Initial analysis of the middle problem in CT image reconstruction

Abstract

The interior and exterior problems have been extensively studied in the field of reconstruction of computed tomography (CT) images, which lead to important theoretical and practical results. In this study, we formulate a middle problem of CT image reconstruction, which is more challenging than either the interior or exterior problems. In the middle problem of CT image reconstruction, projection data are measured through and only through the middle dough-like region, so that each projection profile misses data not only internally but also on both sides. For an object with a radially symmetric exterior, we proved that the middle problem could be uniquely solved if the middle ring-shaped zone is piecewise constant or there is a known sub-region inside this middle region. Then, we designed and evaluated a POCS-based algorithm for middle tomography, which is to reconstruct a middle image only from the available data. Finally, the remaining issues are also discussed for further research.

Keywords

Middle problem middle tomography data truncation limited-angle tomography image reconstruction POCS

1 Introduction

Since its introduction, the x-ray computed tomography (CT) has been widely applied in biomedical and industrial applications. Under an ideal imaging condition, the x-ray CT projections can be modeled as the so-called Radon transform. Let f₀ (x), $x \in ℝ^{2}$ , be a compactly supported objective function on a disk $B_{C} = x \in ℝ^{2} : | x | \leq C}$ , where C > 0 is a real constant and $ℝ$ represents the real domain. The Radon transform of f₀ (x) can be expressed as ${Rf}_{0} (s, φ) = \int_{- \infty}^{\infty} f_{0} (s ω^{⊥} + t ω) dt,$

where $s \in ℝ$ , 0 ≤ φ < π, ω = (cos φ, sin φ) and ω^⊥ = (- sinφ, cosφ). When Rf₀ (s, φ) is measured for all |s| ≤ C and 0 ≤ φ < π, f₀ (x) can be easily reconstructed by using the classical inverse Radon transform and other filtered backprojection algorithms [1].

In practice, it is very common that the data are incomplete. When Rf₀ (s, φ) is only available for 0 < B < |s| ≤ C and 0 ≤ φ < π (see Fig. 1(a)), we have the exterior problem, which is uniquely solvable (see Subsection VI.3 in [1]). This means that the integrals are not available along x-ray paths through the inner region with a radius of B. When Rf₀ (s, φ) is only available for |s| ≤ A < C and 0 ≤ φ < π (see Fig. 1(b)), we have the interior problem, which does not have a unique solution in an unconstrained space (see Subsection VI.4 in [1]). In this case, the x-ray integrals are only available along the x-ray paths through the inner region with a radius of A.

Fig.1

Illustration of (a) the exterior problem, (b) interior problem, and (c) middle problem. For the middle problem, projection data are measured through and only through the middle ring so that each projection profile misses data not only internally but also on both sides.

Since 2007, multiple groups revisited the interior problems to develop interior tomography for theoretically exact local image reconstruction [2]. As a result, the uniqueness and stability of the interior problem were established under the constraint of certain prior knowledge [2]. Assuming a known sub-region inside a region-of-interest (ROI), we proved that the interior problem can be uniquely and stably solved [3]. This exact knowledge based result was also independently reported [4]. Assuming a piecewise constant or polynomial model, we further proved that the interior problem can be uniquely solved by minimizing the total variation (TV) or high-order TV (HOT) of an object function [5 –8].

Recently, we became interested in the middle problem (see Fig. 1(c)), which is “between” the exterior and interior problems. Mathematically, the middle problem is featured by Rf₀ (s, φ) that is only available for 0 < A ≤ |s| ≤ B < C and 0 ≤ φ < π. It is underlined that this mathematical curiosity has a practical background. One example for industrial application is to image a region adjacent to a metal part inside a large object [9]. On one hand, the projections are truncated for the exterior region due to the object size. On the other hand, the projections through the metal are not available due to the photon starvation effect. The clinically meaningful primary example would be tomographic imaging of the neighborhood of metallic implants such as hip prostheses and cardiac stents [10, 11]. On one hand, the metal piece could be too dense to allow decent interior data. On the other hand, features may not be of relevance far away from metals. It is highly desirable to depict interactions between metal and tissue matrix where x-rays can go through metallic surface tangentially, carry information on touching soft issues, and then be recorded with a high-quality small detector for evaluation of healing, inflammation, or restenosis.

Similar scenarios can be found in the field of tissue engineering [12, 13]. Tissue engineering requires differentiation of cellular components into functional structures modulated by signals that guide integration, vascularization, and remodeling. Tissue regeneration is highly complex, hence tomographic characterization are needed of in vitro cell/biomaterial interaction and in vivo engineered tissue/host interaction. For example, cells may be seeded into biomaterial matrices in a bioreactor to proliferate, migrate, and mature over days. During this period, x-ray imaging can noninvasively monitor changes in phenotype and function, and extract biomarkers in attenuation, phase-contrast and dark-field modes [14 –17]. However, the higher resolution demanded for x-ray imaging, and the higher radiation dose will be involved. Similar to interior tomography, middle tomography is another tool for tailored tomographic sampling as a minimized radiation dose.

More generally speaking, many critical physical and chemical phenomena occur at the interface of two regions or two phases [18]. Such surfaces often form membranes, skins, shells, chamber walls, or other local features distributed in both externally and internally. In these scenarios, if the assumption on a radially symmetric exterior (which is the assumption we made in this pilot study; see below) can be removed or relaxed, targeted imaging of the middle zones can be useful for more flexible imaging geometry (namely, a middle region can be focused such as for imaging of a cylindrical wall in an industrial part or a nuclear reactor facility), less hardware requirement (a narrow detector width for coverage of the middle region thickness), shorter data acquisition time (without scanning the interior and exterior parts with the narrow detector so the acquisition time is shortened), and low radiation dose (by avoiding radiating the interior and exterior portions of a biological object) than full-fledged global tomography, being complementary to interior tomography and exterior tomography.

To our best knowledge, the middle problem has not been analyzed before. As the initial effort, we would like to establish the uniqueness of the middle problem with a radially symmetric exterior. The rest of this paper is organized as follows. In Section II, we present our theoretical results. In Section III, we describe a practical algorithm in the compressive sensing framework and present some feasibility results. In section IV, we discuss some related issues and conclude the paper.

2 Theoretical results

Let us assume that an image f₀ (x) is a piecewise smooth function. The middle problem with a radially symmetric exterior can be defined as follows (see Fig. 1): Let 0 < A < B < C. Suppose the object image f₀ (x) is compactly supported in a disk $B_{C} = x \in ℝ^{2} : | x | \leq C}$ , and radially symmetric in the exterior annulus $Ω_{ex} = x \in ℝ^{2} : B < | x | < C}$ . We want to reconstruct the object image in the middle region $Ω = {x \in ℝ^{2} : A < | x | < B}$ , only from the middle data Rf₀ (s, φ) where 0 ≤ φ < π, and A ≤ |s| ≤ B.

An image function f is called a solution to the middle problem with a radially symmetric exterior if it satisfies

suppf ⊆ B_C;

f is radially symmetric in the exterior annulus Ω_ex;

Rf (s, φ) = Rf₀ (s, φ), A < |s| < B.

It is readily known from the presentation about the interior problem [1] that the solution to the middle problem with a radially symmetric exterior is not unique without any constraint. Hence, any solution f is called a candidate function, while f - f₀ is an ambiguous function.

Lemma 1. Any ambiguous function u satisfies that Ru (s, φ) depends only on |s| for B ≤ |s| ≤ C, and u is analytic in Ω.

Proof. Denote Ru (s, φ) by g (s) for B ≤ |s| ≤ C. In light of Cormack’s inversion formula [1], if u is a Schwartz function, for x ∈ Ω, we have $u (x) = - \frac{1}{π} \int_{B}^{C} \frac{g^{'} (s) ds}{\sqrt{s^{2} - | x |^{2}}} = - \frac{1}{π} \int_{B}^{C} \frac{sg (s) ds}{(s^{2} - | x |^{2})^{3 / - 2}} .$ (1)

In order to deal with a piecewise smooth function u, we introduce $k (x) = {(\int_{| x | < 1} e^{1 / - (| x |^{2} - 1)} dx)}^{- 1} {\begin{matrix} e^{1 / - (| x |^{2} - 1)}, & x | < 1 \\ 0, & | x | \geq 1 \end{matrix} .$

Since k (x) is radially symmetric, Rk (s, θ) depends only on |s|, denoted by I (s). Let $k_{ɛ} (x) = ɛ^{- 2} k (ɛ^{- 1} x),$ $I_{ɛ} (s) = ɛ^{- 1} I (ɛ^{- 1} s),$ $u_{ɛ} (x) = (u * k_{ɛ}) (x),$

then $({Rk}_{ɛ}) (s, θ) = ɛ^{- 1} Rk (ɛ^{- 1} s, θ) = I_{ɛ} (s),$

and for |s| ≥ A + ɛ, ${Ru}_{ɛ} (s, θ) = (Ru * {Rk}_{ɛ}) (s, θ) = (g * I_{ɛ}) (s) .$

Applied Eq. (1) to u_ɛ (x), we have $u_{ɛ} (x) = - \frac{1}{π} \int_{B - ɛ}^{C + ɛ} \frac{s (g * I_{ɛ}) (s) ds}{(s^{2} - | x |^{2})^{3 / - 2}}, x \in Ω, ɛ < | x | - A .$

Let ɛ → 0, we have that $u (x) = - \frac{1}{π} \int_{B}^{C} \frac{sg (s) ds}{(s^{2} - | x |^{2})^{3 / - 2}}$

holds almost everywhere x ∈ Ω. This implies that u is analytic in Ω.■

Lemma 2. If an ambiguous function u satisfies u (x) = c, if A ≤ |x| ≤ B, then c = 0.

Proof. Let u (x) = h (|x|) for B ≤ |x| ≤ C. For A < s < B, from Ru (s, φ) =0, we obtain $\sqrt{B^{2} - s^{2}} c + \int_{B}^{C} h (r) d \sqrt{r^{2} - s^{2}} = 0, A < s < B .$

That is, $\sqrt{B^{2} - s^{2}} c + \int_{B}^{C} \frac{h (r) rdr}{\sqrt{r^{2} - s^{2}}} = 0,$ $B \sqrt{1 - (s / - B)^{2}} c + \int_{B}^{C} \frac{h (r) dr}{\sqrt{1 - (s / - r)^{2}}} = 0,$ (2)

Denote (2k - 1) ! ! =1 × 3 ×5 ⋯ × (2k - 1) and k ! =1 × 2 ×3 ⋯ × k, we have the following Taylor’s expansions $\frac{1}{\sqrt{1 - {(s / - r)}^{2}}} = 1 + \sum_{k = 1}^{\infty} \frac{(2 k - 1)!!}{2^{k} k!} {(\frac{s}{r})}^{2 k},$ (3) $\sqrt{1 - {(s / - B)}^{2}} = 1 - \frac{1}{2 k - 1} \sum_{k = 1}^{\infty} \frac{(2 k - 1)!!}{2^{k} k!} {(\frac{s}{B})}^{2 k},$ (4)

Inserting Equations (3) and (4) into Equation (2), we have $\int_{B}^{C} h (r) {(a / - r)}^{2 k} dr = \frac{cB}{2 k - 1}, k \geq 0 .$ (5)

Making the variable transformation t = B/ - r, Equation (5) becomes $\int_{B / - C}^{1} h (B / - t) t^{2 k - 2} dt = \frac{c}{2 k - 1}, k \geq 0 .$ (6)

Let $\tilde{h} (t) = {\begin{matrix} h (B / - t) - c & t \in [B / - C, 1] \\ - c & t \in [0, B / - C) \end{matrix},$ (7)

Equation (6) implies $\int_{0}^{1} \tilde{h} (t) t^{2 k} dt = 0, k \geq 0 .$ (8)

Given the fact that the even function $\hat{h} (t) = \tilde{h} (| t |)$ on the interval [-1, 1], we have $\int_{- 1}^{1} \hat{h} (t) t^{k} dt = 0, k \geq 0 .$ (9)

This means that $\hat{h} (t) = 0$ almost everywhere on [-1, 1], which implies c = 0. ■

Theorem 1. Suppose that an object function f₀ is piecewise constant in Ω, that is, there exists a sub-domain partition of Ω: $Ω = ⋃_{i = 1}^{I_{r}} Ω_{i}$ and a real number sequence $c_{i}}_{i = 1}^{I_{r}}$ , such that $f |_{Ω_{i}} = c_{i}, 1 \leq i \leq I_{r}, then f_{0} = \underset{f is a candidate solution}{arg min} {TV}_{Ω} (f) .$

Proof. For any candidate solution f = f₀ + u, ${TV}_{Ω} (f) = \sum_{1 \leq i < j \leq I_{r}} | c_{i} - c_{j} | | Γ_{i, j} | + \int_{Ω} \sqrt{{(\frac{\partial u}{\partial x_{1}})}^{2} + {(\frac{\partial u}{\partial x_{2}})}^{2}} dx,$

where |Γ_i,j| is the length of the boundary between Ω_i and Ω_j. If $h = f_{0} + \tilde{u} = \underset{f is a candidate solution}{arg min} {TV}_{Ω} (f)$ , then $\frac{\partial \tilde{u}}{\partial x_{1}} |_{Ω} = \frac{\partial \tilde{u}}{\partial x_{2}} |_{Ω} = 0$ . That is, $\tilde{u} |_{Ω} = c$ . By Lemma 2, c = 0. This leads to h = f₀.■

Theorem 2. In the middle problem with a radially symmetric exterior, suppose that the object function f₀ is known in a sub-region Ω_sub of Ω, then f₀ can be determined accurately in Ω.

Proof. For any candidate solution f = f₀ + u, let f|_{Ω
_sub} = f₀|_{Ω
_sub}, then u|_{Ω
_sub} = 0. Combining with the fact that u is analytic in Ω, we have u|_Ω = 0. Therefore, f|_Ω = f₀|_Ω. ■

For practical applications, fan-beam geometry is widely applied. Although the aforementioned theoretical results assume parallel-beam geometry, it is straight forward to extend them to fan-beam geometry after rebinning into parallel-beam geometry or via the Jacobi transform.

3 Numerical results

To test the theoretical result of Theorem 1 in Section II, we developed an algorithm for middle tomography in the projections onto convex set (POCS) framework [19]. Let a digital imagef_m,n = f (mΔ, nΔ) be a candidate image, where Δ represents the sampling interval and m and n are integers. For middle reconstruction of f_m,n, the following five constrains are enforced by our algorithm: (1) The projections data Rf_m,n (s, φ) are measured for A ≤ |s| ≤ B; (2) the radial symmetric property Rf_m,n (s, φ) = g (s) is valid for B < |s| ≤ C; (3) the image f_m,n satisfies the piecewise constant model in the middle region Ω (or a known sub-region); (4) the image f_m,n ≥ 0; and (5) the image f_m,n ≤ C_max, where C_max is the maximum value of f_m,n.

To incorporate the data fidelity, the ordered-subset simultaneous algebraic reconstruction technique (OS-SART) [20] was applied with an area integral projection model [21] to generate the system matrix. To apply the radical symmetry of the object in the exterior region, the forward projections of f_m,n were angularly averaged for B < |s| ≤ C, and the differences in each view were back-projected using OS-SART. A soft-threshold filtering method [22] was employed to minimize the total variation of the middle region. The constraints f_m,n ≥ 0 and f_m,n ≤ C_max were imposed as well. To accelerate the convergence of the algorithm, the fast weighting technique [23] was used in the iterative step for each subset of measured projections. Because the central portion in every projection had no data, the quality of a reconstructed image would be poor due to the discontinuity at the boundary between the middle and interior regions if we assume missing data being zeros [24]. To address this problem, in the first several iterations we estimated the missing interior projections and incorporated them with OS-SART. The radical symmetry of the exterior projections was not utilized until later iterations. Our algorithm was implemented in MatLab on a PC. While the basic structure was in MatLab, all the computationally intensive parts were coded in C, which was linked via a MEX interface. A maximal iteration time was set to stop the main loop. The whole flowchart of the algorithm is in Fig. 2.

Fig.2

Flowchart of the proposed algorithm in the POCS framework to solve the middle problem assuming a radially symmetric exterior. L_st is the main loop number to incorporate the interpolated interior projections.

In our numerical simulation, a 2D phantom of Shepp-Logan-type was designed. This phantom includes a set of ellipses whose parameters are in Table 1, where a, b represent the x, y semi-axes, (x₀, y₀) the center of the ellipse, φ denotes the rotation angle, and υ the relative attenuation coefficient. The units for a, b and (x₀, y₀) are cm. As shown in Fig. 3, the exterior part of the phantom is radially symmetric.

Table 1

Parameters of the 2D modified Shepp-Logan phantom

No.	a	b	x ₀	y ₀	φ	υ
1	9.676	9.676	0	0	0	0.4
2	9.597	9.597	0	0	0	–0.2
3	4.100	1.600	2.20	0	108.0	–0.2
4	3.100	1.100	2.20	0	72.0	–0.2
5	2.100	2.500	0	3.5	0	0.2
6	0.460	0.460	0	1.0	0	0.2
7	0.460	0.230	0.60	–6.5	0	0.1
8	0.460	0.230	–0.80	–6.5	0	0.1

Fig.3

Modified Shepp-Logan phantom with a radially symmetric exterior in the display window [0,0.6]. The middle region is delimited by the small and larger white circles of radii A and B respectively, and the profiles in the reconstructed images will be plotted along the white lines. The red ring-shaped zone is assumed known to verify Theorem 2.

A circular scanning locus of radius 57.0 cm was coupled with a fan-beam imaging geometry. An equi-spatial virtual detector array had a length 20.0 cm. The detector was centered at the system origin and made always perpendicular to the direction from the system origin to the x-ray source. The detector array included 600 elements, with an elemental aperture 0.033 cm. This configuration covered a circular field of view (FOV) of radius 9.85 cm. During a full scan, 720 projections were equi-angularlly collected using an analytic method based on the parameters in Table 1. To simulate different middle problems, we fix B= 7.5 cm, change the values of A, and manually extracted the projection data that directly involved the middle region. The reconstructed images were made in a 256×256 matrix covering an area of 10×10 cm². The subset number was set as S = 8.

The POCS approach was used for middle tomography. The maximum number of iterations was set to 30. In the first 10 iterations, we incorporated the interpolated projections through the interior region to suppress the discontinuity and limited-angle artifacts over the boundary between the interior and middle regions. After 10 iterations, we removed the interpolated interior projections and added the radially symmetric constraint over the exterior region. The soft-thresholding filtering method was used for TV minimization, with a threshold 2×10^–5. The maximum image value was 0.6.

For different values of A (0.1, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0, and 3.5 cm), the reconstructed images from noise-free projections are in Fig. 4, along with representative profiles in Fig. 5. From Figs. 4 and Figs. 5, it is observed that the middle regions can be well reconstructed only from the middle scan consisting of measurements that are angularly limited and truncated in each view on both sides and internally. For a fixed B, the smaller the value of A was, the better the image quality would be. This is because a smaller A means more projections and closer to the interior problem, which is easier and implies a better image quality. This point can be further demonstrated in a magnified view in Fig. 6 along with the root mean square roots (RMSEs) in the middle region listed in Table 2. On the other hand, the smaller the distance between a point to be reconstructed and the interior boundary, the less the projections, and the worse the quality of the reconstructed image.

Fig.4

Reconstructed images from noise-free projections for different values of A in a display window [0,0.6].

Fig.5

Representative profiles along (a) the horizontal and (b) vertical lines indicated in Fig. 3 for the images in Fig. 4. The horizontal color bars indicate the supports of the middle regions for eight different A values respectively.

Fig.6

Magnified images of the sub-regions defined by the white square in Fig. 3. The images (a)-(h) in the display window [0.15, 0.35] correspond to the images (a)-(h) in Fig. 4, while (a’)-(h’) are the corresponding difference images in the display window [–0.1,0.1] relative to the ground truth.

Table 2

RMSEs of the reconstructed images computed in the middle region

A (cm)	0.1	0.5	1.0	1.5	2.0	2.5	3.0	3.5
Noise-free	0.02828	0.02885	0.03074	0.03281	0.03140	0.02898	0.02924	0.03214
Noisy	0.02828	0.02884	0.03074	0.03279	0.03138	0.02895	0.02920	0.03209

To evaluate the robustness of the developed algorithm, 2.0% additive white Gaussian noise was added into the line integrals of the projections. Note that the Gaussian noise model is approximately equivalent to the Poisson noise model with an ideal bowtie filter to reach the same expected photon numbers at all the detector elements. The aforementioned experiments were then repeated. The reconstructed images from noisy projections are in Fig. 7, along with the corresponding RMSEs in Table 2. From Figs. 4 and 7 and Table 2, it can be seen that the reconstructed image quality from noisy projections is very close to that of the images from noise-free projections. This is because the TV minimization can not only remove the data truncation artifacts but also suppress noise strongly.

Fig.7

Reconstructed images from projections with 2% Gaussian noise for different values of A in a display window [0,0.6].

In Theorem 2, we further proved that the middle region can be uniquely determined if there is a known subregion inside the middle region. To verify the correctness of Theorem 2, the algorithm in Fig. 2 was modified by incorporating an additional constraint that there is a known subregion inside the middle region. The implementation is straight forward. During the iteration, the pixels inside the known subregion of an intermediate image are forced to the known values. In the modified Shepp-Logan phantom, we assumed a known ring-shaped subregion marked in red in Fig. 3. The above experiments for verifying Theorem 1 were repeated with both noise-free and noisy projections. The reconstructed images for different values of A (0.1, 0.5, 1.0, and 1.5 cm) are in Fig. 8, and representative profiles along the horizontal and vertical lines in Fig. 3 are plotted in Fig. 9. Compared Figs. 4, 7, 8 and 9, it is observed that there is no obvious differences among those reconstructed images from noise-free projections for the same A value. Regarding the images from noisy projections with the same A value, it seems that the additional constraint of known subregion slightly improved the reconstructed image quality.

Fig.8

Reconstructed images with a known subregion indicated in Fig. 3. The top images were reconstructed from noise-free projections, while the bottom images are the counterparts from projections with 2% Gaussian noise. The display window is [0,0.6].

Fig.9

Representative profiles along (a) the horizontal and (b) vertical lines indicated in Fig. 3 for the top row images in Fig. 8. The horizontal color bars indicate the supports of the middle regions for 4 different A values, respectively.

4 Discussions and conclusion

While our results presented above show the non-trivialness of the middle problem and its solution under some practical constraints, the original middle problem remains open, which drops off the exterior symmetric constraint. The main difficulty is to figure out if the corresponding ambiguous function is in C¹ (Ω), which we suspect so but are unable to prove at this moment.

This middle problem can be extended in several ways. First, the middle zone is not necessarily be delimited by two circles, and can be defined by two closed convex curves, as shown in Fig. 10. Also, there could be several middle regions in an object support. Furthermore, just like interior tomography was studied for both x-ray and nuclear imaging, there should be a SPECT version of the middle problem, as already hinted in the introduction of this paper. Finally, the middle problem in a higher dimensional space would become more complicated, and can be formulated in different contexts. For example, for a short object such as a spherical object in 3D, the middle problem could degrade to the interior problem in 2D; but for a long object such as a cylindrical object, the middle problem is more like the counterpart in 2D.

Fig.10

Illustration of the middle problem in a more general setting, which remains an open problem.

While we seek the middle reconstruction from a middle scan, there was a report on the middle reconstruction from both an exterior scan and an interior scan [9]. This setting is practically useful but the uniqueness of the solution was theoretically already known given the availability of the interior data, aided further by the measured exterior data. Theoretically, it would be interesting to compare systematically the middle reconstruction quality using either the middle scan or the combination of exterior and interior scans. However, what kinds of scans can be performed may depend on applications.

Our POCS algorithm for middle tomography produced encouraging results in this initial study but further improvements are clearly possible. Contemporary methods can be adapted as well such as dictionary learning, nonlocal mean, and deep learning. The essential idea is to use as much a priori knowledge as possible so that missing data can be effectively compensated for, and the reconstruction quality optimized.

In conclusion, we have formulated the middle problem and proved its uniqueness for an object with a radially symmetric exterior. Also, we have designed and evaluated a POCS-based algorithm for middle tomography. There are several interesting opportunities to fully characterize the middle problem and further extend it to other situations. Guided by the theoretical results, major algorithmic efforts can be made for performance optimization and specific applications.

Footnotes

Acknowledgments

This work was partially supported by the National Science Foundation of China (No: 61421062, No: 61520106004) and the National Institute of Biomedical Imaging and Bioengineering (R01 EB016977, EB017140 and R21 EB019074).

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