Contrast transfer function in grating-based x-ray phase-contrast imaging

Nuclear Instruments and Methods in Physics Research A 747 (2014) 13–18

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Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima

Contrast transfer function in grating-based x-ray phase-contrast imaging Jianheng Huang a, Yang Du a,b, Danying Lin a, Xin Liu a, Hanben Niu a,n a b

Key Laboratory of Optoelectronics Devices and Systems of Ministry of Education and Guangdong Province, Shenzhen University, Shenzhen 518060, China Xi0 an Institute of Optics and Precision Mechanics, Chinese Academy of Sciences, Xi0 an 710119, China

art ic l e i nf o

a b s t r a c t

Article history: Received 30 August 2013 Received in revised form 16 January 2014 Accepted 18 January 2014 Available online 27 January 2014

x-Ray grating interferometry is a method for x-ray wave front sensing and phase-contrast imaging that has been developed over past few years. Contrast and resolution are the criteria used to specify the quality of an image. In characterizing the performance of this interferometer, the contrast transfer function is considered in this paper. The oscillatory nature of the contrast transfer function (CTF) is derived and quantified for this interferometer. The illumination source and digital detector are both considered as significant factors controlling image quality, and it can be noted that contrast and resolution in turn depends primarily on the projected intensity profile of the array source and the pixel size of the detector. Furthermore, a test pattern phantom with a well-controlled range of spatial frequencies was designed and imaging of this phantom was simulated by a computer. Contrast transfer function behavior observed in the simulated image is consistent with our theoretical CTF. This might be beneficial for the evaluation and optimization of a grating-based x-ray phase contrast imaging system. & 2014 Published by Elsevier B.V.

Keywords: x-Ray phase contrast imaging Grating interferometry Contrast transfer function Resolution

1. Introduction x-Ray imaging is widely used in medical diagnostics, for example mammography, and chest x-ray. Conventional radiography utilizes the absorption-contrast imaging technique, which records the intensity attenuation due to absorption. This method is not very applicable for a low absorption material, such as soft tissue or polymers. Consequently, various phase contrast techniques have been developed [1,2]. Phase contrast imaging makes use of the contrast from the phase shift in an x-ray wavefield passing through objects. There are basically four different types of phasesensitive methods: crystal interferometry [3,4], the propagationbased method [5–7], diffraction enhanced imaging [8–10] and the grating-based method [11,12]. The grating-based method can be efficiently used to retrieve quantitative phase images with a low brilliance polychromatic x-ray source in a conventional laboratory [13,14]. This method utilizes the Talbot effect and uses a grating interferometer to produce a series of images by the phase stepping scan process. Three different images, i.e., standard absorption image, differential phase image and dark-field image, can be constructed from those detected images. The principles of the grating-based method has n

Corresponding author. Tel.: þ 86 755 26538579; fax: þ 86 755 26538580. E-mail addresses: [email protected] (J. Huang), [email protected] (Y. Du), [email protected] (D. Lin), [email protected] (X. Liu), [email protected] (H. Niu). 0168-9002/$ - see front matter & 2014 Published by Elsevier B.V. http://dx.doi.org/10.1016/j.nima.2014.01.029

been explained and confirmed by experiments in the previous literature [15]. In recent years, the researchers mainly concentrate on the performance evaluation and optimization of the grating-based method, for instance noise analysis [16] and field of view [17]. Contrast and resolution are the criteria used to specify the quality of an image. The optical transfer function in propagation-based imaging with a micro-focus x-ray source was considered by Pogany in 1997, and it had been shown that image resolution depends mainly on lateral coherence, with longitudinal coherence being of lesser importance [18]. Subsequently, similar work was reported by Salditt in 2009 [19]. And spatial resolution characterization of differential phase contrast CT systems via modulation transfer function (MTF) measurements is discussed in 2013 [20]. However, studies on the contrast transfer function (CTF) in grating-based phase imaging have not been reported. In this paper a simple theoretical framework is presented to treat the grating-based phase imaging by x-rays. The theory is based mainly on the approximate Kirchhoff–Fresnel theory, treating the imaging process in terms of contrast transfer function (CTF). Existing literatures have not measured the oscillatory nature of the CTF for phase image yet, so the oscillatory nature of the CTF for phase image will be quantified here. And the projection of the array source and the pixel sampling of the digital detector are considered as two significant factors controlling image quality. We further use a simulated test pattern to quantitatively verify that the CTF shows the predicted oscillatory behavior.

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Fig. 1. Schematic diagram of grating interferometer. A phase specimen in front of the phase grating G1 will cause the incident beam slight refraction, which results in changes of the locally transmitted intensity through the analyzer grating G2.

Fig. 3. Schematic diagram of the Talbot–Lau interferometer. The source grating (G0) is just put in front of the conventional x-ray tube composing the array line source. The fringe displacements are transformed into intensity values by an absorption grating (G2) placed at a distance from the phase grating (G1), and this allows the use of a detector (D) with much larger pixels than the spacing of the fringes.

where Q, M and Φ are the two dimensional Fourier transformations (FTs) of q, μ and φ, respectively. fx and fy are the two dimensional spatial frequencies at the object or image plane. Considering the gratings have symmetric structures and the lines are oriented parallel to each other, the intensity after the second grating G2 can be derived approximately as (see Appendix A) 
 2πxg Iðx; y; xg Þ ¼ exp½  2μðx; yÞ a0 þ a1 cos þ φðx  s; yÞ  φðx þ s; yÞ p2

ð4Þ

where xg is the place where one of the gratings is scanned along the transverse direction. The transverse phase separation s in Eq. (4) between by the two main diffractive beams is given by s¼ Fig. 2. The normalized CTFT,p for the Talbot interferometer(fx0 ¼ sfx).

2. Contrast transfer function derivation The principle of grating interferometry is based on the Talbot effect or a self-imaging phenomenon by a periodic object, for instance a transmission grating, under spatially coherent illumination. It is characteristic of the Talbot effect that one can observe the appearance and disappearance of the grating's self-image along the optical axis. This phenomenon is understood as Fresnel diffraction by the grating. 2.1. Two grating interferometer CTF In the case of monochromatic plane wave illumination, the interferometer is shown diagrammatically in Fig. 1. It consists of two gratings, a phase grating (G1 with a period of p1 and a phase shift of π) and an analyzer absorption grating (G2 with a period of p2). Let a thin object in front of the grating G1 be illuminated with a monochromatic plane wave. The transmission function of the object in the Cartesian coordinate is given by qðx; yÞ ¼ exp½  μðx; yÞ þ iφðx; yÞ

ð1Þ

where μ(x,y) and φ(x,y) are the absorption and phase-shift components of the object (μ is the z projection of half the usual linear attenuation coefficient for the intensity), respectively. For a weak absorption and weak phase object, Eq. (1) can be approximated by the Taylor series as qðx; yÞ ¼ 1  μðx; yÞ þiφðx; yÞ:

ð2Þ

Then we can have in spatial frequency domain of Eq. (2) Q ðf x ; f y Þ ¼ δðf x ; f y Þ  Mðf x ; f y Þ þ iΦðf x ; f y Þ

ð3Þ

λd p1

ð5Þ

where λ is the wavelength of the incident wave and d is the Talbot distance. A first-order Taylor series expansion is applied to the exponential term in Eq. (4), the cosine is expanded using the sum of angle identity and it is assumed that φ(x  s,y)  φ(x þs,y) is close to zero (since the refractive index should usually not change quickly across the sample on the scale of the phase separation between beams). This means cos[φ(x s,y)-φ(x þs,y)] is close to 1 and μ(x,y) sin[φ(x  s,y) φ(xþs,y)] is negligible, and so we find that the intensity distribution can be written as
 
  2πxg 2πxg Iðx; y; xg Þ ¼ a0 þ a1 cos  2 a0 þ a1 cos μðx; yÞ p2 p2   2πxg  a1 sin ½φðx  s; yÞ  φðx þ s; yÞ: ð6Þ p2 Following the treatment given in Ref. [18], we obtain the frequency spectrum of the intensity in Eq. (6) which is as follows:
  2πxg Fðf x ; f y ; xg Þ ¼ a0 þ a1 cos δðf x ; f y Þ p2
  2πxg 2 a0 þ a1 cos Mðf x ; f y Þ p2
  2πxg ð7Þ i sin ð 2πsf x ÞΦðf x ; f y Þ: 2 a1 sin p2 In Eq. (7), the real and imaginary parts of the optical transfer function (OTF) have a simple interpretation in terms of the amplitude and phase components of the object transmission function, which can be retrieved by the phase-stepping approach from the modulated intensity signal [12]. Referring to intensity rather than amplitude, one should more correctly use the term of contrast transfer function (CTF) [18], and the normalized CTF of the

J. Huang et al. / Nuclear Instruments and Methods in Physics Research A 747 (2014) 13–18

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analyzer grating G2 and an image detector; this set-up called the Talbot–Lau interferometer [13]. The function of G0 is to divide the x-ray source into a series of mutually incoherent but individually coherent line sources. An alternative method is using the multiline x-ray source to create the line sources directly [21], which overcomes the defect that G0 cannot absorb the higher x-ray energy completely and thus leads to the deterioration of image quality. To satisfy the requirement of spatial coherence length, the distance l between G0 (with a duty cycle of γ0) and G1 should be [13] lZ

p1 p0 γ 0 : λc

ð9Þ

Due to the effect of a cone-beam x-ray source, the geometric magnification M should be considered as M¼

lþd : l

ð10Þ

Consequently, p2 and d can be written as follows [15]: ( Mp1 2π phase grating p2 ¼ Mp1 π phase grating 2 and Fig. 4. Micrograph of the test pattern. The red number 1, 2, 3, 4, 5 and 6 correspond to the circular period of 400 μm, 320 μm, 160 μm, 80 μm, 40 μm and 20 μm respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

d¼

8 2 < M p2λ1

π 2

: M p1

π phase grating

c

2

8λc

phase grating

:

ð11Þ

ð12Þ

Besides, to ensure that each line source produced by G0 contributes constructively to the image-formation process, the geometry of this setup should satisfy the following condition [13]: p0 l ¼ : p2 d

Fig. 5. Spectrum distribution of the polychromatic x-rays used for simulation. It is carried out with a tungsten target at a potential of 60 kV [25].

ideal Talbot interferometer can be conveniently written as follows: ( CTF T;a ¼ 1 ð8Þ CTF T;p ¼ sin ð  2πsf x Þ where the subscript a and p represent the amplitude and phase component of the object transmission function, respectively. According to Eq. (8), the CTFT,p can be conveniently plotted as shown in Fig. 2. For the convenience of plotting the curve, plotted by fx0 ¼sfx is necessary on the horizontal axis. The CTFT,p for phase acts as a sinusoidal filter to the object transmission function in gratingbased x-ray phase-contrast imaging. The maximal CTFT,p appears at the point of fx0 ¼1/4, 3/4, 5/4,…, and the absolute value of the CTFT,p increases with spatial frequency in the form of a sine curve at low spatial frequencies (fx0 o 1/4). 2.2. Three grating interferometer CTF In the case of a conventional x-ray source, as shown in Fig. 3, the set-up consists of a source grating G0, a phase grating G1, an

ð13Þ

A specimen is placed in front of the phase grating, and the differential phase contrast image can be retrieved by the phase retrieval algorithm [12]. It is important to note that the spatial resolution of the images obtained by the Talbot–Lau interferometer is limited mostly by the array line source and the detector. In the following work, we will discuss the influence of these two components on CTF of the imaging system in detail. Assuming that the array line source consists of numerous individual slits with a rectangular profile, therefore the projected intensity profile on the image plane is written as 8   N=2  1 > x þ ð2n þ 1Þp2 =2 y > > ; if N is even ; ∑ rect > > < n ¼  N=2 p2 γ 0 dly =l f s ðx; yÞ ¼   ðN  1Þ=2 > x þ np2 y > > ; if N is odd rect ; ∑ > > : n ¼  ðN  1Þ=2 p2 γ 0 dly =l ð14Þ where N is the total number of the slits and ly is the length of each slit along the y-axis direction. Because l must be greater than d in the setup, I(x, y; xg) in Eq. (6) can be approximately considered as the point-source intensity profile of the Talbot–Lau interferometer, and the resulting intensity profile after the third grating is the point-source profile convolved with the projected source profile as follows: I s ðx; y; xg Þ ¼ Iðx; y; xg Þ  f s ðx; yÞ

ð15Þ

The image is recorded by an x-ray fluorescence screen coupled with the CCD detector with a pixel size of lp  lp, and the function of each pixel fp(x,y) can be written as 8       <     1; lxp  r 12; lyp  r 12 x y f p ðx; yÞ ¼ rect ¼ ; : ð16Þ : 0; others lp lp

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Fig. 6. Simulation results of the PS test pattern. (a) Transmission image, (b) differential phase-contrast image, (c) the square frame region of transmission image, and (d) the square frame region of differential phase-contrast image. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

Accordingly, the acquired digital intensity Id(x, y; xg) is given by   x y ð17Þ ; I d ðx; y; xg Þ ¼ ½I s ðx; y; xg Þ  hpsf ðx; yÞ  f p ðx; yÞcomb lp lp here hpsf (x,y) in Eq. (17) is the point-spread function of the fluorescence screen mainly determined by its thickness, and the comb function can be written as   x y 1 Np Mp ¼ 2 ∑ ∑ δðx np lp ; y  mp lp Þ ; ð18Þ comb lp lp lp np mp where Mp and Np are the row and column numbers of the pixels respectively. And then according to the Whittaker–Shannon sampling theory [22], we obtain the effective frequency spectrum Fd(u,v;xg) of the resulting digital intensity Id(u,v;xg) in Eq. (17) in the following expression: F d ðf x ; f y ; xg Þ ¼ Fðf x ; f y ; xg ÞH psf ðf x ; f y ÞF s ðf x ; f y ÞF p ðf x ; f y Þrectðlp f x ; lp f y Þ ð19Þ where Hpsf(fx,fy), Fs(fx,fy) and Fp(fx,fy) are the two dimensional FTs of the projected source profile, the effective point-spread function of the fluorescence screen and the profile function of each pixel, respectively. Thus, the normalized CTF of the Talbot–Lau interferometer can be written as (

CTF TL;a ¼ Hpsf ðf x ; f y ÞF s ðf x ; f y ÞF p ðf x ; f y Þrectðlp f x ; lp f y Þ CTF TL;p ¼ sin ð  2πsf x ÞH psf ðf x ; f y ÞF s ðf x ; f y ÞF p ðf x ; f y Þrectðlp f x ; lp f y Þ

a for amplitude : p for phase

ð20Þ

3. Simulated image measurements To analyze the CTF of the Talbot–Lau interferometer, a polystyrene (PS) pattern modeled on the well-known Siemens star is used as a test object with controlled increasing spatial frequencies from the outer to the inner regions. As shown in Fig. 4, the test pattern is designed with a structure height of 9 μm, an outer radius of 30.6 mm, and its finest feature size near the center is 10 μm. From the outer to the inner, the circular period of star-shaped structure at disconnected regions are 400 μm, 320 μm, 160 μm, 80 μm, 40 μm and 20 μm respectively, corresponding to the spatial frequencies of 2.5 lp/mm, 3.125 lp/mm, 6.25 lp/mm, 12.5 lp/mm, 25 lp/mm and 50 lp/mm. At the photon energy of E¼ 31 keV, the intensity transmission of the 9 μm thick PS structure is 0.9982, and the phase shift is 0.3405. Therefore the test structure can be considered as a pure weak phase object at the simulated energy. The simulation is performed on a typical Talbot–Lau interferometric imaging system [21]. The array x-ray source modulated by a source grating or generated by a multi-line x-ray source has an equivalent period of 42 μm with a line width of 10.5 μm corresponding to a duty cycle of γ0 ¼0.25. The total number of the line sources is 20. Polychromatic x-rays are used for simulation (Fig. 5), and the central photon energy is Ec ¼31 keV corresponding to the central wavelength of λc ¼0.04 nm. The π phase shift phase grating G1 for the central wavelength λc has a period of 5.6 μm with a duty cycle of γ1 ¼0.5. A structured scintillator, which functions as both the common scintillator and the analyzer grating, is used in this simulation and its point-spread profile is only 3 um determined by its pixelated structure [21]. The effective pixel size of the CCD

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Fig. 7. The normalized CTF of the designed Talbot–Lau interferometer vs. spatial frequencies. (a) CTFTL,a for amplitude, (b) CTFTL,p for phase, (c) CTFTL,a curve for amplitude vs. spatial frequency fx, and (d) CTFTL,p curve for phase vs. spatial frequency fx. These curves are from the theoretical expression in Eq. (20), and the contrast values in different spatial frequencies fx indicated by the colorized arrowheads are consistent with the simulated images in Fig. 6. The cutoff frequency at 18 lp/mm is limited because of the pixel sampling of detector. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

detector is designed as 27  27 μm2, and the distance of l¼1.47 m and d¼105 mm met the requirements of the geometry condition. The simulation program based on the paraxial Kirchhoff–Fresnel theory is executed by Matlab. The transmission and differential phase-contrast images of the test pattern are retrieved by a five phase-stepping approach [23]. The sampling interval for every stepping image (2048  2048) is 3.15 um to ensure that the simulation results of patterns in the discussed frequency range are effective. Fig. 6 shows the computer simulation results of the PS pattern, and the normalized CTFTL curves are shown in Fig. 7. With increasing the spatial frequency fx, the normalized CTFTL,a value for amplitude decreases quickly. The radial modulation of intensity indicated by a black arrowhead in Fig. 6(c) is suppressed, corresponding to the spatial frequency of zero contrast transfer signed by the black arrowhead in Fig. 7(c). For the phase-contrast image, the CTFTL,p acts as a special nonlinear filter depending strongly on the projected intensity profile of the illumination and the pixel size of the digital detector. Note that the maximum phase contrast transfer appears at the spatial frequency of fx ¼8 lp/mm signed by the blue arrowhead in Fig. 7(d), corresponding to the region indicated by the blue arrowhead in Fig. 6(d). When the frequencies are between fx ¼16 lp/mm (pink arrowhead) and fx ¼18 lp/mm (red arrowhead), the contrast for phase reverses. The zero phase contrast in the frequency of fx ¼16 lp/mm is determined by the projected intensity profile of the array source, while the cutoff frequency of fx ¼ 18 lp/mm is limited because of the pixel sampling of the detector. The radial modulation in the magnified region of the differential phase-contrast image shown in Fig. 6(d) is in good agreement with the CTFTL,p curve.

interferometer. And in the three grating interferometer, with increasing spatial frequencies, the CTF for amplitude decreases quickly, but for the phase contrast image, the CTF acts as a special non-linear filter affected by the illuminating source and the pixel size of the detector. The excellent agreement of the calculated CTF curves with images obtained by simulation indicates that quantitative analysis of contrast and resolution in a designed Talbot–Lau interferometer should be possible. On the basis of these results, we conclude that the projected intensity profile of the illumination and the pixel sampling of the digital detector would be two significant factors determining contrast and resolution. The derived CTFs in this paper would be of use in characterizing the performance of an x-ray grating interferometer, such as contrast and resolution for phase image. It will promote us to establish a well-designed pattern used in experiments, and then realize the quantitative measures of contrast and resolution from the CTF of its experimental results. And the method presented here may open a route potentially through spatial frequencies for the evaluation and optimization of grating-based x-ray phase contrast imaging, and help us to acquire the optimal parameters of an experimental imaging system in the future work.

4. Conclusion

Appendix A

In this article, the oscillatory nature of CTF in grating-based x-ray phase contrast imaging is given by a theoretical explanation and the theoretical CTF shape is consistent with measurements from a computer simulation of the image of the PS test pattern. The CTF for phase acts as a sinusoidal filter in the two grating

The complex transmission function T(x,y) of the phase grating G1 with a Fourier series can be expressed as   1 nx ðA1Þ Tðx; yÞ ¼ ∑ cn exp i2π p1 n ¼ 1

Acknowledgments This study was supported by the National Special Foundation of China for Major Science Instrument (Grant 61227802), the National Natural Science Foundation of China (Grants 60532090, 61001184 and 61101175), and the Science and Technology Bureau of Shenzhen (Grant JC201005280502A).

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J. Huang et al. / Nuclear Instruments and Methods in Physics Research A 747 (2014) 13–18

where cn is the amplitude of the nth harmonic. According to the Kirchhoff–Fresnel theory [24], the complex amplitude at the Talbot distance d of G1 can be written as

here a0 and a1 are the amplitude coefficients of the intensity after the second grating G2, and xg is the place where one of the gratings is scanned along the transverse direction.

Eðx; yÞ ¼

References

" #

Z Z 1 exp i2π 2π ðx  x0 Þ2 þ ðy y0 Þ2 λd dx0 dy0 qðx0 ; y0 ÞTðx0 ; y0 Þexp i iλd 2d λ 1

ðA2Þ where λ is the wavelength. Then we can derive it as

  1 exp i2π n2 πdλ λd cn exp i2π nx Eðx; yÞ ¼ ∑ p1  i p2 1 iλd n ¼ 1
  2   dλ x þ y2 : q xn  exp iπ p1 dλ

[1] [2] [3] [4] [5] [6]

ðA3Þ

For an ideal π phase grating G1, considering the two main diffractive beams would be advisable. Then the complex amplitude in Eq. (A3) can be written approximately as


  exp i2π λd c1 exp i2π px  iπdλ q x  pdλ Eðx; yÞ ¼ p1 2 1 1 iλd      x πdλ dλ þ c  1 exp i2π i 2 q xþ ðA4Þ p1 p1 p1 ignoring the convolution effect of the object in the propagation. Then we substitute q(x,y) with Eq. (1), and the intensity of the self-imaging can be written as 
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ðA5Þ where b0 and b1 are the amplitude coefficients of the intensity in Eq. (A5). When the two gratings G1 and G2 (p2 equals to the half period of the grating G1) have symmetric structures and the lines are oriented parallel to each other, the intensity after the second grating G2 can be written approximately as


     2πxg dλ dλ Iðx; y; xg Þ ¼ exp½ 2μðx; yÞ a0 þ a1 cos þφ x ; y φ xþ ; y p2 p1 p1

ðA6Þ

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