CT Image Quality Characterization

Abstract

This chapter presents mathematical and physics foundations of quantitative CT image quality assessment. The linear systems theory was applied to derive quantitative relations between noise, spatial resolution, and CT number of an idealized linear CT system and the associated hardware specifications, acquisition, and reconstruction parameters. The derivations not only highlighted the physical and mathematical foundations of several widely recognized image quality “laws” (e.g., the inverse scaling law between CT noise variance and radiation exposure level) but also the underlying assumptions behind these laws. Next, the chapter showed how these assumptions can be violated in realistic modern CT systems, even for those employing the linear filtered backprojection (FBP) algorithm. It demonstrated how the classical image quality models can be revised for modern systems. One revision is the introduction of a virtual image object with negative attenuation to compensate for the spatially and directionally varying impacts of the bowtie filter to CT noise power spectrum (NPS). After its revision, linear systems theory can still be used to model CT image quality of FBP-based realistic CT systems. Finally, the chapter demonstrated severe violations of the classical linear systems theory in CT systems with nonlinear model-based iterative reconstruction (MBIR) algorithms and the new challenges in estimating image quality properties for these systems. The chapter presented strategies toward addressing these challenges, including ensemble statistics- and task-based image quality assessment methods as well as empirical modeling of the relationship between image quality performance and system parameters. It also clarified the concept of NPS and justified the applicability of this objective image quality metric in nonlinear MBIR-based systems with nonstationary noise. Throughout this chapter, discussion on the topic of CT image quality was accompanied by illustrations and plots of experimental data, some of which are published for the first time.

Appendices

Appendix I: Autocovariance of Post-log Projection Data

Based on its definition, the autocovariance of R is given by

$$ {\displaystyle \begin{array}{c}{C}_R\left({\rho}_i,{\theta}_m;{\rho}_j,{\theta}_n\right)=\left\langle \Delta R\left({\rho}_i,{\theta}_m\right)\Delta R\left({\rho}_j,{\theta}_n\right)\right\rangle \\ {}=\left\langle \left[\ln N\left({\rho}_i,{\theta}_m\right)-\left\langle \ln N\left({\rho}_i,{\theta}_m\right)\right\rangle \right]\left[\ln N\left({\rho}_j,{\theta}_n\right)-\left\langle \ln N\left({\rho}_j,{\theta}_n\right)\right\rangle \right]\right\rangle \\ {}=\left\langle \ln N\left({\rho}_i,{\theta}_m\right)\ln N\left({\rho}_j,{\theta}_n\right)\right\rangle -\left\langle \ln N\left({\rho}_i,{\theta}_m\right)\right\rangle \left\langle \ln N\left({\rho}_j,{\theta}_n\right)\right\rangle \\ {}={\mathrm{term}}_1-{\mathrm{term}}_2.\end{array}} $$

(6.87)

To facilitate the calculation of term₁ and term₂, we can perform a Taylor expansion of ln N around〈N〉:

$$ {\displaystyle \begin{array}{c}\ln N=\sum \limits_{j=0}^{+\infty}\frac{{\left.{\left(\ln N\right)}^{(j)}\right|}_{N=\left\langle N\right\rangle }}{j!}{\left(N-\left\langle N\right\rangle \right)}^j\\ {}=\ln \left\langle N\right\rangle +\frac{1}{\left\langle N\right\rangle}\left(N-\left\langle N\right\rangle \right)-\frac{1}{2{\left\langle N\right\rangle}^2}{\left(N-\left\langle N\right\rangle \right)}^2+\cdots \end{array}} $$

(6.88)

Based on Eq. (6.88), the expected value of ln N can be expressed as

$$ {\displaystyle \begin{array}{c}\left\langle \ln N\right\rangle =\ln \left\langle N\right\rangle +\frac{1}{\left\langle N\right\rangle}\left\langle N-\left\langle N\right\rangle \right\rangle -\frac{1}{2{\left\langle N\right\rangle}^2}\left\langle {\left(N-\left\langle N\right\rangle \right)}^2\right\rangle +\cdots \\ {}=\ln \left\langle N\right\rangle +0-\frac{\sigma_N^2}{2{\left\langle N\right\rangle}^2}+\cdots \end{array}} $$

(6.89)

With the expanded form of lnN and 〈lnN〉, term₁ in Eq. (6.87) can be approximated by

$$ {\displaystyle \begin{array}{l}\left\langle \ln N\left({\rho}_i,{\theta}_m\right)\ln N\left({\rho}_j,{\theta}_n\right)\right\rangle \\ {}=\left\langle \left\{\ln \left\langle N\left({\rho}_i,{\theta}_m\right)\right\rangle +\frac{1}{\left\langle N\left({\rho}_i,{\theta}_m\right)\right\rangle}\left[N\left({\rho}_i,{\theta}_m\right)-\left\langle N\left({\rho}_i,{\theta}_m\right)\right\rangle \right]-\frac{1}{2{\left\langle N\left({\rho}_i,{\theta}_m\right)\right\rangle}^2}{\left[N\left({\rho}_i,{\theta}_m\right)-\left\langle N\left({\rho}_i,{\theta}_m\right)\right\rangle \right]}^2+\cdots \right\}\right.\\ {}\left.\times \left\{\ln \left\langle N\left({\rho}_j,{\theta}_n\right)\right\rangle +\frac{1}{\left\langle N\left({\rho}_j,{\theta}_n\right)\right\rangle}\left[N\left({\rho}_j,{\theta}_n\right)-\left\langle N\left({\rho}_j,{\theta}_n\right)\right\rangle \right]-\frac{1}{2{\left\langle N\left({\rho}_j,{\theta}_n\right)\right\rangle}^2}{\left[N\left({\rho}_j,{\theta}_n\right)-\left\langle N\left({\rho}_j,{\theta}_n\right)\right\rangle \right]}^2+\cdots \right\}\right\rangle \\ {}\approx \ln \left\langle N\left({\rho}_i,{\theta}_m\right)\right\rangle \ln \left\langle N\left({\rho}_j,{\theta}_n\right)\right\rangle -\ln \left\langle N\left({\rho}_i,{\theta}_m\right)\right\rangle \frac{\sigma_N^2\left({\rho}_j,{\theta}_n\right)}{2{\left\langle N\left({\rho}_j,{\theta}_n\right)\right\rangle}^2}-\ln \left\langle N\left({\rho}_j,{\theta}_n\right)\right\rangle \frac{\sigma_N^2\left({\rho}_i,{\theta}_m\right)}{2{\left\langle N\left({\rho}_i,{\theta}_m\right)\right\rangle}^2}+\frac{C_N\left({\rho}_i,{\theta}_m;{\rho}_j,{\theta}_n\right)}{\left\langle N\left({\rho}_i,{\theta}_m\right)\right\rangle \left\langle N\left({\rho}_j,{\theta}_n\right)\right\rangle }.\end{array}} $$

(6.90)

where we ignored terms higher than the second order since they are much smaller than the zeroth and first order terms. Using similar method, term₂ in Eq. (6.87) can be approximated by

$$ {\displaystyle \begin{array}{l}\left\langle \ln N\left({\rho}_i,{\theta}_m\right)\right\rangle \left\langle \ln N\left({\rho}_j,{\theta}_n\right)\right\rangle =\left[\ln \left\langle N\left({\rho}_i,{\theta}_m\right)\right\rangle -\frac{\sigma_N^2\left({\rho}_i,{\theta}_m\right)}{2{\left\langle N\left({\rho}_i,{\theta}_m\right)\right\rangle}^2}+\cdots \right]\left[\ln \left\langle N\left({\rho}_j,{\theta}_n\right)\right\rangle -\frac{\sigma_N^2\left({\rho}_j,{\theta}_n\right)}{2{\left\langle N\left({\rho}_j,{\theta}_n\right)\right\rangle}^2}+\cdots \right]\\ {}\approx \ln \left\langle N\left({\rho}_i,{\theta}_m\right)\right\rangle \ln \left\langle N\left({\rho}_j,{\theta}_n\right)\right\rangle -\ln \left\langle N\left({\rho}_i,{\theta}_m\right)\right\rangle \frac{\sigma_N^2\left({\rho}_j,{\theta}_n\right)}{2{\left\langle N\left({\rho}_j,{\theta}_n\right)\right\rangle}^2}-\ln \left\langle N\left({\rho}_j,{\theta}_n\right)\right\rangle \frac{\sigma_N^2\left({\rho}_i,{\theta}_m\right)}{2{\left\langle N\left({\rho}_i,{\theta}_m\right)\right\rangle}^2}.\end{array}} $$

(6.91)

By subtracting term₂ in Eq. (6.91) from term₁ in Eq. (6.90), majority of the common terms can be cancelled, leading to a much simplified expression for C _R:

$$ {C}_R\left({\rho}_i,{\theta}_m;{\rho}_j,{\theta}_n\right)\approx \frac{C_N\left({\rho}_i,{\theta}_m;{\rho}_j,{\theta}_n\right)}{\left\langle N\left({\rho}_i,{\theta}_m\right)\right\rangle \left\langle N\left({\rho}_j,{\theta}_n\right)\right\rangle }. $$

(6.92)

Using the formula for C _N in Eq. (6.8), formula above can be written as

$$ {\displaystyle \begin{array}{c}{C}_R\left({\rho}_i,{\theta}_m;{\rho}_j,{\theta}_n\right)\approx \frac{\sigma_N^2\left({\rho}_i,{\theta}_m\right){\delta}_{i,j}{\delta}_{m,n}}{\left\langle N\left({\rho}_i,{\theta}_m\right)\right\rangle \left\langle N\left({\rho}_j,{\theta}_n\right)\right\rangle}\\ {}=\frac{\sigma_N^2\left({\rho}_i,{\theta}_m\right){\delta}_{i,j}{\delta}_{m,n}}{{\left\langle N\left({\rho}_i,{\theta}_m\right)\right\rangle}^2}\\ {}=\frac{\delta_{i,j}{\delta}_{m,n}}{SNR_{N\left({\rho}_i,{\theta}_m\right)}^2}.\end{array}} $$

(6.93)

Appendix II: CT Autocovariance

The autocovariance of $\hat{\mu}$ is defined as

$$ {C}_{\hat{\mu}}\left({x}_i,{y}_i;{x}_j,{y}_j\right)=\left\langle \Delta \hat{\mu}\left({x}_i,{y}_i\right)\Delta \hat{\mu}\left({x}_j,{y}_j\right)\right\rangle . $$

(6.94)

Since $\Delta \hat{\mu}$ is a real number, Eq. (6.94) can also be written as

$$ {C}_{\hat{\mu}}\left({x}_i,{y}_i;{x}_j,{y}_j\right)=\left\langle \Delta \hat{\mu}\left({x}_i,{y}_i\right)\Delta {\hat{\mu}}^{\ast}\left({x}_j,{y}_j\right)\right\rangle . $$

(6.95)

According to the FBP reconstruction formula in Eq. (6.16), $\Delta \hat{\mu}$ is related to the projection data by

$$ {\displaystyle \begin{array}{c}\Delta \hat{\mu}\left(x,y\right)=\hat{\mu}\left(x,y\right)-\left\langle \hat{\mu}\left(x,y\right)\right\rangle \\ {}=\underset{0}{\overset{\pi }{\int }}\mathrm{d}\theta \underset{-\infty }{\overset{+\infty }{\int }} dk\tilde{R}\left(k,\theta \right)G(k)\mid k\mid {\mathrm{e}}^{\mathrm{i}2\pi k\left(x\cos \theta +y\sin \theta \right)}\\ {}-\left\langle \underset{0}{\overset{\pi }{\int }}\mathrm{d}\theta \underset{-\infty }{\overset{+\infty }{\int }} dk\tilde{R}\left(k,\theta \right)G(k)|k|{\mathrm{e}}^{\mathrm{i}2\pi k\left(x\cos \theta +y\sin \theta \right)}\right\rangle \\ {}=\underset{0}{\overset{\pi }{\int }}\mathrm{d}\theta \underset{-\infty }{\overset{+\infty }{\int }} dk\left[\tilde{R}\left(k,\theta \right)-\left\langle \tilde{R}\left(k,\theta \right)\right\rangle \right]G(k)\mid k\mid {\mathrm{e}}^{\mathrm{i}2\pi k\left(x\cos \theta +y\sin \theta \right)}\\ {}=\underset{0}{\overset{\pi }{\int }}\mathrm{d}\theta \underset{-\infty }{\overset{+\infty }{\int }} dk\varDelta \tilde{R}\left(k,\theta \right)G(k)\mid k\mid {\mathrm{e}}^{\mathrm{i}2\pi k\left(x\cos \theta +y\sin \theta \right)}.\end{array}} $$

(6.96)

Based on Eq. (6.96), the autocovariance of $\Delta \hat{\mu}$ in Eq. (6.95) can be expanded as

$$ {\displaystyle \begin{array}{c}{C}_{\hat{\mu}}\left({x}_i,{y}_i;{x}_j,{y}_j\right)=\left\langle \underset{0}{\overset{\pi }{\int }}\mathrm{d}{\theta}_1\underset{-\infty }{\overset{+\infty }{\int }}\mathrm{d}{k}_1\Delta \tilde{R}\left({k}_1,{\theta}_1\right)G\left({k}_1\right)|{k}_1|{\mathrm{e}}^{\mathrm{i}2\pi {k}_1\left({x}_i\cos {\theta}_1+{y}_i\sin {\theta}_1\right)}\right.\\ {}\left.\times \underset{0}{\overset{\pi }{\int }}\mathrm{d}{\theta}_2\underset{-\infty }{\overset{+\infty }{\int }}\mathrm{d}{k}_2\Delta {\tilde{R}}^{\ast}\left({k}_2,{\theta}_2\right){G}^{\ast}\left({k}_2\right)|{k}_2|{\mathrm{e}}^{\hbox{-} \mathrm{i}2\pi {k}_2\left({x}_j\cos {\theta}_2+{y}_j\sin {\theta}_2\right)}\right\rangle \\ {}=\underset{0}{\overset{\pi }{\int }}\mathrm{d}{\theta}_1\underset{-\infty }{\overset{+\infty }{\int }}\mathrm{d}{k}_1G\left({k}_1\right)\mid {k}_1\mid \underset{0}{\overset{\pi }{\int }}\mathrm{d}{\theta}_2\underset{-\infty }{\overset{+\infty }{\int }}\mathrm{d}{k}_2{G}^{\ast}\left({k}_2\right)\mid {k}_2\mid \left\langle \Delta \tilde{R}\left({k}_1,{\theta}_1\right)\Delta {\tilde{R}}^{\ast}\left({k}_2,{\theta}_2\right)\right\rangle \\ {}\times {\mathrm{e}}^{\mathrm{i}2\pi \left[{x}_i{k}_1\cos {\theta}_1-{x}_j{k}_2\cos {\theta}_2+{y}_i{k}_1\sin {\theta}_1-{y}_j{k}_2\sin {\theta}_2\right]},\end{array}} $$

(6.97)

The term $\left\langle \Delta \tilde{R}\left({k}_1,{\theta}_1\right)\Delta {\tilde{R}}^{\ast}\left({k}_2,{\theta}_2\right)\right\rangle$ in Eq. (6.97) is given by

$$ {\displaystyle \begin{array}{l}\left\langle \Delta \tilde{R}\left({k}_1,{\theta}_1\right)\Delta {\tilde{R}}^{\ast}\left({k}_2,{\theta}_2\right)\right\rangle =\left\langle \int d{\rho}_1\right.\Delta R\left({\rho}_1,{\theta}_1\right){\mathrm{e}}^{-i2\pi {k}_1{\rho}_1}\left.\int d{\rho}_2\Delta {R}^{\ast}\left({\rho}_2,{\theta}_2\right){\mathrm{e}}^{i2\pi {k}_2{\rho}_2}\right\rangle \\ {}=\int d{\rho}_1\int d{\rho}_2\left\langle \Delta R\left({\rho}_1,{\theta}_1\right)\Delta {R}^{\ast}\left({\rho}_2,{\theta}_2\right)\right\rangle {\mathrm{e}}^{-i2\pi \left({k}_1{\rho}_1-{k}_2{\rho}_2\right)}\\ {}=\int d{\rho}_1\int d{\rho}_2{C}_R\left({\rho}_1,{\theta}_1;{\rho}_2,{\theta}_2\right){\mathrm{e}}^{-i2\pi \left({k}_1{\rho}_1-{k}_2{\rho}_2\right)}\\ {}=\int d{\rho}_1\int d{\rho}_2\frac{\Delta \rho \Delta \theta \delta \left({\rho}_1-{\rho}_2\right)\delta \left({\theta}_1-{\theta}_2\right)}{\left\langle N\right\rangle }{\mathrm{e}}^{-i2\pi \left({k}_1{\rho}_1-{k}_2{\rho}_2\right)}\\ {}=\frac{\Delta \rho \Delta \theta \delta \left({\theta}_1-{\theta}_2\right)}{\left\langle N\right\rangle}\int d{\rho}_1{\mathrm{e}}^{-i2{\pi \rho}_1\left({k}_1-{k}_2\right)}\\ {}=\frac{\Delta \rho \Delta \theta }{\left\langle N\right\rangle}\delta \left({\theta}_1-{\theta}_2\right)\delta \left({k}_1-{k}_2\right).\end{array}} $$

(6.98)

Based on Eq. (6.98), ${C}_{\hat{\mu}}\left({x}_i,{y}_i;{x}_j,{y}_j\right)$ in Eq. (6.97) can be simplified as follows:

$$ {\displaystyle \begin{array}{c}{C}_{\hat{\mu}}\left({x}_i,{y}_i;{x}_j,{y}_j\right)=\frac{\Delta \rho \Delta \theta }{\left\langle N\right\rangle}\underset{0}{\overset{\pi }{\int }}\mathrm{d}{\theta}_1\underset{-\infty }{\overset{+\infty }{\int }}{dk}_1G\left({k}_1\right)\mid {k}_1\mid \underset{0}{\overset{\pi }{\int }}\mathrm{d}{\theta}_2\underset{-\infty }{\overset{+\infty }{\int }}{dk}_2{G}^{\ast}\left({k}_2\right)\mid {k}_2\mid \delta \left({\theta}_1-{\theta}_2\right)\delta \left({k}_1-{k}_2\right)\\ {}\times {\mathrm{e}}^{\mathrm{i}2\pi \left[{x}_i{k}_1\cos {\theta}_1-{x}_j{k}_2\cos {\theta}_2+{y}_i{k}_1\sin {\theta}_1-{y}_j{k}_2\sin {\theta}_2\right]}\\ {}=\frac{\Delta \rho \Delta \theta }{\left\langle N\right\rangle}\underset{0}{\overset{\pi }{\int }}\mathrm{d}{\theta}_1\underset{-\infty }{\overset{+\infty }{\int }}{dk}_1{\left|G\left({k}_1\right)\right|}^2{\left|{k}_1\right|}^2\underset{0}{\overset{\pi }{\int }}\mathrm{d}{\theta}_2\delta \left({\theta}_1-{\theta}_2\right){\mathrm{e}}^{\mathrm{i}2\pi {k}_1\left[\left({x}_i\cos {\theta}_1-{x}_j\cos {\theta}_2\right)+\left({y}_i\sin {\theta}_1-{y}_j\sin {\theta}_2\right)\right]}\\ {}=\frac{\Delta \rho \Delta \theta }{\left\langle N\right\rangle}\underset{0}{\overset{\pi }{\int }}\mathrm{d}{\theta}_1\underset{-\infty }{\overset{+\infty }{\int }}{dk}_1{\left|G\left({k}_1\right)\right|}^2{\left|{k}_1\right|}^2{\mathrm{e}}^{\mathrm{i}2\pi {k}_1\left[\left({x}_i-{x}_j\right)\cos {\theta}_1+\left({y}_i-{y}_j\right)\sin {\theta}_1\right]}\\ {}=\frac{\Delta \rho \Delta \theta }{\left\langle N\right\rangle}\underset{0}{\overset{\pi }{\int }}\mathrm{d}\theta \underset{-\infty }{\overset{+\infty }{\int }} dk{\left|G(k)\right|}^2{\left|k\right|}^2{\mathrm{e}}^{\mathrm{i}2\pi k\left[\left({x}_i-{x}_j\right)\cos \theta +\left({y}_i-{y}_j\right)\sin \theta \right]}.\end{array}} $$

(6.99)

By switching from polar coordinate system to Cartesian coordinate system , Eq. (6.99) can be written in the following form:

$$ {\displaystyle \begin{array}{c}{C}_{\hat{\mu}}\left({x}_i,{y}_i;{x}_j,{y}_j\right)=\frac{\Delta \rho \Delta \theta }{\left\langle N\right\rangle}\underset{-\infty }{\overset{+\infty }{\int }}\underset{-\infty }{\overset{+\infty }{\int }}{dk}_x{dk}_yJ{\left|G(k)\right|}^2{\left|k\right|}^2{\mathrm{e}}^{\mathrm{i}2\pi \left[\left({x}_i-{x}_j\right){k}_x+\left({y}_i-{y}_j\right){k}_y\right]}\\ {}=\frac{\Delta \rho \Delta \theta }{\left\langle N\right\rangle}\underset{-\infty }{\overset{+\infty }{\int }}\underset{-\infty }{\overset{+\infty }{\int }}{dk}_x{dk}_y\frac{1}{\mid k\mid }{\left|G(k)\right|}^2{\left|k\right|}^2{\mathrm{e}}^{\mathrm{i}2\pi \left[\left({x}_i-{x}_j\right){k}_x+\left({y}_i-{y}_j\right){k}_y\right]}\\ {}=\frac{\Delta \rho \Delta \theta }{\left\langle N\right\rangle}\underset{-\infty }{\overset{+\infty }{\int }}\underset{-\infty }{\overset{+\infty }{\int }}{dk}_x{dk}_y{\left|G(k)\right|}^2\mid k\mid {\mathrm{e}}^{\mathrm{i}2\pi \left[\left({x}_i-{x}_j\right){k}_x+\left({y}_i-{y}_j\right){k}_y\right]},\end{array}} $$

(6.100)

where k _x = k cos θ, k _y = k sin θ, and J = 1/ ∣ k∣ are the Jacobian factor introduced by the polar-to-Cartesian coordinate system change.

The noise variance of $\hat{\mu}$ is related to ${C}_{\hat{\mu}}$ by

$$ {\displaystyle \begin{array}{c}{\sigma}_{\hat{\mu}}^2={C}_{\hat{\mu}}\left({x}_i,{y}_i;{x}_j={x}_i,{y}_j={y}_i\right)\\ {}=\frac{\Delta \rho \Delta \theta }{\left\langle N\right\rangle}\underset{-\infty }{\overset{+\infty }{\int }}\underset{-\infty }{\overset{+\infty }{\int }}\mathrm{d}{k}_x\mathrm{d}{k}_y{\left|G(k)\right|}^2\mid k\mid .\end{array}} $$

(6.101)

Alternatively, the integration in Eq. (6.101) can be written in the polar coordinate system as

$$ {\displaystyle \begin{array}{c}{\sigma}_{\hat{\mu}}^2=\frac{\Delta \rho \Delta \theta }{\left\langle N\right\rangle}\underset{0}{\overset{\pi }{\int }} d\theta \underset{-\infty }{\overset{+\infty }{\int }} d k{J}^{-1}{\left|G(k)\right|}^2\mid k\mid \\ {}=\frac{\Delta \rho \Delta \theta }{\left\langle N\right\rangle}\underset{0}{\overset{\pi }{\int }} d\theta \underset{-\infty }{\overset{+\infty }{\int }} d k{\left|k\Big\Vert G(k)\right|}^2\mid k\mid \\ {}=\frac{\pi \Delta \rho \Delta \theta }{\left\langle N\right\rangle}\underset{-\infty }{\overset{+\infty }{\int }} d k{\left|G(k)\right|}^2{\left|k\right|}^2.\end{array}} $$

(6.102)

Appendix III: CT Image of a Point-Like Object Generated from an Idealized Linear System

According to [33, 34], the radiological path length of an image object μ(x, y) along the (ρ, θ) ray is given by

$$ \underset{L\left(\rho, \theta \right)}{\int }u\left(x,y\right)\mathrm{d}l=\iint u\left(x,y\right)\delta \left(\rho -x\cos \theta -y\sin \theta \right)\mathrm{d}\mathrm{xdy} $$

(6.103)

For the point object in Eq. (6.32), the radiological path length is

$$ {\displaystyle \begin{array}{c}\iint u\left(x,y\right)\delta \left(\rho -x\cos \theta -y\sin \theta \right)\mathrm{dxdy}\\ {}=\iint {u}_0\Delta x\Delta y\delta \left(x-{x}_0\right)\delta \left(y-{y}_0\right)\delta \left(\rho -x\cos \theta -y\sin \theta \right)\mathrm{dxdy}\\ {}={u}_0\Delta x\Delta y\int dx\delta \left(x-{x}_0\right)\int \mathrm{d} y\delta \left(y-{y}_0\right)\delta \left(\rho -x\cos \theta -y\sin \theta \right)\\ {}={u}_0\Delta x\Delta y\int dx\delta \left(x-{x}_0\right)\delta \left(\rho -x\cos \theta -{y}_0\sin \theta \right)\\ {}={u}_0\Delta x\Delta y\delta \left(\rho -{x}_0\cos \theta -{y}_0\sin \theta \right).\end{array}} $$

(6.104)

Based on (6.104) and the Beer-Lambert law , the expected number of photons is

$$ \left\langle N\left(\rho, \theta \right)\right\rangle =\left\langle {n}_0\right\rangle \Delta t\Delta z\Delta \rho {\mathrm{e}}^{-{u}_0 A\delta \left(\rho -{x}_0\cos \theta -{y}_0\sin \theta \right)}. $$

(6.105)

Expected value of the post-log projection data is given by

$$ {\displaystyle \begin{array}{c}\left\langle R\left(\rho, \theta \right)\right\rangle =\left\langle \ln \frac{N_0\left(\rho, \theta \right)}{N\left(\rho, \theta \right)}\right\rangle \\ {}=\left\langle \ln {N}_0\left(\rho, \theta \right)\right\rangle -\left\langle \ln N\left(\rho, \theta \right)\right\rangle \\ {}\approx \ln \left\langle {N}_0\left(\rho, \theta \right)\right\rangle -\ln \left\langle N\left(\rho, \theta \right)\right\rangle \\ {}={u}_0\Delta x\Delta y\delta \left(\rho -{x}_0\cos \theta -{y}_0\sin \theta \right).\end{array}} $$

(6.106)

With 〈R(ρ, θ)〉, the following FBP reconstruction can be performed to get $\left\langle \hat{u}\left(x,y\right)\right\rangle$:

$$ \left\langle \hat{u}\left(x,y\right)\right\rangle =\int \mathrm{d}\theta \int dk\left|k\right|G(k)\left\langle \tilde{R}\left(k,\theta \right)\right\rangle {\mathrm{e}}^{\mathrm{i}2\pi k\left(x\cos \theta +y\sin \theta \right)}, $$

(6.107)

where $\left\langle \tilde{R}\left(k,\theta \right)\right\rangle$ is given by

$$ {\displaystyle \begin{array}{c}\left\langle \tilde{R}\left(k,\theta \right)\right\rangle =\int \mathrm{d}\rho \left\langle R\left(\rho, \theta \right)\right\rangle {\mathrm{e}}^{-\mathrm{i}2\pi k\rho}\\ {}={u}_0\Delta x\Delta y\int \mathrm{d}\rho \delta \left(\rho -{x}_0\cos \theta -{y}_0\sin \theta \right){\mathrm{e}}^{-\mathrm{i}2\pi k\rho}\\ {}={u}_0\Delta x\Delta y{\mathrm{e}}^{-\mathrm{i}2\pi k\left({x}_0\cos \theta +{y}_0\sin \theta \right)}.\end{array}} $$

(6.108)

By putting (6.108) into (6.107), we have

$$ {\displaystyle \begin{array}{c}\left\langle \hat{u}\left(x,y\right)\right\rangle ={u}_0\Delta x\Delta y\int \mathrm{d}\theta \int \mathrm{d}k\left|k\right|G(k){\mathrm{e}}^{-\mathrm{i}2\pi k\left({x}_0\cos \theta +{y}_0\sin \theta \right)}{\mathrm{e}}^{\mathrm{i}2\pi k\left(x\cos \theta +y\sin \theta \right)}\\ {}={u}_0\Delta x\Delta y\int \mathrm{d}\theta \int \mathrm{d}k\left|k\right|G(k){\mathrm{e}}^{\mathrm{i}2\pi \left[\left(x-{x}_0\right)k\cos \theta +\left(y-{y}_0\right)k\sin \theta \right]}.\end{array}} $$

(6.109)

By switching from polar-to- Cartesian coordinate system , Eq. (6.109) becomes

$$ {\displaystyle \begin{array}{c}\left\langle \hat{u}\left(x,y\right)\right\rangle ={u}_0\Delta x\Delta y{\int}_{-\infty}^{+\infty }{\int}_{-\infty}^{+\infty }{dk}_x{dk}_yG(k){e}^{\mathrm{i}2\pi \left[{k}_x\left(x-{x}_0\right)+{k}_y\left(y-{y}_0\right)\right]}\\ {}={u}_0\Delta x\Delta y{IFT}_{2\mathrm{D}}\left\{G(k)\right\}\\ {}={u}_0\Delta x\Delta yg\left(x-{x}_0,y-{y}_0\right),\end{array}} $$

(6.110)

where g(x − x ₀, y − y ₀) denotes the 2D Fourier transform of the window function G(k).