An improved phase shift reconstruction algorithm of fringe scanning technique for X-ray microscopy S. Lian, H. Yang, H. Kudo, A. Momose, and W. Yashiro Citation: Review of Scientific Instruments 86, 023707 (2015); doi: 10.1063/1.4908139 View online: http://dx.doi.org/10.1063/1.4908139 View Table of Contents: http://scitation.aip.org/content/aip/journal/rsi/86/2?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Experimental comparison of grating- and propagation-based hard X-ray phase tomography of soft tissue J. Appl. Phys. 116, 154903 (2014); 10.1063/1.4897225 The second-order differential phase contrast and its retrieval for imaging with x-ray Talbot interferometry Med. Phys. 39, 7237 (2012); 10.1118/1.4764901 High-speed X-ray phase tomography with Talbot interferometer and fringe scanning method AIP Conf. Proc. 1466, 261 (2012); 10.1063/1.4742302 Theoretical aspect of X-ray phase microscopy with transmission gratings AIP Conf. Proc. 1466, 144 (2012); 10.1063/1.4742283 Higher-order phase shift reconstruction approach Med. Phys. 37, 5238 (2010); 10.1118/1.3488888

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REVIEW OF SCIENTIFIC INSTRUMENTS 86, 023707 (2015)

An improved phase shift reconstruction algorithm of fringe scanning technique for X-ray microscopy S. Lian,1 H. Yang,1,a) H. Kudo,2 A. Momose,3 and W. Yashiro3

1

Midorino Research Corporation, 5-15-13 Chuo Rinkan Nishi, Yamato, Kanagawa 242-0008, Japan Division of Information Engineering, Faculty of Engineering, Information and Systems, University of Tsukuba, 1-1-1 Tennoudai, Tsukuba, Ibaraki 305-8573, Japan 3 Institute of Multidisciplinary Research for Advanced Materials, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai, Miyagi 980-8577, Japan 2

(Received 31 May 2014; accepted 2 February 2015; published online 18 February 2015) The X-ray phase imaging method has been applied to observe soft biological tissues, and it is possible to image the soft tissues by using the benefit of the so-called “Talbot effect” by an X-ray grating. One type of the X-ray phase imaging method was reported by combining an X-ray imaging microscope equipped by a Fresnel zone plate with a phase grating. Using the fringe scanning technique, a high-precision phase shift image could be obtained by displacing the grating step by step and measuring dozens of sample images. The number of the images was selected to reduce the error caused by the non-sinusoidal component of the Talbot self-image at the imaging plane. A larger number suppressed the error more but increased radiation exposure and required higher mechanical stability of equipment. In this paper, we analyze the approximation error of fringe scanning technique for the X-ray microscopy which uses just one grating and proposes an improved algorithm. We compute the approximation error by iteration and substitute that into the process of reconstruction of phase shift. This procedure will suppress the error even with few sample images. The results of simulation experiments show that the precision of phase shift image reconstructed by the proposed algorithm with 4 sample images is almost the same as that reconstructed by the conventional algorithm with 40 sample images. We also have succeeded in the experiment with real data. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4908139]

I. INTRODUCTION

Since its discovery in 1895, X-ray has been widely applied to non-destructive inspection and medical imaging field. In most cases, these applications are based on detecting the intensity of X-ray that is attenuated when it travels through the material. However, it is difficult to detect the attenuation of intensity for some objects, such as cartilage and breast cancer made up of light elements, because the attenuation of intensity in these objects is very weak. The X-ray phase imaging method was proposed to solve this problem.1 Just as in the conventional optics, the interaction of X-ray with the material can be described by the complex refractive index which can be written as n = 1 − δ + i β, where the indexes δ and β describe the phase shift and the intensity attenuation in the medium, respectively. Since the ratio δ/ β for light elements is about 1000, it is possible to obtain a phase shift image with higher contrast compared to an intensity image. There exist several approaches to investigate the phase information of the sample, such as crystal interferometry,2,3 grating interferometry,4–7 propagation based imaging,8 and analyzer based imaging.9,10 Recently, X-ray phase-difference microscopy has been proposed by combining a conventional X-ray imaging microscope equipped with a Fresnel zone plate (FZP) with a grating that generates Talbot effect like the grating interferom-

a)Electronic mail: [email protected]

etry.11 The grating pattern was magnified and casted on the image plane through the magnification optics and the Talbot effect. The phase shift at a sample could be sensed as the deformation that stripes in the casted pattern. The fringe scanning technique12 was employed to analyze the deformation. In general, the fringe scanning technique requires several (N) images obtained so that fringe positions shift by 1/N of the fringe spacing by a certain operation in the optical system. Then, the deformation of a fringe pattern can be mapped. In the case of X-ray phase-difference microscopy, the grating is displaced by a step of d/N, where d is the grating period. The fringe scanning technique is dedicated for the analysis of sinusoidal fringes, and then N can be 3 or more. However, because the casted pattern generated by the Talbot effect in the microscope has non-sinusoidal (rectangular) profile, the higher harmonic component causes an error when the fringe scanning technique is applied. To avoid this problem, a larger number for N should be selected.13,14 However, this gives rise to an increase of radiation exposure and a more stringent restriction on the mechanical stability of equipment simultaneously. Moreover, more steps generate larger data sets that may need to develop new algorithm.15 In this paper, we analyze the error related to the fringe scanning technique for the X-ray phase-difference microscopy and propose an improved algorithm that suppresses the error. By using the proposed algorithm, it is possible to obtain highprecision phase information with less number of sample images (that is, fringe-scanning steps). In the simulation experiments,

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we use only 4 sample images with the proposed algorithm to reconstruct almost the same phase information as that reconstructed by the conventional algorithm using 40 sample images obtained by the fringe scanning measurement. Here, the 4 sample images are extracted in uniform interval 10 from the 40 sample images. We also find that the algorithm with the experimental data11 works well. II. FRINGE SCANNING TECHNIQUE

In this section, the conventional fringe scanning technique developed for X-ray phase-difference microscopy is introduced. Throughout this paper, we assume the experimental setup shown in Figure 1. We note that this setup is the same as that studied by Yashiro et al.11 The position of the camera in Fig. 1 coincides with both the image plane of FZP and the self-imaging position of the grating. The phase shift caused by the sample leads to a deformation of the self-image, from which the phase shift information can be reconstructed as a twin image generated by using the fringe scanning technique. We denote the focus length of FZP by f , the distance from the sample to FZP by a and the distance from the FZP to the camera by b. It is required that a, b, and f satisfy 1/a + 1/b = 1/ f , and the magnification ratio is expressed as M = b/a. The grating in Figure 1 is a π/2 phase grating with the Ronchi ruling. In order to use the fractional Talbot effect, the distance z0 from the grating to the camera is determined by the following relation: z0 = p

2

R1 d , λ R1 − p d 2 λ

(1)

where λ is the X-ray wavelength, d is the period of the grating, and p = 1/2 is the fractional Talbot order. By the magnification effect, the period of the self-image at the camera position is given by d 1 = d (R1 + z0)/R1, which can be resolved directly by the camera.

where X-ray attenuation is assumed to be negligible and Φ(x,y) is the phase shift caused by the sample. By the periodic property of the grating, the transmission function of the grating can be expressed as (  n ) (3) T(x) = an exp i2π x , d n where an is the n-th Fourier coefficient. In the fringe scanning technique, it is required to move the grating several times along the x direction with a step size of d/N, which results in measuring N sample images at the camera position. We denote the distance from the source to the sample plane by R0 and the intensity distribution of the sample images on the camera plane by Ik (X,Y )(k = 0, 1, . . . , N − 1). Then, by using the Fresnel diffraction integral repeatedly, we obtain11  Ik (X,Y ) = µ n−m Hk (x n )Hk∗(x m ), (4) n, m

where x n = −(X + npd 1)/M, 2     π(n − m)pd 1σ   µ n−m = exp  −2   M λ R0   (see Appendix A), and  ( ) X k 2 Hk (x n ) = an exp(iπn p) exp i2πn + d1 N ) ( ) ( iπ 2 Y x n S x n, − . × exp − λ R0 M

(5) (6)

(7)

Here, we use σ to represent the standard deviation of Gaussian distribution of the X-ray source in the x direction. B. Fringe scanning technique

For simplicity, we assume that the FZP has an ideal lens characteristic. The amplitude transmission function S(x, y) of the sample is given by

Now, we are going to discuss about the fringe scanning technique that was used by Yashiro et al.11 To simplify the explanation, we first define some new functions as follows: ( ) ( ) iπ 2 Y Sn (X,Y ) = exp − x S x n, − , (8) λ R0 n M ) ( N −1  i2πk . (9) ℑ(X,Y ) = Ik (X,Y ) exp N k=0

S(x, y) = exp(−iΦ(x, y)),

In (9), the zeroth-order term of Fourier expansion vanishes and the resulting function can be rewritten as

A. Phase shift and self-image deformation

(2)

ℑ(X,Y ) = P(X,Y ) + E(Φ),

(10)

where P(X,Y ) and E(Φ) represent the first-order term and the higher-order terms, respectively. By the Ronchi ruling, we have ( )  2πX P(X,Y ) = iN µ−1 exp −i a−1a0∗ S−1 S0∗ − a0a1∗ S0 S1∗ . d1 (11) By direct calculation, we obtain (see Appendix B) ( ) ( )   Y Y P(X,Y ) Φ x 1, − − Φ x −1, − = 2 arg 0 , (12) M M P (X,Y ) where P (X,Y ) = P(X,Y )|Φ≡0. (13) FIG. 1. Setup of X-ray phase-difference microscopy. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP: 130.237.29.138 On: Mon, 09 Mar 2015 10:38:02 0

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We denote the intensity patterns without the sample by Ik0(X,Y ) and define ) ( N −1  i2πk 0 0 ℑ (X,Y ) = . (14) Ik (X,Y ) exp N k=0 Then, by combining (10) and (12) and using the intensity patterns given by (14), the twin image of the phase shift (phasedifference image) can be estimated approximately as follows:11 ) ( ) ( Y Y − Φ x −1, − Φ x 1, − M M     ℑ(X,Y ) − E(Φ) ℑ(X,Y ) = 2 arg ≈ 2 arg 0 ℑ0(X,Y ) − E 0 ℑ (X,Y ) ( ) −1  N   i2πk   Ik (X,Y ) exp N   k=0  = 2 arg  (15) )  , ( N −1    i2πk  Ik0(X,Y ) exp   N  k=0 where

E 0 = E(Φ)|Φ≡0.

This is the resultant image expected by the fringe-scanning technique under the configuration of Fig. 1. Moreover, in order to the sample’s phase shift Φ(x, y) from the phase-difference image given by (15), the boundary conditions Φ(x, y) = 0 are employed.

III. ERROR ANALYSIS

In (15), to reconstruct the phase-difference image of Φ(x, y), the higher order term E(Φ) is ignored by the conventional algorithm of fringe scanning technique. In fact, the error function E(Φ) is composed of the higherorder Fourier terms, which can be written in more detail as follows (see Appendix C): E(Φ) = µ N −1 A N −1 + µ−N −1 A−N −1 + µ2N −1 A2N −1 + µ−2N −1 A−2N −1 + · · ·,

(17)

where

 ∗ ∗ ( ) i al a0∗ Sl S0∗ − a0a−l S0 S−l ,   2πl X    Al (Φ) = N exp i ∗ an a∗n−l Sn Sn−l , d1    n odd 

l = 3, 5, . . . l = 2, 4, . . .

.

(18)

B. Error and spatial coherence length

By direct calculation, we have | Ah N −1| ≤ C, | A−h N −1| ≤ C,

(16)

h = 1, 2, . . . ,

(19)

where C is a constant independent of Φ(x, y). A. Error and size of X-ray source

Combining (10), (11), (17), and (19), we know that the error magnitude is determined by the quantity µ N −1/µ−1 roughly. Therefore, we are going to investigate the magnitude of µ N −1/µ−1 numerically. Here, we assume the geometrical parameters of imaging setup same as those used by Yashiro et al.,11 i.e.,

The above analysis demonstrates that the error is reduced by increasing the number of fringe-scanning. However, this is difficult due to a variety of reasons in practical applications. First, it gives rise to increasing radiation exposure. Second, a higher mechanical stability of equipment is required by decreasing the moving interval. The approximation error can be also investigated from the view point of spatial coherence. By the van Cittert-Zernike theorem, the spatial coherence length can be written as L = λ R0/(2πσ). Therefore, we have )2  (  µl pd 1  l 2 − 1  . = exp −2 µ−1 2M L  

R0 = 245 m, f = 261 mm, a = 272 mm, b = 6461 mm, λ = 0.137 nm, d = 4.3 µm. Synchrotron radiation was used, and the size of the source in the x direction was about 0.4 mm. For comparison, we also compute the magnitude µ N −1/µ−1 for a smaller X-ray source. The results are summarized in Table I. Here, the numbers of fringe-scanning steps are 4, 5, 40, and 90, respectively. From Table I, we observe that the approximation error caused by the conventional algorithm is not ignored for few images. We also observe that the approximation error increases significantly when the sample images are captured with a smaller X-ray source, and it is not ignorable even for N = 40. To achieve an adequate precision, it is necessary to increase the number of steps.

(20)

Roughly speaking, L should be as large as possible to obtain a precise image on the camera. In practical applications, it is

TABLE I. Magnitude of relative error µ N −1/µ −1 for different numbers of steps N . Size of X-ray source 0.4 mm N =4 N =5 N = 40 N = 90

≈ 0.76 ≈ 0.60 ≈ 2.65 × 10−23 ≈ 2.31 × 10−118

Size of X-ray source 0.04 mm ≈ 0.99 ≈ 0.99 ≈ 0.59 ≈ 0.07

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FIG. 4. Reconstructed phase-difference image (a) using the conventional algorithm (gray scale window [−0.12, 0.12]) and thereby phase shift image (b) (gray scale window [1.0, 1.04]) with N = 5. FIG. 2. Shepp-Logan phantom with a gray scale window [1.0, 1.04].

required that11 pd 1 . (21) M However, from (20) and the above error analysis, we know that the larger L gives rise to larger errors. Generally, the period of the grating is limited to micron scale, so that the interval to move the grating by d/N may reach a nanometer scale. Thus, for an X-ray source with a small size, high mechanical stability is required. L>

FIG. 5. Reconstructed phase-difference image (a) using the conventional algorithm (gray scale window [−0.12, 0.12]) and thereby phase shift image (b) (gray scale window [1.0, 1.04]) with N = 40.

IV. IMPROVED ALGORITHM AND EXPERIMENTS

From the analysis in Sec. III, we know that, though higher precise sample image can be obtained with a larger spatial coherence length system, the approximation error caused by the conventional algorithm is not ignored. New algorithm is necessary to decrease the number of fringe-scanning steps. A. Improved algorithm

To solve the problem mentioned above, we propose an improved algorithm that can suppress the approximation error in applying (15) and obtain more accurate phase shift information with few sample images. The proposed algorithm is iterative and its processing steps are summarized as follows.

FIG. 6. Error profile of Figure 4(b) along the central line.

Initialization: Using the conventional algorithm given by (15) with E(Φ) = 0, compute the phase-difference image of phase shift. Then, the initial single image Φ0(X,Y ) is generated with the condition that the sample is completely within the field of view. Set the iteration number as i ← 0. Define a stopping criterion parameter ε > 0. FIG. 7. Error profile of Figure 5(b) along the central line.

FIG. 8. Reconstructed phase-difference image (a) (gray scale window [−0.12, 0.12]) and thereby phase shift image (b) (gray scale window [1.0, 1.04]) using FIG. 3. A simulated sample image. the proposed algorithm with N = 4. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:

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FIG. 9. Error profile of Figure 8(b) along the central line.

FIG. 11. Reconstructed phase-difference image (a) (gray scale window [−0.3π, 0.3π]) and thereby phase shift image (b) (gray scale window [−0.2π, 0.3π]) from real data using the proposed algorithm with N = 6.

Iteration: [STEP1] : Compute the error function E(Φi ) by (17) using the current solution Φi (X,Y ). [STEP2] : Compute a new solution Φi+1(X,Y ) by (12) and the image conversion process from the phase-difference image with P(X,Y ) = ℑ (X,Y ) − E(Φi ).  i+1  [STEP3] : If Φ (X,Y ) − Φi (X,Y ) ≥ ε, then set i ← i + 1 and return to [STEP1]. Otherwise, terminate the iteration and output Φi+1(X,Y ) as the final phase shift image.

that 5 sample images are not enough to obtain satisfactory results by the conventional algorithm. Figure 8 shows the phase-difference image and the phase shift image reconstructed by the proposed algorithm from only four sample images. The error profile along the central line in Fig. 8(b) is shown in Figure 9, which is almost the same as that by the conventional algorithm with 40 sample images (Fig. 7).

Experiments were done to validate effectiveness of the proposed algorithm. We assume that the pixel size of camera is 4.34 µm and use the geometrical parameters same as those described in Sec. III.

We also have done experiment with the real data obtained for polystyrene spheres by Yashiro et al.11 Here, the experiment was performed at BL20XU in SPring-8 where the Xray beam was monochromatized using Si 111 double-crystal monochromator. A tantalum FZP which fabricated on a SiC membrane was used as an X-ray lens. A gold Ronchi grating fabricated on a glass substrate was used as a π/2 phase grating for 9 keV X-rays. An X-ray camera consisting of a phosphor screen, a relay lens, and a cooled CCD camera was used as the detector in the setup. To suppress the approximation error due to the higherorder terms in Fourier expansion, 43-step fringe scanning was performed,11 and the phase-difference image shown in Fig. 10(a) was reconstructed using the conventional algorithm. The phase shift image converted from Fig. 10(a) is shown in Figure 10(b) by using the boundary conditions Φ(x, y) = 0. For the reconstruction using the proposed algorithm, we used 6 sample images that were selected from these 43 sample images. To suppress the effect of the noise, we used a simple weight smoothing filter [1/4, 1/2, 1/4] in the reconstruction

B. Simulation

We use the Shepp-Logan phantom (Figure 2) as an original phase sample. Then, N sample images can be computed from (4). One of the sample images is shown in Figure 3. Note again that the phase shift images are reconstructed from the phase-difference image using the boundary conditions that Φ(x, y) = 0. The reconstructed phase-difference images, converted phase shift images, and their corresponding error profiles along the central horizontal line are shown in the following figures. Figures 4 and 5 show images obtained by using the conventional algorithm with 5 and 40 sample images, respectively. The error profiles along the central lines in Figs. 4(b) and 5(b) are shown in Figures 6 and 7, respectively. It is clear

C. Experiment with real data

FIG. 10. Reconstructed phase-difference image (a) using the conventional FIG. 12. Reconstructed phase-difference image (a) using the conventional algorithm (gray scale window [−0.3π, 0.3π]) and thereby phase shift image algorithm (gray scale window [−0.3π, 0.3π]) and thereby phase shift image (b) (gray scale window [−0.2π, 0.3π]) from real data with N = 43. (b) (gray scale window [−0.2π, 0.3π]) from real data with N = 6. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:

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procedure. Figure 11 shows the phase-difference image and phase shift image using the proposed algorithm. Figure 12 shows the reconstructed phase-difference image using the conventional algorithm and thereby phase shift image from the same 6 sample images. It is clear that the effect of higher-order terms in Fourier expansion that results the approximation error in Figure 12 is greatly suppressed in Figure 11.

 sin πn 2 an = (−1 − i), πn

1 a0 = (1 − i), 2

n = ±1, ±2, . . . . (B2)

So, a−1a0∗ = −a0a1∗ and by (11), we have S−1 S0∗ + S0 S1∗ P(X,Y )  . = P0(X,Y ) S−1 S0∗ + S0 S1∗ Φ≡0

(B3)

Since S−1 S0∗ + S0 S1∗

V. SUMMARY

In this paper, we performed an error analysis of the fringe scanning technique in the X-ray phase-difference microscopy and proposed an improved algorithm. The experiments of simulation and real data showed that the approximation error is significantly suppressed by the proposed algorithm, and a precise phase shift image can be reconstructed with few sample images. This contributes to decreasing radiation exposure and also reducing a requirement on the precision of mechanical operation by using the proposed algorithm. Since we supposed that the FZP has an ideal lens characteristic, the expression in (4) is analogous to that in the case of the x-ray projection microscope with a grating.11 We expect that the improved algorithm might make contributions to these fields.

= cos(−ϕ˜−1 + ϕ˜0) + cos(−ϕ˜0 + ϕ˜1) + i [sin(−ϕ˜−1 + ϕ˜0) + sin(−ϕ˜0 + ϕ˜1)] ( ) ϕ˜1 + ϕ˜−1 − 2ϕ˜0 = 2 cos 2  ( ϕ˜ − ϕ˜ ) ( ϕ˜ − ϕ˜ )  1 −1 1 −1 × cos + i sin 2 2 ) ( ( ϕ˜ − ϕ˜ ) ϕ˜1 + ϕ˜−1 − 2ϕ˜0 1 −1 exp i , (B4) = 2 cos 2 2 we have 

 P(X,Y ) 2 arg = ϕ˜1 − ϕ˜−1 − (ϕ˜1 − ϕ˜−1)|Φ≡0 P0(X,Y ) Y Y = Φ(x 1, − ) − Φ(x −1, − ). M M

(B5)

APPENDIX A: THE EXPRESSION IN (6) APPENDIX C: THE EXPRESSIONS OF ERROR IN (17) AND (18)

From the Cittert-Zernike theorem, we have11 µ n−m = µ′(x m − x n , 0)    J0(ξ, η) exp λi 2π R 0 ξ (x m − x n ) dξdη  , (A1) ≡ J0(ξ, η)dξdη where J0(ξ, η) is the intensity distribution of X-ray source. For the X-ray source of Gaussian distribution, denote

For any integer n, denote bn (p) = an exp(iπ n2 p),

(C1)

Then, Ik (X,Y ) =



µ n−m bn (p)b∗m (p)

n, m

1 η2 + ξ2 J0(ξ, η) = exp *− 2 − . 2πσσ1 2σ12 , 2σ

 ( ) X k × exp i2π (n − m) + d1 N ∗ × Sn (X,Y )Sm (X,Y ).

(A2)

Then, 

(

)

ξ i 2π (n−m)pd 1 dξ exp − 2σ 2 exp λ R 0 ξ M µ n−m = √ 2π σ 2     π(n − m)pd 1σ   . = exp  −2 (A3)  M λ R0   Here, we used that ( 2)   √ α exp −ξ 2 exp (iαξ) dξ = π exp − , for real α. (A4) 4 2





APPENDIX B: THE PHASE-DIFFERENCE IN (12)

Introduce the notation Y π 2 )+ x , n = 0, ±1. (B1) M λ R0 n Note that the Fourier coefficients of the transmission function of the grating are ϕ˜ n (X,Y ) = Φ(x n , −

(C2)

Therefore, ℑ(X,Y ) =

N −1  k=0

=

(

i2πk Ik (X,Y ) exp N

)

  X µ n−m bn (p)b∗m (p) exp i2π(n − m) d1 n, m   N −1     k  ∗  × Sn (X,Y )Sm (X,Y )  exp i2π(n − m + 1)  . N  k=0  (C3) 

Since N −1 

  k exp i2π(n − m + 1) N k=0   N, =  0, 

n − m + 1 = hN(h = 0, ±1, ±2, . . .), otherwise (C4)

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and a2h = 0,

h = ±1, ±2, . . . .

(C5)

Combining (10), (11), (C3), (C4), and (C5), we obtain the expressions in (17) and (18). 1A.

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An improved phase shift reconstruction algorithm of fringe scanning technique for X-ray microscopy.

The X-ray phase imaging method has been applied to observe soft biological tissues, and it is possible to image the soft tissues by using the benefit ...
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