The trigonometric orthogonality of phase‐stepping curves in grating‐based x‐ray phase‐contrast imaging: Integral property and its implications for noise optimization

Wu, Chengpeng; Zhang, Li; Chen, Zhiqiang; Xing, Yuxiang; Li, Xinbin; Zhu, Xiaohua; Arboleda, Carolina; Wang, Zhentian; Gao, Hewei

doi:10.1002/mp.13957

Cited by 4 publications

(6 citation statements)

References 30 publications

(79 reference statements)

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“…To overcome these limitations of the original deconvolution‐based MMA (named DB‐MMA in this paper), some analytic MMA methods are developed recently 18,19 . The first one is referred as the direct MMA (D‐MMA) 18 that is applicable to EI and ABI at first, and it is extended to the GBI with some preprocessing operations 23 .…”

Section: Methodsmentioning

confidence: 99%

“…The first one is referred as the direct MMA (D‐MMA) 18 that is applicable to EI and ABI at first, and it is extended to the GBI with some preprocessing operations 23 . Then, a generalized D‐MMA (GD‐MMA) is developed by using the trigonometric orthogonality of PSCs, 19 which can be utilized to improve the noise performance of the original D‐MMA. The typical contrasts retrieved by D‐MMA and GD‐MMA can be expressed as:

A \to M_{0} false(g false) = \int_{- π}^{π} g false(ϕ false) d ϕ = \frac{\int_{- π}^{π} s false(ϕ false) d ϕ}{\int_{- π}^{π} f false(ϕ false) d ϕ},

\begin{matrix} true \\ right P \to {\bar{M}}_{1} (g) & = & left \frac{M_{1} (g)}{M_{0} (g)} = \frac{\int_{- π}^{π} ϕ \cdot g (ϕ) d ϕ}{\int_{- π}^{π} g (ϕ) d ϕ} \\ = & left \frac{\int_{- π}^{π} h_{1} (ϕ) \cdot s (ϕ) d ϕ}{\int_{- π}^{π} s (ϕ) d ϕ} - \frac{\int_{- π}^{π} h_{1} (ϕ) \cdot f (ϕ) d ϕ}{\int_{- π}^{π} f (ϕ) d ϕ}, \end{matrix}

\begin{matrix} true \\ right D \to {\tilde{M}}_{2} (g) & = \end{matrix}

…”

Section: Methodsmentioning

confidence: 99%

“…In the original D-MMA, 18 the kernel functions are h 1 (𝜙) = 𝜙 and h 2 (𝜙) = 𝜙 2 , while h 1 (𝜙) = 2 sin(𝜙) and h 2 (𝜙) = 𝜋 2 ∕3 − 4 cos(𝜙) in the noise-optimal GD-MMA for the GBI. 19 From the analytic expressions in D-MMA and GD-MMA above, it is found that multiple contrasts can be calculated by summing together the weighted PSCs with some kernel functions. These nonuniform weighting kernel functions suggest that the raw data across the steps has different contributions for retrieved contrasts and so do their corresponding noises.…”

Section: The Information Retrieval Algorithmsmentioning

confidence: 99%

“…To overcome these limitations of the original deconvolution-based MMA (named DB-MMA in this paper), some analytic MMA methods are developed recently. 18,19 The first one is referred as the direct MMA (D-MMA) 18 that is applicable to EI and ABI at first, and it is extended to the GBI with some preprocessing operations. 23 Then, a generalized D-MMA (GD-MMA) is developed by using the trigonometric orthogonality of PSCs, 19 which can be utilized to improve the noise performance of the original D-MMA.…”

Section: The Information Retrieval Algorithmsmentioning

confidence: 99%

“…To the best of our knowledge, there is lack of literature focusing on the effects of step-varying fluence distributions during the PS acquisition. Inspired by recent analytic multiorder moment analyses (MMAs), 18,19 we have found that the raw PSCs will be weighted across steps in the postprocessing information retrieval. Intuitively, the weighted PSCs have different noise contributions compared with the original acquired PSCs, which suggests that there may be some rooms for further noise optimization.…”

Section: Introductionmentioning

confidence: 99%

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Fluence adaptation for contrast‐based dose optimization in x‐ray phase‐contrast imaging

Xing

Zhang

et al. 2021

Medical Physics

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Purpose: X-ray phase-contrast imaging (XPCI) can provide multiple contrasts with great potentials for clinical and industrial applications, including conventional attenuation, phase contrast, and dark field. Grating-based imaging (GBI) and edge-illumination (EI) are two promising types of XPCI as the conventional x-ray sources can be directly utilized. For the GBI and EI systems, the phasestepping acquisition with multiple exposures at a constant fluence is usually adopted in the literature.This work, however, attempts to challenge such a constant fluence concept during the phase-stepping process and proposes a fluence adaptation mechanism for dose reduction. Method: Given the importance of patient radiation dose for clinical applications, numerous studies have tried to reduce patient dose in XPCI by altering imaging system designs, data acquisition, and information retrieval. Recently, analytic multiorder moment analysis has been proposed to improve the computing efficiency. In these algorithms, multiple contrasts can be calculated by summing together the weighted phase-stepping curves (PSCs) with some kernel functions, which suggests us that the raw data at different steps have different contributions for the noise in retrieved contrasts. Therefore, it is possible to improve the noise performance by adjusting the fluence distribution during the phasestepping process directly. Based on analytic retrieval formulas and the Gaussian noise model for detected signals, we derived an optimal adaptive fluence distribution, which is proportional to the absolute weighting kernel functions and the root of original sample PSCs acquired under the constant fluence. Considering that the original sample PSC might be unavailable, we proposed two practical forms for the GBI and EI systems, which are also able to reduce the contrast noise when comparing with the constant fluence distribution. Since the kernel functions are target contrast-dependent, our proposed fluence adaptation mechanism provides a way of realizing a contrast-based dose optimization while keeping the same noise level. Results: To validate our analyses, simulations and experiments are conducted for the GBI and EI systems. Simulated results demonstrate that the dose reduction ratio between our proposed fluence distributions and the typical constant one can be about 20% for the phase contrast, which is consistent with our theoretical predictions. Although the experimental noise reduction ratios are a little smaller than the theoretical ones, low-dose experiments observe better noise performance by our proposed method. Our simulated results also give out the effective ranges of the parameters of the PSCs, such as the visibility in the GBI, the standard deviation, and the mean value in the EI, providing a guidance for the use of our proposed approach in practice.

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Section: Methodsmentioning

confidence: 99%

A \to M_{0} false(g false) = \int_{- π}^{π} g false(ϕ false) d ϕ = \frac{\int_{- π}^{π} s false(ϕ false) d ϕ}{\int_{- π}^{π} f false(ϕ false) d ϕ},

\begin{matrix} true \\ right P \to {\bar{M}}_{1} (g) & = & left \frac{M_{1} (g)}{M_{0} (g)} = \frac{\int_{- π}^{π} ϕ \cdot g (ϕ) d ϕ}{\int_{- π}^{π} g (ϕ) d ϕ} \\ = & left \frac{\int_{- π}^{π} h_{1} (ϕ) \cdot s (ϕ) d ϕ}{\int_{- π}^{π} s (ϕ) d ϕ} - \frac{\int_{- π}^{π} h_{1} (ϕ) \cdot f (ϕ) d ϕ}{\int_{- π}^{π} f (ϕ) d ϕ}, \end{matrix}

\begin{matrix} true \\ right D \to {\tilde{M}}_{2} (g) & = \end{matrix}

…”

Section: Methodsmentioning

confidence: 99%

Section: The Information Retrieval Algorithmsmentioning

confidence: 99%

Section: The Information Retrieval Algorithmsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Fluence adaptation for contrast‐based dose optimization in x‐ray phase‐contrast imaging

Xing

Zhang

et al. 2021

Medical Physics

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Fourier-based interpretation and noise analysis of the moments of small-angle x-ray scattering in grating-based x-ray phase contrast imaging

Xing

Zhang

et al. 2021

Opt. Express

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In grating-based x-ray phase contrast imaging, Fourier component analysis (FCA) is usually recognized as a gold standard to retrieve the contrasts including attenuation, phase and dark-field, since it is well-established on wave optics and is of high computational efficiency. Meanwhile, an alternative approach basing on the particle scattering theory is being developed and can provide similar contrasts with FCA by calculating multi-order moments of deconvolved small-angle x-ray scattering, so called as multi-order moment analysis (MMA). Although originated from quite different physics theories, the high consistency between the contrasts retrieved by FCA and MMA implies us that there may be some intrinsic connections between them, which has not been fully revealed to the best of our knowledge. In this work, we present a Fourier-based interpretation of MMA and conclude that the contrasts retrieved by MMA are actually the weighted compositions of Fourier coefficients, which means MMA delivers similar physical information as FCA. Based on the recognized cosine model, we also provide a truncated analytic MMA method, and its computational efficiency can be hundreds of times faster than the original deconvolution-based MMA method. Moreover, a noise analysis for our proposed truncated method is also conducted to further evaluate its performances. The results of numerical simulation and physical experiments support our analyses and conclusions.

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Talbot and Talbot—Lau X-ray interferometers

Лидер

2022

Phys. Usp.

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The trigonometric orthogonality of phase‐stepping curves in grating‐based x‐ray phase‐contrast imaging: Integral property and its implications for noise optimization

Cited by 4 publications

References 30 publications

Fluence adaptation for contrast‐based dose optimization in x‐ray phase‐contrast imaging

Fluence adaptation for contrast‐based dose optimization in x‐ray phase‐contrast imaging

Fourier-based interpretation and noise analysis of the moments of small-angle x-ray scattering in grating-based x-ray phase contrast imaging

Talbot and Talbot—Lau X-ray interferometers

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