Angular signal radiography

Panyun Li; Kai Zhang; Yuan Bao; Yuqi Ren; Zaiqiang Ju; Yan Wang; Qili He; Zhongzhu Zhu; Wanxia Huang; Qingxi Yuan; Peiping Zhu

doi:10.1364/OE.24.005829

1. Introduction

In accordance with the classical description on signal processing, the recorded experimental data can often be expressed as the convolution of initial signal function with the impulse response function of linear-time-invariant (LTI) system. Similarly, this classical description also applies to optical imaging process, which can be considered as linear-space-invariant (LSI) system, such as microscopy techniques using visible photons, x-rays, neutrons, and electrons. In x-rays imaging field, the attenuation-based x-ray radiography has made significant impact to its applications in various fields for the past few years while the phase contrast imaging, as a complementary approach, provides considerable advantages when soft matters are studied by taking the advantages of penetration capability and sensitivity to the chemical species offered by hard x-rays [1]. X-ray grating interferometry (XGI) is one of the most popular phase-sensitive radiography methods that provides multiple contrast mechanisms, including the x-ray absorption, refraction gradient and ultra-small angle scattering [2–7 ]. Fortunately, an x-ray Talbot-Lau interferometer has recently been installed and operated in hospitals for phase imaging [7]. To separate different properties, many algorithms have been proposed [4,5,8–20 ]. While the phase stepping (PS) method [4,5 ], the most widely adapted method in different experimental configurations, can retrieve the physical information of the specimen by calculating pixel by pixel the differences between two measured intensity curves with and without sample in XGI system, the analytical theoretical framework in XGI system has not yet to be described in detail.

In this contribution we present a systematic study of a theoretical frame work, termed “angular signal radiography” (ASR), for describing the imaging process in the classical way. Accordingly, the physical properties of sample and the intrinsic response of optical system in XGI are analytically described using ASR. Meanwhile, signals associated with different contrast mechanisms can be retrieved using ASR. Further, by ASR, the absorption, refraction, and scattering information of sample can be recovered from only three images required at specific configuration of the grating system. If one or two of these three types of sample-beam interactions can be neglected, the proposed ASR method can be simplified allowing full-quantification with even few input images. Simple analytical phase retrieval algorithm, low radiation dose and fast image acquisition speed, are the main contribution of ASR method. The limitation of the proposed ASR method will also be discussed in this work.

2. Principle and method

In general, the goal of the grating-based differential phase contrast imaging (DPCI) technique is to detect the angle-modulated x-ray induced by specimens’ properties, then to extract specimens’ inner structure by analyzing the projection images. Through the ASR framework proposed in this study, we apply the classical imaging description to the grating-based DPCI system as expressed by Eq. (1).

I (ψ; x, y) = f (ψ; x, y) \otimes S (ψ),

Where

\otimes

denotes the convolution;

I

is the detected imaging function;

f

is the angle-modulated function (or object function) that is related to object’s properties, such as absorption, refraction and scattering;

S

is the angular signal response function (ASRF, similar to impulse response function), representing the relationship of detected intensity in grating-based DPCI optical system via the x-ray deviation angle

ψ

.

2.1 Angle-modulated function

It is well known that, sample interacts with the incoming x-rays through several processes including absorption, refraction and scattering. These three types of interactions come down to different mechanisms of angle deviation to incoming x-rays, and are determined by specimens’ properties. Figuratively illustrations of these three types of sample-beam interactions are presented in panels (a), (b) and (c) in Fig. 1 , respectively. The absorption produces zero angular signal, and is relative to an energy consuming process. The refraction generates angular signal and the scattering effect is related to the angular distribution signal. The corresponding analytical expressions of these three types of angular signals can be written as Eqs. (2)-(4) , respectively.

Fig. 1 The pictures of (a) absorption, (b) refraction, and (c) scattering.

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In general, the scattering angle distribution is assumed to follow the Gaussian distribution [21] and the standard deviation $σ$ of the Gaussian function serves as a measurement of the scattering capability.

I = I_{0} \tilde{δ} (ψ) \cdot \exp (- M (x, y)),

and

I = I_{0} \tilde{δ} (ψ - θ (x, y)),

and

I = I_{0} \frac{1}{\sqrt{2 π} σ (x, y)} \exp [- \frac{1}{2} \frac{{(ψ)}^{2}}{σ^{2} (x, y)}],

where

I_{0}

is the intensity of the x-ray beam incident on the sample plane,

\tilde{δ}

is Dirac function and

M

is the absorption of the sample as described by Eq. (5).

M (x, y) = \int_{- \infty}^{\infty} μ (x, y, z) d z,

where

μ

is the linear absorption coefficient,

θ

is the refraction angle induced by the sample, which can be calculated using Eq. (6).

θ (x, y) = - \int_{- \infty}^{\infty} \frac{\partial δ (x, y, z)}{\partial x} d z,

where

δ

is the decrement of the real part of the refractive index,

σ^{2}

is the scattering variance of the sample. In addition to the scattering variance [22,23 ], the scattering angle distribution can also be represented by dark-field image [6]. The quantitative relationship between scattering variance and dark-field can be established using Eq. (7) [24].

σ^{2} (x, y) = \sum_{i} σ_{i}^{2} (x, y) = \sum_{i} ω_{i} (x, y, z) Δ z_{i} = \int_{- \infty}^{\infty} ω (x, y, z) d z,

where

Δ z_{i}

is the thickness of the thin layer of the sample in the integral path,

σ_{i}^{2}

is the scattering variance produced by a certain thin layer,

ω_{i}

is the scattering coefficient of the thin layer,

ω

is the linear scattering coefficient of the sample.

Taking Eqs. (2)-(4) into account, the angle-modulated function $f$ is composed of absorption, refraction and scattering effect, and can be described as [25]:

f (ψ; x, y) = I_{0} \frac{\exp (- M (x, y))}{\sqrt{2 π} σ (x, y)} \exp [- \frac{1}{2} \frac{{(ψ - θ (x, y))}^{2}}{σ^{2} (x, y)}] .

2.2 Angular signal response function (ASRF)

In addition to the angular-modulated function, the ASRF is the other dominate component in the theoretical framework of grating-based ASR. The ASRF, $S (ψ)$ , characterizes the optical system’s intensity response, and is a function of the incoming x-ray deviation angle $ψ$ . It also determines the angular sensitivity of grating-based DPCI setup.

The typical x-ray grating-based Talbot interferometer setup is shown in Fig. 2(a) . It is composed of an x-ray source, a rotational sample stage, a phase grating G1, an analyzer grating G2 and a detector. The phase grating acts as an angular collimator (or splitter), while analyzer grating serves as an angular filter. In grating interferometer setup, G2 is usually placed at the self-image distance $D$ downstream of G1. The detected intensity changes as the G2 is shifted, by a small amount of displacement $x_{g}$ , with respect to G1 in the direction that is perpendicular to the incoming beam. The variation in intensity as a function of the displacement is usually called the Shifting Curve (SC), which is shown in Fig. 2(b).

Fig. 2 (a) Schematic imaging principle of the grating interferometer, (b) measured Shifting curve.

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Equivalent to incoming x-ray angular deviation, angular deviation $x_{g} / D$ is produced in XGI setup by shifting the relative displacement between G1 and G2. In this case, ASRF in XGI can be substituted by SC, and with the expression as

S (ψ) = S (x_{g} / D) .

As an unbound and periodical curve, SC can be fitted with cosine function [5,15 ]. In this work, we analyze the imaging data by investigating four typical sections on the SC, respectively named as valley-SC, upslope-SC, peak-SC and downslope-SC. By choosing different offsets in the phase stepping, we can center different points (the ‘V’, ‘U’, ‘P’ and ‘D’ marked on Fig. 2(b)) in the SC. And the ASRF in XGI can be depicted using Eq. (10).

S (ψ) = \bar{S} [1 - V_{0} \cos (\frac{2 π D}{p} (ψ + ψ_{0}) + η \frac{π}{2})],

where

p

is the period of the analyzer grating G2;

ψ_{0}

is the initial deviation angle, which is induced by the local imperfection of two gratings or incoming x-ray wavefront;

\bar{S} = (S_{\max} + S_{\min}) / 2

, the mean value of the SC;

V_{0} = (S_{\max} - S_{\min}) / (S_{\max} + S_{\min})

, the visibility of the SC, while

S_{\max}

and

S_{\min}

are the maximum and minimum value of SC, respectively.

η

is modulation parameter. The four typical SCs can be respectively obtained by setting

η = 0; 1; 2; 3

.

2.3 Angular signal imaging eqs. in XGI

When an object is placed before or after the phase grating, the absorption, refraction and scattering caused by the sample deform the periodic intensity pattern and, therefore, affects the intensity measured by the area detector after the filtering process of the analyzer grating. Based on ASR in XGI, substituting Eqs. (8) and (10) into Eq. (1), the four imaging Eqs., valley-, upslope-, peak- and downslope-imaging Eqs. $I_{V}$ , $I_{U}$ , $I_{P}$ and $I_{D}$ can be respectively expressed as Eqs. (11)-(14) [26,27 ] (see the Appendix 1 While the phase stepping (PS) method).

I_{V} (x, y) = I_{0} \bar{S} e^{- M (x, y)} [1 - V_{0} e^{- 2 π^{2} D^{2} σ^{2} (x, y) / p^{2}} \cos (\frac{2 π D}{p} (θ (x, y) + ψ_{0}))],

and

I_{U} (x, y) = I_{0} \bar{S} e^{- M (x, y)} [1 + V_{0} e^{- 2 π^{2} D^{2} σ^{2} (x, y) / p^{2}} \sin (\frac{2 π D}{p} (θ (x, y) + ψ_{0}))],

and

I_{P} (x, y) = I_{0} \bar{S} e^{- M (x, y)} [1 + V_{0} e^{- 2 π^{2} D^{2} σ^{2} (x, y) / p^{2}} \cos (\frac{2 π D}{p} (θ (x, y) + ψ_{0}))],

and

I_{D} (x, y) = I_{0} \bar{S} e^{- M (x, y)} [1 - V_{0} e^{- 2 π^{2} D^{2} σ^{2} (x, y) / p^{2}} \sin (\frac{2 π D}{p} (θ (x, y) + ψ_{0}))] .

To eliminate constant angular deviation $ψ_{0}$ , which could be induced by imperfections in the grating system or in the incoming x-ray wavefront, the effective absorption, refraction and scattering image of the specimen will be calculated by taking the projection images with and without sample in the x-ray beam path at the same positions along SC [28]. Then combing Eqs. (11)-(14) , the absorption, refraction and scattering information of sample can be extracted respectively as:

M (x, y) = \ln (\frac{I_{U}^{b} (x, y) + I_{D}^{b} (x, y)}{I_{U}^{s} (x, y) + I_{D}^{s} (x, y)}),

and

θ (x, y) = - \frac{p}{2 π D} (arc \tan (\frac{I_{U}^{s} (x, y) - I_{D}^{s} (x, y)}{I_{V}^{s} (x, y) - I_{P}^{s} (x, y)}) - arc \tan (\frac{I_{U}^{b} (x, y) - I_{D}^{b} (x, y)}{I_{V}^{b} (x, y) - I_{P}^{b} (x, y)})),

and

\begin{array}{l} σ^{2} (x, y) = - \frac{p^{2}}{2 π^{2} D^{2}} \ln (\frac{\sqrt{{(I_{U}^{s} (x, y) - I_{D}^{s} (x, y))}^{2} + {(I_{V}^{s} (x, y) - I_{P}^{s} (x, y))}^{2}}}{V_{0} (I_{U}^{s} (x, y) + I_{D}^{s} (x, y))}) \\ + \frac{p^{2}}{2 π^{2} D^{2}} \ln (\frac{\sqrt{{(I_{U}^{b} (x, y) - I_{D}^{b} (x, y))}^{2} + {(I_{V}^{b} (x, y) - I_{P}^{b} (x, y))}^{2}}}{V_{0} (I_{U}^{b} (x, y) + I_{D}^{b} (x, y))}) . \end{array}

where

I_{V}^{b}

,

I_{U}^{b}

,

I_{P}^{b}

and

I_{D}^{b}

are projection raw images measured at the position valley, upslope, peak and downslope without sample in system, respectively, while

I_{V}^{s}

,

I_{U}^{s}

,

I_{P}^{s}

and

I_{D}^{s}

are measured with sample in system.

After further study on comparing the Eqs., the Eqs. (15)-(17) given by ASR are similar to the Eqs. in PS for extracting different sample-beam interaction using four images equally spaced over one period of the SC (see the Appendix 2 for the detailed derivation process). It worth to emphasize that there is some redundancy in the four images collected. This redundancy can be expressed in Eq. (18).

I_{P} + I_{V} = I_{U} + I_{D} .

According to Eqs. (15)-(18) , the absorption, refraction and scattering images can be extracted from the three raw images collected at three arbitrary positions chosen from

I_{V}

,

I_{U}

,

I_{P}

and

I_{D}

. In real applications, it is possible that only one or two sample-beam interactions is/are significant. For example, for weakly absorption low-Z samples, refraction and scattering provide the most detectable signals, while the absorption is negligible. To extract physical information of a specimen with the minimum number of projective images (with minimum radiation dose), the ASR can be further simplified.

2.3.1 Negligible scattering for ASR (NS-ASR)

When scattering information can be neglected, a simpler ASR method can be used and named as “Negligible scattering for ASR” (NS-ASR). In this case,

σ^{2} (x, y) = 0.

Combing the Eqs. (12) and (14) with the Eq. (19), by acquiring upslope-image and downslope-image, the absorption image and the refraction angle image can respectively be written as

M (x, y) = \ln (\frac{I_{U}^{b} (x, y) + I_{D}^{b} (x, y)}{I_{U}^{s} (x, y) + I_{D}^{s} (x, y)}),

and

\begin{array}{l} θ_{N S} (x, y) = \frac{p}{2 π D} arc \sin (\frac{1}{V_{0}} \frac{I_{D}^{s} (x, y) - I_{U}^{s} (x, y)}{I_{D}^{s} (x, y) + I_{U}^{s} (x, y)}) \\ - \frac{p}{2 π D} arc \sin (\frac{1}{V_{0}} \frac{I_{D}^{b} (x, y) - I_{U}^{b} (x, y)}{I_{D}^{b} (x, y) + I_{U}^{b} (x, y)}) . \end{array}

At this stage, it is worthy to note that: the absorption is a scalar and therefore rotational-invariant; the refraction angle strongly depends on the direction along which it is measured [14,29,30 ]. In this contribution, for perfect gratings $ψ_{0} = 0$ , considering the basic characteristics of absorption and refraction angle, the relationship between the projected image $I_{U}$ (or $I_{D}$ ) at the rotation angle $φ$ and the reverse image of $I_{D}$ (or $I_{U}$ ) at $φ + π$ can be written as

I_{U (o r D)} (x, y, φ; ψ_{0} = 0) = I_{D (o r U)} (- x, y, φ + π; ψ_{0} = 0) .

According to the Eq. (15-1) and the Eqs. (19)-(21) , we found that NS-ASR can be further simplified into the reverse projection (RP) algorithm for perfect gratings we already proposed and discussed in [14].

2.3.2 Negligible absorption for ASR (NA-ASR)

For samples with a negligible absorption, the simplified ASR is dubbed as “negligible absorption for ASR” (NA-ASR), and we have

M (x, y) = 0.

Combing Eqs. (12)-(13) with Eq. (22), the refraction angle image and scattering image can be obtained with upslope-image or downslope-image and peak-image or valley-image.

2.3.3 Negligible refraction for ASR (NR-ASR)

For samples with negligible refraction information, the simplified ASR can be identified as “negligible refraction for ASR” (NR-ASR). We then have

θ (x, y) = 0.

Combining the Eqs. (11) and (13) with the Eq. (23), the absorption image and scattering image can be extracted by acquiring peak-image and valley-image.

2.3.4 Negligible absorption and refraction for ASR (NANR-ASR)

For samples with both negligible absorption and refraction information, the simplified ASR becomes the “Negligible absorption and refraction for ASR” (NANR-ASR). Combing with the Eqs. (22)-(23) , the scattering variance image can be extracted by acquiring peak-image or valley-image.

2.3.5 Negligible absorption and scattering for ASR (NANS-ASR)

For samples with both negligible absorption and scattering information, the simplified ASR is the “negligible absorption and scattering for ASR” (NANS-ASR). Combing with the Eqs. (19) and (22) , the refraction angle image can be extracted by acquiring upslope-image or downslope-image.

2.3.6 Negligible refraction and scattering for ASR (NRNS-ASR)

For samples with negligible refraction and scattering information, the simplified ASR is the “negligible refraction and scattering for ASR” (NRNS-ASR). Combing with the Eqs. (19) and (23) , the absorption image can be extracted by acquiring arbitrary projection image choosing from the above four positions.

3. Experimental studies

The experiments were carried out using the grating-based Talbot interferometer (as schematically illustrated in Fig. 2(a)) available at the beam line BL13W1 at the Shanghai Synchrotron Radiation Facility (SSRF). The distance from the phase grating G1 to the absorption grating G2 is set to the first fractional Talbot distance at 4.64 cm. The $π / 2$ phase shift grating is made of Ni with a period at 2.396 μm. The analyzer grating is made of Au with period at 2.4 μm. The x-ray energy was set to 20keV. A CCD with a pixel size of 9 μm was used and placed at ~16 cm downstream of the analyzer grating. This imaging system was used to collect images of a hamster front toe. To compare this method with the eight-step-PS method, eight projective images were collected at equally spaced positions over one period of the SC. The exposure time was set to 1.5s for each image.

We first demonstrated the extraction of the different physical information using the ASR method. These four projection images, including the valley-image $I_{V}$ , the upslope-image $I_{U}$ , the peak-image $I_{P}$ and the downslope-image $I_{D}$ , are shown in Figs. 3(a)-3(d) , respectively. As described in the theoretical framework of ASR, the specimens’ valley-image, upslope-image, peak-image and downslope-image, are convolution of the angle-modulated function and valley-ASRF, upslope-ASRF, peak-ASRF and downslope-ASRF, respectively. As shown in Fig. 3, for the fluff, valley-image exhibits brightness while peak-image shows darkness; for the front toe, the right side of front toe has exhibited much brighter than left side in upslope-image, while it is totally opposite for downslope-image. After further analysis, we have deduced that, upslope-image and downslope-image contain rich refraction signal response but with little scattering signal response because of the antisymmetric response of upslope-ASRF and downslope-ASRF. On the other hand, the valley-image and peak-image contain rich scattering information but with little refraction signal because of the symmetric response of valley-ASRF and peak-ASRF. These findings can be verified by analyzing the four SCs. The first-order derivative of the upslope-SC and the downslope-SC near the points ‘U’ and ‘D’ are quite large; while the first-order derivative of the peak-SC and the valley-SC near the points ‘V’ and ‘P’ are relatively small. The second-order derivative of the valley-SC and the peak-SC near the points ‘V’ and ‘P’ quite large; while the second-order derivative of the upslope-SC and the downslope-SC near the points ‘U’ and ‘D’ are relatively small. Considering these findings, the upslope-image and/or the downslope-image are used to extract refraction; while the valley-image and/or the peak-image are used to extract scattering in the evaluation of experimental results below.

Fig. 3 Projection images for (a) valley-image, (b) upslope-image, (c) peak-image, and (d) downslope-image.

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The absorption, refraction and scattering maps are extracted and presented in Figs. 4(a)-4(c) . Comparing the three extracted maps, the bone structure is better depicted in the absorption images, since bone has the highest absorption coefficient comparing to the other structures; the soft tissue as well as fur exhibit a greatly improved contrast in the refraction and scattering maps.

Fig. 4 Comparison between ASR and eight-step-PS methods. In the panels (a)-(c) we show absorption, refraction angle, and scattering variance images, respectively, extracted with the ASR method. The corresponding images extracted with the eight-step-PS method are shown in panels (d)-(f).

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To further evaluate the quality of reconstruction through ASR, comparisons between ASR and eight-step-PS are shown in Fig. 4 while the plots of line profiles are shown in Fig. 5 . It is found that results of both methods match quite well, as confirmed by the value of the calculated Pearson product-moment correlation coefficient (PPMCC), defined as the covariance of the variables divided by their standard deviations: 0.9991, 0.9901 and 0.9892, respectively.

Fig. 5 Comparison of profiles between ASR and eight-step-PS methods for (a) absorption, (b) refraction angle, and (c) scattering variance. The profiles correspond to the horizontal white dotted line across the sample in Fig. 4(a).

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In addition, we verified the simplified ASR methods. Detailed investigations of the applications of the simplified ASR methods in the selected regions of interest (ROI) are presented in the following section.

Three regions of interest (ROIs, highlighted in Fig. 4(a)) are selected for detailed analysis. Magnified views of these ROIs are shown in Fig. 6 , Fig. 7 , and Fig. 8 , highlighting the consistency of the overall shape of the reconstructed maps. Line profiles shown in Fig. 9 , Fig. 10 , and Fig. 11 , demonstrate a good degree of agreement between the simplified ASR and eight-step PS. In additional to the PPMCC analysis of the difference between the simplified ASR and the eight-step-PS shown in Table 1 , the analysis of signal-to-noise ratio (SNR) has also been presented in Table 2 . Based on the proposed algorithms to calculate SNR by some published work [31,32 ], the SNR of ROI in the extracted absorption or scattering image with respect to the surrounding region can be defined as:

S N R_{A / S} = \frac{\sqrt{A} (I_{o b j} - I_{b a c k})}{\sqrt{s t d^{2} (I_{o b j}) + s t d^{2} (I_{b a c k})}},

where

I_{o b j}

and

s t d (I_{o b j})

are respectively the mean intensity value and standard deviation of the given ROI with sample in system; while

I_{b a c k}

and

s t d (I_{b a c k})

are respectively the mean intensity value and standard deviation of the given ROI without sample in system.

Fig. 6 Comparison of the extracted images of ROI 1 between eight-step-PS, NS-ASR and NANS-ASR. Panels (a)-(b) contain absorption and refraction angle obtained by using eight-step-PS. The corresponding extracted images obtained by NS-ASR are shown in panels (c)-(d). The extracted refraction angle by NANS-ASR is shown in panel (e).

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Fig. 7 Comparison of extracted images of ROI 2 between eight-step-PS, NA-ASR and NANR-ASR. Panels (a)-(b) show the refraction angle and scattering variance by using the eight-step-PS method. The corresponding extracted images of the refraction angle and scattering variance obtained by NA-ASR are shown in panels (c)-(d). The extracted scattering variance by NANR-ASR is shown in panel (e).

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Fig. 8 Comparison of extracted images of ROI 3 among eight-step-PS, NR-ASR and NRNS-ASR. Panels (a)-(b) are absorption and scattering variance by using the eight-step-PS method. The corresponding extracted images of absorption and scattering variance obtained by NR-ASR are shown in panels (c)-(d). The extracted absorption by NRNS-ASR is shown in panel (e).

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Fig. 9 Comparison of the profiles, corresponding to the horizontal white solid line of ROI 1 in Fig. 6(a), between eight-step-PS, NS-ASR and NANS-ASR for (a) absorption, and (b) refraction angle.

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Fig. 10 Comparison of the profiles, corresponding to the horizontal white solid line of ROI 2 in Fig. 7(a), between eight-step-PS, NA-ASR and NANR-ASR for: (a) refraction angle, and (b) scattering variance.

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Fig. 11 Comparison of the profiles, corresponding to the horizontal white solid line of ROI 3 in Fig. 8(a), among eight-step-PS, NR-ASR and NRNS-ASR for: (a) absorption, and (b) the scattering variance.

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Table 1. Calculated PPMCC between the new methods and eight-step-PS indicated by “—” mean ignorable information, A, R and S are the absorption, refraction angle and scattering variance, respectively.

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Table 2. Calculated ratio of SNR and ratio of projection images’ number needed to extract information, between the new methods and eight-step-PS indicated by “—” mean ignorable information, A, R and S are the absorption, refraction angle and scattering variance, respectively.

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While for the extracted refraction image, the calculation of SNR is different to Eq. (24) and can be expressed as [32,33 ]:

S N R_{R} = \frac{\int_{y_{0}}^{y_{1}} d y \int_{x_{0}}^{x_{1}} d x | I (x, y) - I_{b a c k} |}{\sqrt{2 \int_{y_{0}}^{y_{1}} d y \int_{x_{0}}^{x_{1}} d x I_{b a c k}}},

where

| x_{1} - x_{0} |

,

| y_{1} - y_{0} |

are respectively the width of image on the direction of parallel and perpendicular to the refraction sensitivity direction.

3.1 NS-ASR experimental studies

To verify NS-ASR, we choose the ROI at the front toe, marked as ‘1’ in Fig. 4(a), where the scattering is relatively weak. $I_{U}$ and $I_{D}$ were used to extract absorption and refraction information by NS-ASR. The extracted information and the corresponding comparisons between NS-ASR and eight-step-PS are shown in Figs. 6 and 9 , respectively. The calculated PPMCC and SNR between the NS-ASR and eight-step-PS methods are shown in Table 1 and Table 2, respectively.

3.2 NA-ASR experimental studies

To verify NA-ASR, we choose the fur, marked as ‘2’ in Fig. 4(a), which has a fairly small absorption coefficient respect to the whole sample, thus the absorption information can be neglected. $I_{V}$ and $I_{U}$ were collected to extract refraction angle and scattering variance in this case by NA-ASR. In mathematics, the response to scattering signal of peak-image $I_{P}$ and that of valley-image $I_{V}$ are anti-symmetrical, and have equivalent function to extract scattering information for NA-ASR method. However, the absorption of sample has lower effect on valley-image than peak-image. So it might be better to choose valley-image to verify NA-ASR. The extracted information and the corresponding comparisons between NA-ASR and eight-step-PS are shown in Figs. 7 and 10 , respectively. The calculated PPMCC and SNR between NA-ASR and eight-step-PS methods are also respectively shown in Table 1 and Table 2.

3.3 NR-ASR experimental studies

For the bone structure, marked as ‘3′ in Fig. 4(a), the refraction contribution is significantly weaker than absorption and scattering effect. So the bone structure was selected to verify NR-ASR. In this experiment, $I_{P}$ and $I_{V}$ were collected to extract absorption and scattering variance. The extracted information and the corresponding comparisons between NR-ASR and eight-step-PS are shown in Figs. 8 and 11 , respectively. The calculated PPMCC and SNR between NR-ASR and eight-step-PS methods are respectively shown in Table 1 and Table 2.

3.4 NANR-ASR experimental studies

We define the “no refraction” as the phenomenon that the refraction effect is not large enough to cause deviation the beam by one or more pixel at the detector plane. Generally, the deviation of the photons within one detector pixel is included in the scattering, which is the result of local refraction caused by sub-pixel structural heterogeneity.

In this experiment, the fur region, marked as ‘2’ in Fig. 4(a) has been chosen to verify NANR-ASR. As fur region is mainly consisted of low-Z elements, thus, the small absorption attenuation can be ignored. In addition, multiple-refraction effects can happen when incoming x-ray went through between abundant furs. So for the ROI 2 fur region, scattering information is dominating.

In mathematics, either rich-scattering information projection image of $I_{P}$ or $I_{V}$ can be used to extract scattering information of sample. However, zero angular signal were all transmitted in $I_{P}$ , but were all stopped in $I_{V}$ . So the SNR of extracted scattering image by $I_{V}$ should be higher by $I_{P}$ . As a result, $I_{V}$ was used to extract scattering information for NANR-ASR in this experiment. The extracted information and the corresponding comparisons between NANR-ASR and eight-step-PS are shown in Figs. 7 and 10 , respectively. The calculated PPMCC and SNR between the NANR-ASR and eight-step-PS methods are respectively shown in Table 1 and Table 2.

3.5 NANS-ASR experimental studies

As proved in NS-ASR experimental studies, the scattering effect of the front toe region, marked as ‘1’ in Fig. 4(a) is quite weak. Besides, as shown in Figs. 4(a) and 4(d), the absorption effect of ROI 1 is quite small. Thus, not only the scattering effect of the front toe is negligible, but also the absorption effect could be ignored for further simplification. Thus the refraction information of the front toe is vital. $I_{U}$ was acquired to extract refraction information of ROI 1. The latter and the corresponding comparisons between NANS-ASR and eight-step-PS are shown in Figs. 6 and 9 , respectively. The calculated PPMCC and SNR between NANS-ASR and eight-step-PS methods are respectively shown in Table 1 and Table 2.

3.6 NRNS-ASR experimental studies

As seen in the NR-ASR experiments, the refraction information is negligible for the bone structure, marked as ‘3′ in Fig. 4(a). Besides, as shown in Figs. 4(c) and 4(f), the scattering effect of ROI 3 is also quite small. Moreover, because the energy conservation principle is fulfilled by scattering effect and absorption results in the energy loss [34], neglecting scattering information of the strong-absorption-bone structure may have only a minor effect on the extraction of the absorption information. In this experiment, the ROI 3 was used to verify the NRNS-ASR. In mathematics, one arbitrary projection image from $I_{U}$ , $I_{D}$ , $I_{P}$ and $I_{V}$ , can be used to extract absorption information of sample by NRNS-ASR. However, zero angular signal were all transmitted in $I_{P}$ . So the SNR of extracted absorption image by $I_{P}$ should be higher than the other projection images. As a result, the $I_{P}$ was used to extract absorption information of ROI 3. The latter and the corresponding comparisons between NRNS-ASR and eight-step-PS are shown in Figs. 8 and 11 , respectively. The calculated PPMCC and SNR between NRNS-ASR and eight-step-PS methods are respectively shown in Table 1 and Table 2.

4. Discussion

As discussed above, we have developed the concept of ASRF in XGI in this work to investigate the imaging system with angular sensitivity. The four typical ASRFs in a grating interferometer system were discussed, namely the valley-ASRF, the upslope-ASRF, the peak-ASRF, and the downslope-ASRF. We then integrate the ASRF and the angle-modulated function to establish the theoretical frame work of ASR. Although the angle-modulated function has been proposed and applied in the diffraction-enhanced imaging (DEI) setup [25,35 ], the theoretical framework that combines the angle-modulated function and ASRF, namely the ASR, is established for the first time.

As a prerequisite of ASR, the effective angular signal induced by sample should be no bigger than the angle subtended by the resulting cell, such as ultra-small- and small-angle. Otherwise, the independency of two adjacent resolution cells is no longer valid. Fortunately, the ultra-small- or small-angle dominates the signal induced by sample, so the theory of ASR is applicable to most of the real-life applications. In fact, the imaging sensitivity of DPCI depends on the detected minimum angle, and detecting small-angle is an attractive but difficult research [36].

By comparing the calculated PPMCC in Table 1, another experimental result should be discussed. As shown in Table 1, for low-Z samples such as ROI 1 and ROI 2, characterized by a negligible absorption, the extracted refraction or scattering information by simplified ASR do not match perfectly with the information extracted by PS. However, compared to the result of the PS method, neglecting refraction or scattering information, the extracted remaining information has a much smaller margin of error for low-Z samples. This is possibly due to the fact that absorption results in an energy loss process while refraction or scattering effect fulfills the law of conservationofenergy. Also this mechanism has to be verified and discussed with much care in a future work. What’s more, by analyzing and comparing the calculated the ratio of SNR and the ratio of projection number on extracted image between ASR and eight-step-PS in Table 2, it has shown that SNR of ASR may be about same or may be not higher than that of PS, but ASR has same or better properties on SNR under the same dose in this experiment.

This paper only described four typical ASRFs, but the ASRF obtained in an XGI set-up in practice is generally not a pure typical ASRF due to the imperfection in the grating. Because $ψ_{0}$ in Eq. (10) can be an arbitrary value, ASR does not necessarily require the full-field uniformity, but the local-field uniformity. In other words, ASR can work with arbitrary three of four independent projection images collected with the grating displacement quadrant on one period of the SC regardless the starting position. These four independent projection images collected in imperfect gratings system are equivalent to those collected in perfect gratings system, the analytical formula proposed in this manuscript to extract different properties are still applicable.

5. Conclusion

In this manuscript, the theoretical framework of ASR was established and systematically evaluated for grating-based DPCI technique. Through ASR, the imaging process in XGI can build an analytical relationship between the angular sensitivity of XGI setup and specimens’ different properties, including absorption, refraction and scattering properties. Further, we have shown that the ASRF, representing angular sensitivity of XGI setup, is composed of four typical ASRFs, respectively namely valley-ASRF, upslope-ASRF, peak-ASRF and downslope-ASRF. Accordingly, the four raw specimens’ specific projection images, valley-image, upslope-image, peak-image and downslope-image have been acquired. This theoretical framework is then successfully applied to synchrotron based experiments, achieving comparable result with reduced radiation dose to the sample.

It worth mentioning that the XGI formation process achieved by grating interferometer is very similar to other DPCI methods [37]. As a result, the theory of ASR may provide a unified, classical and analytical description for general DPCI technique that has angular collimator and filter.

Appendix 1

To simplify the written expression, the following formulas have been obtained pixel by pixel. Defining $\frac{x_{g}}{D} = ψ$ while $ℱ$ and $ℱ^{- 1}$ are the Fourier transformation and its inversion, respectively, we may write

\begin{array}{l} ∵ ℱ (f (ψ)) = I_{0} e^{- M} e^{- i 2 π υ θ} e^{- 2 π^{2} υ^{2} σ^{2}}, \\ a n d \\ ℱ (S (ψ)) = \frac{1}{2} [δ (v - \frac{D}{p}) + δ (v + \frac{D}{p})] e^{j v (ψ_{0} + \frac{η p}{4 π D})} . \\ ∴ f (ψ) \otimes S (ψ) = {ℱ^{- 1} (ℱ (f (ψ)) • ℱ (S (ψ))) |}_{ψ = 0} \\ = I_{0} \bar{S} e^{- M} [1 - V_{0} e^{- 2 π^{2} D^{2} σ^{2} / p^{2}} \cos (\frac{2 π D}{p} (θ + ψ_{0}) + η \frac{π}{2})] . \\ ∴ I_{V} = {f (ψ) \otimes S (ψ) |}_{η = 0} = I_{0} \bar{S} e^{- M} [1 - V_{0} e^{- 2 π^{2} D^{2} σ^{2} / p^{2}} \cos (\frac{2 π D}{p} (θ + ψ_{0}))], \\ a n d \\ I_{U} = {f (ψ) \otimes S (ψ) |}_{η = 1} = I_{0} \bar{S} e^{- M} [1 + V_{0} e^{- 2 π^{2} D^{2} σ^{2} / p^{2}} \sin (\frac{2 π D}{p} (θ + ψ_{0}))], \\ a n d \\ I_{P} = {f (ψ) \otimes S (ψ) |}_{η = 2} = I_{0} \bar{S} e^{- M} [1 + V_{0} e^{- 2 π^{2} D^{2} σ^{2} / p^{2}} \cos (\frac{2 π D}{p} (θ + ψ_{0}))], \\ a n d \\ I_{D} = {f (ψ) \otimes S (ψ) |}_{η = 3} = I_{0} \bar{S} e^{- M} [1 - V_{0} e^{- 2 π^{2} D^{2} σ^{2} / p^{2}} \sin (\frac{2 π D}{p} (θ + ψ_{0}))] . \end{array}

Appendix 2

By PS, the Eqs. to extract information by collecting four images equally spaced positions over one period of the SC can be expressed as:

{\begin{cases} M (x, y) = \ln (\frac{\sum_{k = 1}^{4} I_{k}^{b} (x, y)}{\sum_{k = 1}^{4} I_{k}^{s} (x, y)}) = \ln (\frac{I_{1}^{b} (x, y) + I_{2}^{b} (x, y) + I_{3}^{b} (x, y) + I_{4}^{b} (x, y)}{I_{1}^{s} (x, y) + I_{2}^{s} (x, y) + I_{3}^{s} (x, y) + I_{4}^{s} (x, y)}), \\ θ (x, y) = - \frac{p}{2 π D} \arg (\frac{\sum_{k = 1}^{4} I_{k}^{s} (x, y) \exp (\frac{i 2 π k}{4})}{\sum_{k = 1}^{4} I_{k}^{b} (x, y) \exp (\frac{i 2 π k}{4})}) = - \frac{p}{2 π D} \arg (\frac{i I_{1}^{s} (x, y) - I_{2}^{s} (x, y) - i I_{3}^{s} (x, y) + I_{4}^{s} (x, y)}{i I_{1}^{b} (x, y) - I_{2}^{b} (x, y) - i I_{3}^{b} (x, y) + I_{4}^{b} (x, y)}) \\ = - \frac{p}{2 π D} (arc \tan (\frac{I_{1}^{s} (x, y) - I_{3}^{s} (x, y)}{I_{4}^{s} (x, y) - I_{2}^{s} (x, y)}) - arc \tan (\frac{I_{1}^{b} (x, y) - I_{3}^{b} (x, y)}{I_{4}^{b} (x, y) - I_{2}^{b} (x, y)})), \\ σ^{2} (x, y) = \frac{p^{2}}{2 π^{2} D^{2}} \ln (\frac{| \sum_{k = 1}^{4} I_{k}^{b} (x, y) \exp (\frac{i 2 π k}{4}) |}{\sum_{k = 1}^{4} I_{k}^{b} (x, y)} • \frac{\sum_{k = 1}^{4} I_{k}^{s} (x, y)}{| \sum_{k = 1}^{4} I_{k}^{s} (x, y) \exp (\frac{i 2 π k}{4}) |}) \\ = \frac{p^{2}}{2 π^{2} D^{2}} \ln (\frac{| i I_{1}^{b} (x, y) - I_{2}^{b} (x, y) - i I_{3}^{b} (x, y) + I_{4}^{b} (x, y) |}{I_{1}^{b} (x, y) + I_{2}^{b} (x, y) + I_{3}^{b} (x, y) + I_{4}^{b} (x, y)} • \frac{I_{1}^{s} (x, y) + I_{2}^{s} (x, y) + I_{3}^{s} (x, y) + I_{4}^{s} (x, y)}{| i I_{1}^{s} (x, y) - I_{2}^{s} (x, y) - i I_{3}^{s} (x, y) + I_{4}^{s} (x, y) |}) \\ = \frac{p^{2}}{2 π^{2} D^{2}} \ln (\frac{\sqrt{{(I_{1}^{b} (x, y) - I_{3}^{b} (x, y))}^{2} + {(I_{4}^{b} (x, y) - I_{2}^{b} (x, y))}^{2}}}{I_{1}^{b} (x, y) + I_{2}^{b} (x, y) + I_{3}^{b} (x, y) + I_{4}^{b} (x, y)} • \frac{I_{1}^{s} (x, y) + I_{2}^{s} (x, y) + I_{3}^{s} (x, y) + I_{4}^{s} (x, y)}{\sqrt{{(I_{1}^{s} (x, y) - I_{3}^{s} (x, y))}^{2} + {(I_{4}^{s} (x, y) - I_{2}^{s} (x, y))}^{2}}}) . \end{cases}

When

I_{1} = I_{V}, I_{2} = I_{U}, I_{3} = I_{P}, I_{4} = I_{D}

, and

I_{V} + I_{P} = I_{U} + I_{D}

Then the Eqs. to extract different information by PS can be rewritten as:

{\begin{cases} M (x, y) = \ln (\frac{I_{U}^{b} (x, y) + I_{D}^{b} (x, y)}{I_{U}^{s} (x, y) + I_{D}^{s} (x, y)}), \\ θ (x, y) = - \frac{p}{2 π D} (arc \tan (\frac{I_{U}^{s} (x, y) - I_{D}^{s} (x, y)}{I_{V}^{s} (x, y) - I_{P}^{s} (x, y)}) - arc \tan (\frac{I_{U}^{b} (x, y) - I_{D}^{b} (x, y)}{I_{V}^{b} (x, y) - I_{P}^{b} (x, y)})), \\ σ^{2} (x, y) = \frac{p^{2}}{2 π^{2} D^{2}} \ln (\frac{\sqrt{{(I_{U}^{b} (x, y) - I_{D}^{b} (x, y))}^{2} + {(I_{V}^{b} (x, y) - I_{P}^{b} (x, y))}^{2}}}{I_{U}^{b} (x, y) + I_{D}^{b} (x, y)} • \frac{I_{U}^{s} (x, y) + I_{D}^{s} (x, y)}{\sqrt{{(I_{U}^{s} (x, y) - I_{D}^{s} (x, y))}^{2} + {(I_{V}^{s} (x, y) - I_{P}^{s} (x, y))}^{2}}}) . \end{cases}

Acknowledgments

The authors are grateful to Yijin Liu (SLAC, USA) and Claudio Marcelli (LNF-INFN, Italy) for many fruitful discussions and manuscript revisions. This work was partly supported by the National Basic Research Program of China (Grant No. 2012CB825800), the National Natural Science Foundation of China (Grant No. 11205189, 11375225, U1332109 and 11535015), the project supported by Institute of High Energy Physics, Chinese Academy of Sciences (Grant No. Y4545320Y2).

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Angular signal radiography

Abstract

1. Introduction

2. Principle and method

2.1 Angle-modulated function

2.2 Angular signal response function (ASRF)

2.3 Angular signal imaging eqs. in XGI

2.3.1 Negligible scattering for ASR (NS-ASR)

2.3.2 Negligible absorption for ASR (NA-ASR)

2.3.3 Negligible refraction for ASR (NR-ASR)

2.3.4 Negligible absorption and refraction for ASR (NANR-ASR)

2.3.5 Negligible absorption and scattering for ASR (NANS-ASR)

2.3.6 Negligible refraction and scattering for ASR (NRNS-ASR)

3. Experimental studies

3.1 NS-ASR experimental studies

3.2 NA-ASR experimental studies

3.3 NR-ASR experimental studies

3.4 NANR-ASR experimental studies

3.5 NANS-ASR experimental studies

3.6 NRNS-ASR experimental studies

4. Discussion

5. Conclusion

Appendix 1

Appendix 2

Acknowledgments

References and links

Cited By

Figures (11)

Tables (2)

Equations (29)

Optics Express

Method	ROI	PPMCC
Method	ROI	A	R	S
ASR	whole sample	0.9991	0.9901	0.9892
NS-ASR	ROI 1	0.9531	0.9652	—
NA-ASR	ROI 2	—	0.8232	0.9636
NR-ASR	ROI 3	0.9977	—	0.9224
NANR-ASR	ROI 2	—	—	0.9611
NANS-ASR	ROI 1	—	0.8513	—
NRNS-ASR	ROI 3	0.9743	—	—

Method	ROI	Ratio of SNR/Ratio of projection number
Method	ROI	A	R	S
ASR	ROI 1	0.695/0.25	0.630/0.375	0.988/0.375
	ROI 2	0.741/0.25	0.737/0.375	0.889/0.375
	ROI 3	0.968/0.25	0.606/0.375	1.001/0.375
NS-ASR	ROI 1	0.695/0.25	0.562/0.25	—
NA-ASR	ROI 2	—	0.882/0.25	0.508/0.25
NR-ASR	ROI 3	0.968/0.25	—	0.808/0.25
NANR-ASR	ROI 2	—	—	0.521/0.125
NANS-ASR	ROI 1	—	0.382/0.125	—
NRNS-ASR	ROI 3	0.859/0.125	—	—