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Some simple rules for contrast, signaltonoise
and resolution in inline xray phasecontrast
imaging
Timur E. Gureyev *, Yakov I. Nesterets, Andrew W. Stevenson, Peter R. Miller,
Andrew Pogany and Stephen W. Wilkins
CSIRO Materials Science and Engineering, Private Bag 33, Clayton South 3169, Australia
* Corresponding author: Tim.Gureyev@csiro.au
Abstract: Simple analytical expressions are derived for the spatial
resolution, contrast and signaltonoise in Xray projection images of a
generic phase edge. The obtained expressions take into account the
maximum phase shift generated by the sample and the sharpness of the
edge, as well as such parameters of the imaging setup as the wavelength
spectrum and the size of the incoherent source, the sourcetoobject and
objecttodetector distances and the detector resolution. Different
asymptotic behavior of the expressions in the cases of large and small
Fresnel numbers is demonstrated. The analytical expressions are compared
with the results of numerical simulations using Kirchhoff diffraction theory,
as well as with experimental Xray measurements.
©2008 Optical Society of America
OCIS codes: (340.7440) Xray imaging; (100.2960) Image analysis; (110.2990) Image
formation theory; (110.4980) Partial coherence in imaging.
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1. Introduction
Xray phasecontrast imaging is a rapidly developing technique that shows great promise in
medical, scientific and industrial applications [1]. Among different types of Xray phase
contrast imaging, the socalled inline imaging method is the simplest in principle and the
easiest in practical implementation [24]. It typically involves an Xray source that can
provide high degree of spatial coherence in the incident illumination, while high chromatic
coherence is often not required [3]. Suitable sources that have been used in this imaging
modality include synchrotron insertion devices [2, 4], microfocus laboratory sources [3], and
more recently plasma Xray sources generated either by highpower optical lasers [5] or X
pinch [6]. It has been shown that inline phasecontrast imaging is capable of providing
greatly enhanced image contrast compared to conventional absorptionbased imaging,
especially for hard Xrays (~10100 keV) and samples consisting predominantly of lowZ
chemical elements. As a consequence, this technique is considered particularly promising for
medical diagnostic applications [3, 7].
It is wellknown [8, 3, 9] that inline Xray phase contrast in the case of weakly absorbing
samples and moderate propagation distances is proportional to the second derivatives
(Laplacian) of the projected electron density distribution in the sample. As a consequence,
this imaging method is particularly sensitive to edges and interfaces in the sample providing a
natural edgeenhancement effect in the images. From the point of view of most applications
this edge enhancement is considered the main advantage of the method. Therefore, it is
particularly important to find quantitative dependencies of the degree of edge enhancement,
which is conventionally characterised by suitably defined maximum image contrast, spatial
resolution and signaltonoise ratio in the vicinity of the geometric image of the edge, as a
function of the relevant properties of the source, the sample, the detector and the imaging
layout. Some results in this direction have been obtained previously (see e.g. [10] and
references therein). However, the need still exists in simple analytical expressions describing
the main characteristics of the images of edgelike features, which on one hand are general
enough to cover most experimental conditions of interest, and on the other hand are simple
enough to allow one to easily estimate the expected degree of edge enhancement without
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resorting to computer simulations. The present paper aims at providing such simple formulae
which can be used for rapid evaluation of inline Xray imaging conditions.
The structure of the paper is as follows. In the next section we derive analytical
expressions for contrast, spatial resolution and signaltonoise ratio (SNR) in inline images of
a generic phase edge. In the case of large Fresnel numbers (short propagation distances), the
formulae are derived for both monochromatic and polychromatic incident radiation, while in
the case of small Fresnel numbers (long propagation distances) only monochromatic incident
radiation is treated. In section 3 we analyze the expressions describing the contrast, spatial
resolution and SNR and present simple "rules" for estimation of these quantities in inline
images. Section 4 contains the results of numerical simulations verifying the accuracy of the
proposed "rules", while in section 5 we describe the experimental tests performed with an in
line imaging system based on a laboratory microfocus Xray source and Imaging Plates.
Finally, brief conclusions are given in section 6.
2. Analytical formulae for inline phase contrast
In this section we derive analytical expressions for contrast, spatial resolution and SNR in in
line images of an edgelike feature in a nonabsorbing object. We consider three different
cases: (1) monochromatic incident radiation and short propagation distances; (2)
polychromatic incident radiation and short propagation distances; and (3) monochromatic
incident radiation and long propagation distances. We show that our analytical formulae
obtained in these three different cases are consistent which each other, as well as with the
expected qualitative physical behavior of Xray image contrast in the relevant regimes.
Let a sample be located immediately before the "object" plane z = 0 transverse to the
optic axis z (Fig. 1). The sample is illuminated by an Xray beam emanating from an
extended spatially incoherent source located near the point
ray transmission through the sample can be characterised by the complex transmission
function
),,(
ν
yxQ
,
] exp[
μ ϕ −≡
iQ
, where (x, y) are the Cartesian coordinates in the object
plane and ν is the radiation frequency (
/
λν
c
=
transmitted beam is registered by a positionsensitive detector located immediately after the
"detector" plane
2
Rz =
.
1 R
−
z
=
. We assume that the X
)2 /(
π
kc
=
, where k is the wavenumber). The
Fig. 2. Phase shift distribution in the object
plane after transmission of the incident plane
Xray wave through a blurred phase edge.
1 0.50
x
0.51
1
0.75
0.5
0.25
0
ϕ/ϕmax
σ
Fig. 1. Inline imaging geometry.
Let us consider a simple but instructive example. Consider a generic edgelike feature in
a nonabsorbing sample (μ = 0), in a vicinity of which the distribution of transmitted phase
can be modeled as (Fig. 2)
)(
ν
  )
x
)(*(),
ν
,(
maxobj
ϕϕ
PHyx
−=
, (1)
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where the edge is assumed to be extended along the y coordinate, ϕ max(ν) is a constant
representing the local maximum of the absolute value of the phase shift,
Heaviside "step" function (which is equal to 0 for negative x and is equal to 1 for positive x),
)]2 /()( exp[) 2 (),(
obj objobj
σ πσ
yxyxP
+−=
is a function describing the "sharpness" of the
edge and the asterisk denotes twodimensional convolution.
Let
(exp{ )]( 2 [),,(
syssys
xMMyxP
πσ−=
function (PSF) of the imaging system referred to the object plane (we assume for simplicity
that the PSF is the same at all Xray energies),
variance of the PSF, σsrc and σdet are the standard deviations of the source intensity
distribution and the detector PSF, respectively, and
magnification. The particular form of the above expression for σsys is a direct consequence of
the projection imaging geometry (Fig. 1).
)(xH
is the
22212
−
)]}(2 /[)
2
sys
2212
My
σ+
−
be the pointspread
2
det
22
src
222
sys
) 1
−
()(
σσσ
−−
+=
MMMM
is the
121
/ )(
RRRM
+=
is the geometric
2.1. Monochromatic nearfield inline contrast for a phase edge
It is well known that at sufficiently short propagation distances the spatial distribution of the
spectral density,
),,,(
ν
zyxS
, in inline images can be described by the Transport of Intensity
equation (TIE) [8, 11, 12]. Let us substitute Eq. (1) into the monochromatic TIE for pure
phase objects [3, 12],
)],(*),
ν
,()/(1 )[(
ν
),
ν
,,(
sys
2
in
2
2
yxPyxkRSMR MyMxS
ϕ∇
′
−=
−
, (2)
where
distribution in the image plane
("defocus") distance. As the derivative of the Heaviside function is the Dirac deltafunction,
we can write:
(*)( *
objmaxsys
δϕϕ
x
PxP
∂−=∇
where
)(
sys obj
M
M
σσσ+≡
. Substituting this expression into Eq. (2), we obtain
)(
inν
S
is the spectral density of the incident beam,
),,,
is the effective propagation
(
2ν
RyxS
M
is the spectral density
2
Rz =
and
RR
/
2
=
′
)2 /()]2 /(exp[)*
3
M
max
2
M
2
sys
2
πσϕσ
xxP
−=
,
222
)]}2 /(exp[  )2 /(1){(
ν
),
ν
,,(
2
M
2
max
3
M
in
2
2
xxkRSMR My MxS
σϕπσ−
′
−=
−
. (3)
15 105051015
0.8
0.9
1.0
1.1
1.2
+σM
σM
S/Sin
x, μm
NF=5
NF=1
NF=0.2
Fig. 3. Plot of the spectral density distribution in the vicinity of the geometric
image of the edge feature for different values of the Fresnel number. The
hatched area indicates the location of the object.
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From Eq. (3) we can easily find the locations,
the spectral density in the vicinity of an edge. These positions correspond to the centre of the
"positive" and "negative" Fresnel fringes near the geometric shadow of the edge (Fig. 3) (note
that the TIE allows for only one positive and one negative Fresnel fringe).
In the monochromatic case we define the image signal, Σ(ν, A), as the absolute value of
the difference between the image spectral density distribution and the corresponding spectral
density, S0(ν) ≡ M 2 Sin(ν), in the background image (without the edge feature), integrated
over the area A = 2Ma×MLy in the image plane, where Ly is the "length" of the edge feature in
the object plane in the direction parallel to the edge and
the x coordinate. Note that Ly is assumed to be sufficiently large (in particular, it is much
larger than σsys and σobj). Using Eq. (3) we obtain:
M
x
σ
±=
, of local maximum and minimum of
),(
aa
−
is a vicinity of the edge along
()
. )]
2
exp(1 [)(
ν

2
)(
ν
2d
2
exp
2
)(
ν
)(
ν
2d)(
ν
,
ν
,,),(
ν
2
M
2
max in
2
M
2
2
0
3
M
max002
M
y
Ma
∫
y
Ma
∫
−
Ma
y
a
σ
k
R
LSx
M
x
M
x
k
R
MLSxSRyxSMLA
ϕ
πσ
σ
σπ
ϕ
−−
′
=
′
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
′
−×
′′
=
′
−
′′
≡Σ
The noise is then calculated assuming the Poisson statistics as
where
y
aLSAD
)(4),(
ν
=
is the sum of the total Xray spectral densities incident on the
region of interest in the images with and without the edge feature. The SNR is then equal to,
R
LSASNR
νϕ
πσ
),(
A SNR ν
as a function of a reaches its maximum at
approximately equal to the width of the first Fresnel fringes in the TIE regime, it is natural to
choose
M
a
σ
2
=
for the calculation of the SNR corresponding to the phase contrast generated
by the edge. Such a choice leads to almost maximal SNR, while also being convenient for
practical evaluation of SNR in experimental phasecontrast images. We therefore define
) 4 ,()(
yM
MLM SNRSNR
×≡
σνν
and obtain:
),(
ν
),(
ν
ADAN
=
,
inν
)]
2
162
exp(1 [)(
2
])(
ν
[),(
ν
2
M
σ
2
2 / 1
−
max
2/ 1
in
M
y
a
σ
a
k
−−
′
=
. It can be verified that
M
a
. 2
≅
. As the quantity
M
σ
2
is
TIE
)(
ν
 )](
ν
[)(
ν
max
12 / 1 TIE
1
TIE
ϕ
−
F
=
NDCSNR
, (4)
where
yMLSD
σ
)
ν
(
ν
(8)
in
TIE
=
is the corresponding total incident Xray spectral density,
RkN
MF
′
=
/
2
σ
is the minimal Fresnel number (which corresponds to the size, 2
M
σ
, of the
smallest resolvable detail in the image [13]) and
absolute constant.
We define the
)([),(
min maxmax
νν
SSAC
−=
maximum and minimum of the image spectral density in region A. Given the locations,
x
σ±=
, of local maximum and minimum of spectral density in the vicinity of an edge and
using Eq. (3) we obtain the following expression for the maximum contrast
)4 ,()(
maxmax
yM
MLMCC
×≡
σνν
inside the area A = 4MσM × MLy :
0862. 0])2 ( 4/[)e1 (
2/ 12
1
≅−=
−
π
C
is an
maximum
/[)](
ν
contrast
S
+
in
, where
a region
S
of
and )
ν
an image
are the
as
)](
ν
)(
ν
min max
S
)(
ν
(
minmax
S
M
TIE
2/ 1
−
TIE TIE
3max

1
2
TIE
max
)](
ν
[ )(
ν
)(
ν
)(
ν
−
F
==ϕ
DSNRCNCC
, (5)
where
Equation (5) describes the dependence of the contrast on such parameters as the wavelength
2420 . 0e) 2 (
1/2
2
≅=π
C
and
806 . 2)]e1 (e/[4
2 1/2
3
≅−=
C
are absolute constants.
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