4Transmission and emission tomography

This book considers real imaging systems in this chapter. If the radiation source is outside the patient, the imaging system acquires transmission data. If the radiation sources are inside the patient, the imaging system acquires the emission data. For transmission scans, the image to be obtained is a map (or distribution) of the attenuation coefficients inside the patient. For the emission scans, the image to be obtained is the distribution of the injected isotopes within the patient. Even for emission scans, an additional transmission scan is sometimes required in order to compensate for the attenuation effect of the emission photons. Some attenuation compensation methods for emission imaging are discussed in this chapter.

4.1X-ray computed tomography

In this chapter, we relate transmission and emission tomography measurements to line-integral data so that the reconstruction algorithms mentioned in the previous chapters can be used to reconstruct practical data in medical imaging.

X-ray computed tomography (CT) uses transmission measurements to estimate a cross-sectional image within the patient body. X-rays have very high energy, and they are able to penetrate the patient body. However, not every X-ray can make it through the patient’s body. Some X-rays get scattered within the body, and their energy gets weakened. During X-ray scattering, an X-ray photon interacts with an electron within the patient, transfers part of its energy to that electron, and dislodges the electron (see Figure 4.1). The X-ray is then bounced to a new direction with decreased energy.

Some other X-rays completely disappear within the body, converting their energy to the tissues in the body, for example, via the photoelectric conversion. The photo-electric effect is a process in which the X-ray photon energy is completely absorbed by an atom within the patient. The absorbed energy ejects an electron from the atom (see Figure 4.2).

Energy deposition within the body can damage DNA if the X-ray dose is too large. Let the X-ray intensity before entering the patient be I0, and the intensity departing the patient be Id; I0 and Id follow the Beer’s law (see Figure 4.3):

$\frac{I_d}{I_0}=\exp \left(-p\right),$

where p is the line integral of the linear attenuation coefficients along the path of the X-rays. A line integral of the attenuation coefficients is obtained by

$p=\text{In}\left(\frac{I_0}{I_d}\right),$

which is supplied to the image reconstruction algorithm for image reconstruction.

Fig. 4.1: Schematic representation of Compton scattering. The incident photon transfers part of its energy to an electron and is scattered in a new direction.

Fig. 4.2: Schematic representation of the photoelectric effect. The incident photon transfers all its energy to an electron and disappears.

Fig. 4.3: The X-ray intensity is reduced after going through the object.

Fig. 4.4: An X-ray CT image.

The goal of X-ray CT is to obtain a cross-sectional image of various attenuation coefficients within the body. A typical X-ray CT image is shown in Figure 4.4. The attenuation coefficient (commonly denoted by notation μ) is a property of a material; it is the logarithm of the input/output intensity ratio per unit length. Bones have higher μ values, and soft tissue has lower μ values. The attenuation coefficient of a material varies with the incoming X-ray energy; it becomes smaller when the X-ray energy gets higher.

Fig. 4.5: In the first-generation CT, the X-ray tube and the detector translate and rotate.

The first-generation CT, which is no longer in use, had one small detector (see Figure 4.5). The X-ray source and the detector have two motions: linear translation and rotation. The X-ray source sends out a narrow pencil beam to obtain parallel-beam projections. The scanning time was rather long (about 25 min).

The second-generation CT used narrow fan-beam geometry, consisting of 12 detectors (see Figure 4.6). Like the first-generation CT, it has two motions: linear translation and rotation. Due to the fan-beam geometry, the scan time was shortened to about 1 min.

Fig. 4.6: The second-generation CT uses narrow fan-beam X-rays. The X-ray tube and the detector translate across the field of view and rotate around the object.

Fig. 4.7: The third-generation CT uses wide fan-beam X-rays. The X-ray tube and the detector rotate around the object; they do not translate anymore.

Fig. 4.8: In the fourth-generation CT, the ring detector does not rotate. The X-ray source rotates around the object.

The third-generation CT uses wide fan-beam geometry, consisting of approximately 1,000 detectors (see Figure 4.7). No linear translation motion is necessary, and the scanning time was further reduced to about 0.5 s. The third-generation CT is currently very popular in medical imaging.

The fourth-generation CT has a stationary ring detector. The X-ray source rotates around the subject (see Figure 4.8). This scanning method forms a very fast fan-beam imaging geometry; however, it is impossible to collimate the X-rays on the detector, which causes this geometry to suffer from high rates of scattering.

Modern CT can perform helical scans, which is implemented as translating the patient bed in the axial direction as the X-ray source and the detectors rotate. The modern CT has a 2D multi-row detector, and it acquires cone-beam data (see Figure 4.9). Image reconstruction methods for the cone-beam geometry will be covered in the next chapter.

Fig. 4.9: The modern CT can perform cone-beam helix scans with a 2D detector. Some systems have multiple X-ray sources and 2D detectors. The helix orbit is implemented by translating the patient bed while the source and detector rotate.

4.2Positron emission tomography and single-photon emission computed tomography

In the last section, transmission imaging was discussed. In transmission imaging, the radiation source is placed outside the patient body. The radiation rays (either X-rays or gamma rays) enter the patient from outside, pass through the patient body, exit the patient, and finally get detected by a detector outside.

This section will change the subject to emission imaging, where the radiation sources are inside the patient body. Radiation is generated inside the patient body, emitted from within, and detected by a detector after it escapes from the patient body.

Radioactive atoms with a short half-life are generated in a cyclotron or a nuclear reactor. Radiopharmaceuticals are then made and injected into a patient (in a peripheral arm vein) to trace disease processes. The patient can also inhale or ingest the radiotracer. Radiopharmaceuticals are carrier molecules with a preference for a certain tissue or disease process. The radioactive substance redistributes itself within the body after the injection. The goal of emission tomography is to obtain a distribution map of the radioactive substance.

Unstable atoms emit gamma rays as they decay. Gamma cameras are used to detect the emitted gamma photons (see Figure 4.10). The cameras detect one photon at a time. These measurements approximate the ray sums or line integrals. Unlike the transmission data, we do not need to take the logarithm. SPECT (single-photon emission computed tomography) is based on this imaging principle.

Some isotopes, for example, O-15, C-11, N-13, and F-18, emit positrons (positive electrons) during radioactive decay. A positron exists in nature only for a very short time before it collides with an electron. When the positron interacts with an electron, their masses are annihilated, creating two gamma photons of 511 keV each. These photons are emitted 180° apart. A special gamma camera is used to detect this pair of photons, using coincidence detection technology. Like SPECT, the measurements approximate ray sums or line integrals; no logarithm is necessary to convert the data. This is the principle of PET (positron emission tomography) imaging (see Figure 4.11).

Fig. 4.10: Preparation for a nuclear medicine emission scan.

Fig. 4.11: Principle of PET imaging

The imaging geometry for SPECT is determined by its collimator, which is made of lead septa to permit gamma rays oriented in certain directions to pass through and stop gamma rays with other directions. If a parallel-beam or a fan-beam collimator is used, then the data are acquired in the same corresponding form (see Figure 4.12). Similarly, if a cone-beam or a pinhole collimator is used, the imaging geometry is cone beam (see Figure 4.13). Convergent beam geometries magnify the object so that an image larger than the object can be obtained on the detector.

In PET, each measured event determines a line. The imaging geometry is made by sorting or grouping the events according to some desired rules. For example, we can group them into parallel sets (see three different sets in Figure 4.14). We can also store each event by itself as the list mode.

Fig. 4.12: SPECT uses collimators to selected incoming projection ray geometry.

Fig. 4.13: SPECT collimators can be parallel, convergent, or divergent. They produce different sizes of images.

Fig. 4.14: PET data can be grouped into parallel sets.

A typical SPECT image and a PET image are shown in Figure 4.15. When comparing X-ray CT, PET, and SPECT, we observe that X-ray CT has the best resolution and is least noisy, and SPECT has the worst resolution and most noise. Image quality is directly proportional to the photon counts.

Fig. 4.15: SPECT cardiac images and a PET torso image. The PET image is displayed in the inverse gray scale.

4.3Noise propagation in reconstruction

In this section, we will see how the noisy data make the image noisy. A filtered back-projection (FBP) algorithm can be considered as a linear system. The input is the projection data and the output is the reconstructed image. The noise in the output is related to the noise in the input.

4.3.1Noise variance of emission data

Measured projections are noisy. Due to the nature of photon counting, the measurement noise roughly follows the Poisson statistics. A unique feature of Poisson random variable p is that the mean value λ = E(p) and the variance σ2 =Var(p) are the same. In practice, the mean value E(p) is unknown for a measurement p, and we often use the current measurement p to approximate the mean value E(p) and the noise variance Var(p).

$\text{Emission: Var}(p)\approx p\text{.}$

4.3.2Noise variance of transmission data

For transmission imaging, the line integral of the attenuation coefficient is calculated as p =ln(I0/Id), where Id is assumed to be Poisson and I0 can be approximated as a constant. The noise distribution for the post-log data p is complicated. The common practice is to assume the noise in p obeys the Gaussian distribution. We are interested in finding the noise variance of p. Using the linear approximation of the Taylor expansion,

$\begin{array}{l}p=f(I_d)\approx f(E(I_d))+\frac{f^{\prime }(E(I_d))}{1!}{I}_d\\ \text{with}f(x)=\ln I_0-\ln x.\end{array}$

Thus, the variance of p can be approximated as

$\begin{array}{l}\text{Var}(p)\approx {\left(\frac{f^{\prime }(E(I_d))}{1!}\right)}^2\text{Var}(I_d)=\frac{1}{{(E(I_d))}^2}E(I_d)=\frac{1}{E(I_d)}\approx \frac{1}{I_d}=\frac{1}{I_0{e}^{-p}}.\\ \text{Transmission:}\text{Var}(p)\approx \frac{1}{I_0{e}^{-p}}.\end{array}$

4.3.3Noise propagation in an FBP algorithm

We assume that the projection measurements are independent random variables. A general FBP algorithm can be symbolically expressed by the following convolution backprojection algorithm:

$f(x,y)={\displaystyle \sum _s{\displaystyle \sum _{\theta }b(x,y,s,\theta )h(s-\widehat{s})p(}}\widehat{s},\theta ),$

where p is the projection, h is the convolution kennel, and b is the combined factor for all sorts of interpolation coefficients and weighting factors. Then the noise variance of the reconstructed image f(x, y) can be approximately estimated as

$\text{Var(}f(x,y))\approx {\displaystyle \sum _s{\displaystyle \sum _{\theta }{b}^2(x,y,s,\theta )h^2(s-\widehat{s})Var(p(}}\widehat{s},\theta )),$

where Var(p(s, θ)) can be substituted by the results in Section 4.3.1 or 4.3.2 depending on whether the projections are emission data or transmission data.

4.4Attenuation correction for emission tomography

In emission tomography, the gamma ray photons are emitted from within the patient’s body. Not all the photons are able to escape from the patient body; thus they are attenuated when they propagate. The attenuation follows Beer’s law, which we have seen in Section 4.1.

4.4.1PET

In PET, two detectors are required to measure one event with coincidence detection. An event is valid if two detectors simultaneously detect a 511 keV photon. Let us consider the situation depicted in Figure 4.16, where the photons are emitted at an arbitrary location in a nonuniform medium. The photons that reach detector 1 are attenuated by path L1 with an attenuation factor determined by Beer’s law. We symbolically represent this attenuation factor as exp $(-{\displaystyle \underset{L_1}{\int }\mu }).$ Similarly, the attenuation factor for path L2 is exp $(-{\displaystyle \underset{L_2}{\int }\mu }).$ The attenuation factor is a number between 0 and 1 and can be treated as a probability. The probability that a pair of photons will be detected by both detectors is the product of the probabilities that each photon is detected; that is,

$\exp \left(-{\displaystyle {\int }_{L_1}\mu }\right)\times \exp \left(-{\displaystyle {\int }_{L_2}\mu }\right)=\exp \left(-{\displaystyle {\int }_{L_1+L_2}\mu }\right)=\exp \left(-{\displaystyle {\int }_L\mu }\right).$

Therefore, the overall attenuation factor is determined by the entire path L, regardless of where the location of the gamma source is along this path.

To do an attenuation correction for PET data, a transmission measurement is required with an external transmission (either X-ray or gamma ray) source. This transmission measurement gives the attenuation factor exp $\left(-{\displaystyle {\int }_L\mu }\right).$

The attenuation-corrected line integral or ray sum of PET data is obtained as

$p(s,\theta )=\exp \left({\displaystyle {\int }_{L(s,\theta )}\mu }\right)\times [\text{emission}\text{measurement}\text{of}\text{the path}L(s,\theta )],$

where the reciprocal of the attenuation factor is used to compensate for the attenuation effect. Note that there is no need to reconstruct the attenuation map for PET data attenuation correction.

Fig. 4.16: The attenuation of the PET projection is the effect of the total length L = L1 + L2.

4.4.2SPECT: Tretiak–Metz FBP algorithm for uniform attenuation

Attenuation correction for SPECT data is much more complicated than the PET data, because an event in SPECT is single photon detection (see Figure 4.17). On the same imaging path, the emission source at a different location has a different attenuation factor. This makes SPECT attenuation correction very difficult. We still do not know how to compensate for the attenuation by processing the projections as easily done as in PET. Some people have tried to perform attenuation compensation in the 2D Fourier domain of the sinogram using the “frequency–distance principle,” which is an approximate relationship: The angle in the Fourier domain is related to the distance from the detector in the spatial domain. However, the attenuation effect can be corrected during image reconstruction.

In SPECT, if the attenuator is uniform (i.e., μ = constant within the body boundary), the FBP image reconstruction algorithm is similar to that for the regular unattenuated data. The attenuation-corrected FBP algorithm, developed by Tretiak and Metz, consists of three steps:

(i)Pre-scale the measured projection $p(s,\theta )\text{by}{e}^{\mu d(s,\theta )},$ where the definition of distance d(s, θ) is given in Figure 4.18. We denote this scaled projection as p(s, θ).

(ii)Filter the pre-scaled data with a notched ramp filter (see Figure 4.19).

(iii)Backproject the data with an exponential weighting factor e–μt, where t is defined in Figure 4.20 and is dependent on the location of the reconstruction point as well as the backprojection angle θ.

Fig. 4.17: In SPECT, the attenuation is a mixed effect of all lengths.

Fig. 4.18: The distance d(s, θ) is from the boundary of the uniform attenuator to the central line parallel to the detector. A central line is a line passing through the center of rotation.

Fig. 4.19: The notched ramp-filter transfer function for image reconstruction in the case of a uniformly attenuated Radon transform. A transfer function is a filter expression given in the Fourier domain, and its inverse Fourier transform is the convolution kernel.

Fig. 4.20: The distance t is from the reconstruction point to the central line parallel to the detector.

Similar to the μ = 0 case, the notched ramp filtering can be decomposed into a derivative and a notched Hilbert transform. In the spatial domain, the Hilbert transform is a convolution with a convolution kernel 1/s. The notched Hilbert transform, on the other hand, is a convolution with a convolution kernel (cos μs)/s.

In the Fourier domain, the notched Hilbert filter function is shown on the left-hand side of the graphic equation in Figure 4.21. The cosine function can be decomposed as cos(μs)= (eiμs + e–iμs) /2. The Fourier transform has a property that multiplication by eiμs in the s domain corresponds to shifting by μ/(2π) in the ω domain (i.e., the Fourier domain). Therefore, the Fourier transform of (cos μs)/s is the combination of two shifted versions (one shifted to the left by μ/(2π) and the other one shifted to the right by μ/(2π)) of the Fourier transform of 1/s (see Figure 4.21).

If μ is nonuniform, an FBP-type algorithm exists but is rather sophisticated and will be briefly discussed in Section 4.5.4.

Fig. 4.21: A notched Hilbert transform transfer function can be decomposed as two shifted Hilbert transform transfer functions.

4.4.3SPECT: Inouye’s algorithm for uniform attenuation

The Tretiak–Metz algorithm is in a beautiful FBP form. However, this algorithm generates very noisy images. The significant noise propagation from the projections to the reconstructed image is mainly contributed by the backprojector. In the Tretiak–Metz FBP algorithm, the backprojector has an exponential factor e–μt, where t can be positive or negative. This exponential factor gives a noise variance amplification factor of e–2μt. In other words, if the noise variance in the filtered sinogram is σ2, the noise will propagate into the reconstructed image with a variance approximately e–2μtσ2. For example, let μ = 0.15/cm and t = –10 cm; these result in e–2μt =20. A 20-fold noise amplification is huge! In the regions farther away from the center (i.e., |t | is larger) the image noise dramatically becomes larger.

The main idea in Inouye’s algorithm is to convert the attenuated projections into unattenuated projections, and then use a regular FBP algorithm to reconstruct the image. The regular FBP algorithm does not have an exponential factor in the backprojector. Inouye’s algorithm is summarized as follows.

(1)Pre-scale the measured projection p(s, θ) by eμd(s,θ), where the definition of distance d(s, θ) is given in Figure 4.18. We denote this scaled projection as (s, θ). [This step is the same as inthe Tretiak–Metz algorithm.]

(2)Take the 2D Fourier transform of (s, θ), obtaining (ω, k). [Note that (s, θ) is periodic with respect to θ, and the Fourier transform in the θ dimension is actually the Fourier series expansion.]

Fig. 4.22: FBP image reconstruction with uniformly attenuated SPECT projections. Left: Reconstruction by the Tretiak–Metz algorithm. Right: Reconstruction by the Inouye algorithm.

(3)Convert (ω, k) to P(ω, k) according to

$P(\omega ,k)={\gamma }^k\widehat{P}\left(\sqrt{{\omega }^2+{\left(\frac{\mu }{2\pi }\right)}^2},k\right).$

with

$\gamma =\frac{\left|\omega \right|}{\sqrt{{\omega }^2+{\left(\frac{\mu }{2\pi }\right)}^2}+\frac{\mu }{2\pi }}$

(4)Find the inverse 2D Fourier transform and obtain pnew(s, θ).

(5)Use a regular FBP to reconstruct the image using the newly estimated projections pnew(s, θ).

Figure 4.22 shows the comparison of the images reconstructed by the Tretiak–Metz algorithm and the Inouye algorithm, using noisy computer-simulated uniformly attenuated emission projections. The Tretiak–Metz algorithm gives a much noisier image, and the noise gets more severe as the location is farther away from the center.

4.5Mathematical expressions

4.5.1Expression for Tretiak–Metz FBP algorithm

Now we give the mathematical expression of the FBP algorithm for SPECT with uniform attenuation correction. This algorithm has been outlined in Section 4.4.2 After the pre-scaling step, the scaled projection (s, θ) can be related to the original image f(x, y) as

$\widehat{p}(s,\theta )={\displaystyle \underset{-\infty }{\overset{\infty }{\int }}{e}^{\mu t}f(s\stackrel{\rightharpoonup }{\theta }+t}{\stackrel{\rightharpoonup }{\theta }}^{\perp })dt.$

You can refer to Section 1.5 for notation definitions. The following gives an FBP algorithm using the derivative and the notched Hilbert transform:

$f(x,y)=\frac{1}{4{\pi }^2}{\displaystyle \underset{0}{\overset{2\pi }{\int }}{e}^{-\mu (-x\sin \theta +y\cos \theta )}}{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}\frac{\cos (\mu (s-x\cos \theta -y\sin \theta ))}{s-x\cos \theta -y\sin \theta }\frac{\partial \widehat{p}(s,\theta )}{\partial s}dsd\theta .}$

4.5.2Derivation for Inouye’s algorithm

Let f(x, y) be a density function in the x–y plane and fθ be the function f rotated by θ clockwise. After the pre-scaling step, the scaled projection (s, θ) can be related to the original image f(x, y) as

$\widehat{p}(s,\theta )={\displaystyle \underset{-\infty }{\overset{\infty }{\int }}{e}^{\mu t}{f}_{\theta }(s,t)dt.}$

Taking the 1D Fourier transform with respect to s yields

$\begin{array}{l}{\widehat{p}}_{s\to \omega }(\omega ,\theta )={\displaystyle \underset{-\infty }{\overset{\infty }{\int }}{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}{e}^{\mu t}{f}_{\theta }(s,t)e^{-2\pi is\omega }}dtds}\\ ={\displaystyle \underset{-\infty }{\overset{\infty }{\int }}{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}{f}_{\theta }(s,t)e^{-2\pi i(s\omega +t\frac{\mu }{-2\pi i})}}dsdt.}\end{array}$

Let

$\begin{array}{l}\omega =\widehat{\omega }\cos \alpha ,\\ \frac{\mu }{-2\pi i}=\widehat{\omega }\sin \alpha ,\end{array}$

which give

$\begin{array}{l}\widehat{\omega }=\sqrt{{\omega }^2-{\left(\frac{\mu }{2\pi }\right)}^2},\\ \alpha =\frac{i}{2}\ln \frac{\omega +\frac{\mu }{2\pi }}{\omega -\frac{\mu }{2\pi }}.\end{array}$

Changing the s–t coordinate system to the x–y coordinate system with s = x cos θ + y sin θ and t =–x sin θ + y cos θ, we have

$\begin{array}{l}{\widehat{p}}_{s\to \omega }(\omega ,\theta )={\displaystyle \underset{-\infty }{\overset{\infty }{\int }}{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}f(x,y)e^{-2\pi i\widehat{\omega }[x\cos (\theta +\alpha )+y\sin (\theta +\alpha )]}}dxdy}\\ =F(\widehat{\omega }\cos (\theta +\alpha ),\widehat{\omega }\cos (\theta +\alpha )).\end{array}$

Here F is the 2D Fourier transform of the image f. Since F(ŵ cos(θ + α), ŵ cos(θ + α)) is a periodic function in θ with a period of 2π, F has a Fourier series expansion. Thus, we can use the Fourier series expansion on the right-hand side of eq. (4.19):

${\widehat{p}}_{s\to \omega }(\omega ,\theta )={\displaystyle \sum _n{F}_n(\widehat{\omega })e^{in(\theta +\alpha )}},$

where the Fourier coefficients are defined as

$F_n(\widehat{\omega })=\frac{1}{2\pi }{\displaystyle \underset{0}{\overset{2\pi }{\int }}F(\widehat{\omega }\cos \theta ,}\widehat{\omega }\sin \theta )e^{-in\theta }d\theta .$

Then,

$\begin{array}{l}{\widehat{p}}_{s\to \omega }(\omega ,\theta )={\displaystyle \sum _n{F}_n(\widehat{\omega })e^{in(\theta +\alpha )}}\\ ={\displaystyle \sum _n{F}_n\left(\sqrt{{\omega }^2-{\left(\frac{\mu }{2\pi }\right)}^2}\right)}{e}^{in\theta }{e}^{in\frac{i}{2}\ln \frac{\omega +\frac{\mu }{2\pi }}{\omega -\frac{\mu }{2\pi }}}\\ ={\displaystyle \sum _n{\left(\frac{\omega -\frac{\mu }{2\pi }}{\omega +\frac{\mu }{2\pi }}\right)}^{\frac{n}{2}}{F}_n\left(\sqrt{{\omega }^2-{\left(\frac{\mu }{2\pi }\right)}^2}\right)}{e}^{in\theta }.\end{array}$

On the other hand, s→ω(ω, θ) is also a periodic function with a period of 2π. It has a Fourier series expansion:

${\widehat{p}}_{s\to \omega }(\omega ,\theta )={\displaystyle \sum _n\widehat{P}(\omega ,n)e^{in\theta }}.$

By comparing the expansion coefficients on both sides, we have

$\widehat{P}(\omega ,n)={\left(\frac{\omega -\frac{\mu }{2\pi }}{\omega +\frac{\mu }{2\pi }}\right)}^{\frac{n}{2}}{F}_n\left(\sqrt{{\omega }^2-{\left(\frac{\mu }{2\pi }\right)}^2}\right).$

If we use P(ω, n) to represent the 2D Fourier transform of the unattenuated projections p(s, θ), the above relationship is

$\widehat{P}(\omega ,n)={\left(\frac{\omega -\frac{\mu }{2\pi }}{\omega +\frac{\mu }{2\pi }}\right)}^{\frac{n}{2}}P\left(\sqrt{{\omega }^2-{\left(\frac{\mu }{2\pi }\right)}^2},n\right)$

or

$\begin{array}{l}P(\omega ,n)={\left(\frac{\sqrt{{\omega }^2+{\left(\frac{\mu }{2\pi }\right)}^2}+\frac{\mu }{2\pi }}{\sqrt{{\omega }^2+{\left(\frac{\mu }{2\pi }\right)}^2}-\frac{\mu }{2\pi }}\right)}^{\frac{n}{2}}\widehat{P}\left(\sqrt{{\omega }^2+{\left(\frac{\mu }{2\pi }\right)}^2},n\right)\\ ={\left(\frac{\omega }{\sqrt{{\omega }^2+{\left(\frac{\mu }{2\pi }\right)}^2}-\frac{\mu }{2\pi }}\right)}^n\widehat{P}\left(\sqrt{{\omega }^2+{\left(\frac{\mu }{2\pi }\right)}^2},n\right).\end{array}$

The measured attenuated data are used to compute (ω, n). The above Inouye’s relationship is used to estimate P(ω, n), which gives the estimated unattenuated projections p(s, θ). The final reconstruction is obtained from p(s, θ) by a regular FBP algorithm. We would like to clarify that when we say 2D Fourier transform of p(s, θ), it really is a combination of the 1D Fourier transform with respect to s and the Fourier series expansion with respect to θ.

4.5.3Rullgård’s derivative-then-backprojection algorithm for uniform attenuation

There are many other ways to reconstruct SPECT data with uniform attenuation corrections. We can still play the game of switching the order of filtering and backprojection. For example, we can first take the derivative and then backproject. This results in an intermediate image (x, y) that is closely related to the original image f(x, y):

$\widehat{f}(x,y)=f(x,y)\times \frac{\cosh (\mu x)}{x},$

which is a 1D convolution (i.e., line-by-line convolution in the x direction). The deconvolution of this expression to solve for f(x, y) is not an easy task, because the function $\frac{\cosh (\mu x)}{x}$ tends to infinity as x goes to infinity. It is impossible to find a function u(x) such that

$\delta (x)=u(x)\times \frac{\cosh (\mu x)}{x}$

for – ∞ < x < ∞. However, it is possible to find such a function u(x) to make the above expression hold in a small interval, say (–1, 1). Outside this small interval, $u(x)\times \frac{\cosh (\mu x)}{x}$ is undefined. This “second best” solution is useful in image reconstruction because our objects are always supported in a small finite region. Unfortunately, we don’t yet have a closed-form expression for such a function $u(x),\text{and}\widehat{f}(x,y)=f(x,y)\times \frac{\cosh (\mu x)}{x}$can only be deconvolved numerically.

The advantage of this derivative-then-backprojection algorithm is its ability to exactly reconstruct a region of interest with truncated data.

4.5.4Novikov–Natterer FBP algorithm for nonuniform attenuation SPECT

In SPECT imaging with a nonuniform attenuator μ(x, y), if attenuation correction is required, a transmission scan should be performed in addition to the emission scan. The transmission projections are used to reconstruct the attenuation map μ(x, y). The following is a reconstruction algorithm that can correct for the nonuniform attenuation. Despite its frightening appearance, it is merely an FBP algorithm:

$f(x,y)=\frac{1}{4{\pi }^2}Re\left\{{{\displaystyle \underset{0}{\overset{2\pi }{\int }}\frac{\partial }{\partial q}\left[e^{a_{\theta }(q,t)-g(q,\theta )}{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}\frac{(e^{g}p)(l,\theta )}{q-l}dl}\right]|}}_{q=s}d\theta \right\},$

where Re means taking the real part, p(s, θ) is the measured attenuated projection, $s=x\cos \theta +y\sin \theta ,t=-x\sin \theta +y\cos \theta ,a_{\theta }(s,t)={\displaystyle \underset{t}{\overset{\infty }{\int }}\mu (s\stackrel{\rightharpoonup }{\theta }+\tau {\stackrel{\rightharpoonup }{\theta }}^{\perp }})d\tau ,g(s,\theta )=\frac{1}{2}[(R+$$i\text{HR)}\mu ](l,\theta ),i=\sqrt{-1},\text{R}$is the Radon transform operator, and H is the Hilbert transform operator with respect to variable l. This algorithm was independently developed by Novikov and Natterer.

Unfortunately, this algorithm contains an exponential factor in the backprojector, which amplifies the noise and makes the reconstruction very noisy. The Tretiak–Metz algorithm is a special case of this algorithm.

4.6Worked examples

Example 4: In radiology, did the X-ray CT scanners provide images of the distribution of the linear attenuation coefficients within the patient body?

Answer

Not quite. The reconstructed linear attenuation coefficients μ are converted to the so-called CT numbers, defined as

$\text{CT}\text{ number}h=1,000\times \frac{\mu -{\mu }_{\text{water}}}{{\mu }_{\text{water}}}.$

The CT numbers are in Hounsfield units (HU). For water, h =0 HU; for air, h = –1, 000 HU; and for bone, h = 1, 000 HU.

Example 5: Is there a central slice theorem for the exponential Radon transform?

Answer

In 1988, Inouye derived a complex central slice theorem to relate the uniformly attenuated projections to the object in the Fourier domain. He used a concept of “imaginary” frequency, which was attenuation coefficient. Let the exponential Radon transform be

$\widehat{p}(s,\theta )={\displaystyle \underset{-\propto }{\overset{\infty }{\int }}{e}^{\mu t}f(s\stackrel{\rightharpoonup }{\theta }+t{\stackrel{\rightharpoonup }{\theta }}^{\perp })}dt.$

Let the 1D Fourier transform of (s, θ) with respect to s be P,(ω, θ) and the 2D Fourier transform of the original object be F(ωx, ωy). Inouye’s complex central slice theorem is expressed as follows:

$\begin{array}{l}{P}_{\mu }(\omega ,\theta )=F(\omega \cos (\theta +v),\omega \cos (\theta +v)),\\ \text{where}v=\frac{\mathrm{i}}{2}1\text{n}\frac{\omega -\mu /(2\pi )}{\omega +\mu /(2\pi )}\text{is}\text{an}\text{imaginary}\text{frequency}\text{.}\\ \end{array}$

Example 6: In Figure 4.19, the frequency components below μ/(2π) are discarded during image reconstruction. How do the low-frequency components of the image get reconstructed?

Answer

We will use Inouye’s result to answer this question. Let the attenuation-free Radon transform be

$p(s,\theta )={\displaystyle \underset{-\infty }{\overset{\infty }{\int }}f(s\stackrel{\rightharpoonup }{\theta }+t{\stackrel{\rightharpoonup }{\theta }}^{\perp })}dt$

and the exponential Radon transform be

$\widehat{p}(s,\theta )={\displaystyle \underset{-\infty }{\overset{\infty }{\int }}{e}^{\mu t}}f(s\stackrel{\rightharpoonup }{\theta }+t{\stackrel{\rightharpoonup }{\theta }}^{\perp })dt.$

The exponential Radon transform is the attenuated projection scaled by the factor $e^{\mu d(s,\theta )}.$

Then we take the 2D Fourier transform of p(s, θ) and (s, θ), and obtain P(ω, k) and (ω, k), respectively. Inouye’s relationship is given as

Fig. 4.23: Concentrations are different in the two compartments.

$\begin{array}{l}p(\omega ,k)={\left(\frac{\sqrt{{\omega }^2+{\left(\frac{\mu }{2\pi }\right)}^2}+\frac{\mu }{2\pi }}{\sqrt{{\omega }^2+{\left(\frac{\mu }{2\pi }\right)}^2}-\frac{\mu }{2\pi }}\right)}^{\frac{k}{2}}\widehat{P}\left(\sqrt{{\omega }^2+{\left(\frac{\mu }{2\pi }\right)}^2,k}\right)\\ \text{for }\omega \ge \text{0 and }k\ge 0.\end{array}$

This relationship implies that the frequency ω has been shifted to $\sqrt{{\omega }^2+{(\mu /(2\pi ))}^2}.$ The attenuation procedure during data generation actually shifts the “frequency.” The lowest frequency in the image corresponds to the frequency at μ/(2π) in the projections. Therefore, all frequency components are preserved.

Example 7: Consider a 2D PET imaging problem shown in Figure 4.23, where the object consists of two compartments R1 and R2. The radionuclide concentration is ρ1 in R1 and ρ2 in R2. The attenuation coefficient μ is the same for both compartments. Calculate the PET coincident measurement p and the attenuation-corrected measurement pc.

Solution

The coincidence measurement is

$p=(L_1{\rho }_1+L_2{\rho }_2)e^{-\mu (L_1+L_2)}.$

The attenuation-corrected measurement is

$p_c=L_1{\rho }_1+L_2{\rho }_2.$

4.7Summary

–The working principle for X-ray CT measurement is Beer’s law. One needs to take the logarithm to convert the CT measurements into line integrals.

–PET and SPECT measure line integrals directly. However, these measurements suffer from photon attenuation within the patient.

–Attenuation correction for PET can be achieved by the pre-scaling method. The scaling factor is obtained by transmission measurements.

–Attenuation correction for SPECT is difficult and cannot be done by pre-scaling. Attenuation correction is a built-in feature in SPECT reconstruction algorithms. FBP algorithms exist for uniform attenuator and for nonuniform attenuator as well. However, the FBP algorithm for the nonuniform attenuator is very complicated to implement.

–The readers are expected to understand the differences between transmission and emission tomography, and understand the different attenuation effects in PET and SPECT.

Problems

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