Noise properties of grating-based x-ray phase contrast computedtomography

Thomas Kohlera)

Philips Technologie GmbH Forschungslaboratorien Rontgenstraße, 24–26 D–22335 Hamburg, Germany

Klaus Jurgen EngelPhilips Technologie GmbH Forschungslaboratorien Weißhausstraße, 2 D–52066 Aachen, Germany

Ewald RoesslPhilips Technologie GmbH Forschungslaboratorien Rontgenstraße, 24–26 D–22335 Hamburg, Germany

(Received 1 September 2010; revised 29 November 2010; accepted for publication 2 December

2010; published 20 July 2011)

Purpose: To investigate the properties of tomographic grating-based phase contrast imaging with

respect to its noise power spectrum and the energy dependence of the achievable contrast to noise

ratio.

Methods: Tomographic simulations of an object with 11 cm diameter constituted of materials of

biological interest were conducted at different energies ranging from 25 to 85 keV by using a wave

propagation approach. Using a Monte Carlo simulation of the x-ray attenuation within the object, it

is verified that the simulated measurement deposits the same dose within the object at each energy.

Results: The noise in reconstructed phase contrast computed tomography images shows a maxi-

mum at low spatial frequencies. The contrast to noise ratio reaches a maximum around 45 keV for

the simulated object. The general dependence of the contrast to noise on the energy appears to be

independent of the material. Compared with reconstructed absorption contrast images, the recon-

structed phase contrast images show sometimes better, sometimes worse, and sometimes similar

contrast to noise, depending on the material and the energy.

Conclusions: Phase contrast images provide additional information to the conventional absorption

contrast images and might thus be useful for medical applications. However, the observed noise

power spectrum in reconstructed phase contrast images implies that the usual trade-off between noise

and resolution is less efficient for phase contrast imaging compared with absorption contrast imag-

ing. Therefore, high-resolution imaging is a strength of phase contrast imaging, but low-resolution

imaging is not. This might hamper the clinical application of the method, in cases where a low spa-

tial resolution is sufficient for diagnosis. VC 2011 American Association of Physicists in Medicine.

[DOI: 10.1118/1.3532396]

Key words: phase contrast tomography, grating-based phase contrast imaging, Talbot interferometer

I. INTRODUCTION

In conventional x-ray imaging, the contrast is generated by

the attenuation of the x-ray beam by the photoelectric effect

and the Compton effect. An alternative way to generate con-

trast is to measure the phase shift that an object imposes on

the “wave front” of an x-ray beam.

Although a number of phase contrast imaging (PCI) tech-

niques exist,1 PCI plays only a minor role in the domain of

medical imaging, mainly because of the stringent require-

ments of known techniques on coherence and monochroma-

ticity of the x-ray beams. These are met by synchrotron

x-ray sources but unmet by conventional x-ray tubes as

almost exclusively used in the medical domain. Recently, a

technique well known in the visible range of wavelengths

has been adopted to the x-ray domain, namely so-called Tal-

bot-Lau interferometry.2–5 This technique permits the use of

conventional tubes and functions with partially coherent

beams generated by gold gratings placed close to the x-ray

tube’s exit window. Among all known techniques of phase-

contrast imaging in the hard (medical) x-ray domain, Talbot-

Lau interferometry requires the smallest coherence volume

for phase-sensitive imaging.4,41

The grating-based setup allows tomographic reconstruc-

tion of the three-dimensional distribution of the refractive

index.6–8 The resulting images of biological samples like a

rat brain6 or a rat heart9 show excellent soft tissue contrast

that would likely add clinical value in diagnostic imaging if

it is applicable to human imaging.

Currently, experimental setups for the grating-based PCI

use x-ray energies in the range from 20 to 60 keV (Ref. 10)

meaning that a wide range of medical applications are

already in reach from an energy perspective.

Noise in phase contrast x-ray imaging is different to that

of absorption contrast imaging as shown, e.g., for the case of

propagation-based PCI.11,12 In fact the noise shows a broad

spatial auto-covariance profile, or, in other words, there is a

significant amount of low-frequency noise.

The purpose of the present work is to investigate the noise

features of tomographic grating-based PCI, viz., the noise

S106 Med. Phys. 38 (7), July 2011 0094-2405/2011/38(7)/S106/11/$30.00 VC 2011 Am. Assoc. Phys. Med. S106

power spectrum and the energy dependence of the contrast

to noise ratio (CNR) that can be achieved.

II. METHOD

This section starts with a brief summary of the main

aspects of grating-based differential phase contrast imaging

(DPCI). An in-depth description can be found in Ref. 13.

The setup for grating-based DPCI is shown in Fig. 1, where

only the most important optical elements required for phase

contrast imaging are shown.3,5 A plane wave of coherent

x-rays hits the sample. Due to refraction, the phase front

behind the object is distorted. The distorted wavefront passes

a beam-splitter grating G1, which creates a characteristic inter-

ference pattern at the location of the analyzer grating G2. The

interference pattern is imaged by measuring the x-ray inten-

sity using a detector D immediately behind G2 at several rela-

tive transverse positions of the gratings G1 and G2.

The pitch g1 of the beam-splitter grating G1 is typically in

the order of a few micrometers, and the interference pattern

at the position of G2 changes on the same length scale. By

analyzing the interference pattern using an absorber grating

G2, the fine changes which the object imposes on the inter-

ference pattern can still be resolved, despite the fact that the

pixel size of the detector D is much larger than the character-

istic length scale of the interference pattern. This is true,

however, only as long as the interference pattern is still peri-

odic with the spatial frequency matching the pitch of G2,

which is the case as long as the gradient of the phase front

just before G1 is constant over the pixel area.

II.A. Simulation

In phase contrast imaging, the interaction of the object

with the x-ray beam is described by wave optics.14 The com-

plex refractive index of the material is

n ¼ 1� dþ ib; (1)

where d describes a phase shift that the object imposes onto

the wavefront and b is related to the linear attenuation coeffi-

cient l by

l ¼ 4pbk

(2)

with k being the wavelength of the x-rays, which is related to

the energy E of the photons by the relation E¼ hc=k where his the Planck constant and c is the speed of light. The propaga-

tion of a monochromatic x-ray beam with wave vector~k ¼ ðkx; ky; kzÞ represented by a complex wave field wx is

mathematically described by the Helmholtz equation

ðr2 þ k2Þwxðx; y; zÞ ¼ 0; k ¼ j~kj ¼ x=c ¼ 2p=k: (3)

If we assume that the wave field is known at the plane y¼ 0

before the object and that the wave propagates along the yaxis, i.e., jkxj; jkzj � jkyj, then the wave field behind the

object, i.e., at y¼D, can be calculated according to the pro-

jection approximation as

wxðx;D; zÞ � e�ikDe�ikÐ D

0ðdðx;y;zÞ�ibðx;y;zÞÞdywxðx; 0; zÞ: (4)

For later reference, it is convenient to introduce here the two

projections of interest, namely

aðx; zÞ ¼ðD0

2kbðx; y; zÞdy (5)

Uðx; zÞ ¼ðD0

kdðx; y; zÞdy: (6)

The projection a(x,z) is used to reconstruct an attenuation

contrast image, whereas the projection U(x,z) is used to

reconstruct the phase contrast image.

In our simulations, we applied the projection approxima-

tion to calculate the propagation of the initial wave front

through the object to the position of G1, i.e., the grating G1

is located at y¼D. For the further propagation from G1 to

G2, separated by a distance d, we used the Fresnel propaga-

tion operator

DFd ¼ expðikdÞF�1 exp � id

2kðk2

x þ k2z Þ

� �F ; (7)

where F and F�1 denotes the two-dimensional Fourier

transform with respect to the coordinates x and z and its

inverse, respectively.

In summary, the simulation of the measurement process

consists of the steps

(1) Setup of a plane wave wx(x, 0, z) at the plane y¼ 0 as

wx(x,0,z) ¼ 1.

(2) Application of the projection approximation, Eq. (4),

using analytically calculated projections a(x,z) and

U(x,z) to obtain the wave field wx(x,D,z) at the location

of the grating G1.

(3) Simulating the effect of the phase grating G1 by modulat-

ing the phase of wx(x,D,z) with a rectangular function with

pitch g1, resulting in a modulated wave field ~wxðx;D; zÞ.(4) Forward propagation of ~wxðx;D; zÞ to the position of

grating G2 by application of the Fresnel operator result-

ing in ~wxðx;Dþ d; zÞ.FIG. 1. Grating-based phase contrast CT setup.

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Medical Physics, Vol. 38, No. 7, July 2011

(5) Simulating for each relative position of the analyzer gra-

ting G2 with respect to the phase grating G1 a detector

reading by:

(a) Simulating the effect of the analyzer grating G2 by

modulating the amplitude of ~wxðx;Dþ d; zÞ with a

rectangular function with pitch g2 resulting in

wxðx;Dþ d; zÞ.(b) Simulating an integrating detector by integrating

jwxðx;Dþ d; zÞj2 over the detector pixel area.

II.B. System setup

Figure 2 shows the mathematical phantom used in the

study. The material compositions were taken from,15 and the

optical properties of these compositions were calculated

based on the formulas described in Ref. 16 using atomic

form factors as tabled in Ref. 17.

Most system parameters were taken from the experimen-

tal setup used by Pfeiffer et al.:6 The total of 721 parallel

beam projections over 180 with 1024 detector pixels cover-

ing 13 cm field of view were simulated (128 lm pixel size).

Since we simulated only a single line system, the coordinate

z was dropped in Eq. (4), which is equivalent to simulating

an object that is homogeneous in z. For each projection

angle, eight different relative positions of G1 with respect to

G2 were simulated in order to retrieve the absorption contrast

and phase contrast images as described in Sec. II C. The

scans were simulated for energies from 25 to 85 keV in steps

of 5 keV assuming monochromatic, coherent x-rays in order

to evaluate the energy dependence of the CNR. The pitches

of the gratings G1 and G2 were g1 = 4 lm and g2¼ 2 lm and

kept constant. G1 was simulated as a pure phase grid impos-

ing 0 or p phase shift to the wave field and G2 was a perfect

analyzer grid, i.e., the amplitude was either not or com-

pletely damped. The distance from G1 to G2 was adjusted for

each energy to the first fractional Talbot distance given by

the relation

d ¼ g21

8k; (8)

resulting in distances ranging from d¼ 40.3 mm for 25 keV

to d¼ 137.1 mm for 85 keV.

For some evaluations, the results are also compared with

a conventional CT system having the same pixel size.

Results from this setup are labeled as obtained by incoherent

absorption contrast imaging (IACI) whereas absorption con-

trast results from the grating-based setup are labeled as

obtained by coherent absorption contrast imaging (CACI).

II.C. Data processing and reconstruction

Absorption and differential phase contrast projections

were obtained from the eight detector readings at different

relative positions of G1 with respect to G2 by the standard

processing as described in Ref. 3: The intensity variation of

the measured intensity at detector pixel j as a function of the

displacement n of the grating G2 in x-direction is modeled as

fjðnÞ ¼ Ijð1þ Vj cosð2pn=g2 þ ujÞÞ; (9)

where Ij is the mean transmitted intensity, Vj is the visibility,

and uj is a phase parameter. All three parameters or deriva-

tives of them can be used for tomographic imaging,3,18 but

here, we restrict ourselves to absorption contrast imaging

(ACI, related to the parameter Ij) and to differential phase

contrast imaging (related to the parameter uj).The two quantities needed for DPCI and CACI were

derived from the l¼ 1, …, N samples acquired at relative dis-

placements nl¼ lg2=N of the transmitted intensity fjl¼ fj(nl)

(which are obtained by the step 5(b) as described in Sec. II A)

for each individual detector pixel j according to:

Ij ¼1

N

XN

l¼1

fjl (10)

uj ¼ argXN

l¼1

fjl cosð2pl=NÞþ iXN

l¼1

fjl sinð2pl=NÞ !

; (11)

where arg(c) denotes the main branch of the argument of a

complex number c, which is sometimes also written as

arg(c)¼ atan2(=(c), <(c)). It is important to note that the

phase uj cannot be calculated unambiguously from the

measurements since the codomain of arg is (�p, p]. This is

reflected in Eq. (11) by the use of the symbol uj rather than

uj in Eq. (9). Phases uj outside the interval (�p, p] are

wrapped into this interval.42

From the first quantity, the conventional absorption pro-

jection is computed by calculating pixel by pixel

aj ¼ � logIj

�I

� �; (12)

FIG. 2. Phantom used for simulations. The diameter of the object is 110

mm, the background material is breast mammary gland adult 1. The objects

have a diameter of 5.1 mm. All objects are invariant in z direction.

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Medical Physics, Vol. 38, No. 7, July 2011

where �I is the intensity of the blank scan, which was constant

for the entire detector in our simulation. The set of all aj

derived from one projection is a discrete version of the con-

tinuous projection as introduced in Eq. (5).

The values uj are proportional to the derivative of the pro-

jection of the real part of the complex refractive index with

respect to x, in the following denoted as the phase gradient,

with the exception that phase gradient wrapping can occur.

Specifically, if we start from the relation given in Ref. 3

(dropping the index j for the sake of compact notation and

stressing that the left hand side is u and not u)

u ¼ kd

g2

@U@x

; (13)

with k being the wavelength of the x-rays, and insert for the

distance d the mth fractional Talbot distance

dm ¼ mg2

1

8k; (14)

where m is an odd positive integer as well as the relation g1

¼ 2g2 for the parallel beam geometry used in this study, the

phase is

u ¼ mg2

2

@U@x) @U

@x¼ 2

mg2

u: (15)

Now we define the dynamic range of the measurement

setup as the range of @U=@x that can be correctly recovered

from a measurement of u, i.e., as the range of @U=@x,

where u ¼ u. For a setup using the first fractional Talbot

distance (m¼ 1) we infer from Eq. (15) that the dynamic

range of the estimated phase gradient @U=@x is (�p, p)

phase difference per one half of the grating pitch of G2. For

setups using a higher order fractional Talbot distance, the

dynamic range of @U=@x is even smaller. Gradients outside

this interval are wrapped into this interval because the arg

in Eq. (11) wraps phases outside the interval (�p, p) into

this interval.

In a previous work,19 we observed strong cupping or

capping artifacts in tomographic phase contrast CT recon-

structions, which we attributed to phase wrapping and to

the fact that near the object border, the approximation that

the gradient of the phase front over the physical detector

width is constant becomes poor since the differential pro-

jection is singular here. We first assumed that this effect

has not been observed previously since in former experi-

ments the object was either thin4,7,13,20,21 or embedded in a

cup with fluid and the blank scan was taken with the cup

entirely filled with fluid,6,8,9 whereas in the simulations

conducted in Ref. 19 and here, the object is rather large.

However, this effect was not observed in a recent study by

Bartl et al.22 although a large object (17 cm diameter) was

embedded in air. After a careful reinvestigation, our hy-

pothesis turned out to be wrong: The phase wrapping was

not the cause of the artifacts. In order to discuss the possi-

ble appearance of phase wrapping, let us consider a cylin-

drical object consisting of a material that creates a phase

shift of a per unit length. The phase front U behind the

object in the projection approximation is

UðxÞ ¼ 0 for jxj > R

2affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2 � x2p

for jxj � R

�(16)

resulting in a phase gradient for jxj<R

@U@x¼ � 2xaffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

R2 � x2p : (17)

Now we can calculate from Eqs. (15) and (17) the positions

xw, where the gradient leaves the dynamic range of the sys-

tem, which happens when the phase u hits the border of the

dynamic range of the arg function, i.e., if the true phase ubecomes �p or þp. Thus we need to solve the equation

6p¼! u ¼ mg2

2

�2xaffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2 � x2p (18)

for x, which turns out to be

xw ¼ �Rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ ðmg2a=pÞ2q : (19)

For the parameters in our setup (m¼ 1, R¼ 55 mm,

g2¼ 2 lm, and a � 46 rad=mm at 25 keV), we obtain

xw ¼ �54:976 mm, i.e., only the outermost 24 lm of the

object’s projection on each side are affected. In other words,

the problem of phase wrapping will not be visible for the

system with a simulated pixel size of 128 lm. However, it is

still true that the phase gradient @U=@x in this outermost

area varies strongly since it is approaching a singularity

and thus, an accurate estimate of the phase gradient is

impossible.

Finally we found that the cupping and capping artifacts

disappeared almost completely after decreasing the sample

spacing used when the Helmholtz equation was applied [cf.,

Eq. (4)] to 8 nm, meaning that wx(x, 0, 0) was discretized on

a grid in x with spacing 8 nm and this spacing was kept until

the final step of integrating the square of the wave function

over the pixel area. Additionally, we placed the object

slightly off-center to create a variablity in the sampling of

the edges.

Reconstruction was performed using filtered back-projec-

tion using a conventional ramp-filter for the absorption con-

trast image23 and a Hilbert filter for the phase contrast

image24 with the additional trick of using a half-pixel-shift as

described in Ref. 25 for antialiasing purposes. In order to sup-

press aliasing in both absorption and phase contrast imaging,

cubic spline interpolation was used during back-projection.

II.D. CNR measurement

The CNR for each object o was calculated as

CNRo ¼So � SBG

ro(20)

where So is the reconstructed image value of the object o,SBG is the reconstructed image value of the background ma-

terial (i.e., breast mammary gland adult 1) and ro is the esti-

mated noise level in the object o. Note that in our definition,

the CNR may be positive or negative, which is unusual but it

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Medical Physics, Vol. 38, No. 7, July 2011

is useful for the discussion of the results to be able to distin-

guish between negative and positive contrasts.

For each object o, the signal level So was determined as

the average reconstructed valued within a square of 2.56 mm

edge size centered within the object, using a pixel size of

20 lm. The background signal was determined as the aver-

age reconstructed valued within a square of 2.56 mm edge

size located at the center of the object.

As will be seen in the results section, the phase contrast

images contain a considerable amount of low-frequency

noise, which makes a contrast measurement difficult. There-

fore, the contrast was measured in noise-free reconstructions.

Furthermore, since we still observe minor cupping artifacts

in the reconstructed phase contrast images, in particular for

low energies, we determined the noise level in the difference

images between the noisy and the noiseless case.

II.E. Relative contrast measurement

The relative contrast for each object o was calculated as

RCo ¼So � SBG

SBG

(21)

where So and SBG are the same quantities as defined in

Sec. II D. For the phase contrast images, where the contrast

is exclusively caused by differences in the electron density,

this quantity should equal for each object the relative differ-

ence in electron density

Dno

nBG

¼ no � nBG

nBG

; (22)

where no is the electron density of the object and nBG is the

electron density of the background.

II.F. Modulation transfer function measurements

One aspect of this study is to compare the CNR of DPCI

and ACI. Of course, a fair comparison is only possible if the

CNR is measured in images with the same resolution. In

order to measure the spatial resolution, the modulation trans-

fer function (MTF) was determined.

Typically, the MTF is measured using a bead or a wire

with a size much smaller than the detector pixel size.26,27

This approach is, however, impractical for DPCI since the

system is actually insensitive to objects much smaller than

the pixel size. This is due to the fact that the phase gradient

is averaged over the pixel area. For objects being projected

completely onto a single pixel, the resulting mean gradient is

zero and thus, the object is invisible in DPCI.43 A second

aspect that needs to be kept in mind is that we performed

simulations with coherent x-rays which may result in edge-

enhancement effects in the reconstructed absorption contrast

images. This effect is well visible in the absorption images

obtained by the grating-based measurement setup, e.g., in

Ref. 3 which were obtained using synchrotron radiation.

Thus, this effect needs to be considered.

Therefore, the MTF was determined from the edge

response function measured directly in the phantom where

the CNR evaluation is done with the exception that the

noise-free data were used: For each object, 360 radial lines

starting from the center of the object were reconstructed on a

5 lm grid and averaged. Averaging was required to become

insensitive to remaining aliasing artifacts in the recon-

structed images. The point spread function was estimated by

numerical differentiation of the resulting edge-response fol-

lowed by normalization with the step height. Finally, the

MTF was calculated as the absolute value of the one-dimen-

sional Fourier-transformation of the point spread function.

This procedure of calculating the MTF neglects the influence

of the discrete calculation of the derivative, which is a minor

effect since the sampling density of 5 lm is much smaller

than the pixel size (128 lm). Second, the finite curvature of

the edge is neglected. However, since all lines are perpendic-

ular to the edge, its curvature has only a second order effect.

In any case, the main purpose of the MTF calculation is to

compare the different imaging methods which are all influ-

enced equally.

II.G. Noise evaluation

For each calculated scenario, we first adapted the number

of source photons such that the total energy deposited into

the object was constant, thus allowing for a meaningful com-

parison of the CNR in the different reconstructed images.

The dose calculation was performed using a Monte Carlo

simulation considering photoelectric absorption and Comp-

ton scattering. For this dose calculation, the 3D structure of

the object was assumed to be a cylinder of infinite length in zdirection and the energy deposited within the cylinder was

taken as a dose measure. Thereafter, we deduced analytically

the expected mean number of detected photons for each de-

tector pixel, and created white noise according to the Poisson

distribution.

For the calculation of the noise power spectrum (NPS), a

central square of 50 mm edge size was reconstructed using a

10242 image matrix. Object structures were removed by sub-

tracting the noise-free reconstruction. The NPS is deter-

mined by a 2D FFT followed by a binning into several

frequency bands of 3 lp/cm width and a calculation of the

total signal intensity in each frequency bin. The resulting

spectra are further normalized to the total noise power in the

image.

For the calculation of the noise ro required for CNR

measurements, a similar procedure was performed: The

noise-free reconstruction was subtracted from the noisy

reconstruction and ro was calculated as the square root of

the variance of all image values in the object. A single noise

realization was not sufficient for a stable noise estimation

since the objects are relatively small and the NPS shows sub-

stantial low-frequency noise for the DPCI case. Therefore,

the noise estimate of ten independent noise realizations was

averaged.

III. RESULTS

Figure 3 shows an overview of the reconstruction results

for the absorption and phase contrast images for selected

energies. The detailed evaluation of MTF and CNR are

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Medical Physics, Vol. 38, No. 7, July 2011

performed on the central part containing four objects of in-

terest. This region of interest (ROI) is shown in Fig. 4.

III.A. Modulation transfer function

Results for the MTF measured at the object edges of the

four objects within the ROI are shown in Fig. 5 for DPCI,

CACI, and IACI. Exemplarily, we selected the results at

25 keV.

The following observations can be made: First of all, the

MTF of DPCI matches well the MTF of IACI for all cases.

Second, the MTF of CACI shows some variability. We

attribute this variation to so-called edge-enhancement

effects, which are well known in the field of x-ray imaging

using coherent sources.

In the following, we will give a qualitative reasoning for

the observed MTFs in the CACI case. To begin with, we

recall a basic formula for the edge-enhancement phenom-

enon in propagation-based x-ray imaging. Let I0(x) denote

the intensity directly behind the object, i.e., in the so-call

contact plane. For a weakly absorbing object, the intensity

Id(x) at a distance d behind the object can be calculated as28

IdðxÞ ¼ I0ðxÞ 1� kd

2pr2UðxÞ

� �; (23)

where U is the phase of the wavefront as introduced in

Sec. II A.

For a qualitative discussion of the effect, we further

assume that kd2pr2UðxÞ � 1, so that the estimation a(x) of

the projection of the linear attenuation coefficient (which is

used for CACI) is approximately

~aðxÞ ¼ � logIdðxÞ

�I¼ � log

I0ðxÞ�Iþ log 1� kd

2pr2UðxÞ

� �

� aðxÞ � kd

2pr2UðxÞ; (24)

meaning that the estimated projection is perturbed by an

additive term � kd2pr2UðxÞ.

The effect of edge-enhancement is often seen in

propagation-based phase contrast imaging at the object

FIG. 3. Sample reconstruction results of the phantom

simulated for selected energies. The top row shows the

absorption contrast images, the bottom row the recon-

structed phase contrast images. Level and window are

set as in Fig. 4.

FIG. 4. Central region of interest of the images shown

in Fig. 3. The top row shows the absorption contrast

images and the bottom row the reconstructed phase

contrast images. For each individual image, the level is

set to the background material and window is selected

such that the full dynamic range of the region of inter-

est is shown.

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Medical Physics, Vol. 38, No. 7, July 2011

boundary, i.e., at the air–tissue interface. At this interface,

the object has a positive contrast in both b and d compared

to the background.

Now we turn again to our simulation results. It is impor-

tant to keep the contrasts of the different objects as shown in

Fig. 4 (leftmost column) in mind: Water and adipose tissue

lipoma create the least contrast in the phase contrast image

among the four objects and the coherent absorption contrast

image is only to a very small extent affected by edge-

enhancement and the MTF of CACI matches the other two

MTFs. The material breast mammary gland adult 2 shows

decent positive contrast in phase contrast and the absorption

contrast images. The projection of this object is subject to

edge-enhancement resulting in an improvement of the MTF

at higher frequencies. A very special situation is present for

the material breast (whole) 50/50, which shows positive con-

trast in the absorption image but negative contrast in

the phase contrast image. Furthermore, the contrast in the

absorption image is rather low whereas the contrast in the

phase contrast image is relatively strong, which leads to a

strong influence of the phase contrast on the MTF of CACI.

The fact that the material has opposite contrasts in the

absorption contrast and the phase contrast image is key to

understand the observed effect of a degradation of the MTF:

As argued in the discussion of Eq. (24), the additive perturb-

ance � kd2pr2UðxÞ leads to the edge-enhancement. If the con-

trast of d changes its sign, it also changes the sign of the

disturbing term and the edge-enhancement turns into a

smoothing.

We need to stress here that the change of the MTF for

CACI due to edge-enhancement can give only a qualitative

indication about how the MTF is affected by the propagation

of the coherent wave field in a real experiment. For a correct

quantitative prediction of this effect it would be required to

take into account the finite size of the object along the x-ray

beam, since the object size (110 mm) is comparable to the

distance d between the two gratings (40.3 mm for the shown

case of 25 keV). Therefore, edge-enhancement of the same

magnitude will already take place while the beam propagates

through the object to the grating G1. Furthermore, the

approximation that the object is only weakly absorbing is

not very accurate for the object studied here.

III.B. Noise power spectrum

The visual appearance of noise can be seen already in

Fig. 4, showing that phase contrast images show predomi-

nantly low-frequency noise. The noise power spectra are

shown in Fig. 6. For comparison, the NPS graph in Fig. 6

shows additionally the NPS of IACI. Evidently and not sur-

prising, the NPS for CACI and IACI are identical.

In order to illustrate one important aspect of the different

noise power spectra, the ROI as shown in Fig. 4 was once

reconstructed as described in Sec. II C and for comparison,

this image was further smoothed with a Gaussian kernel of

full-width-half-maximum of 0.3 mm. The resulting images

are shown in Fig. 7. The noise level in the absorption con-

trast images decreases by the smoothing by a factor of 3.0,

FIG. 5. MTFs obtained at 25 keV for the objects within the ROI. The MTF graphs are ordered according to the position of the objects in the ROI shown in the

center.

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Medical Physics, Vol. 38, No. 7, July 2011

whereas the noise reduction for the phase contrast image is

only a factor of 1.4.

III.C. Energy dependence of CNR

A detailed analysis of the CNR as a function of x-ray

energy was performed on the four objects shown in the ROI,

see Fig. 4. The analysis is shown in Fig. 8. The curves of

CACI and IACI differ in their scale by a factor offfiffiffi2p

reflect-

ing the absorption of half the photons in the analyzer grating

G2 in CACI.

As stated before, a fair comparison of CNR between dif-

ferent imaging methods requires that the methods provide

the same resolution. The results of MTF measurements as

shown in Fig. 5, however, indicate that CACI and DPCI do

not provide the same resolution. Strictly speaking, only the

CNR curves of IACI and DPCI shown in Fig. 8 should be

compared, but not the CACI curve.

III.D. Contrast measurements

Table I sumarizes for the material within the ROI the rela-

tive difference in electron density as reported in Ref. 15 and

the measured relative contrasts in the phase contrast images.

Since the electron densities are given with three significant

digits in Ref. 15, the uncertainty in the relative differences in

the electron density is in the order of 0.1% and the measured

relative contrasts match well the expected values.

IV. DISCUSSION

The noise power spectrum of the phase contrast image

shows a maximum at low frequencies, see Fig. 6. This can

be easily understood from the reconstruction kernel used in

the reconstruction: The ramp filter used in absorption con-

trast imaging damps low frequencies and boosts the higher

frequencies leading to the well-known peak in the noise

power spectrum near the Nyquist frequency of the detection

system (actually, due to the smoothing effect of interpolation

during back-projection, the peak is well below the Nyquist

frequency of 39 lp=cm). On the other hand, the Hilbert filter

used in phase contrast imaging treats all frequencies equally,

leading to a higher noise power at lower frequencies.44 One

of the important implications of this feature for tomographic

medical imaging is that the superior CNR of phase contrast

images compared with absorption contrast as demonstrated

in Refs. 6 and 9 will diminish for applications, where a lower

spatial resolution is required: In absorption contrast imaging,

noise and resolution can be traded off very efficiently since

smoothing also removes noise in the frequency range, where

most of the noise is. However, for phase contrast imaging,

smoothing only suppresses the tail of the noise power spec-

trum and does not lead to a substantial increase in CNR. For

the example shown in Fig. 7, the CNR of the absorption con-

trast image after smoothing improves by a factor of 3.0 in

CACI whereas it improves only by a factor of 1.4 in the

DPCI. This effect should always be considered if the two

modalities are compared with respect to the achieved CNR.

Regarding CNR, it should also be noted that the CNR in the

phase contrast image can be increased by increasing the dis-

tance between G1 and G2 to a higher fractional Talbot order

because the sensitivity to the gradient of the phase front is

increased, see Eq. (15). This was for example used to obtain

the brain images in Ref. 6, where the 9th fractional Talbot

order was used, resulting in a sensitivity increase by a factor

of nine. The price to pay for this increase in CNR is that the

phase wrapping as discussed in Sec. II C will appear and the

problem of unwrapping the phase gradient will become a

demanding task. Finally, we need to stress that we simulated

a perfect DPCI system, i.e., perfect gratings have been used

and a perfectly coherent and monochromatic x-ray beam was

simulated. In a real clinical system, the CNR of the DPCI

would be reduced due to a decrease in the visibility caused

FIG. 6. Noise power spectra of absorption and phase contrast imaging.

FIG. 7. ROI reconstruction at 85 keV at system resolution, i.e., reconstructed with a pure ramp and Hilbert filter, respectively (a and c) and the same images

smoothed using a Gaussian kernel of 0.3 mm full-width-half-maximum (b and d) for absorption contrast (a and b) and phase contrast (c and d).

S113 Kohler, Engel, and Roessl: Noise properties of phase contrast CT S113

Medical Physics, Vol. 38, No. 7, July 2011

by imperfections of the grating, the limited coherence of the

x-ray source, and the finite bandwidth of the x-rays.

The MTF measurements show that a direct and fair com-

parison of the CNR obtained by DPCI and CACI is difficult

since the MTF for CACI shows a dependency on the object’s

material. In our simulations, we found that the MTF in

CACI is improved whenever the contrasts of the object have

the same sign in d and b. Only for a single object (and even

for this only for energies below 35 keV), the contrasts have

opposite signs leading to a degradation in the MTF. Thus,

we may state that the loss of CNR in CACI due to the ana-

lyzer grating G2 is typically compensated partially by a

slight improvement of the MTF.

With respect to the noise properties of reconstructed

phase contrast images, it is also worth comparing the fea-

tures observed here with previous investigations for other ac-

quisition methods for phase contrast x-ray imaging: For the

in-line phase contrast imaging method, the auto-covariance

of the noise was studied by Chou and Anastasio.12 Similar to

our results for the grating-based method, they found that the

auto-covariance function is rather broad, which is equivalent

to predominantly low frequency noise. While in Ref. 12, the

reconstruction involved an explicit retrieval of the projection

of the real part of the refractive index, there are also other

reconstruction algorithms, which do not need this intermedi-

ate step. Interestingly, these algorithm are found to be stable

at high frequencies and unstable at low frequencies,28–30

where the instability originates from a singularity, viz., a first

order pole, in the reconstruction kernel at zero frequency.

This feature implies at least qualitatively the same noise

behavior.

The CNR of phase contrast images measured at constant

dose for the sample phantom are qualitatively very similar:

For all objects, the magnitude of the CNR in DPCI shows a

maximum around 45 keV. The existence of a maximum at a

distinct energy independent of the object is caused by several

effects: First, the phase shift introduced by the objects is

exclusively created by the electron density, see Sec. III D,

with an energy dependence of E�1: the real part of the re-

fractive index d has an energy dependence of E�2, and one

factor of E is included in the wave vector k in the projection

approximation, see Eqs. (4) and (6). Thus, it is expected

that the energy dependence of the CNR of all objects in the

phantom is qualitatively the same. The second cause is the

different contributions of the photoelectric effect and

FIG. 8. CNR measurements for the objects within the ROI as a function of energy.

TABLE I. Relative difference in electron density of the materials in the ROI

as derived from (Ref. 15) and measured relative contrast as defined in

Sec. II E. The measurements for RCo are independent of the energy.

Material o Dno=nBG RCo

Adipose tissue lipoma �0.9% �1.0%

Breast mammary gland adult 2 þ2.7% þ2.7%

Breast (whole) 50=50 �2.4% �2.5%

Water þ1.2% þ1.1%

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Medical Physics, Vol. 38, No. 7, July 2011

Compton scatter to the total attenuation. For lower energies,

the photoelectric effect dominates. However, it decreases

approximately as E�3, whereas the phase contrast decreases

only as E�1. Thus, going to higher energies decreases the

contrast in the phase image, but this is more than compen-

sated by the fact that at constant dose more photons can be

used for imaging since fewer of them are absorbed. How-

ever, as soon as the Compton effect contributes significantly

to the attenuation at higher energies, the total transmission

decreases again and the decreasing phase contrast is—in the

constant dose scenario—no longer compensated by an even

faster decreasing absorption.

Herzen et al.31 showed in experiments using several dif-

ferent liquids that phase and absorption contrast images

show complementary information in the sense that the CNR

is sometimes larger in the phase contrast image and some-

times in the absorption contrast image. Our simulation using

material compositions of biological tissues confirms that this

feature is relevant for medical imaging.

Conceptually, the grating-based DPCI setup invests half

the dose, namely the dose that is absorbed in the analyzer

grating G2, to obtain additional information about the object.

Qi et al.,21 for instance, derived quantitative images of the

electron density and the effective atomic number. It should

be acknowledged that there are also other setups which can

provide the same information without investing dose in the

first place: These methods are based on dual energy acquisi-

tions or spectrally resolved detection followed by a separa-

tion of the total attenuation into the contributions of the

photoelectric effect and Compton scattering.32–34 In these al-

ternative methods, there is no dose penalty to pay in the first

place. Furthermore, it has been demonstrated that spectral

CT can also detect specifically materials which have a

K-edge in the used spectral range, providing even further

useful information.35,36 Although it should be theoretically

possible to detect these materials also in the phase contrast

signal (this is due to the relationship between d and b given

by the Kramers–Kronig relation), it has not been shown

experimentally yet. From a system point of view, this alter-

native option to obtain the additional information might be

preferred since the required system modifications are moder-

ate compared with the system modifications required for a

DPCI system. On the other hand, it is unclear currently,

whether these methods can provide the additional informa-

tion at a CNR-level that is comparable to that obtained by a

DPCI setup, which is not a-priori clear since the material

decomposition from dual energy acquisitions suffers from

noise correlation.37 Finally, as mentioned before, the gra-

ting-based DPCI setup can provide a third reconstructed

image, the so-called decoherence image, based on the visi-

bility function, see Eq. (9). Such an image cannot be

obtained by dual energy or spectral CT. Thus, the potential

clinical benefit of the decoherence image might be finally a

key argument whether a grating-based DPCI setup is worth

being used for clinical applications.

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