A fast method based on NESTA to accurately reconstruct CT image from highly undersampled projection measurements

Abstract

BACKGROUND: The CT image reconstruction algorithm based compressed sensing (CS) can be formulated as an optimization problem that minimizes the total-variation (TV) term constrained by the data fidelity and image nonnegativity. There are a lot of solutions to this problem, but the computational efficiency and reconstructed image quality of these methods still need to be improved.

OBJECTIVE: To investigate a faster and more accurate mathematical algorithm to settle TV term minimization problem of CT image reconstruction.

METHOD: A Nesterov’s algorithm (NESTA) is a fast and accurate algorithm for solving TV minimization problem, which can be ascribed to the use of most notably Nesterov’s smoothing technique and a subtle averaging of sequences of iterates, which has been shown to improve the convergence properties of standard gradient-descent algorithms. In order to demonstrate the superior performance of NESTA on computational efficiency and image quality, a comparison with Simultaneous Algebraic Reconstruction Technique-TV (SART-TV) and Split-Bregman (SpBr) algorithm is made using a digital phantom study and two physical phantom studies from highly undersampled projection measurements.

RESULTS: With only 25% of conventional full-scan dose and, NESTA method reduces the average CT number error from 51.76HU to 9.98HU on Shepp-Logan phantom and reduces the average CT number error from 50.13HU to 0.32HU on Catphan 600 phantom. On an anthropomorphic head phantom, the average CT number error is reduced from 84.21HU to 1.01HU in the central uniform area.

CONCLUSIONS: To the best of our knowledge this is the first work that apply the NESTA method into CT reconstruction based CS. Research shows that this method is of great potential, further studies and optimization are necessary.

Keywords

Compressed sensing total variation NESTA CT

1 Introduction

The development of compressed sensing (CS) technique in recent years enable high quality and high computational efficiency in CT reconstruction from highly undersampled projections [1 –6]. If the original image is sparse or can be sparsified by a given transformation, it can be accurately reconstructed by minimizing its L₁ norm or the L₁ norm of its sparsified transformation under the measurement constraint [7 –9]. The CS-based reconstruction has been demonstrated to keep the same image quality using less projection data [10, 11]. As such, the CS based iterative reconstruction is promising for advanced clinical applications where radiation dose is a critical concern [12 –14].

The prerequisite for the application of CS theory is that the image has a sparse or compressible representation in a basis such that most entries of its sparse representation are zero or close to zero [15]. For most medical images, we can usually derive their sparse representations by some sparsifying transform such as fast Fourier transform, discrete cosine transform or wavelet transform. Since most CT images are piecewise constant due to small variation within the area of their organs, their discrete gradient transformation are sparse [16]. Thus, total variation (TV) minimization has been one of the most extensively used techniques to obtain an accurate image [7]. The image reconstruction is then formulated as an optimization problem that minimizes the TV term constrained by the data fidelity and image nonnegativity. Where the framework for the optimization problem in the context of CS is based on L₁ norm minimization and can be formulated as the constrained convex optimization problem [17, 18], $\min | | x | |_{TV}, s . t ., | | b - Ax | |_{2} \leq ɛ, x \geq 0,$ (1) or the unconstrained convex optimization problem [4, 19] $\min λ | | x | |_{TV} + \frac{1}{2} | | Ax - b | |_{2}^{2}, x \geq 0,$ (2) where the vector b with a length of M (i.e., number of detector pixels) represents the line integral measurements, x is the vectorized image to be reconstructed with a length of N (i.e., number of image pixels), A is the M×N system matrix modeling the forward projection with M<<N, λ is a penalty weighting factor for TV regularization term, ∥ • ∥ ₂ calculates the L₂ norm in the projection space, ∥ • ∥ _TV is the TV term defined as the L₁ norm of the spatial gradient image.

There are plenty of solutions to this problem, but the computational efficiency and reconstructed image quality of these methods is not perfect. NESTA is a fast, accurate, robust algorithm based on Nesterov’s smoothing technique and suited for solving a wide range of large-scale problem, especially sparse recovery problem [20]. Therefore, we attempt to apply NESTA to CT image reconstruction.

In this paper, both digital phantom study and physical validation study are carried out so as to demonstrate the main features of NESTA algorithm such as image quality improvement and convergence speed augment. As we know, SART [21] is a conventional and classical iterative algorithm and its performance is much better than the performance of ART, SIRT and other iterative algorithms [21, 22]. What’s more, SART-TV [23] is a algorithm combine SART algorithm and TV-Regularization approach which is a classical compressed sensing algorithm and is often used as compared algorithm [24 –26]. Besides, SpBr algorithm is an effective method to solve L₁ norm minimization problem and have been used in CT reconstruction [5, 27]. On the digital Shepp–Logan phantom, we first compare NESTA with SART-TV, SpBr on its computational efficiency and image quality from highly undersampled projection measurements in MATLAB. Then, more comparison studies are carried out between NESTA and SART-TV, SpBr to reveal its superior performance on the Catphan 600 phantom and an anthropomorphic head phantom. These algorithms are implemented on a Hewlett-Packard (HP) workstation not using parallel computational capability. The HP workstation consists of 4 processing cores with 2.8 GHz clock speed and 16GB memory.

The paper is organized as follows. In Section II, firstly we describe the method for NESTA, then the pseudocode and evaluation methodology for this method is proposed, and in Section III, we demonstrate the accuracy of the method through digital phantom study, Catphan 600 phantom study and anthropomorphic head phantom study. In Section IV, we make a summary, and put forward several points needed to be improved.

2 Method

2.1 Framework of NESTA algorithm

The major step of NESTA algorithm is the calculation of optimal value of the objective function which is formulated as equation (2). In order to express conveniently, the TV term of a two-dimensional image is defined as follows. ${∥ x ∥}_{TV} : = \sum_{i, j} ∥ \nabla x (i, j) ∥, \nabla x (i, j) = [\begin{matrix} (D_{1}) x (i, j) \\ (D_{2}) x (i, j) \end{matrix}],$ (3) where D₁ and D₂ is horizontal difference and vertical difference respectively which is derived as ${\begin{matrix} (D_{1}) x (i, j) = x (i + 1, j) - x (i, j) \\ (D_{2}) x (i, j) = x (i, j + 1) - x (i, j) \end{matrix},$ (4)

The TV term is approximated in a similar form as shown in the Ref. [28] $f (x) = {∥ x ∥}_{TV} = max_{u \in Q_{d}} 〈 u, Dx 〉,$ (5) where u = [u₁, u₂] ^T ∈ Q_d is for any (i, j) meet the conditions of $u_{1}^{2} (i, j) + u_{2}^{2} (i, j) \leq 1$ and D = [D₁, D₂] ^T. Among this, T is transpose operator, Q_d is referred as dual feasible set, and assuming it is convex.

It is difficult to solve Equation (5) due to the nonsmooth in the calculation of the convex function. In this paper, we use a smooth approximation approach, and Equation (5) is rewritten as similar as Ref. [28] $f_{u} (x) = max_{u \in Q_{p}} 〈 u, Dx 〉 - {up}_{d} (u),$ (6) where Q_p ={ x : ∥ b - Ax ∥ ₂ ≤ ɛ }, and p_d (u) is a Prox-function for Q_p; that is, p_d (u) is continuous and strongly convex on Q_p, with convexity parameter σ_d (we shall assume that p_d (u) vanishes at some point in Q_p). Nesterov have proved that f_u (x) is continuously differentiable [20], and its gradient obeys $\nabla f_{u} (x) = D^{T} u_{u} (x),$ (7) where the form of u_u (x) is [u₁, u₂] ^T, for any α∈ { 1, 2 }, $u_{α} (i, j) = {\begin{matrix} u^{- 1} (D_{α}) x (i, j), & if ∥ \nabla x (i, j) ∥ < u \\ {∥ \nabla x (i, j) ∥}^{- 1} (D_{α}) x (i, j) & otherwise \end{matrix},$ (8)

Furthermore, ∇f_u (x) is shown to be Lipschitz with constant $L_{u} = \frac{1}{u σ_{d}} {∥ D ∥}^{2},$ (9)

In order to solve the convex optimization problem of Equation (2), two auxiliary iterates, named y_k and z_k is updated as follows using Lagrangian theory and KKT (Karush Kuhn Tucker) condition [28]. $y_{k} = \underset{x \in Q_{p}}{arg min} \frac{L_{u}}{2} {∥ x_{k} - x ∥}_{2}^{2} + 〈 \nabla f_{u} (x_{k}), x - x_{k} 〉,$ (10)

$z_{k} = \underset{x \in Q_{p}}{arg min} \frac{L_{u}}{σ_{d}} p_{p} (x) + 〈 \sum_{i = 0}^{k} β_{i} \nabla f_{u} (x_{i}), x - x_{i} 〉,$ (11) where P_p (x) is the primal Prox-function. The point z_k differs from y_k since it is computed from a weighted cumulative gradient $\sum_{i = 0}^{k} β_{i} \nabla f_{u} (x_{i})$ , making it less prone to zig-zagging, which typically occurs when we have highly elliptical level sets. We set the value of β_i as 0.5*(i+1) in all the studies shown in this paper.

A good primal Prox-function is a smooth and strongly convex function that is likely to have some positive effect near the solution. In the setting of Equation (2), a suitable smoothing Prox-function may be $P_{p} (x) = \frac{1}{2} {∥ x - x_{0} ∥}_{2}^{2},$ (12) where x₀ ∈ Rⁿ is an initial predicted value of the solution and the strong convexity parameter σ_d of P_p (x) is equal to 1.

After the y_k and z_k of Equations (10) and (11) is calculated respectively, the optimal solution of Equation (6) can be updated according the following equation. $x_{k} = τ_{k} z_{k} + (1 - τ_{k}) y_{k},$ (13) where τ_k is the iterate weights which is set as 2/(k+3) in all the studies shown in this paper. Besides, in order to extend NESTA algorithm for use in practice, a correction factor named δ is imported to calculate two significant parameters λ and u as follows $λ = 0.01 \times δ \times \sqrt{2 ln (N)}$ (14) $u = {0.1 \times δ / ∥ A ∥}_{2}$ (15) where δ is correction factor which is directly proportional to the standard of noise.

2.2 Pseudocode for the NESTA algorithm

In summary, we present the pseudocode of the NESTA algorithm as below. The symbol “=” means assignment. Both image and data space variables are denoted by a vector sign, e.g., x and b.

Algorithm I. Pseudocode for NESTA (the comments are shown in italic).

N = 256²; δ = 0.5; $λ = 0.01 \times δ \times \sqrt{2 ln (N)}$ ; u = 0.1 × δ/AA^T; σ_d = 1;

x = zeros(1,N);

for i_num = 0:i_max % iteration times, i_max is the max iteration times

${\begin{matrix} (D_{1}) x (i, j) = x (i + 1, j) - x (i, j) \\ (D_{2}) x (i, j) = x (i, j + 1) - x (i, j) \end{matrix}$ ; $P_{p} (x) = \frac{1}{2} {∥ x - x_{0} ∥}_{2}^{2}$ ; Q_p ={ x : ∥ b - Ax ∥ ₂ ≤ ɛ };

∇f_u (x) = D^Tu_u (x); $u_{α} (i, j) = {\begin{matrix} u^{- 1} (D_{α}) x (i, j), & if ∥ \nabla x (i, j) ∥ < u \\ {∥ \nabla x (i, j) ∥}^{- 1} (D_{α}) x (i, j) & otherwise \end{matrix}$ ;

$L_{u} = \frac{1}{u σ_{d}} {∥ D ∥}^{2}$ ;

$y_{k} = \underset{x \in Q_{p}}{arg min} \frac{L_{u}}{2} {∥ x_{k} - x ∥}_{2}^{2} + 〈 \nabla f_{u} (x_{k}), x - x_{k} 〉$ ; % where k = 1:N

$z_{k} = \underset{x \in Q_{p}}{arg min} \frac{L_{u}}{σ_{d}} p_{p} (x) + 〈 \sum_{i = 0}^{k} β_{i} \nabla f_{u} (x_{i}), x - x_{i} 〉$ ; % where k = 1:N, β_i = 0.5 (i + 1)

x_k = τ_kz_k + (1 - τ_k) y_k; % where k = 1:N, τ_k = 2/(k + 3)

if(x_k < 0) % nonnegative constrains

x_k = 0; % where k = 1:N,

end

x^i_num⁺¹ = x_k; % where k = 1:N

end

img_out = reshape(x^i_max,256,256) ^T ;

return img_out;

The parameters in line 1 control the whole algorithm. Their typical values are shown in the code, which are used to acquire the results in this paper except the setting of correction factor δ. The initial guess image is generated using zeros function. Zero initial may increases the computation time but it is ease of operation. The main loop, lines 3–13, solves Equation (2) using NESTA method with a smooth approximation approach (lines 4–8). The updated image (line 9) is enforced to be non-negative (line 10–12). The iteration stops with a returned optimal image img_out (line 17).

2.3 Evaluation

The performance of the algorithm has been evaluated on a digital phantom and two physical phantoms.

The Shepp-Logan phantom consists of several ellipses to mimic different structures, including soft tissues, bones, gas pockets and etc. The projections were simulated using mono-energetic x-ray beams, covering 360° with an equal angular spacing. We only choose 60 angles of projections in this paper. The reconstructed image has a dimension of 256×256 with a pixel size of 0.5×0.5 mm². Besides, the original Shepp-Logan image is saved as reference for a quantitative error analysis.

Both the physical phantom data and an anthropomorphic head phantom data were acquired on our tabletop CBCT system at Georgia Institute of Technology. The geometry of this system exactly matches that of a Varian On-Board Imager (OBI) CBCT system on the True-beam radiation therapy machine. The details of system geometry and hardware parameters are described in Ref. [29]. A total of 655 projections are acquired for the conventional FBP reconstruction. Few-view projection data are generated from the 655 projections with an evenly distributed angular spacing. The estimated dose reduction ratio is calculated based on the number of measured projection lines. In both physical phantom studies, we use164 projections and block 25% of illuminated area in each projection, resulting in the dose reduction ratio of around 75% [30]. Besides, an additional set of projections is acquired with a fan-beam geometry, which narrows the collimator open width to around 10 mm on the detector for inherent scatter suppression. The resultant images are used as references for a quantitative error analysis [31].

In all the studies presented in this paper, we compared the results of NESTA with those of the SART-TV and the SpBr algorithm. Image quality metrics are used to quantitatively evaluate the performance of every reconstruction method. The CT number error is calculated as the square root of the mean square error (RMSE) which could be found in Refs. [29, 30].

3 RESULTS

3.1 Digital phantom study

To demonstrate the superior performance of NESTA, digital phantom study is implemented. The RMSE of reconstructed images are calculated as significant evaluation criterions of the performance with different algorithms under the condition of same reconstruction time.

Figure 1 shows the CT images reconstructed from 60 projection views over 360° under a clinical radiation level (a flat-field intensity of I₀ = 50000 photons per ray) using different algorithms. In addition, the reconstruction time of every method is 60 seconds (s). Striped artifacts are observed in the image from conventional SART-TV reconstruction [Fig. 2(b)], and are well suppressed in the reconstructions using SpBr and NESTA [Fig. 2(c), (d)]. Besides, the three elliptic areas in the bottom of the reconstruction image using SpBr are difficult to be distinguished. Above all, the RMSEs of the images reconstructed by NESTA vary significantly. For a comparable reconstruction time (60 s), the RMSEs of SART-TV is 33.7 HU, SpBr is 51.7 HU and NESTA is 9.9 HU. The RMSEs of the image reconstructed by NESTA is 29.4% of that reconstructed by SART-TV and 19.1% of that reconstructed by SpBr.

From Fig. 2 we can see that the reconstructed images using SART-TV and SpBr methods contain a lot of noise and artifacts, while the reconstructed images using NESTA method contain less noise and artifacts and have clearer edges. What’s more, the CT image reconstructed by NESTA is about the same as the ground truth image.

To further evaluate the computational efficiency, we plot the data fidelity error and TV term as a function of computation time shown in Fig. 3. NESTA converge much faster than SART-TV and SpBr. Note that, our implementation of SpBr is exactly the version used in Ref. [27]. SART-TV algorithm alternates between SART iteration process and TV minimization process, so the data fidelity error and TV term present serrated undulation. SpBr method first finds a solution that satisfies the TV minimization constraint and then searches for the image with a minimum data fidelity error by means of iteration [9]. In the process of iteration, ensure the stability of TV value, using shrink operator. NESTA algorithm acquires smaller data error and the undulation between 15 and 30 in the Fig. 3(a) will be settled on next paper.

As seen in the above results, NESTA can reconstruct higher quality image with the same reconstruction time. As such, the NESTA is promising for advanced clinical applications where image quality and computational efficiency is a critical concern.

3.2 Catphan 600 phantom study

Figure 4 shows the reconstruction images of Catphan 600 phantom. Only 164 projections and block 25% of illuminated area in each projection are used to reconstruct the images from Fig. 4(b)–(d), resulting in the dose reduction ratio of around 75%. The line pairs in zoom-in inserts of Fig. 4 (d) are clearer than that of Fig. 4(c) which could be observed intuitively. In addition, the background regions of Fig. 4 (d) are smoother than that of Fig. 4 (b). Above all, the CT number errors of the selected ROIs for Fig. 4(b)–(d) are 50.1, 5.0, and 0.3 HU under the condition of identical reconstruction time (420 s).

3.3 Head phantom study

The superior performance of NESTA is further demonstrated in CT reconstruction using anthropopathic head projection data. Fig. 5(b)–(d) shows the CT images reconstructed by SART-TV, SpBr, and NESTA method and the reconstruction time is 300 s. By comparing Fig. 5(c) and (d) we find that NESTA achieves much higher image equality and have more detail information especially at the regions marked with solid rectangles. The view aliasing artifacts as in the SART-TV reconstruction [Fig. 5(b)] are not present in the NESTA reconstruction, and the image noise is well suppressed without losing fine structures in the image (see the zoom-in inserts). What’s more, the CT number errors for Fig. 5(b)–(d) are 84.2, 2.6, and 1.0 HU, respectively. The CT number errors of the reconstruction CT image using NESTA are far lower than that using SART-TV and SpBr methods.

As a final remark, the NESTA algorithm should be considered as an improved CS based CT reconstruction. Therefore, NESTA also inherits the advantages of the conventional CS based reconstruction, such as dose reduction and spatial resolution improvement which have already been demonstrated in many literatures. Above all, NESTA provides superior performance in CT reconstruction such as image quality improvement. Besides, both digital phantom study and physical phantom study demonstrate the effectiveness of NESTA algorithm in practical application.

4 Discussion and conclusions

In this paper, we mainly apply the new NESTA algorithm which is a mathematic method proposed for sparse recovery to CT image reconstruction. During the process of the implementation of the NESTA algorithm, a data error fluctuation phenomenon was found in the Fig. 3(a). We will make an attempt to improve the convergence property of this method and solve the problem of error fluctuation in our further research. Besides, the thought of Bregman iteration will be used to resolve the contradiction between noise suppression and loss of details [8].

Then, we demonstrate the performance of NESTA method by simulation study and physical study. On the Shepp-Logan phantom, the NESTA reconstruction reduces reconstruction errors in the SART-TV reconstruction from 33.73HU to 9.98HU with 60 projections, and reduces reconstruction errors in the SpBr reconstruction from 51.75HU to 9.98HU with 60 projections. With 164 projections on the Catphan 600 phantom, our method reduces CT number error in selected ROI from 50.13HU in the SART-TV reconstruction to 0.32HU, from 5.04HU in the SpBr reconstruction to 0.32HU. Using 164 projections on the anthropomorphic head phantom, the NESTA method reduces CT number error in selected ROI from 84.2HU in the SART-TV result to 1.01HU, from 2.64HU in the SpBr result to 1.01HU. It is important to indicate that all of the three algorithms are implemented with a same time. In addition, the criteria to stop iterating the reconstruction in this paper is reconstruction time. We used different reconstruction time for optimal reconstruction result about every algorithm.

In summary, our method achieves a satisfactory image quality at the condition of simulation or physical experiment. Meanwhile the convergence property of NESTA method is also satisfactory. Therefore, NESTA is an excellent algorithm ascribed to its image quality and computation time which is critically concerned in clinical applications. In conclusion, NESTA is more efficient for practical applications in CT image reconstruction. However, there are still some drawbacks in NESTA algorithm which is convergence curve jitter and convergence instability.

Footnotes

Acknowledgments

This work is partially supported by the National Natural Science Foundation of China under Grant No. U1401255, 81227901. The authors would like to thank Dr. L. Zhu (Nuclear and Radiological Engineering and Medical Physics Programs, The George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, Georgia) for generous offer of the Catphan 600 phantom and the anthropomorphic head phantom. They would also like to thank the anonymous reviewers for their constructive suggestions which substantially strengthen the paper.

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