Optimization of image recording distances for quantitative X-ray in-line phase contrast imaging

Yuqi Ren; Can Chen; Rongchang Chen; Guangzhao Zhou; Yudan Wang; Tiqiao Xiao

doi:10.1364/OE.19.004170

1. Introduction

Due to its simple setup and no mirror or crystal needed, X-ray in-line phase contrast imaging (XIPCI) has received broadly investigation and application in recent years, providing that the beam with sufficiently spatial coherence [1–12]. XIPCI is actually an outline imaging to the edges in which the density inside the sample changes suddenly, and also could be used for qualitative investigation. Combined with phase retrieval, quantitative information could be obtained, namely quantitative X-ray in-line phase contrast imaging (QXIPCI), which found increasing applications in a variety of research fields, such as pathology [7], material science [8], paleontology [9], botany [10] and metrology [11, 12] etc..

Several algorithms for phase retrieval in QXIPCI have been proposed, such as the Gerchberg-Saxton-Fienup (GSF) iterative method [13, 14], methods based on the transport of intensity equation (TIE) [15, 16], the contrast transfer function (CTF) [17, 18], the mixed approach [1, 19], and the first Born approximation [20, 21], and so on. With GSF algorithm, high retrieval accuracy could be achieved as long as enough iterations is used, but there is always the intrinsic problem of iterative stagnation. TIE method is only valid for short SDD and works well for noise free data due to fewer assumptions imposed on the sample. In order to study the situation of long SDD, the algorithm of Born approximation, which is based on assumption of weak absorption and weak phase, is always adopted. The method based on CTF could accurately retrieve the phase of the object using multiple SDD images. The mixture of CTF and TIE could be applied to strongly absorbing objects and performs best in terms of quantitative results and robustness against noise.

Phase retrieval from single image is feasible, and has found applications in many fields up to now, due to its recording simplicity and post-processing flexibility [13–16, 20–23]. However, it has suffered from some disadvantages when it was used in the conditions of complicated samples or with high precision required, for instance, the quantitative retrieval of biological tissues. On the other hand, in principle, phase retrieval with images recorded at multiple SDD could improve the precision apparently, among which phase retrieval with two images recorded at different SDD, could make a very good compromise between precision and dose [24–27].

Usually images recorded at different SDD will affect the retrieval precision a lot. However, how to select the right recording SDD for the multiple images to achieve higher retrieving precision, remains unsolved. In this paper, this problem was investigated systematically with digital simulation and related experiments, using phase retrieval algorithm based on the first-order Born approximation [21]. As to the contents, the retrieval algorithm and related theory was introduced firstly. Then numerical simulation was carried out at section 3. Experimental results and related analysis was given in section 4. Finally, a summarization and some related discussions were given in section 5.

2. Principle

A conceptual diagram of the imaging technique is shown in Fig. 1 , supposing that a coherent monochromatic plane scalar wavefront with a wavelength λ and a complex wave-field distribution $U_{i}$ , propagates along the positive z direction. The object is located immediately before the object plane $z = 0$ and thin enough, i.e. its dimension of z-direction is much smaller than the propagation distance. The complex refractive index of the object is designated as

n = 1 - δ + i β

where δ and β represent the phase shift and absorption properties introduced by the object respectively.

Fig. 1 Schematic diagram for the quantitative in-line phase contrast imaging system, where the distances between the object plane and the two image planes are $z_{1}$ and $z_{2}$ , respectively.

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With paraxial approximation of the incident X-ray beam, the complex amplitude distribution at the image plane is given in Fresnel-Kirchhoff diffraction integral [28]

U_{z} (x, y) \equiv \frac{\exp (i k z)}{i λ z} \iint U_{o} (x^{'}, y^{'}) \exp {i \frac{k}{2 z} [{(x - x^{'})}^{2} + {(y - y^{'})}^{2}]} d x^{'} d y^{'}

where

k = \frac{2 π}{λ}

denotes the wave number,

U_{o}

is the complex amplitude on object plane.

According to the Fourier properties of linear imaging system [29], Eq. (2) is rewritten as a convolution form and the intensity distribution at the image plane could be obtained

I_{z} (x, y) = {| h (x, y; x^{'}, y^{'}) \otimes U_{o} (x^{'}, y^{'}) |}^{2}

where

h (x, y; x^{'}, y^{'})

is the impulse response function of the system, ⊗denotes a two-dimensional convolution. Equation (3) represents the phase retrieval with respect to unknown phase shift

φ_{o}

. By solving the non-linear equation, the original phase of the object could be retrieved, in which

I_{z} (x, y)

could be recorded by experiments. In order to find out the solution in a straightforward way, Eq. (3) can be simplified using first order Born approximation,

F {(I_{z} / I_{o} - 1) / 2} = \cos (χ z) F {Re φ_{o}} + \sin (χ z) F {Im φ_{o}}

where

F {*}

denotes the Fourier transform,

χ = π λ (u^{2} + v^{2})

,

(u = \frac{x}{λ z}, v = \frac{y}{λ z})

are the coordinates in spectral space,

I_{o}

is the average intensity of the transmitted beam,

Re φ_{o}

and

Im φ_{o}

represents absorption and phase shift respectively. Equation (4) is the formula for the single-SDD-image phase retrieval under Born approximation, under the assumption that

I_{o} = constant

(phase object with weak absorption) and

Re φ_{o} = 0

(pure phase object) or

Im φ_{o} = γ Re φ_{o}

(homogenous object), γ is a real constant.

In order to retrieve both phase and amplitude simultaneously in the object plane, at least two images at different SDD are needed, as shown in Fig. 1. According to Eq. (4), the solution with respect to $Re φ_{o}$ and $Im φ_{o}$ could be written as

(\begin{array}{l} F {Re φ_{o}} \\ F {Im φ_{o}} \end{array}) = \frac{1}{\sin [(z_{2} - z_{1}) χ]} \times [\begin{array}{l} \sin (z_{2} χ) \\ - \cos (z_{2} χ) \end{array} \begin{array}{l} - \sin (z_{1} χ) \\ \cos (z_{1} χ) \end{array}] \times (\begin{array}{l} F {(I_{z_{1}} / I_{o} - 1) / 2} \\ F {(I_{z_{2}} / I_{o} - 1) / 2} \end{array})

According to Eq. (5), the amplitude and phase information in the object plane could be retrieved. However, it is obvious that the equation is ill-posed, because that there are some isolated singularities in Eq. (5) in Fourier domain. These irrecoverable frequencies are defined by the following equation

(z_{2} - z_{1}) λ (u^{2} + v^{2}) = m, (m = 0, \pm 1, \pm 2, ...)

where m is an arbitrary integer.

Conventionally, Tikhonov’s regularization [30] can be used to handle the ill-posed problem. In particular, the regularized formula is as following:

(\begin{array}{l} F {Re φ_{o}} \\ F {Im φ_{o}} \end{array}) = \frac{\sin [(z_{2} - z_{1}) χ]}{\sin^{2} [(z_{2} - z_{1}) χ] + α} \times [\begin{array}{l} \sin (z_{2} χ) \\ - \cos (z_{2} χ) \end{array} \begin{array}{l} - \sin (z_{1} χ) \\ \cos (z_{1} χ) \end{array}] \times (\begin{array}{l} F {(I_{z_{1}} / I_{o} - 1) / 2} \\ F {(I_{z_{2}} / I_{o} - 1) / 2} \end{array})

where α is an infinite small positive constant, which will affect obviously the retrieving accuracy and noise level [21, 31]. As depicted in Eq. (7), the critical point that determines the retrieval efficiency is the optimization of SDD

z_{1}

and

z_{2}

, which should satisfy the following inequality relationship

Δ z = z_{2} - z_{1} \neq \frac{m}{λ (u^{2} + v^{2})}, (m = 0, \pm 1, \pm 2, ...)

where

Δ z

is the distance between the two image planes.

With the wavelength of incident X-ray and detector resolution fixed, $Δ z$ only depends on the space frequency $(u, v)$ , i.e., the combination of $z_{1}$ and $z_{2}$ . In general, the selection of the optimized combination set for the best retrieval is rather complicated.

3. Simulation and analysis

3.1 Simulation

In this section, the effect of the recording distance pairs on the retrieval precision was investigated by digital simulation. A nylon fiber with homogeneous composition and density was adopted as the phantom, which has weak absorption to X-rays. The parameters for the simulation are listed as follows: photon energy 15keV (corresponding wavelength $λ = 8.26562 \times 10^{- 11}$ m), diameter of the phantom was 100μm, in which the real and imaginary parts of the complex refractive index were $δ = 9.84657 \times 1 0^{- 7}$ , $β = 3.72017 \times 1 0^{- 10}$ respectively [32]. The image field is 512×512 pixels with the pixel size of 1 micron. Figure 2(a) presents the picture of the phantom, while Fig. 2(b) shows the profile at the white line marked in Fig. 2(a). A series of XIPCI images were obtained accordingly by increasing the SDD with an interval of 10cm ranging from 0 to 600cm. Consequently, the retrieved images were achieved from one pair of images according to Eq. (7).

Fig. 2 Phantom for simulation (a) nylon fiber with a diameter of 100μm and (b) corresponding profile at the white line marked in Fig. 2(a).

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In order to evaluate the relative errors of retrieved images to real one, the root-mean-square (RMS) criterion was introduced by the following formula [20, 21]

R M S = {(\sum_{i, j} {| g (i, j) - f (i, j) |}^{2} / \sum_{i, j} {| f (i, j) |}^{2})}^{\frac{1}{2}} \times 100 %

where

f (i, j)

denotes the original image and

g (i, j)

is the retrieved image.

3.2 Optimization of imaging distances

In order to achieve the best retrieved result, optimization of the SDD of $z_{1}$ and $z_{2}$ was needed. The choice of the SDD-pair was implemented through following steps: Firstly, the distance $z_{1}$ was fixed at a certain value that matched the Born-type approximation [21]. Secondly, the distance $z_{2}$ was changed from $z_{1}$ to 4 $z_{1}$ step by step with an increment of 10cm. Thus, a data set of SDD-pair could be obtained. Finally, the images were retrieved and then the retrieving errors could be evaluated as shown in Fig. 3 .

Fig. 3 Retrieving errors vs. z₂, while z₁ was set to (a) 50cm, (b) 100cm, (c) 120cm and (d) 150cm, respectively. The photon energy is set to 15keV and the diameter for the fiber sample is 100μm.

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In Fig. 3, the RMS curves vs. $z_{2}$ are given, where SDD $z_{1}$ for Fig. 3(a), (b), (c) and (d) were set to 50cm, 100cm, 120cm and 150cm respectively. From Fig. 3, it is obvious that the RMS errors reach a minimum while $z_{2} = 3 z_{1}$ , in spite of different $z_{1}$ values. During simulation, the value of parameter α remains constant and was set to $10^{- 6}$ . Though the value of parameter α could impose impact on the retrieving precision, it would not affect the trend of the curves. This means that there is indeed an optimized SDD for the second image, which equals to three times that for the first image.

The results from Fig. 3 could be confirmed by Fig. 4 , in which the intensity profiles of the optimized retrievals at the white line, similar to that in Fig. 2(a), were plotted. As a comparison, the correspondent profile for the ideal phantom is also given.

Fig. 4 Retrieved image profiles by the triple rules at the white line as marked in Fig. 2, (a) profiles with different SDD-pair, (b) profiles at different z₂ while z₁ fixed to 120cm. The photon energy is set to 15keV and the diameter for the fiber sample is 100μm.

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As shown in Fig. 4(a), among SDD pairs of 50-150cm, 100-300cm, 120-360cm and 150-450cm, the profile retrieved with SDD-pair of 120-360cm is the best one, which is the closest one to the ideal profile. This means that the highest retrieval precision could be achieved while $z_{1} = 120 c m$ and $z_{2} = 360 c m$ . RMS errors for the retrieved images at more SDD-pairs are listed in Table 1 , which confirm the best retrieval at SDD-pair of 120-360cm. Figure 4(b) presents the retrieved profiles at different $z_{2}$ while $z_{1}$ was fixed to 120cm. The results are in accordance with Fig. 4(a).

Table 1. Relative RMS Errors for the Retrieved Images at a Series of SDD-Pairs Following the Triple Relations

View Table

3.3 Spectral correlation degree

To understand the results achieved in the above section, further analysis is given, based on Fourier Spectrum Analysis (FSA). Spectral correlation degree (SCD) was introduced to evaluate the similarity between the two images recorded at different SDD, which is expressed as [33]

ρ_{12} = \frac{\int_{- \infty}^{\infty} R_{1} (u, v) R_{2} (u, v) d u d v}{{[\int_{- \infty}^{\infty} R_{1}^{2} (u, v) d u d v \int_{- \infty}^{\infty} R_{2}^{2} (u, v) d u d v]}^{\frac{1}{2}}}

where

R_{1} (u, v)

and

R_{2} (u, v)

are the Fourier spectrum of the two images respectively. SCD value between the two images vs.

z_{2}

with

z_{1}

fixed to certain value, was given in Fig. 5 .

Fig. 5 Spectral correlation degree vs. z₂ when z₁ was set to (a) 50cm, (b) 100cm, (c) 120cm and (d) 150cm, respectively. The photon energy is set to 15keV and the diameter for the fiber sample is 100μm.

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From Fig. 5, the SCD value reaches to a maximum at $z_{2} = 2 z_{1}$ and $z_{2} = 3 z_{1}$ . When $z_{1}$ is set to 50cm, SCD value decreases almost monotonously from 0.9 to 0.7 approximately, with $z_{2}$ varying from 60cm to 180cm, as shown in Fig. 5(a), from which, it is hard to find out the two maxima when $z_{2} = 2 z_{1}$ and $z_{2} = 3 z_{1}$ . However, this is not the case for that when $z_{1}$ is set to 100, 120 and 150cm, as shown in Fig. 5(b), (c) and (d). For these cases, the SCD value goes down quickly to a minimum when $z_{2}$ increase to a point close to $2 z_{1}$ , and then oscillates around a fixed value approximately. It is obvious that the first maximum could be found at $z_{2} = 2 z_{1}$ for all the three cases. The second maximum appears at $z_{2} = 3 z_{1}$ . For the case in Fig. 5(c) while $z_{1}$ is set to 120cm, the SCD curve is oscillatory damped to a flat state when $z_{2}$ gets close to $3 z_{1}$ . But still the second maximum could be found at $z_{2} = 3 z_{1}$ , though the amplitude is much less than that of the first one. In Fig. 5(b) with $z_{1} < 120 c m$ and Fig. 5(d) with $z_{1} > 120 c m$ , the amplitude damping at the second maximum is not so clear as that in Fig. 5(c) with $z_{1} = 120 c m$ . During the above simulation, $z_{2}$ is not extended too much larger than $3 z_{1}$ , considering the actual limit on the length of experimental stage inside the radiation shielding hutch.

Based on the above analysis, the spectral correlation degree has a maximum at points where $z_{2}$ equals to integer times of $z_{1}$ , and the SDD of the first image $z_{1}$ will obviously affect the damping oscillation trends. From section 3.2, at points with $z_{2} = 3 z_{1}$ minimum

RMS errors could be achieved, which means that best retrieval precision could be obtained when the distance of the second image to sample is three times that of the first image. Except the three times rule, different $z_{1}$ value will also contribute to the retrieving precision a lot. Among all the $z_{1}$ - $z_{2}$ pairs, there is one pair which has the minimum RMS error, i.e., the highest precision, as shown in Table 1. All these imply that the two images employed for the retrieval should be not only well spectrum correlated, but also spectrally complementary in order to achieve high retrieval precision.

4. Experiments

The experiments were carried out at the X-ray imaging beamline BL13W at Shanghai Synchrotron Radiation Facility (SSRF), as shown in Fig. 6 is the experimental set up. The sample is about 34m downstream the wiggler source, which provides not only high spatial coherence beam, characterized by the transversal coherence length which is 31μm at photon energy 30keV in the vertical direction, but also almost parallel beam incident on the sample. Two stages were installed at the end-station, with the first stage 3.0 meters long made of granite and intended for high stable and high resolution imaging, and the second stage 5.0 meters long made of stainless steel with alveolate structure and intended for experiments needing long free space. The sample was fixed on one end of the first stage, the detector was installed on a rail which could be driven by a motor remotely controlled. In this way, the sample to detector distance could be easily adjusted according different purposes.

Fig. 6 Picture for experimental facilities at SSRF BL13W beamline (a) sample stage, (b) CCD and (c) slide rails.

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Similar to that in simulation, nylon fiber was also employed as samples in the experiments, with a diameter of 128μm. The photon energy was also set to 15keV. An X-ray CCD from PCO (Germany) was employed for the experiments, which is integrated with a series of amplification lens (2×, 4×, 10×, 20×, 40× ). In the following experiments, the lens of 4 × was selected and the effective pixel size of the detector is 1.85×1.85μm² with a field of view of 3.5×3.5 mm².

The two separate sliding rails at SSRF BL13W end-station are mounted in a fore-and-aft way along the optical path. Due to actual limit of the stages mounted on the rails, the allowed ranges of SDD are from 0 to 160cm for the anterior one and from 300cm to 800cm for the posterior one. Hence, the data between 160cm to 300cm were missed in the experiments. For convenience, in the following experiments only the fore rail is exploited where continuous SDD were need. Different from simulation, the images recorded in the experiments need to be preprocessed [34–39] before phase retrieval could proceed, because that the SR beam used for the experiments are slightly divergent. The experimental results are shown in Fig. 7 .

Fig. 7 In-line phase contrast images at SDD of 50cm in (a), retrieved image with single image at SDD of 50cm is shown in (b) and retrieved image with images at SDD pair of 50cm and 150cm in (c). Profiles of retrieved images with single image at 50cm, SDD-pair 50-150cm and 120-360cm are displayed in (d).

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Figure 7(a) is in-line phase contrast image at 50cm. For reference, retrieval from single image recorded at SDD of 50cm, is also given in Fig. 7(b). Figure 7(c) is the retrieved image with SDD pair of 50cm and 150cm. Shown in Fig. 7(d) are the profiles of (b) and (c), and for the retrieved image with SDD pair of 120cm and 360cm. For the convenience of comparison, the profile for the sample is also given. During data processing, the effect of divergent SR beam on the image enlargement and intensity change has been taken into account [40]. From this figure, it is obvious that the retrieved image with the 50-150cm SDD pair deviated a lot from the original sample especially in the central part of the fiber, while the retrieved image with the 120-360cm SDD pair meet well with the sample. All these results are consistent with that drawn from the simulation investigation. This means that the conclusion achieved in section 3 could be partly confirmed by experiments. One more thing need to be mentioned here is that the pixel size and sample dimension used in the experiments are a little bit larger than that used in the simulation.

For comparison, the simulation results of phase retrieval at different SDD pairs, with the same parameters as that employed in the experiment, i.e. nylon fiber of 128 micron in diameter and effective pixel size of 1.85 micron, were given in Fig. 8 . In overall, these results are consistent with that given in section 3. However, as shown in Fig. 8, the profiles of SDD pairs of 120cm-360cm and 130cm-390cm are both very close to that of the phantom, which means that the predicted best SDD $z_{1}$ could slightly differ from the value as predicted in section 3. This implies that value of the best SDD for the first image could be changed slightly around a fixed value. The minute dissimilarity between Fig. 7 and Fig. 8 could also be found, which may owe to the post-processing errors of experimental data.

Fig. 8 The simulation results for the retrieved image profiles at different SDD pairs with the triple rules, with the same parameters as that employed in the experiment, i.e., the photon energy 15keV, the diameter for the fiber sample 128μm and the effective pixel size 1.85μm.

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Furthermore, with the SDD of the first image fixed and the SDD of the second image increasing, the varying trends of the retrieving precision and spectral correlation degree of the two images employed for the higher precision retrieval should also be verified by experiments. Similar to that in simulation, RMS error was also applied to evaluate the accuracy of the phase retrieval during the experiments. Considering the practical limit of the experimental stage, only the experiments of $z_{1} = 50 c m$ was brought out. The experimental results were shown in Fig. 9 .

Fig. 9 Experimental results with SDD for the first image fixed to 50cm, where (a) RMS errors of the retrieved images vs. z₂. (b) Spectral correlation degree vs. z₂.

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In Fig. 9(a), the RMS errors of the retrieved images reach to a minimum at $z_{2} = 3 z_{1}$ with $z_{2}$ increasing from 50cm to 160cm, which is consistent with the simulation. Accordingly, the SCD vs. $z_{2}$ was shown in Fig. 9(b). In this figure, it is clear that the SCD value reaches to a maximum when $z_{2} = 2 z_{1}$ and $z_{2} = 3 z_{1}$ , which is almost identical to the simulation results.

5. Conclusions

According to all the above simulation and experimental results, we could conclude that the optimized sample-to-detector distances could be found to achieve higher precision on the two-image-based phase retrieval. With the recording distance of the first image fixed, the highest retrieving precision could be obtained when the recording distance of the second image is three times that of the first image. Furthermore, the sample-to-detector distance for the first image will evidently affect the retrieval, though the three-time law should always be followed. With the sample size, X-ray wavelength and detector pixel size given, there is an optimized recording distance for the first image, while the spectral correlation degree of the two images vs. the recording distance of the second image occurs with a typical damping oscillation, according to the simulations.

The simulation investigation shows that the retrieving precision of less than 8% could be achieved with only two images, as long as the optimized recording distances are used for both of the images. This implies that the method developed in this paper could find practical applications in many fields, such as biomedicine, material science, geology, paleontology etc., where relatively higher precision quantitative information is required. Compared to multiple images and single image phase retrieval, the two-image-based retrieval with optimized recording distances meet the very good compromise between dose and precision. Combined with microtomography, higher density resolution should be achievable.

Phase retrieval with three or more images is not investigated in this paper. However, we believe that there should be optimized recording distances for all the images if the best retrieval is required, though the optimizing process would be far more complicated.

Acknowledgments

This research was supported in part by National Basic Research Program of China grant 2010CB834301, the External Cooperation Program of Chinese Academy of Sciences grant GJHZ09058, the National Natural Science Foundation of China grant 10805071 and 10705020.

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z₁(cm)	z₂(cm)	RMS(%)	z₁(cm)	z₂(cm)	RMS(%)
50	150	63.46	110	330	11.49
60	180	58.42	120	360	7.42
70	210	42.54	130	390	8.28
80	240	39.49	140	420	9.08
90	270	22.22	150	450	17.76
100	300	14.69	160	480	23.01

Optimization of image recording distances for quantitative X-ray in-line phase contrast imaging

Abstract

1. Introduction

2. Principle

3. Simulation and analysis

3.1 Simulation

3.2 Optimization of imaging distances

3.3 Spectral correlation degree

4. Experiments

5. Conclusions

Acknowledgments

References and links

Cited By

Figures (9)

Tables (1)

Equations (10)

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