Revisit of combined parallel-beam/cone-beam or fan-beam/cone-beam imaging

Abstract

Purpose: This aim of this paper is to revisit the parallel-beam/cone-beam or fan-beam/cone-beam imaging configuration, and to investigate whether this configuration has any advantages.

Methods: Twenty years ago, it was suggested to simultaneously use a parallel-beam (or a fan-beam) collimator and a cone-beam collimator to acquire single photon emission computed tomography data. The motivation was that the parallel-beam (or the fan-beam) collimator can provide sufficient sampling, while the cone-beam collimator is able to provide higher photon counts. Even with higher total counts, this hybrid system does not give significant improvement (if any) in terms of image noise and artifacts reduction. If a conventional iterative maximum-likelihood expectation-maximization algorithm is used to reconstruct the image, the resultant reconstruction may be worse than the parallel-beam-only (or fan-beam-only) system. This paper uses the singular value decomposition (SVD) analysis to explain this phenomenon.

Results: The SVD results indicate that the parallel-beam-only and the fan-beam-only system outperform the combined systems.

Conclusions: The optimal imaging system does not necessary to be the one that generates the projections with highest signal-to-noise ratio and best resolution.

INTRODUCTION

The cone-beam (CB) imaging geometry is able to magnify the object, so that single photon emission computed tomography (SPECT) using CB collimation exhibits increased sensitivity with the same spatial resolution (or improved resolution with the same sensitivity), compared with the parallel-beam (PB) or fan-beam (FB) collimation. However, if the collimators rotate in a planar orbit (say, a circular orbit), the CB collimator alone is unable to provide sufficient measurements for artifact-free image reconstruction, because Tuy's condition is not satisfied.¹ By including a PB (or a FB) collimator, the combined circular-orbit data are sufficiently sampled. Unfortunately, when the iterative maximum likelihood-expectation maximization (ML-EM) algorithm is directly used to reconstruct the image, the CB artifacts remain in the reconstruction.² It has been observed that an improved reconstruction can be obtained from the CB and PB hybrid data if a special reconstruction strategy is applied. The strategy lets the PB data dominate the reconstruction and lets the CB data play a minor role.² Thus the CB data are hardly utilized.

Our group made the similar observation when we used one FB and two CB collimators in a three-detector SPECT system to acquire projection data and used the hybrid data for image reconstruction with the ML-EM algorithm.^{3, 4, 5, 6, 7} Convergent collimators can magnify the image, and the larger detection solid angle results in higher counts in the data set. It was shown that for the Jaszazck cardiac torso phantom experiment, the combined CB/FB collimation provided the sensitivity that is 1.7 times higher than the standard PB collimation (low-energy high-resolution collimators).⁷ However, a direct application of the ML-EM algorithm to the combined data gave noisy images with typical CB artifacts. We found that the combined CP/FP imaging geometry with appropriate weighting of the two data sets provides improved image quality. The appropriate weighting factor for the FB data was approximately 0.9 and was approximately 0.1 for the CB data.

We have been wondering for over 20 years about the reason why a direct application of the ML-EM algorithm to the hybrid data results in poor images. Why must the higher count CB data be de-emphasized while the lower count PB data be emphasized in order to produce a less noisy image? This seems against the common sense of noise weighting strategy when mixed data sets are used for an imaging task. The goal of this paper is to provide an answer to the above questions via the singular value decomposition (SVD) method.

METHODS

Let the CB and PB imaging equations be

$$F_{c}X=P_c\mathrm{and}{F}_{p}X=P_p,$$ (1)

respectively. Here X is a vector representation of the image array, and P_c and P_p are arrays of projection data for the CB and PB geometries, respectively. Combining these two sets of equations yields

$$\left[\begin{array}{c}{F}_c\\ F_p\end{array}\right]X=\left[\begin{array}{c}{P}_c\\ P_p\end{array}\right].$$ (2)

If weighting is introduced between CB and PB equations, a weighted least-squares problem can be set up as

$$\begin{array}{ccc}& & \left[\begin{array}{cc}\hfill F_c^T& \hfill F_p^T\end{array}\right]·\left[\begin{array}{cc}\hfill \left(1-\beta \right)I_c& \hfill 0\\ \hfill 0& \hfill \beta I_p\end{array}\right]·\left[\begin{array}{c}{F}_c\\ F_p\end{array}\right]X=\left[\begin{array}{cc}\hfill F_c^T& \hfill F_p^T\end{array}\right]\hfill \\ & & ·\left[\begin{array}{cc}\hfill \left(1-\beta \right)I_c& \hfill 0\\ \hfill 0& \hfill \beta I_p\end{array}\right]·\left[\begin{array}{c}{P}_c\\ P_p\end{array}\right],\hfill \end{array}$$ (3)

or

$$AX=B,$$ (4)

with $A=(1-\beta )F_c^T{F}_c+\beta F_p^T{F}_p$, $B=(1-\beta )F_c^T{P}_c+\beta F_p^T{P}_p$, and 0 ⩽ β ⩽ 1. Here I_c and I_p are the identity matrices, and matrix A is a symmetric, positive, or semi-positive definite, square matrix.

A measurement perturbed system can be written as⁸

$$A{X}_{\varepsilon }=B+\varepsilon \widehat{B}.$$ (5)

It is well known that⁸

$$\frac{\Vert X_{\varepsilon }-X\Vert }{\Vert X\Vert }\le \kappa \left(A\right)\frac{\varepsilon \Vert \widehat{B}\Vert }{\Vert B\Vert }+O\left({\varepsilon }^2\right),$$ (6)

where the κ is the condition number of matrix A, and is defined as κ(A) = ||A|| · ||A⁻¹||. If the L₂-norm is used, then

$$\kappa \left(A\right)=\frac{{\sigma }_1\left(A\right)}{{\sigma }_n\left(A\right)},$$ (7)

where σ₁(A) and σ_n(A) are the maximum and minimum singular values of matrix A, respectively. Equation 6 implies that in order to obtain a stable solution one needs a small value of $\kappa \left(A\right)\varepsilon \Vert \widehat{B}\Vert /\Vert B\Vert$.

Next, we estimate the normalized perturbation factor $\varepsilon \Vert \widehat{B}\Vert /\Vert B\Vert$, which according to the definition in Eq. 5 can be approximated by $\sqrt{\mathrm{var}\left(B\right)}/\Vert B\Vert$ for the Poisson noise model. Let us make some assumptions: The image is one large pixel, the total number of counts in CB measurements is C_c, and the total number of counts in PB measurements is C_p.

From the definition of B: $B=(1-\beta )F_c^T{P}_c+\beta F_p^T{P}_p$, the one-large-pixel model gives

$$B=(1-\beta )C_c+\beta C_p.$$ (8)

The one-large-pixel model also implies

$$\Vert B\Vert =B=(1-\beta )C_c+\beta C_p.$$ (9)

From Eq. 8 and the Poisson noise assumption, the variance of B can be approximated as

$$\begin{array}{ccc}\hfill \mathrm{var}\left(B\right)& =& {(1-\beta )}^2\mathrm{var}\left(C_c\right)+{\beta }^2\mathrm{var}\left(C_p\right)\hfill \\ & \approx & {(1-\beta )}^2{C}_c+{\beta }^2{C}_p,\hfill \end{array}$$ (10)

which immediately gives

$$\begin{array}{ccc}\hfill \frac{\sqrt{\mathrm{var}\left(B\right)}}{\Vert B\Vert }& \approx & \frac{\sqrt{{(1-\beta )}^2{C}_c+{\beta }^2{C}_p}}{(1-\beta )C_c+\beta C_p}\hfill \\ & =& \frac{1}{\sqrt{C_p}}\frac{\sqrt{{(1-\beta )}^2{C}_c/C_p+{\beta }^2}}{(1-\beta )C_c/C_p+\beta }.\hfill \end{array}$$ (11)

The first term in the right-hand-side of Eq. 6 can be estimated as

$$\begin{array}{ccc}\hfill \kappa \left(A\right)\frac{\varepsilon \Vert \widehat{B}\Vert }{\Vert B\Vert }& \approx & \kappa \left(A\right)\frac{\sqrt{\mathrm{var}\left(B\right)}}{\Vert B\Vert }\hfill \\ & \approx & \kappa \left(A\right)\frac{\sqrt{{(1-\beta )}^2{C}_c/C_p+{\beta }^2}}{(1-\beta )C_c/C_p+\beta }\frac{1}{\sqrt{C_p}}\hfill \\ & =& \kappa \left((1-\beta )F_c^T{F}_c+\beta F_p^T{F}_p\right)\hfill \\ & & \times \frac{\sqrt{{(1-\beta )}^2{C}_c/C_p+{\beta }^2}}{(1-\beta )C_c/C_p+\beta }\frac{1}{\sqrt{C_p}}.\hfill \end{array}$$ (12)

We notice in Eq. 12 that the factor $1/\sqrt{C_p}$ is a constant that is independent from the variable β. Our goal is to select an optimal β such that the quantity can reach to its minimum. Therefore, by discarding the factor $1/\sqrt{C_p}$ a figure-of-merit (FoM) can be defined as

$$\begin{array}{c}\hfill \mathrm{FoM}=\kappa \left((1-\beta )F_c^T{F}_c+\beta F_p^T{F}_p\right)\frac{\sqrt{{(1-\beta )}^2{C}_c/C_p+{\beta }^2}}{(1-\beta )C_c/C_p+\beta }.\end{array}$$ (13)

The FoM defined in Eq. 13 is able to compare SPECT systems with two types of collimators. A better and more stable system should have a smaller FoM. This FoM considers the tradeoff between the system matrix's condition number and the system's sensitivity.

In this paper, we consider three imaging geometries; they are CB, FB, and PB. The detector matrix size is 128 × 128. The image volume is 64 × 64 × 48, with the voxel's side length being twice as long as the detector pixel's side length. If the voxel's side length is used as the unit length, the detector pixel has the dimension of 0.5 × 0.5 unit². In our computer simulations, the focal length of the CB and FB collimators takes different values: 84, 168, 336, 672, 6720, and 67 200 units, respectively, in different studies. The image reconstruction region is an ellipsoid with the semiaxes of 30, 30, and 25 units in the x, y, and z directions, respectively, containing 84 872 voxels. The axis of rotation is the z axis.

We change the sensitivity ratio of CB vs. PB (or CB vs. FB) so that various cases of collimator designs and system configurations can be covered. The condition number is the ratio of the largest singular value over the smallest singular value of a matrix. A larger condition number indicates a more ill-conditioned system when an inverse problem is to be solved. In this paper, the condition number is calculated using the Lanczos method.⁸ The Lanczos method is an iterative algorithm that estimates the extreme singular values. This is a powerful tool to approximately estimate the condition number of a huge symmetric matrix, which is too large for any exact SVD algorithm to run on a regular computer.

When we consider the CB/FB combination, F_p and C_p in Eq. 13 are replaced by their FB counterparts: F_f and C_f, respectively.

RESULTS

Table 1 shows the FoM results for a combined CB/PB system where the CB focal-length is 168 units. When β = 0, it is a CB-only system; when β = 1, it is a FB-only system. When β is between 0 and 1, it represents a combined CB/PB system. In Table 1, a smaller FoM indicates a more stable system, which may contain lower noise and fewer artifacts in the reconstructed images.

Table 1.

The FoM for a CB/PB system (focal length = 168 units).

			FoM
Imaging geometry	Combination parameter β	Condition number κ	Cc/Cp = 1	Cc/Cp = 2	Cc/Cp = 3	Cc/Cp = 4	Cc/Cp = 5
CB only	0.0	258 186	2390.4	1690.3	1380.1	1195.2	1069.0
Combined system	0.2	276	2.1	1.6	1.4	1.2	1.1
	0.4	331	2.2	1.8	1.6	1.4	1.3
	0.6	432	2.9	2.4	2.0	1.8	1.7
	0.8	829	6.3	5.4	4.8	4.3	3.9
PB only	1.0	108	1.0	1.0	1.0	1.0	1.0

It is not surprising to see in Table 1 that the FoM of the CB-only system is much larger than that of the PB-only system. However, it is surprising to observe that in all combined CB/PB systems the FoMs are worse (i.e., larger) than that of the PB-only system. One would naively expect that the FOM of a combined system would be smaller than 1, but this did not happen.

Table 2 shows the FoM results for a combined CB/FB system where the CB and FB's focal-length is 168 units. The results in Table 2 are similar to those in Table 1, indicating that there is no advantage to use a combined CB/PB system if the sensitivity gain for CB over FB C_c/C_p is less than 5. If the sensitivity gain gets larger, there may be some advantages to use the combined system.

Table 2.

The FoM for a CB/FB system (focal length = 168 units).

			FoM
Imaging geometry	Combination parameter β	Condition number κ	Cc/Cp = 1	Cc/Cp = 2	Cc/Cp = 3	Cc/Cp = 4	Cc/Cp = 5
CB only	0.0	258 186	2717.7	1921.7	1569.1	1358.9	1215.4
Combined system	0.2	261	2.3	1.8	1.5	1.3	1.2
	0.4	261	2.0	1.6	1.4	1.2	1.1
	0.6	260	2.0	1.6	1.4	1.2	1.1
	0.8	316	2.7	2.4	2.1	1.9	1.7
FB only	1.0	95	1.0	1.0	1.0	1.0	1.0

The setup of Tables 3, 4 is the same as that of Tables 1, 2, except that the focal length of the CB and FB collimators is changed from 168 units to 336 units. From the results in Tables 3, 4, we make the same observation that the FoMs reach the minimum for the PB-only system or the FB-only system. The combined systems do not show any advantages in these examples, even though they acquire higher photon counts. Therefore, it makes sense not to use the combined CB/PB or CB/FB collimation in a SPECT system.

Table 3.

The FoM for a CB/PB system (focal length = 336 units).

			FoM
Imaging geometry	Combination parameter β	Condition number κ	Cc/Cp = 1	Cc/Cp = 2	Cc/Cp = 3	Cc/Cp = 4	Cc/Cp = 5
CB only	0.0	3998	37.0	26.2	21.4	18.5	16.6
Combined system	0.2	467	3.6	2.8	2.3	2.1	1.9
	0.4	432	2.9	2.3	2.0	1.8	1.6
	0.6	689	4.6	3.8	3.2	2.9	2.6
	0.8	1113	8.5	7.3	6.4	5.8	5.2
PB only	1.0	108	1.0	1.0	1.0	1.0	1.0

Table 4.

The FoM for a CB/FB system (focal length = 336 units).

			FoM
Imaging geometry	Combination parameter β	Condition number κ	Cc/Cp = 1	Cc/Cp = 2	Cc/Cp = 3	Cc/Cp = 4	Cc/Cp = 5
CB only	0.0	3998	42.1	29.8	24.3	21.0	18.8
Combined system	0.2	429	3.7	2.9	2.4	2.1	1.9
	0.4	417	3.2	2.6	2.2	2.0	1.8
	0.6	386	2.9	2.4	2.1	1.8	1.7
	0.8	426	3.7	3.2	2.8	2.5	2.3
FB only	1.0	95	1.0	1.0	1.0	1.0	1.0

As the focal length changes, the condition number κ changes, too; see Table 5 for some examples. It is interesting to notice that in a certain focal-length range the FB system can be better (i.e., have a smaller condition number κ) than the PB system. Using a FB-only system may give the optimal results.

Table 5.

The condition numbers κ for different focal lengths.

Focal length	CB	FB
84 units	3 221 478	97
168 units	258 186	95
336 units	3998	95
672 units	3143	98
6720 units	109	110
67 200 units	108	108
∞	108 (PB)

CONCLUSIONS

In this paper, we have revisited a two-decade-old problem, in which combined CB/PB (or CB/FB) collimators are used in a multidetector SPECT system. Twenty years ago, it was observed that if the iterative ML-EM algorithm was directly applied to the hybrid data, the resultant image was worse than the PB-only (or FB-only) image, in the sense that the reconstruction from CB/FB (or CB/PB) data still contained the CB artifacts. Improved images with reduced artifacts could be obtained if a special scheme was applied to de-emphasize the CB data's contribution to the reconstruction procedure.

This paper introduces a new FoM which is a combined quantity including the condition number and sensitivity gain. This new FoM is able to explain the two-decade-old problem. We do not see obvious advantages of using the hybrid CB/PB or CB/FB system, because the sensitivity gain of CB is not large enough to overcome the increase of the condition number.

The goal of this paper is to investigate the problem that under the condition of circular-orbit data acquisition, whether it is helpful to add some CB data to the FB or PB data set. When Tuy's condition is satisfied,¹ the condition number for the CB system is significantly decreased, and different orbits have different performance.¹¹

Figure 1 shows a general imaging system that contains a data acquisition part and an image reconstruction part. A common way of developing a new imaging system is to adjust the data acquisition subsystem A such that the measurements have the best possible spatial resolution and the lowest noise. Based on the discussion above, this traditional design approach may not result in an optimal overall system, because the associated inverse operator B can be extremely ill-conditioned. A higher sensitivity system may be more ill-conditioned. Some compromises must be made, for example, using our proposed FoM as a criterion to achieve the optimal overall performance. The optimal system may not select the data acquisition system that generates the lowest noise or the best resolution. In the spirit of this observation, one can re-evaluate many other imaging systems that aim to acquire high photon counts. Such systems can be a combined SPECT system with a high resolution collimator and a high sensitivity collimator,⁹ or a coded aperture system that tries to achieve a large opening percentage.¹⁰ A high sensitivity system may not give us the most stable overall system.

Figure 1.

A general imaging system consists of a data acquisition subsystem and an image reconstruction subsystem.