Is high sensitivity always desirable for a grating-based differential phase contrast imaging system?

Abstract

Purpose

In grating-based x-ray differential phase contrast (DPC) imaging, the measured signal amplitude of the phase shift induced by an image object is proportional to the so-called system sensitivity. Therefore, to achieve a better signal-to-noise (SNR) for improved imaging performance, it is generally believed that one should increase the system sensitivity by reducing the period of the analyzer grating or increasing the distance between the phase grating and analyzer grating. The purpose of this work is to theoretically and experimentally demonstrate that there is an optimal system sensitivity to attain the highest SNR for a given task provided that the standard phase-stepping acquisition and phase retrieval methods are used. When system sensitivity goes beyond this optimal value, SNR decreases and the imaging performance deteriorates.

Methods

Due to the fundamental fact that the measured phase signal is a cyclic variable, the phase wrapping effect is inevitable in DPC imaging when the system sensitivity increases. The phase wrapping effect appears in both signal and noise measurements. The effect in the signal measurement is manifested in the so-called signal statistical bias and such effect often impacts the accuracy of the measurement. The phase wrapping effect also appears in the noise variance measurement and impacts the precision of the measurement. A thorough theoretical analysis was performed in this work to demonstrate the quantitative impacts of phase wrapping on both signal bias and noise variance and thus on the actual system SNR. The joint effect of phase wrapping in both the signal bias and noise variance yields an optimal system sensitivity to achieve the highest SNR. Both extensive numerical simulation studies and experimental studies were performed to validate the theoretical analysis.

Results

Both theoretical analysis and experimental studies show that the SNR of the DPC signal is not always proportional to the sensitivity due to the cyclic nature of the signal and the phase wrapping effect. For a given refraction angle and exposure level, there exists an optimal sensitivity factor that maximizes the SNR, beyond which, increasing the sensitivity will decrease the SNR.

Conclusions

Increase of system sensitivity does not always improve x-ray DPC imaging performance provided that the standard phase-stepping acquisition and phase retrieval methods are used.

I. INTRODUCTION

For the past two decades, grating-based x-ray phase contrast imaging has shown promising applications in medical imaging and material science^1–12. In grating-based phase contrast imaging, the signal, which is the local moiré fringe shift ϕ for a given refraction angle θ, is usually given by

$$\varphi ={\varphi }^o-{\varphi }^b=S\theta .$$ (1.1)

Here, ϕ^o, ϕ^b are the extracted phase signals from the object scan (object present in x-ray beam) and air scan (object absent from x-ray beam), respectively. S is defined as the sensitivity of the grating interferometer¹³. For a Talbot-Lau interferometer, $S=\frac{2\pi d}{p}$, where d is the distance between the phase grating and the analyzer grating and p is the period of the analyzer grating. When using the phase stepping technique^1,2 for phase information extraction, the noise of the extracted signal can be calculated as^14,15:

$$\sigma =\sqrt{\frac{2}{M{N}_0{ϵ}^2}},$$ (1.2)

which is related to the total number of phase steps M, average number of photons in each phase step N₀ and the fringe visibility ϵ. According to Eqs. (1.1–1.2), increasing S will always improve the signal-to-noise ratio (SNR) of the resulting differential phase contrast images.

In the past several years, one of the major efforts in improving the performance of the Talbot-Lau interferometer based phase contrast imaging system has been to increase the sensitivity of the system by using higher Talbot orders (increasing d)¹⁶ or by using gratings with submicron period (decreasing p)¹¹. However, this conclusion is based on the assumptions that the fringe shift ϕ is always linearly proportional to the sensitivity S and the measured image noise is always independent of the sensitivity. This linear assumption can be violated in practice. As a result, the SNR of the phase contrast image may have a more complex relationship with the sensitivity and lead to a different conclusion when one wants to optimize the system imaging performance. The purpose of this work was to quantitatively analyze how the SNR of the phase contrast image depends on the sensitivity when the linear assumption shown in Eq. (1.1) is violated. Both numerical simulations and experimental studies were performed to verify the theoretical results.

II. MATERIALS AND METHODS

A. Theoretical analysis of SNR of the phase shift

In phase stepping acquisitions, the number of photons per pixel detected in each phase step is denoted as N^(k), k = 1, 2, 3, …, M, where M is the number of phase steps. In the absence of noise, N^(k) is related to the fringe visibility ϵ and the average number of photons in each phase step N₀ as follows

$$N^{(k)}=N_0\left[1+ϵ\cos \left(\frac{2\pi k}{M}+\varphi \right)\right],$$ (2.1)

where ϕ is the phase shift of the moiré fringe induced by the object. The widely-used phase retrieval equation was applied to estimate the phase signal

$$\varphi =\mathrm{Arg}\left(\sum _{k=1}^M{N}^{(k)}\cos \frac{2\pi k}{M}-i\sum _{k=1}^M{N}^{(k)}\sin \frac{2\pi k}{M}\right),$$ (2.2)

where Arg(z) takes the principle value of the argument of a complex number z. As a result, the value of the phase signal is restricted to the first Riemann branch, i.e., ϕ ∈ [−π,π).

In an experiment, a phase retrieval process involves an air scan and an object scan. Both signals are extracted based on Eq. (2.2). The difference of the phase signals between the two scans is the so-called phase shift, denoted as ϕ. Therefore, as a routine procedure, ϕ^b is subtracted from ϕ^o to get the phase shift ϕ introduced by the object:

$$\varphi ={\varphi }^o-{\varphi }^b+2m\pi ,m\in \mathbb{Z}.$$ (2.3)

The 2mπ factor is added to wrap the phase shift into the range of ϕ ∈ [−π,π). In the absence of noise, ϕ is determined by the sensitivity S and the refraction angle θ as follows:

$$\varphi =S\theta .$$ (2.4)

The above discussion is based on the assumption that the measurement is free of noise. In reality, although the acquisition of the background phase signal ϕ^b is often performed with a higher dose towards meeting such an assumption, limited dose is usually applied to the object scan, especially when the object is an animal or human¹⁷. In this case, for the measurement of N^(k), the signal is always contaminated by noise, which is denoted as

$${\widehat{N}}^{(k)}=N^{(k)}+{\text{noise}}^{(k)}.$$ (2.5)

The phase extraction stays the same except that N^(k) in Eq. (2.2) is substituted by ${\widehat{N}}^{(k)}$, namely,

$${\widehat{\varphi }}^o=\mathrm{Arg}\left(\sum _{k=1}^M{\widehat{N}}^{(k)}\cos \frac{2\pi k}{M}-i\sum _{k=1}^M{\widehat{N}}^{(k)}\sin \frac{2\pi k}{M}\right),$$ (2.6)

where ${\widehat{\varphi }}^o$ denotes extracted phase signal for the object scan. As a result, the phase shift is no longer free of noise

$$\widehat{\varphi }={\widehat{\varphi }}^o-{\varphi }^b+2m\pi ,m\in \mathbb{Z}.$$ (2.7)

To rigorously determine the signal-to-noise ratio, one needs to perform repeated scans to obtain an ensemble of measurements for $\widehat{\varphi }$. In the following analysis, we assume that the experiment is repeated L times. Then the mean signal value is given by the following ensemble average

$$\langle \widehat{\varphi }\rangle =\underset{L\to \infty }{\lim }\frac{{\sum }_{i=1}^L{\widehat{\varphi }}_i}{L},$$ (2.8)

and the standard deviation of the signal (noise) is given as

$$\sigma (\widehat{\varphi })=\sqrt{\underset{L\to \infty }{\lim }\frac{{\sum }_{i=1}^L{\left[{\widehat{\varphi }}_i-\langle \widehat{\varphi }\rangle \right]}^2}{L-1}}.$$ (2.9)

In the following, we will perform analysis to address the following two questions: First, is $\langle \widehat{\varphi }\rangle$ always equal to Sθ? Second, is $\sigma (\widehat{\varphi })$ always equal to $\sqrt{\frac{2}{M{N}_0{ϵ}^2}}?$ If so, then the signal-to-noise ratio increases monotonically with respect to S as discussed in the introduction section. However, if not, how would the new result modify our current thinking of system optimization?

In previous studies, it has been found that the estimated ${\widehat{\varphi }}^o$ should follow a wrapped normal distribution as a result of the Arg(z) operation^18,19. The wrapped normal distribution is given as

$$g({\widehat{\varphi }}^o)=\sum _{k=-\infty }^{+\infty }\frac{1}{\sqrt{2\pi {\sigma }^2}}\exp \left[-\frac{{\left({\widehat{\varphi }}^o-{\varphi }^o+2k\pi \right)}^2}{2{\sigma }^2}\right],{\widehat{\varphi }}^o\in [-\pi ,\pi ),$$ (2.10)

where σ is the standard deviation of ${\widehat{\varphi }}^o$ of the original normal distribution before the phase wrapping effect is taken into account (See Eq. (1.2)). Based on the distribution of ${\widehat{\varphi }}^o$ and the relationship between $\widehat{\varphi }$ and ${\widehat{\varphi }}^o$ shown in Eq. (2.7), the distribution of $\widehat{\varphi }$ is given as follows:

$$f(\widehat{\varphi };S)=\sum _{k=-\infty }^{+\infty }\frac{1}{\sqrt{2\pi {\sigma }^2}}\exp \left[-\frac{{(\widehat{\varphi }-S\theta +2k\pi )}^2}{2{\sigma }^2}\right],\widehat{\varphi }\in [-\pi ,\pi ),$$ (2.11)

which is independent of the background phase signal. A detailed derivation of the distribution is given in the appendix. For a relatively small σ (for example, σ < 1), it was found that the mean of the wrapped normal distribution can be approximated as¹⁹

$$\langle \widehat{\varphi }(S)\rangle ={\int }_{-\pi }^{\pi }\widehat{\varphi }f(\widehat{\varphi })\text{d}\widehat{\varphi }\approx S\theta +\pi \left[\mathrm{erf}\left(A_{-}\right)-\mathrm{erf}\left(A_{+}\right)\right],$$ (2.12)

where erf(·) is the so-called error function and variable A_± is defined as $A_{\pm }=\frac{\pi \pm S\theta }{\sqrt{2}\sigma }$. The additional item π [erf(A₋) − erf(A₊)] is referred to as the signal bias¹⁹. Due to the existence of bias, $\langle \widehat{\varphi }\rangle$ may no longer be proportional to S. The variance of the wrapped normal distribution, denoted as ${\sigma }_w^2$, can be approximated as

$$\begin{gathered} {\sigma }_w^2(\widehat{\varphi };S)={\int }_{-\pi }^{\pi }(\widehat{\varphi }-\langle \widehat{\varphi }\rangle {)}^{2}f(\widehat{\varphi })\text{d}\widehat{\varphi } \\ \approx {\sigma }^2-2\sqrt{2\pi }\sigma (e^{-A_{+}^2}+e^{-A_{-}^2})+2{\pi }^2\left[2-\mathrm{erf}\left(A_{-}\right)-\mathrm{erf}\left(A_{+}\right)\right] \\ -{\pi }^2{\left[\mathrm{erf}\left(A_{-}\right)-\mathrm{erf}\left(A_{+}\right)\right]}^2. \end{gathered}$$ (2.13)

Detailed derivations of the mean and variance of the wrapped normal distribution are shown in the appendix. Despite the relatively complex form of $\langle \widehat{\varphi }\rangle$ and ${\sigma }_w^2(\widehat{\varphi })$, one can immediately find that when phase wrapping is taken into consideration, $\langle \widehat{\varphi }\rangle$ differs from ϕ and ${\sigma }_w^2(\widehat{\varphi })$ differs from σ², due to the extra A_± related terms. The signal-to-noise ratio is defined as

$$\mathrm{SNR}(S)=\frac{|\langle \widehat{\varphi }(S)\rangle |}{{\sigma }_w(\widehat{\varphi };S)}\ne \frac{S\left|\theta \right|}{\sigma },$$ (2.14)

which means that the signal-to-noise ratio may not be proportional to sensitivity.

B. Validation studies using a numerical simulation method

Numerical simulations were performed as the first step in validating the theoretical analysis. The procedure of the numerical simulation is summarized in Fig. 1. Six phase steps were used (M = 6) and the visibility (ϵ) was assumed to be 0.2 to represent our benchtop grating-based phase contrast system¹⁹. The refraction angle (θ) was assumed to be 1 × 10⁻⁶π. Sensitivity (S) varied from 2 × 10⁴ to 9.8 × 10⁵, resulting in a phase shift from 0.02π to 0.98π. The maximum phase shift is selected to be smaller than π. As shown in the previous subsection, the SNR of the phase shift is independent of the background phase angle, so the background phase angle is assumed to be 0. The expected number of photons received in each pixel per phase step (N₀) was between 50 and 300 to represent different dose levels. The number of photons in each phase step was calculated based on the sinusoidal model given by Eq. (2.1). Poisson noise was added to each phase step and the signal retrieval was performed based on Eq. (2.6). For a given set of parameters (M, ϵ, S, N₀), the Poisson noise addition and signal retrieval process was repeated so that an ensemble of $\widehat{\varphi }$ was obtained. The mean, variance and SNR of $\widehat{\varphi }$ were calculated.

FIG. 1.

Flow chart of the numerical simulations.

Additionally, given the parameters (M, ϵ, S, N₀), the mean, variance and SNR of $\widehat{\varphi }$ can be theoretically calculated based on Eqs. (2.12)–(2.14). The theoretically calculated values were compared with those from numerical simulations to validate the theory.

C. Validation studies using experimental methods

To experimentally study the impact of sensitivity on the SNR of phase contrast information, some technical challenges needed to be overcome. Note that it is almost impossible to perform experimental studies on multiple systems with a broad range of sensitivity, while keeping other factors such as visibility e and the magnification factor fixed. Therefore, the phase map determined from the background phase stepping scan usually covers a full range of signal values from −π to π, which can be used to represent signals obtained from systems with different sensitivity values. The experimental setups were similar to those of our previous works; therefore, detailed experimental parameters are directly quoted from one of our published papers¹⁹: “Experiments were performed on a benchtop-based x-ray Talbot-Lau interferometer, which uses a medical grade rotating-anode x-ray tube with 1.0 mm nominal focal spots and a CdTe-based photon counting detector, as shown in Fig. 2. The source grating G0 has a pitch size of 20.7 μm and a duty cycle of 35%; the phase grating G1 has a pitch size of 4.3 μm and a duty cycle of 50%. The analyzer grating G2 has a pitch size of 2.4 μm and a duty cycle of 50%. The G1 grating is a π-phase grating designed for 28 keV x-rays. The distance between G0 and G1 was set to 151 cm, and the distance between G1 and G2 was 17.5 cm. During data acquisition, the tube potential was fixed at 40 kVp (mean energy: 28 keV). The mean fringe visibility ϵ of the system is approximately 0.2 under the above experimental conditions” The tube current was set to 15 mA, and the exposure time per phase step ranged from 33 ms to 333 ms. With this acquisition setup, the value of N₀ ranged from 30 to 300. The total number of phase steps (M) was 6.

FIG. 2.

Photo of the benchtop-based x-ray Talbot-Lau interferometer system.

For a given N₀ (a given exposure level), the measurements were repeated 200 times. The data processing method is shown in Fig. 3. The estimated $\widehat{\varphi }$ corresponding to each individual set of phase stepping data were generated based on Eq. (2.6). The ensemble of $\widehat{\varphi }$ was statistically analyzed to get the mean and standard deviation. To obtain the ground truth of ϕ, the repeated measurements in each phase step were first averaged and the phase retrieval process was performed afterwards, as shown in Fig. 3(b). With the knowledge of N₀, ϵ and ϕ, the wrapped normal distribution can be theoretically determined so that $\langle \widehat{\varphi }\rangle$, ${\sigma }_w(\widehat{\varphi })$ and SNR can be theoretically predicted. The predicted values were compared with those from experimental measurements.

FIG. 3.

Flow chart showing the data processing method for the experimental studies.

D. Phantom studies

Phantom studies were performed to directly study the impact of sensitivity on the SNR of the DPC images. In this study, a PMMA wedge with PMMA spheres attached was used as the image object, as shown in Fig. 4. The PMMA wedge was carefully positioned so that a large refraction angle can be generated using our benchtop system. Most of the data acquisition conditions were similar to what were described in Sec. II C, including the tube potential, tube current and the number of phase steps used. For each phase step in data acquisitions, the exposure time was 0.4 s and the total air kerma was measured to be 0.058 mGy. The phase image was acquired at two different sensitivity levels (7 × 10⁴ and 5 × 10⁴) by changing the location of the image object between the G1 grating and the G2 grating. For each sensitivity level, the data acquisitions were repeated 25 times to quantify the pixel-by-pixel mean signal values and noise variances, and finally a pixel-by-pixel SNR map was then generated. The phase image and the SNR map at different sensitivity levels were then compared.

FIG. 4.

(a) Picture of the PMMA wedge with PMMA spheres attached. (b) The wedge is carefully placed so that the angle between one of its side and the x-ray traveling direction (the angle is denoted as α) is relatively small (a few degrees) to create a large refraction angle.

In order to study the impact of sensitivity on SNR for an object with relatively complex anatomical structures, one would ideally prefer to directly image real biological tissues. Unfortunately, the sensitivity of experimental system used in this work is not high enough to generate significant phase wrapping effects for bioloigical samples to investigate the targeted phenomenon. Therefore, numerical simulations with biological tissue were performed to investigate the effect in this work. The flow chart of the simulation is shown in Fig. 5. Repeated scans of the bovine udder specimens (preserved in wax) with high exposure level were first performed using our benchtop system. The scans were averaged to get the relatively noise-free phase images and absorption images of the specimens. The phase images were normalized by the system sensitivity to estimate the corresponding refraction angles at each image pixel. The refraction angle and the absorption images were used as the ground truths for the simulation studies. The x-ray beam was simulated as a monochromatic beam with energy of 28 keV, which represents the mean energy of our system. The photon fluence was simulated to be 3.7×10⁵ mm⁻², resulting in an air kerma of 0.01 mGy which is consistent with our experimental measurements. The absorption caused by the gratings after the image object was assumed to be 50% of the post-object photons which is also consistent with the transmission coefficient of the G₂ grating used in our experimental system. The visibility of the grating interferometer was assumed to be 0.2 which is similar to the average visibility of our experimental system. The pixel pitch was simulated to 100 μm by 100 μm. For a given sensitivity level, the phase shift can be determined by multiplying the simulated sensitivity level and the refraction angle map. Then a phase stepping procedure with six phase steps was simulated according to Eq. (2.1). For each phase step, Poisson noise was added according to the mean number of photons received for each pixel of the x-ray detector. Then a DPC image with noise can be extracted from the noisy phase steps. The sensitivity was simulated in the range of [5 × 10⁴, 6 × 10⁵] and the phase images corresponding to different sensitivity levels were compared.

FIG. 5.

Flow chart of the simulation to study the impact of the sensitivity on the phase image quality of biological objects.

III. RESULTS

A. Numerical simulation validation results

The distributions of the estimated phase signal $\widehat{\varphi }$ that corresponds to S = 2 × 10⁵, 8 × 10⁵ or 9 × 10⁵ and N₀ = 57 or 228 are shown in Fig. 6. When N₀ is fixed, for a relatively small S, little of the original normal distribution falls outside the range [−π,π), as shown in Fig. 6 (a). The wrapped normal distribution is almost the same as the original distribution. However, when S increases, the corresponding distribution is truncated abruptly at π without the continuous tail of the original normal distribution, as shown in Fig. 6 (b)–(c). Due to the phase wrapping effect, the missing part was wrapped to the left, resulting in the small tail around −π. For a fixed sensitivity, increasing N₀ will decrease the standard deviation of the original normal distribution, so that a smaller fraction of the normal distribution is wrapped to the other side, as seen in the comparison of Fig. 6 (c) and (d). The theoretical wrapped normal distribution calculated from Eq. (2.11) is also shown in Fig. 6 as a comparison of the distribution of the simulated results.

FIG. 6.

Distributions of $\widehat{\varphi }$ under the following simulation conditions: (a) S = 2 × 10⁵ (ϕ = 0.2π), N₀ = 57; (b) S = 8 × 10⁵ (ϕ = 0.9π), N₀ = 57; (c) S = 9 × 10⁵ (ϕ = 0.9π), N₀ = 57; (d) S = 9 × 10⁵ (ϕ = 0.9π), N₀ = 228. The bars represent the measurement of statistical distribution of $\widehat{\varphi }$ from the numerical simulation. The solid curves represents the theoretical wrapped normal distributions shown in Eq. (2.11).

As a result of the phase wrapping effect, the mean of the distribution is not necessarily the mean of the original normal distribution. Fig. 7 shows the relationship between $\langle \widehat{\varphi }\rangle$ and S. When ϕ = Sθ is much smaller than π, the phase wrapping effect is negligible so that $\langle \widehat{\varphi }\rangle \approx \varphi \propto S$. However, when S becomes very large and ϕ approaches π, the relationship between $\langle \widehat{\varphi }\rangle$ and S is no longer linear. In an extremal case, $\langle \widehat{\varphi }\rangle$ may decrease when S increases, as indicated in the right part in Fig. 7(a). In this case, the measured mean phase shift severely deviates from the ground truth ϕ = Sθ, which may impact the signal to noise ratio. Increasing the number of photons per phase step decreases the phase wrapping effect, so that the region where $\langle \widehat{\varphi }\rangle$ is proportional to S becomes broader.

FIG. 7.

Mean, standard deviation and SNR of $\widehat{\varphi }$ versus S. The discrete data points represent measurement of $\widehat{\varphi }$ from the numerical simulation. The continuous curves represent the theoretical values of the properties of $\widehat{\varphi }$ from the wrapped normal distribution.

The relationship between the standard deviation of the phase shift and the sensitivity S is shown in Fig. 7(b). For relatively small S, the standard deviation is approximately a constant independent of S, as the phase wrapping effect did not provide much contribution. For higher sensitivity factors, however, the standard deviation increases as S increases. Increasing N₀ results in broader region where ${\sigma }_w(\widehat{\varphi })$ is a constant independent of S.

As the result of the dependence of the mean and standard deviation of the $\widehat{\varphi }$ distribution on S, the relationship between SNR and S is shown in Fig. 7(c). The SNR is approximately proportional to the sensitivity for relatively small sensitivities, with which the phase wrapping is less likely to occur. For larger sensitivity factors, the dependence of SNR on S is non-linear. For a given set of experimental parameters (M, N₀, ϵ) and image object (θ), there is an optimal sensitivity value at which SNR can reach a maximum.

B. Experimental validation results

Figure 8 shows the experimentally retrieved phase images from the benchtop system (M = 6, N₀ = 57). As shown in Fig. 8 (a), the ϕ image from the background scan covers a phase signal ranging from −π to π and the value of ϕ is used to represent signals from systems with different sensitivity factors. The measured phase signal $\widehat{\varphi }$, as shown in Fig. 8 (b), is different from that of Fig. 8 (a), particularly in the brightest or darkest regions, where ϕ → ±π. From the standard deviation map shown in Fig. 8 (c), one can observe that the standard deviation is no longer a constant for a given exposure level, but has a dependence on ϕ. In the region where ϕ approaches ±π, the standard deviation significantly increases. As a result, the calculated SNR map is not proportional to |ϕ|. Especially when ϕ → ±π, SNR dramatically drops, almost to zero, due to the phase wrapping analyzed previously.

FIG. 8.

Image of the experimentally retrieved (a) ϕ, (b) $\langle \widehat{\varphi }\rangle$, (c) ${\sigma }_w(\widehat{\varphi })$, and (d) SNR. The display ranges are: [−π,π] for (a) and (b); [0,3] for (c); [0,10] for (d).

The quantitative dependence of the mean, standard deviation and SNR of the distribution as a function of ϕ (sensitivity factor S) are shown in Fig. 9 (a)–(c). The observations are similar to those from the numerical simulations. When ϕ is much smaller than π (indicating relatively low sensitivity), $\langle \widehat{\varphi }\rangle$ is equal to ϕ, and ${\sigma }_w(\widehat{\varphi })$ is a constant independent of ϕ. As a result, the SNR of the phase shift is proportional to ϕ (therefore proportional to the sensitivity). However, when ϕ approaches π, increasing ϕ leads to more severe phase wrapping effect so that the SNR is no longer proportional to ϕ and may decrease as ϕ increases. There is an optimal ϕ (or optimal sensitivity) where SNR is maximized. Increasing N₀ (photons received in one pixel per phase step) decreases the variance of the original normal distribution, which alleviates the phase wrapping effect. Therefore, a larger N₀ leads to an expanded region where SNR is proportional to ϕ (or proportional to sensitivity factor), as shown in Fig. 9(c). The optimal ϕ value to maximize the SNR as a function of N₀ is shown in Fig. 9(d).

FIG. 9.

(a) Mean, (b) standard deviation and (c) SNR of $\widehat{\varphi }$ versus ϕ. (d) The optimal ϕ (sensitivity factor) to achieve the highest SNR as a function of N₀. (a) (b) and (c) share the same legend. The discrete data points represent experimental measurements. The continuous curves represent theoretical values derived based on the properties of $\widehat{\varphi }$ from the wrapped normal distribution.

C. Phantom study results

Fig. 10 shows the phase contrast image of PMMA phantom scanned using our benchtop system. As one can tell, for regions with small refraction angles, e.g. interior regions of PMMA spheres, the SNR corresponding to the high sensitivity is higher since the phase signal is amplified more while the noise remains almost the same. For regions with large refraction angles, the phase wrapping effect comes to play and the SNR decreases for higher sensitivity.

FIG. 10.

Phase contrast image and SNR map the PMMA phantom. (a) and (b) Differential phase contrast image acquired under the high sensitivity and the low sensitivity, respectively. Display range is [−4.3,4.3] × 10⁻⁵. (c) and (d) SNR map acquired under the high sensitivity and the low sensitivity, respectively. Display range is [0,4].

Fig. 11 shows the simulated phase contrast image of the bovine udder specimen under different sensitivity levels. The images were normalized by their corresponding sensitivity level so that they can be compared using the same display window and level. For lower sensitivity levels (Fig. 11 (a) (b)), the image quality improved as the sensitivity increases. However, when the sensitivity became higher (Fig. 11 (c) (d)), the phase wrapping effect came into play. The relatively large phase signals became imprecise and the overall image quality deteriorated.

FIG. 11.

Simulated phase contrast image of the bovine udder specimen under different sensitivity levels. (a) 5 × 10⁴, (b) 1 × 10⁵, (c) 3 × 10⁵, (d) 6 × 10⁵. Display range is [−2,2] × 10⁻⁵.

IV. DISCUSSION

In this work, the dependence of the signal-to-noise ratio of the phase signal on interferometer sensitivity was analyzed both theoretically and experimentally. The most important result is that SNR of the phase signal may not always be proportional to the sensitivity.

The result can be primarily attributed to the phase wrapping procedure intrinsic to the phase retrieval process. As a result, the measured phase signal or phase shift can be modeled by a wrapped normal distribution. The measured mean phase signal (denoted as ⟨ϕ⟩) is not only determined by the sensitivity factor and the refraction angle of the object, but is also related to the noise of the unwrapped phase signal (denoted as σ), as shown in Eq. (2.12), where σ is further related to the system and data acquisition parameters, e.g. fringe visibility and exposure level, as indicated in Eq. (1.2). Meanwhile the variance of the wrapped normal distribution (denoted as σ_w in this paper) differs from that of the unwrapped normal distribution (σ). All the quantititative relationships are summarized in Eqs. (2.12–2.14). When the sensitivity is relatively low (0 is not close to ±π) and when σ is small, there is little difference between the wrapped distribution and the normal distribution so Eqs. (1.1)–(1.2) can still be used to predict ⟨ϕ⟩ and ${\sigma }_w(\widehat{\varphi })$. However, with relatively high sensitivity and relatively low dose (large σ), the phase wrapping effect becomes more prominent. The standard deviation of the phase signal increases while measured mean phase signal may even decrease as the sensitivity is further increased. As a result, SNR may decrease for very high sensitivity. For real phase contrast imaging systems, for a given image object, if the phase shift (the multiplication of the sensitivity and the refraction angle) is much smaller than π, a higher sensitivity will lead to improved image quality under the same dose constraint, or in other words, higher dose efficiency. But the sensitivity cannot be pushed to the high end without limitation to increase the dose efficiency due to the phase wrapping effect and the intrinsic noise (e.g. Poisson noise) effects. The optimal sensitivity depends on several parameters, including the dose level, fringe visibility and refraction angle of the image object.

A standard phase retrieval process involves an air scan and an object scan, and the difference in the phase signal from the two scans is referred to as the phase shift. For a fixed phase shift, the phase signal from the air scan can impact the distribution of the phase signal from the object scan. Therefore, one might think that the distribution of the phase shift also depends on the air scan. However, our theoretical analysis demonstrated that, as long as noise from the air scan can be neglected, the phase signal from the air scan does not impact statistical properties of the phase shift.

The paper has several limitations. The origin of the noise of the phase signal is only attributed to the quantum noise of the detected photons. This is valid for photon counting detector-based imaging systems. For energy integrating detector-based systems, the impact of other additional noise sources (e.g. electronic noise) should also be considered. In addition, the analysis is based on point statistics and is limited to the standard phase stepping data acquisition and phase retrieval methods. When other non-local data acquisition schemes and the corresponding phase retrieval methods were used, for example single shot methods^20–22 or conjugate ray methods²³, it remains open to study the potential interplay between system sensitivity and the phase wrapping effect. Finally, it is unclear how the interplay between the system sensitivity and phase wrapping effects in this paper would influence the signal and noise properties of the dark field imaging. Particularly, although the correlation length related to the dark field signal was found to be proportional to the system sensitivity, the relationship between the dark field signal and the correlation length is non-linear^24,25. Therefore, the quantitative relationship between the dark field signal and the sensitivity factor needs to be carefully investigated in future works.

When the projection images are used to reconstruct the differential phase contrast CT images, the signal bias caused by the phase wrapping effect will inevitably be propagated into the reconstructed CT images. The noise properties of the phase contrast CT images has been discussed in several papers^14,26,27. The discussions were based on the assumption that the detected phase signal is proportional to the sensitivity factor, without the statistical bias and the wrapped normal distribution being considered. Therefore, it would be a very interesting future topic to extend the results in this paper to CT acquisitions. Given the fact that the wrapped normal distribution in differential phase contrast signal has strong dependence on both the signal and noise level of the unwrapped phase signal, one may anticipate that some denoising techniques can be introduced into the CT reconstruction processes, such as using an iterative reconstruction method with an appropriate regularization term, to mitigate the phase wrapping effects^22,28. However, it remains an open question to investigate how the system sensitivity quantitatively impacts the image quality of the phase contrast CT images provided that different reconstruction methods are used and the phase wrapping effect is taken into account.

Finally, the phase wrapping effects discussed in this paper call for reliable techniques to alleviate the phase wrapping effects. In this aspect, a number of phase unwrapping techniques were proposed to mitigate the phase wrapping effect and may impact the results if these techniques were applied to x-ray differential phase contrast imaging. Some techniques were specifically developed for x-ray phase contrast imaging^29,30, while other algorithms^31–34 were developed for general purpose phase unwrapping. Most of the algorithms either require additional measurements (e.g. measurements of phase signals under different x-ray spectra), or are based on some assumptions (e.g. the slowly varying phase signals in the spatial domain). Meanwhile, as discussed in this work, the phase wrapping effect is not only related to the signal but also depends on the noise, since the variance of the original normal distribution determines how much the distribution gets wrapped to other side (shown in Fig. 6(c)(d)). When the measured data are corrupted by noise, it is hard to unambiguously recover the unknown absolute phase³³. Given a noisy x-ray projection phase image of an object acquired from a phase stepping procedure, currently there is no widely used algorithm to generate the unwrapped phase. Therefore, how the relationship between SNR and sensitivity is affected by which phase wrapping techniques are applied remains an open question.

V. CONCLUSIONS

While it is a common belief that higher sensitivity of a grating-based phase-contrast system always leads to better image quality and dose efficiency, as shown in this work, this conclusion may not always hold true if the phase wrapping effect intrinsic to the phase signal extraction procedure is considered. Theoretical analysis shows that the phase shift signal is not always proportional to the sensitivity and the noise may increase as the sensitivity increases. Both numerical simulations and experimental studies were performed to validate the theoretical results. For a given refraction angle, there exists an optimal sensitivity which maximizes the SNR of the phase shift signal, beyond which increasing the sensitivity factor actually decreases the SNR.

APPENDIX

Mean and variance of a wrapped normal distribution

A wrapped normal distribution is given as

$$f(x)=\sum _{k=-\infty }^{+\infty }\frac{1}{\sigma \sqrt{2\pi }}{e}^{-\frac{{(x-\mu +2k\pi )}^2}{2{\sigma }^2}},x\in [-\pi ,\pi )$$ (5.1)

where μ ∈ [−π,π) and σ² are the mean and variance of the normal distribution before being wrapped, respectively. The mean of the wrapped normal distribution is calculated as

$$\langle x\rangle ={\int }_{-\pi }^{\pi }x\sum _{k=-\infty }^{+\infty }\frac{1}{\sigma \sqrt{2\pi }}{e}^{-\frac{{(x-\mu +2k\pi )}^2}{2{\sigma }^2}}\text{d}x$$ (5.2)

$$=\sum _{k=-\infty }^{+\infty }{\int }_{(2k-1)\pi -\mu }^{(2k+1)\pi -\mu }\frac{\xi +\mu -2k\pi }{\sigma \sqrt{2\pi }}{e}^{-\frac{{\xi }^2}{2{\sigma }^2}}\text{d}\xi (\xi =x-\mu +2k\pi )$$ (5.3)

$$\begin{gathered} =\sum _{k=-\infty }^{+\infty }{\int }_{(2k-1)\pi -\mu }^{(2k+1)\pi -\mu }\frac{\xi }{\sigma \sqrt{2\pi }}{e}^{-\frac{{\xi }^2}{2{\sigma }^2}}\text{d}\xi +\sum _{k=-\infty }^{+\infty }{\int }_{(2k-1)\pi -\mu }^{(2k+1)\pi -\mu }\frac{\mu }{\sigma \sqrt{2\pi }}{e}^{-\frac{{\xi }^2}{2{\sigma }^2}}\text{d}\xi \\ -\sum _{k=-\infty }^{+\infty }{\int }_{(2k-1)\pi -\mu }^{(2k+1)\pi -\mu }\frac{2k\pi }{\sigma \sqrt{2\pi }}{e}^{-\frac{{\xi }^2}{2{\sigma }^2}}\text{d}\xi \end{gathered}$$ (5.4)

$$=\mu -\sum _{k=-\infty }^{+\infty }{\int }_{(2k-1)\pi -\mu }^{(2k+1)\pi -\mu }\frac{2k\pi }{\sigma \sqrt{2\pi }}{e}^{-\frac{{\xi }^2}{2{\sigma }^2}}\text{d}\xi$$ (5.5)

$$=\mu -\pi \sum _{k=-\infty }^{+\infty }k\left[\mathrm{erf}\left(\frac{(2k+1)\pi -\mu }{\sqrt{2}\sigma }\right)-\mathrm{erf}\left(\frac{(2k-1)\pi -\mu }{\sqrt{2}\sigma }\right)\right],$$ (5.6)

where

$$\mathrm{erf}(x)=\frac{2}{\sqrt{\pi }}{\int }_0^x{e}^{-t^2}dt$$ (5.7)

is the error function. By changing the dummy variable k and using the property erf(x) = −erf(−x), Eq. (5.6) can be further simplified as

$$\langle x\rangle =\mu +\pi \sum _{k=0}^{+\infty }\left[\mathrm{erf}\left(\frac{(2k+1)\pi -\mu }{\sqrt{2}\sigma }\right)-\mathrm{erf}\left(\frac{(2k+1)\pi +\mu }{\sqrt{2}\sigma }\right)\right]$$ (5.8)

$$\approx \mu +\pi \left[\mathrm{erf}\left(\frac{\pi -\mu }{\sqrt{2}\sigma }\right)-\mathrm{erf}\left(\frac{\pi +\mu }{\sqrt{2}\sigma }\right)\right].$$ (5.9)

The approximation in Eq. (5.9) is based on the assumption that σ < 1. Most often this assumption is valid and the proof is given below. σ is related to data acquisition and the system parameters as

$${\sigma }^2=\frac{2}{M{N}_0{ϵ}^2}$$ (5.10)

Typical, the visibility ϵ is 0.2 for a given system. If we assume M = 5 and N₀ = 20 (A very low dose setting), σ² = 2/(5 × 20 × 0.2²) = 1/2. In a regular dose setting, σ will be even smaller.

The variance of the wrapped normal distribution is given by

$${\sigma }_w^2(x)=\langle {(x-\langle x\rangle )}^2\rangle$$ (5.11)

Since ⟨x⟩ have a relatively complex form as shown in Eq. (5.9), μ is added as an intermediate variable

$${\sigma }_w^2(x)=\langle {(x-\mu +\mu -\langle x\rangle )}^2\rangle$$ (5.12)

$$=\langle {(x-\mu )}^2+2(x-\mu )(\mu -\langle x\rangle )+{(\mu -\langle x\rangle )}^2\rangle$$ (5.13)

$$=\langle {(x-\mu )}^2\rangle -{(\mu -\langle x\rangle )}^2$$ (5.14)

The above derivation uses the fact that

$$\langle (x-\mu )\rangle =(\langle x\rangle -\mu )$$ (5.15)

Now, we relate the variance to the mean of (x − μ)² and (μ − ⟨x⟩)². The reason we do this is that ⟨(x − μ)²⟩ is easier to calculate and calculation of (μ − ⟨x⟩)² is straightforward from Eq. (5.9). Mean of (x − μ)² is given by its definition as

$$\langle {(x-\mu )}^2\rangle ={\int }_{-\pi }^{\pi }{(x-\mu )}^{2}f(x)\text{d}x$$ (5.16)

$$=\sum _{k=-\infty }^{\infty }{\int }_{-\pi }^{\pi }\frac{{(x-\mu )}^2\exp \left[-\frac{{(x-\mu +2k\pi )}^2}{2{\sigma }^2}\right]}{\sigma \sqrt{2\pi }}\text{d}x$$ (5.17)

Assuming that ξ = x − μ + 2kπ, Eq. (5.17) becomes

$$\langle {(x-\mu )}^2\rangle =\sum _{k=-\infty }^{\infty }{\int }_{(2k-1)\pi -\mu }^{(2k+1)\pi -\mu }\frac{{(\xi -2k\pi )}^2\exp \left[-\frac{{\xi }^2}{2{\sigma }^2}\right]}{\sigma \sqrt{2\pi }}\text{d}\xi$$ (5.18)

$$\begin{gathered} =\sum _{k=-\infty }^{\infty }{\int }_{(2k-1)\pi -\mu }^{(2k+1)\pi -\mu }\frac{{\xi }^2\exp \left[-\frac{{\xi }^2}{2{\sigma }^2}\right]}{\sigma \sqrt{2\pi }}\text{d}\xi \\ +\sum _{k=-\infty }^{\infty }{\int }_{(2k-1)\pi -\mu }^{(2k+1)\pi -\mu }\frac{-4k\pi \xi \exp \left[-\frac{{\xi }^2}{2{\sigma }^2}\right]}{\sigma \sqrt{2\pi }}\text{d}\xi \\ +\sum _{k=-\infty }^{\infty }{\int }_{(2k-1)\pi -\mu }^{(2k+1)\pi -\mu }\frac{4k^2{\pi }^2\exp \left[-\frac{{\xi }^2}{2{\sigma }^2}\right]}{\sigma \sqrt{2\pi }}\text{d}\xi \end{gathered}$$ (5.19)

As shown in Eq. (5.19), the variance can be split into three items. The first item is

$$\sum _{k=-\infty }^{\infty }{\int }_{(2k-1)\pi -\mu }^{(2k+1)\pi -\mu }\frac{{\xi }^2\exp \left[-\frac{{\xi }^2}{2{\sigma }^2}\right]}{\sigma \sqrt{2\pi }}\text{d}\xi ={\sigma }^2$$ (5.20)

The second item is

$$\sum _{k=-\infty }^{\infty }{\int }_{(2k-1)\pi -\mu }^{(2k+1)\pi -\mu }\frac{-4k\pi \xi \exp \left[-\frac{{\xi }^2}{2{\sigma }^2}\right]}{\sigma \sqrt{2\pi }}\text{d}\xi$$ (5.21)

$$=-4\pi \sum _{k=-\infty }^{\infty }k{\int }_{(2k-1)\pi -\mu }^{(2k+1)\pi -\mu }\frac{{\sigma }^2\exp \left[-\frac{{\xi }^2}{2{\sigma }^2}\right]}{\sigma \sqrt{2\pi }}\text{d}\frac{{\xi }^2}{2{\sigma }^2}$$ (5.22)

$$=-2\sigma \sqrt{2\pi }\sum _{k=-\infty }^{\infty }k\left\{e^{\frac{-{[(2k-1)\pi -\mu ]}^2}{2{\sigma }^2}}-e^{\frac{-{[(2k+1)\pi -\mu ]}^2}{2{\sigma }^2}}\right\}$$ (5.23)

$$=-2\sigma \sqrt{2\pi }\sum _{k=-\infty }^{\infty }{e}^{\frac{-{[(2k-1)\pi -\mu ]}^2}{2{\sigma }^2}}$$ (5.24)

$$\approx -2\sigma \sqrt{2\pi }\left[e^{\frac{-{(\pi +\mu )}^2}{2{\sigma }^2}}+e^{\frac{-{(\pi -\mu )}^2}{2{\sigma }^2}}\right]$$ (5.25)

The approximation in Eq. (5.25) is based on the assumption that σ < 1. The third item is

$$\sum _{k=-\infty }^{\infty }{\int }_{(2k-1)\pi -\mu }^{(2k+1)\pi -\mu }\frac{4k^2{\pi }^2\exp \left[-\frac{{\xi }^2}{2{\sigma }^2}\right]}{\sigma \sqrt{2\pi }}\text{d}\xi$$ (5.26)

$$=4{\pi }^2\sum _{k=-\infty }^{\infty }{k}^2{\int }_{(2k-1)\pi -\mu }^{(2k+1)\pi -\mu }\frac{\exp \left[-\frac{{\xi }^2}{2{\sigma }^2}\right]}{\sigma \sqrt{2\pi }}\text{d}\xi$$ (5.27)

$$=2{\pi }^2\sum _{k=-\infty }^{\infty }{k}^2\left[\mathrm{erf}\left(\frac{(2k+1)\pi -\mu }{\sqrt{2}\sigma }\right)-\mathrm{erf}\left(\frac{(2k-1)\pi -\mu }{\sqrt{2}\sigma }\right)\right]$$ (5.28)

$$\approx 2{\pi }^2\left[\mathrm{erf}\left(\frac{-\pi -\mu }{\sqrt{2}\sigma }\right)-\mathrm{erf}\left(\frac{-3\pi -\mu }{\sqrt{2}\sigma }\right)+\mathrm{erf}\left(\frac{3\pi -\mu }{\sqrt{2}\sigma }\right)-\mathrm{erf}\left(\frac{\pi -\mu }{\sqrt{2}\sigma }\right)\right]$$ (5.29)

$$\approx 2{\pi }^2\left[2-\mathrm{erf}\left(\frac{\pi +\mu }{\sqrt{2}\sigma }\right)-\mathrm{erf}\left(\frac{\pi -\mu }{\sqrt{2}\sigma }\right)\right]$$ (5.30)

The approximation in Eqs. (5.29)–(5.30) is also based on σ < 1. When the results from Eqs. (5.9), (5.14), (5.25) and (5.30) are combined, the variance is given as

$$\langle {(x-\langle x\rangle )}^2\rangle =\langle {(x-\mu )}^2\rangle -{(\mu -\langle x\rangle )}^2$$ (5.31)

$$\begin{gathered} \approx {\sigma }^2-2\sigma \sqrt{2\pi }[\exp \left[\frac{-{(\pi +\mu )}^2}{2{\sigma }^2}\right]+\exp \left[\frac{-{(\pi -\mu )}^2}{2{\sigma }^2}\right]] \\ +2{\pi }^2\left[2-\mathrm{erf}\left(\frac{\pi +\mu }{\sqrt{2}\sigma }\right)-\mathrm{erf}\left(\frac{\pi -\mu }{\sqrt{2}\sigma }\right)\right] \\ -{\pi }^2{\left[\mathrm{erf}\left(\frac{\pi +\mu }{\sqrt{2}\sigma }\right)-\mathrm{erf}\left(\frac{\pi -\mu }{\sqrt{2}\sigma }\right)\right]}^2 \end{gathered}$$ (5.32)

If we define $A_{\pm }=\frac{\pi \pm \mu }{\sqrt{2}\sigma }$, the mean and variance of the wrapped normal distribution will have a less complex form

$$\langle x\rangle \approx \mu +\pi \left[\mathrm{erf}\left(A_{-}\right)-\mathrm{erf}\left(A_{+}\right)\right]$$ (5.33)

$$\begin{gathered} {\sigma }_w^2(x)\approx {\sigma }^2-2\sqrt{2\pi }\sigma (e^{-A_{+}^2}+e^{-A_{-}^2})+2{\pi }^2\left[2-\mathrm{erf}\left(A_{-}\right)-\mathrm{erf}\left(A_{+}\right)\right] \\ -{\pi }^2{\left[\mathrm{erf}\left(A_{-}\right)-\mathrm{erf}\left(A_{+}\right)\right]}^2. \end{gathered}$$ (5.34)

Distribution of the phase shift

The phase shift is defined in Eq. (2.7) based on an additional phase wrapping procedure

$$\widehat{\varphi }={\widehat{\varphi }}^o-{\varphi }^b+2m\pi ,m\in \mathbb{Z}.$$ (5.35)

Therefore, ${\widehat{\varphi }}^o=\widehat{\varphi }+{\varphi }^b-2m\pi$. As mentioned in section II A, the distribution of object phase signal ${\widehat{\varphi }}^o$ is given as

$$g({\widehat{\varphi }}^o)=\sum _{k=-\infty }^{+\infty }\frac{1}{\sigma \sqrt{2\pi }}{e}^{-\frac{{\left({\widehat{\varphi }}^o-{\varphi }^o+2k\pi \right)}^2}{2{\sigma }^2}},{\widehat{\varphi }}^o\in [-\pi ,\pi ).$$ (5.36)

So the distribution of $\widehat{\varphi }$, denoted as $g(\widehat{\varphi })$, is given as

$$f(\widehat{\varphi })=g({\widehat{\varphi }}^o(\widehat{\varphi }))$$ (5.37)

$$=\sum _{k=-\infty }^{+\infty }\frac{1}{\sigma \sqrt{2\pi }}{e}^{-\frac{{\left(\widehat{\varphi }+{\varphi }^b-2m\pi -{\varphi }^o+2k\pi \right)}^2}{2{\sigma }^2}}$$ (5.38)

$$=\sum _{k=-\infty }^{+\infty }\frac{1}{\sigma \sqrt{2\pi }}{e}^{-\frac{{\left[\widehat{\varphi }-\left({\varphi }^o-{\varphi }^b\right)+2(k-m)\pi \right]}^2}{2{\sigma }^2}}$$ (5.39)

$$=\sum _{k=-\infty }^{+\infty }\frac{1}{\sigma \sqrt{2\pi }}{e}^{-\frac{[\widehat{\varphi }-S\theta +2k\pi {]}^2}{2{\sigma }^2}},\widehat{\varphi }\in [-\pi ,\pi ).$$ (5.40)

Such distribution depends on Sθ (the expected phase shift introduced by the object) and has no dependence on ϕ^b (the background reference phase signal).