Simulation study of phase retrieval for hard X-ray in-line phase contrast imaging

Transcription

1 450 Science in China Ser. G Physics, Mechanics & Astronomy 2005 Vol.48 No Simulation study of phase retrieval for hard X-ray in-line phase contrast imaging YU Bin 1,2, PENG Xiang 1,2, TIAN Jindong 1, NIU Hanben 1, DIAO Luhong 3 & LI Hua 3 1. Key Laboratory of Optoelectronic Devices and Systems of Ministry of Education, Institute of Optoelectronics, Shenzhen University, Shenzhen , China; 2. State Key Laboratory of Precision Measuring Technology and Instruments, Tianjin University, Tianjin ,China; 3. Key Laboratory of Intelligent Information Processing, Institute of Computing Technology, Chinese Academy of Sciences, Beijing ,China Correspondence should be addressed to Yu Bin ( yubin@szu.edu.cn) Received March 21, 2005; revised April 2, 2005 Abstract Two algorithms for the phase retrieval of hard X-ray in-line phase contrast imaging are presented. One is referred to as Iterative Angular Spectrum Algorithm (IASA) and the other is a hybrid algorithm that combines IASA with TIE (transport of intensity equation). The calculations of the algorithms are based on free space propagation of the angular spectrum. The new approaches are demonstrated with numerical simulations. Comparisons with other phase retrieval algorithms are also performed. It is shown that the phase retrieval method combining the IASA and TIE is a promising technique for the application of hard X-ray phase contrast imaging. Keywords: X-ray optics, phase contrast imaging, phase retrieval, angular spectrum, hard X-ray. DOI: / X-ray phase contrast imaging (XPCI) is a novel method that exploits the phase shift for the incident X-ray to form an image. For light elements such as carbon, hydrogen and oxygen, the phase-shift term can be up to 1000 times greater than the absorption term in the hard X-ray energy region. So XPCI has attracted much attention in recent years. Various methods for XPCI have been proposed and demonstrated on synchrotron devices and other X-ray sources [1 13], particularly the in-line method [7 8] proposed by Wilkins et al., which simply exploits Fresnel diffraction to provide contrast. The in-line hard X-ray phase contrast imaging has become one of the promising XPCI techniques. A key problem of X-ray phase contrast imaging is how to retrieve the phase from the image intensity. Nugent et al. established a method based on the transport of intensity equation (TIE) [9 13], which can be applied successfully to retrieve the phase information in near-field region. Cheng J. et al. [4] also presented a general treatment of X-ray image

2 Simulation study of phase retrieval for hard X-ray in-line phase contrast imaging 451 formation by direct Fresnel diffraction with partially coherent hard X- rays. In this article, two algorithms for the phase retrieval of hard X-ray in-line phase contrast imaging are presented. One is referred to as Iterative Angular Spectrum Algorithm (IASA) and the other is a hybrid algorithm that combines IASA with TIE. The new approaches are demonstrated with numerical simulations. Comparisons with other phase retrieval algorithms are also performed. It is shown that the phase retrieval method combining the IASA and TIE is a promising technique for the application of hard X-ray phase contrast imaging. 1 Algorithm for phase retrieval The simplified schematic setup for in-line hard X-ray phase contrast imaging is shown in Fig. 1. We consider the imaging setup as a plane monochromatic incident wave propagating along the direction of the optic axis z. Fig. 1. Schematic set-up of hard X-ray in-line phase contrast imaging. According to angular spectrum of plane-waves diffraction theory [14], the complex wave field U ( x, y ) ρ ( x, y ) exp i φ ( x, y ) = in the image plane can be described as the result of propagation of the complex wave field (, ) ρ ( x, y ) exp i φ ( x, y ) U x y = in the object plane to the distance z, with the Fourier transforms of these two (initial and propagated) wave functions given by A 1 (α, β, 0) and A 2 (α, β, z) correspondingly. In the Fresnel approximation, the Fourier transform of the diffracted wave function is related to the Fourier transform of the initial function via the frequency transfer characteristic of the free space H(α, β, z), given by: ( αβ,, z) A H = = A exp ( ) 1 ikz 1 α β ( αβ,,0) ( αβ,,0) = (,,0exp ) { i ( α + β )} (1a) (1b) A U x y k x y dx dy

3 452 Science in China Ser. G Physics, Mechanics & Astronomy 2005 Vol.48 No ( αβ ) = ( ) { ( α + β )} 2 2 (1c) A,, z U x, y, z exp i k x y dx dy, 2π where α, β is the x and y direction cosine, k = is wave number, λ is the radiation λ wavelength. The intensity distribution in the image plane is proportional to the intensity of this field as ( ) ( ) 2 I x, y = U x, y. (2) Thus the problem of numerical reconstruction of X-ray phase contrast imaging can be described as follows: if we know the data of intensity measurements, I 2 (x 2, y 2 ), collected U x, y, in one or more planes orthogonal to the optic axis z, and the amplitude ρ 1 of ( ) how can we recover the phase φ 1 of (, ) U x y where the recovered wave functions satisfy expression (1) with high accuracy? Expressions (1) provide a symmetrical relation between the initial and diffracted wave functions in the Fresnel approximation. So we can reconstruct the phase information based on the iterative angular spectrum approach (IASA) proposed by Mellin and Nordin [15]. The IASA algorithm is based on Gerchberg-Saxton (GS) [16 18] and applied in the order (1b) (1a) (1c), resulting in the diffracted wave function; while being applied in the reversed order, they allow for reconstruction of the initial wave function from the result of diffraction. We shall denote the forward and the reversed propagation operations defined by expressions (1a), (1b), (1c) respectively. The algorithm described above can be implemented numerically by using Fast Fourier Transform (FFT) on a finite rectangular grid under periodic border conditions. Thus an iterative phase retrieval algorithm from two intensity planes in the Fresnel domain is simply obtained by using free space calculation, as shown in Fig. 2, instead of a single FFT block. Fig. 2. Free space propagation using angular spectrum calculation. Fig. 3 shows the flow chart of the iterative procedure based IASA algorithm. First, the phase object can be used as the initial value of φ 1 to start the iteration. The algorithm starts with the grabbed object plane and a random phase. The obtained wavefront is then transformed to the second plane using free space propagation calculation. Secondly, the phase at this point is maintained unchanged while the amplitude is forced to be the one grabbed at the image plane. The obtained wavefront is transformed back using free space propagation backwards where the obtained phase is again maintained unchanged and the amplitude is forced to be the one grabbed at the object plane. The algorithm is

4 Simulation study of phase retrieval for hard X-ray in-line phase contrast imaging 453 Fig. 3. IASA algorithm used for phase retrieval. running back and forth between the object and image planes until the sum-squared error (SSE) between the grabbed and the calculated amplitudes, defined by 2 2 (3) ( n) SSE = ρ ρ ρ ( n) where ρ2 expresses the result of the nth iteration for initial guess phase φ 1, reaches a given small quantity for determining the calculation accuracy, or until the number of iterations exceeds a given value N max. At this point, the calculated phase of the object and image planes has converged towards the values of the physical phase. 2 Numerical Simulation In this section we solve the phase-and-amplitude retrieval problem, which occurs in quantitative in-line hard X-ray phase contrast imaging. In the following calculations we use the simulated object-plane phase distribution shown in Fig. 4.The well-known Lena image shown here has a variety of features on different resolution scales that tend to be quite helpful for visual assessment of the reconstruction accuracy. The image had pixels and was assigned a physical size of µm. The radiation wavelength was set to λ = 0.1 nm which corresponds to hard X-rays. The phase values in Fig. 4 varied in the interval [ 1, 0] rad. These conditions are realistic for quantitative in-line hard X-ray phase contrast imaging. The object-plane intensity distribution was uniform, I 1 (x 1, y 1 ) 1, which corresponds to the case of a plane incident Fig. 4. Simulated phase distribution in the wave and a non- absorbing object. object plane.

5 454 Science in China Ser. G Physics, Mechanics & Astronomy 2005 Vol.48 No We calculated the intensity distribution, (, ) I x y, in the image planes for different propagation distances z = 0.1, 1 and 10 m (Fig. 5), which was calculated by numerically evaluating expression (1). Fig. 5. Intensity distributions in the different image plans: (a) z = 0.1 m; (b) z = 1 m; (c) z = 10 m. From Fig. 5, we can see that the image is an edge-enhanced image in near-field region. Then we used the TIE to recover the object phase φ 1 (Figs. 6 (a) (c)) from the image intensities (, ) I x y at z = 0.1, 1 and 10 m. Obviously, the reconstruction quality is very good at shorter distances, and quite bad at the longest one where only the low spatial frequencies have been reconstructed successfully. Fig. 6. Reconstructed phase with the TIE algorithm from intensity distributions in Fig. 5. Next we applied the GS and IASA algorithms for phase reconstruction from the im- I x, y at z = 0.1, 1 and 10 m, respectively. We applied the two algo- age intensities ( ) rithms with N =100 iterations between the object and the image planes, and with the random phases used as an initial approximation. As assumed the experimental intensity distributions in the object and image planes is known, we forced the intensity distribu-, I x, y in the object and image planes respec- tions to be equal to I ( x y ) and ( ) tively at each iteration of the algorithm. The results of the phase retrievals using the two new methods described above are presented in Figs. 7 8

6 Simulation study of phase retrieval for hard X-ray in-line phase contrast imaging 455 Fig. 7. Reconstructed phase with the GS algorithm from intensity distributions in Fig. 5. Fig. 8. Reconstructed phase with the IASA algorithm from intensity distributions in Fig. 5. From above images, at the two longer distances the two methods produced the better results. But the results produced by IASA algorithm were better than by GS algorithm, because the IASA method is based on angular spectrum propagation. In view of expression (1), the transfer function has no approximation, so the IASA method can be used in any field. Nevertheless, because the random of originally guess phase and iteration of the algorithm, the reconstructed results had some errors. We can improve it by combining the iterative algorithm with TIE method or adding iterative numbers. Then we applied IASA or GS algorithm with N = 100 iterations between the object and the image planes, and with the corresponding TIE-reconstructed phases used as an initial approximation (IASA+TIE algorithm or IASA+TIE algorithm). The results of the phase retrievals using the two new methods described above are presented in Figs One can see that the IASA+TIE algorithm produced the best results at shorter distances, but all the results show some improvement over the original TIE-reconstructed phase, GS-reconstructed phase and the original IASA-reconstructed phase. A quantitative estimation of the phase retrieval accuracy can be calculated with the following relative root-mean-square (RMS) error formula:

7 456 Science in China Ser. G Physics, Mechanics & Astronomy 2005 Vol.48 No Fig. 9. Reconstructed phase with the GS+TIE algorithm from intensity distributions in Fig. 5 Fig. 10. Reconstructed phase with the IASA+TIE algorithm from intensity distributions in Fig. 5 where RMS = 100% φ φ φ rec true 2 true 2 ij ij ij i, j i, j (4) true φ ij was the value of the phase at pixel (i, j) of the original simulated phase distribution and rec φ ij was the reconstructed phase value at the same pixel. The results of phase retrieval are presented in Table 1 for different values of the object-to-detector distances. These error values are in good agreement with the visual impression of the reconstruction quality. Table 1 Phase reconstruction errors RMS for varying object-to-image distance z Method/distance z = 0.1 m z = 1 m z = 10 m TIE 8.03% 15.41% 24.90% GS 35.73% 31.46% 21.92% IASA 35.06% 29.41% 19.28% GS+TIE 2.58% 5.05% 9.52% IASA+TIE 2.45% 4.24% 6.70% 1 2

8 Simulation study of phase retrieval for hard X-ray in-line phase contrast imaging 457 Fig. 11 illustrates the relationship between the SSE and the number of iterations for four iterative algorithms at z = 1 m. As only 100 iterations are used, the SSE are for IASA+TIE algorithm. It can be seen that the SSE is reduced faster when the IASA+TIE algorithm is used. We spectulate that better results could be obtained with the IASA+TIE algorithm if more iteration were used. Fig. 11. Relationship between the SSE and the number of iterations at z =1 m. 3 Conclusions We have demonstrated that IASA algorithm can be used to phase retrieval. It is also shown that the new method combining the TIE method has improved the quality of phase retrieval from hard X-ray in-line phase contrast imaging. The algorithms delivered substantially more accurate retrieved phase and amplitude compared to the results of the IASA and TIE methods. Numerical examples show the effectiveness of our methods. Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant No ), and the Natural Science Foundation of Guangdong Province (Grant Nos and ), and Shenzhen Bureau of Science, Technology & Information (Grant No ). References 1. Fitzgerald, R., Phase sensitive X-ray imaging, Phys. Today, 2000, 53(7): Gao, D. C., Pogany, A., Stevenson, A.W. et al., Hard X-ray phase contrast imaging, Acta Phys. Sin., 2000, 49: Chen Min, Xiao Tiqiao, Luo yuyu et al., Phase contrast imaging with microfocus X-ray source, Acta Phys. Sin., 2004, 53:

9 458 Science in China Ser. G Physics, Mechanics & Astronomy 2005 Vol.48 No Cheng Jing, Han Shensheng, Shao Wenwen et al., Study of partial coherent X-ray phase imaging, Acta Optica Sinica, 1999, 19(5): Cheng, J., Han, S., Phase imaging with partially coherent X rays, Opt. Lett., 1999, 24(4): Yu Hong, Zhu Pinpin, Han Shensheng, A study of diffractive phase imaging and phase restoration with partially coherent X-ray, Acta Optica Sinica, 2003, 23 (4): Wilkins, S. W., Gureyev, T. E., Gao, D. et al., Phase-contrast imaging using polychromatic hard X-rays, Nature, 1996, 384: Pogany, A., Gao, D., Wilkins, S. W., Contrast and resolution in imaging with a microfocus X-ray source, Rev. Sci. Insirum., 1997, 68(7): Nugent, K. A., Gureyev, T. E., Cookson, D. F. et al., Quantitative phase imaging using hard X-rays, Phys. Rev. Lett., 1996, 77(14): Gureyev, T. E., Wilkins, S. W., On X-ray phase imaging with a point source, J. Opt. Soc. Am. A, 1998, 15(3): Gureyev, T. E., Composite techniques for phase retrieval in the Fresnel region, Optics Communications, 2003, 220: Gureyev, T. E., Pogany, A., Paganin, D. M. et al., Linear algorithms for phase retrieval in the Fresnel region, Optics Communications, 2004, 231: Gureyev, T. E., Nugent, K. A., Rapid quantitative phase imaging using the transport of intensity equation, Optics Communications, 1997, 133: Goodman, J., Introduction to Fourier Optics (New York: McGraw-Hill), 1968, Stephen, D., Mellin, Gregory, P., Nordin, Limits of scalar diffraction theory and an iterative angular spectrum algorithm for finite aperture diffractive optical element design, Opt. Express, 2001, 8(13): Gerchberg, R. W., Saxton, W. O., A practical algorithm for the determination of phase from image and diffraction plane pictures, Optik (Stuttgart), 1972, 35: Fienup, J. R., Reconstruction of an object from the modulus of its Fourier transform, Opt. Lett., 1978, 3(1): Fienup, J. R., Phase retrieval algorithms: a comparison, Appl. Opt., 1982, 21: