On the Conditioning of Spectral Channelization (Energy Binning) and Its Impact on Multi-Material Decomposition Based Spectral Imaging in Photon-Counting CT

Ren, Yan; Xie, Huiqiao; Long, Wenting; Yang, Xiaofeng; Tang, Xiangyang

doi:10.1109/tbme.2020.3048661

Cited by 14 publications

(35 citation statements)

References 42 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Take the binning detector depicted in Figure 2(c) as an example, when Bin 1 and Bin 3 do not overlap, the system is invertible for

N = 3

(when imaging materials with no K‐edges). However, the overlap between bins would result in a reduction in the Jacobian and hence more uncertainty in the coefficient estimation, which has also been observed in other works 36,37 . One can employ the CRLB to optimize bin boundaries for a given set of AM coefficients

bold-italicA

.…”

Section: Discussionmentioning

confidence: 67%

“…However, the overlap between bins would result in a reduction in the Jacobian and hence more uncertainty in the coefficient estimation, which has also been observed in other works. 36,37 One can employ the CRLB to optimize bin boundaries for a given set of AM coefficients A. As the CRLB varies with A, the optimum bin boundaries depend on the prior distribution of the objects.…”

Section: Discussionmentioning

confidence: 99%

See 1 more Smart Citation

Invertibility of multi‐energy X‐ray transform

2021

View full text Add to dashboard Cite

The goal is to provide a sufficient condition for the invertibility of a multi-energy (ME) X-ray transform. The energy-dependent X-ray attenuation profiles can be represented by a set of coefficients using the Alvarez-Macovski (AM) method. An ME X-ray transform is a mapping from N AM coefficients to N noise-free energy-weighted measurements, where N ≥ 2. Methods: We apply a general invertibility theorem to prove the equivalence of global and local invertibility for an ME X-ray transform. We explore the global invertibility through testing whether the Jacobian of the mapping J(A) has zero values over the support of the mapping. The Jacobian of an arbitrary ME X-ray transform is an integration over all spectral measurements. A sufficient condition for J(A) ≠ 0 for all A is that the integrand of J(A) is ≥ 0 (or ≤ 0) everywhere. Note that the trivial case of the integrand equals 0 everywhere is ignored. Using symmetry, we simplified the integrand of the Jacobian to three factors that are determined by the total attenuation, the basis functions, and the energy-weighting functions, respectively. The factor related to the total attenuation is always positive; hence, the invertibility of the X-ray transform can be determined by testing the signs of the other two factors. Furthermore, we use the Cramér-Rao lower bound (CRLB) to characterize the noise-induced estimation uncertainty and provide a maximum-likelihood (ML) estimator. Results:The factor related to the basis functions is always negative when the photoelectric/Compton/Rayleigh basis functions are used and K-edge materials are not considered. The sign of the energy-weighting factor depends on the system source spectra and the detector response functions. For four special types of X-ray detectors, the sign of this factor stays the same over the integration range. Therefore, when these four types of detectors are used for imaging non-K-edge materials, the ME X-ray transform is globally invertible. The same framework can be used to study an arbitrary ME X-ray imaging system, for example, when K-edge materials are present. Furthermore, the ML estimator we presented is an unbiased, efficient estimator and can be used for a wide range of scenes. Conclusions: We have provided a framework to study the invertibility of an arbitrary ME X-ray transform and proved the global invertibility for four types of systems. K E Y W O R D Sinvertibility, multi-energy X-ray imaging, spectral X-ray imaging, X-rayThis is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs License, which permits use and distribution in any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made.

show abstract

“…Take the binning detector depicted in Figure 2(c) as an example, when Bin 1 and Bin 3 do not overlap, the system is invertible for

N = 3

bold-italicA

.…”

Section: Discussionmentioning

confidence: 67%

Section: Discussionmentioning

confidence: 99%

Invertibility of multi‐energy X‐ray transform

2021

View full text Add to dashboard Cite

show abstract

“…We anticipate that the FOMs supported by observer or model observer studies [29][30][31][32][33][34] may have to be employed for image quality assessment and comparison among the three detector materials in photon-counting CT for spectral imaging of biological tissues. [4][5][6][7] Finally, we indicate that only head is considered in this study, but the investigation can be readily extended to other anatomic locations, such as thorax, abdomen, and pelvis, in which the spectral channelization needs to be optimized to accommodate the anatomic locations at larger dimensions.…”

Section: Discussionmentioning

confidence: 99%

“…The underlying reason is most likely that the inter‐channel correlation plays an important and dominant role in determining the performance of material specific (or basis material) imaging. It should be noted herein that the inter‐channel correlation has to be perceived as not only the inter‐channel spectral overlapping, that is, the spectral conditioning 7 or simply the difference between the effective energies of the two spectral channels in data acquisition 11,12 (see Table 1), but also the similarity in the basis materials’ attenuating property over the channels (i.e., the conditioning of basis materials 6 ). From our point of view, this observation deserves further investigation.…”

Section: Discussionmentioning

confidence: 99%

“…It is found that, except for the two schemes (ch 1 , ch 2 ) and ((ch 1 + ch 4 ), (ch 2 + ch 3 )), all cases listed in Table 1 can go through material decomposition and generate material specific images for virtual monochromatic imaging at acceptable accuracy (i.e., no artifact). The major F I G U R E 8 Material specific images (soft tissue: top row; cortical bone: bottom row) acquired under realistic detector spectral response, corresponding to the benchmark scheme ((ch 1 +ch 2 ), (ch 3 +ch 4 )) and other schemes (ch 2 , ch 4 ), (ch 1 , ch 4 ), ((ch 1 +ch 3 ), (ch 2 +ch 4 )) and (ch 1 , ch 3 )+(ch 2 , ch 4 ) (left to right) reason behind the infeasibility of scheme (ch 1 , ch 2 ) is that the energy of the x-ray photons in channels 1 and 2 is relatively too low to penetrate the head phantom for adequate material decomposition, while that associated with the scheme ((ch 1 + ch 4 ), (ch 2 + ch 3 )) is that the effective energy of the two synthesized spectral channels are too close to each other ((62.6, 57.7) keV, see Table 1), which may lead to uncertainty (poor condition in spectral channelization 7 ) in material decomposition.…”

Section: Verification Of Materials Decomposition and Imaging Formationmentioning

confidence: 99%

See 1 more Smart Citation

Photon‐counting CT via interleaved/gapped spectral channels: Feasibility and imaging performance

2022

Self Cite

View full text Add to dashboard Cite

Purpose Compared to energy integration, photon‐counting x‐ray detection facilitates the spectral channelization (energy binning) in spectral CT and thus offers the opportunity to implement data acquisition via sophisticated schemes, for example, gapping or interleaving in spectral channels. In this article, we report our investigation of the performance of material decomposition based spectral imaging in photon‐counting CT implemented in such data acquisition schemes, and their comparison with the benchmark scheme and other schemes without spectral gapping or interleaving. Materials and methods Using a deliberately designed anthropomorphic head phantom that mimics the intracranial soft tissues and bony structures, a simulation study is carried out with the focus on two‐material decomposition based spectral imaging in photon‐counting computed tomography (CT), under both ideal and realistic detector spectral responses. The projection data are acquired in four spectral channels, and then are sorted to implement the schemes of gapping ((ch1, ch3); (ch2, ch4); (ch1, ch4)) and interleaving ((ch1, ch3) + (ch2, ch4); (ch1, ch4) + (ch2, ch3); ((ch1 + ch3), (ch2 + ch4)); ((ch1 + ch4), (ch2 + ch3))) in spectral channels, in addition to the benchmark scheme ((ch1 + ch2), (ch3 + ch4)) and other conventional schemes (ch1, ch2), (ch2, ch3) and (ch3, ch4), where ‘‘ch’’ denotes channel, ‘‘+’’ denote addition, and (·,·) the operation of material decomposition and image reconstruction. Using the contrast‐to‐noise ratio between targeted regions of interest as the figure of merit, we study the performance of spectral imaging (material specific and virtual monochromatic) associated with these spectral channelization schemes. Results Under ideal detector spectral response, the scheme (ch1, ch4) outperforms the benchmark scheme ((ch1 + ch2), (ch3 + ch4)) and others in gapped and/or interleaved spectral channelization in material specific imaging, while the interleaved scheme (ch1, ch4) + (ch2, ch3) performs the best in virtual monochromatic imaging. Notably, only about half x‐ray dose is utilized in the scheme (ch1, ch4) for image formation. Under realistic detector spectral response, the difference in imaging performance over all spectral channelization schemes diminishes, along with degradation in each scheme's individual performance. The results suggest that (i) different strategy in spectral channelization may have to be exercised in material specific imaging and virtual monochromatic imaging, respectively, and (ii) the spectral distortion in realistic detector's response due to charge‐sharing, Compton scattering, and fluorescent escaping should be mitigated as much as possible. Conclusion The spectral channelization schemes and associated imaging performance reported herein are novel and thus informative to the community, which may further the understanding of physical fundamentals and design principles for material decomposition based spectral imaging in photon‐counting CT and other x‐ray related imaging modalities.

show abstract

Noise correlation and its impact on the performance of multi‐material decomposition‐based spectral imaging in photon‐counting CT

Tang

Ren

Xie

2022

J Applied Clin Med Phys

Self Cite

View full text Add to dashboard Cite

Purpose It has been known that noise correlation plays an important role in the determination of the performance of spectral imaging based on two‐material decomposition (2‐MD). To further understand the basics of spectral imaging in photon‐counting CT toward optimal design and implementation, we study the noise correlation in multi‐MD ( m ‐MD) and its impact on the performance of spectral imaging. Method We derive the equations that characterize the noise and noise correlation in the material‐specific (basis) images in m ‐MD, followed by a simulation study to verify the derived equations and study the noise correlation's impact on the performance of spectral imaging. Using a specially designed digital phantom, the study of noise correlation runs over the cases of two‐, three‐, and four‐MD (2‐MD, 3‐MD, and 4‐MD). Then, the noise correlation's impact on the performance of spectral imaging in photon‐counting CT is investigated, using a modified Shepp–Logan phantom. Results The results in 2‐MD show that, in‐line with what has been reported in the literature, the noise correlation coefficient between the material‐specific images corresponding to the basis materials approaches −1. The results in m ‐MD ( m ≥ 3) are more complicated and interesting, as the noise correlation coefficients between a pair of the material‐specific images alternate between ±1, and so do in the case of 4‐MD. The m ‐MD data show that the noise in virtual monochromatic imaging (a form of spectral imaging) is moderate even though the noises in material‐specific (basis) images vary drastically. Conclusions The observation of noise correlation in 3‐MD, 4‐MD, and beyond (i.e., m ‐MD) is informative to the community. The relationship between noise correlation and the performance of spectral imaging revealed in this work may help clinical medical physicists understand the fundamentals of spectral imaging based on MD and optimize the performance of spectral imaging in photon‐counting CT and other X‐ray imaging modalities.

show abstract

On the Conditioning of Spectral Channelization (Energy Binning) and Its Impact on Multi-Material Decomposition Based Spectral Imaging in Photon-Counting CT

Cited by 14 publications

References 42 publications

Invertibility of multi‐energy X‐ray transform

Invertibility of multi‐energy X‐ray transform

Photon‐counting CT via interleaved/gapped spectral channels: Feasibility and imaging performance

Noise correlation and its impact on the performance of multi‐material decomposition‐based spectral imaging in photon‐counting CT

Contact Info

Product

Resources

About