · Y D_{} F_{XY}{∆I(x, y)} 2E (2.28)
where I(x, y) is a homogeneous image with dimensions X × Y , ¯I is the
average of the gray levels of the entire image and the symbols hi indicate the average over an ensemble of realizations needed to reduce NPS fluctuations.
If σ2 is the variance measured on the image I(x, y), NPS respect the
condition σ2 = Z ∞ −∞ Z ∞ −∞ N P S(u, ν) du dν (2.29)
NPS is a metric which measures not only the magnitude of the noise in SD (σ2), but also the spatial correlation of the image noise i.e. the noise texture in FD.
Practical implementation of the NPS
A digital detection system has finite dimensions, and images are composed by discrete distributions of N pixels with ∆x and ∆y spacings.
The derivation (2.28) must be adapted to the discrete case by replacing the
integrals with the summation (R →P), the continuous Fourier’s transform
with the discrete Fourier transform FXY → F Tnk and the continuous
frequency with discrete frequency bin u → un = n/(N ∆x). Under these
conditions, the expression of the NPS is8
N P S(un, νk) = lim Nx,Ny→∞ (NxNy∆x∆y) D_{} F T_{nk}{I(x, y) − S(x, y)} 2E (2.30)
In (2.30) the mean of the image ¯I can be optionally substituted by S(x, y)
i.e. the 2D second-order polynomial fit of the image I(x, y) to remove any
background signal in order to analyze only the stochastic noise removing the structured noise.
Theoretically the mean on the ensemble hi must be performed over an infinite number of noise realizations. In practice, this condition is approx-imated acquiring a set of flat field images, each image is then divided in ‘m’ subregions (ROI) obtaining a total of M realizations from all the acquired images. In this approximation, the mean over the ensemble hi
becomes limM →∞(1/M )
P
m, and the (2.30) can be written in a practical
implementable form N P S(un, νk) = lim Nx,Ny→∞ lim M →∞ (NxNy∆x∆y) M M X m=1 F Tnk{I(x, y) − S(x, y)} 2 (2.31)
Since NPS is proportional to the pixel area ∆x × ∆y is usually measured
in mm2.
Normalized Noise Power Spectrum
The N P S(u, ν) is often normalized for the square of the large area signal (LAS) which is the average of pixel values of the image. The normalized
NPS is denoted as NNPS and is
N N P S(u, ν) = N P S(u, ν)
(LAS)2 (2.32)
The LAS is dimensionless, hence the NNPS is measured in mm2 as the NPS.
2.4.4 NEQ
The NPS and the MTF describe respectively the amplitude variance and the signal response of the system at different frequencies. The ratio of these two quantities, normalized with the LAS, gives informations about the maximum available SNR in FD [20]. The square of this ratio is called
NEQ and is frequency dependent: N EQ(u, ν) = SN Rreal2 = LAS2 · M T F2_{(u, ν)} N P S(u, ν) = M T F2(u, ν) N N P S(u, ν) (2.33)
The NEQ function leads to various physical interpretations, for instance in planar imaging it is interpreted as the number of Poisson-distributed quanta that would produce the same SNR in case of an ideal detector. However, various studies have shown that NEQ concept can be generalized to other imaging modalities such as CT reconstructions, magnetic resonance imaging (MRI) or ultrasound images [22, 23].
In fact, an image with greater NEQ corresponds ever to a better image. The NEQ gives an universal figure of merit for all digital images allowing the comparisons of the overall quality of single images as well as the perfor-mances of the imaging systems.
The great advantage of the NEQ respect the other metrics such as SNR, CNR etc. is that it summarizes the informations of both the spatial resolu-tion and the noise variance in frequency domain giving a deep knowledge of the properties of the images.
In this work, we mainly use NEQ to compare the performances of various reconstruction algorithms.
Materials and methods
3.1
Overview of the chapter
The quality of X-ray images depends from the hardware (source and de-tector), the experimental setup (energy, intensity and shape of the beam, distance source-detector, time of acquisition etc.) and from the software which can include the preprocessing, the reconstruction algorithm for CT images and the post processing operations.
In this chapter, the experimental apparatus, the preprocessing of the im-ages, the reconstruction software and the measurements performed to the assessment of the quality of both planar images and CT-reconstructions are described in detail.
3.2
Description of the experimental apparatus
3.2.1 Beamline
All the experimental data have been acquired at the SYRMEP1 beamline
of Elettra synchrotron (Trieste). Elettra is a second generation synchrotron with 28 beamlines designed for specific tasks as photoelectron emission, imaging, lithography etc.
The storage ring can work at two energies: 2GeV or 2.4GeV .
1_{Acronym of SYnchrotron Radiation for MEdical Physics}