WAVELET TIGHT FRAME AND PRIOR IMAGE-BASED IMAGE RECONSTRUCTION FROM LIMITED-ANGLE PROJECTION DATA

. The limited-angle projection data of an object, in some practical applications of computed tomography (CT), are obtained due to the restric- tion of scanning condition. In these situations, since the projection data are incomplete, some limited-angle artifacts will be presented near the edges of reconstructed image using some classical reconstruction algorithms, such as ﬁltered backprojection (FBP). The reconstructed image can be ﬁne approximated by sparse coeﬃcients under a proper wavelet tight frame, and the quality of reconstructed image can be improved by an available prior image. To deal with limited-angle CT reconstruction problem, we propose a minimization model that is based on wavelet tight frame and a prior image, and perform this min- imization problem eﬃciently by iteratively minimizing separately. Moreover, we show that each bounded sequence, which is generated by our method, con- verges to a critical or a stationary point. The experimental results indicate that our algorithm can eﬃciently suppress artifacts and noise and preserve the edges of reconstructed image, what’s more, the introduced prior image will not miss the important information that is not included in the prior image.


(Communicated by Hui Ji)
Abstract. The limited-angle projection data of an object, in some practical applications of computed tomography (CT), are obtained due to the restriction of scanning condition. In these situations, since the projection data are incomplete, some limited-angle artifacts will be presented near the edges of reconstructed image using some classical reconstruction algorithms, such as filtered backprojection (FBP). The reconstructed image can be fine approximated by sparse coefficients under a proper wavelet tight frame, and the quality of reconstructed image can be improved by an available prior image. To deal with limited-angle CT reconstruction problem, we propose a minimization model that is based on wavelet tight frame and a prior image, and perform this minimization problem efficiently by iteratively minimizing separately. Moreover, we show that each bounded sequence, which is generated by our method, converges to a critical or a stationary point. The experimental results indicate that our algorithm can efficiently suppress artifacts and noise and preserve the edges of reconstructed image, what's more, the introduced prior image will not miss the important information that is not included in the prior image.
industrial fields, CT is utilized to detect a pipeline in service or a large object size [42], and the straight-line trajectory CT [17,12] is utilized to detect the defects of a long object. In the medical domain, there are several examples, such as C-arm CT [1], imaging of the breasts [39], and short exposure time [22]. In these situations, the filtered backprojection (FBP) algorithm (Chap.7 in [5]) does not work well because the projection data are incomplete and the reconstruction problem is severely illposed [27] (see the NCAT phantom [36] for limited-angle reconstruction shown in Figure 2). Figure 1 shows the scanning geometry of limited-angle CT.  In limited-angle CT reconstruction problem, the goal is to reconstruct the attenuation coefficient f (a CT image) from the incomplete projection data with noisy where η ∈ R M denotes the noise, g ∈ R M , f ∈ R N , and A ∈ R M ×N (M N ) denotes the system matrix of X-ray transform in a limited scanning range.
Due to the ill-posed for the limited-angle problem, the regularization has to be considered to stabilize the procedure of limited-angle reconstruction. Thus, in the reconstruction procedure, a priori knowledge about the solution has to be incorporated.
The classical approach is the so-called Tikhonov type regularization, where a regularized solution f λ is obtained from the following optimization problem (2) f λ ∈ arg min f ∈Ω where Ω denotes a convex set, λ > 0 is a regularization parameter, and R : Ω → [0, +∞] is a regularization function (may be non-convex and non-continuous). The first term of (2) is the data fidelity term that controls the error of data. The second term is the penalty term or prior term that includes the object's prior information.
According to the goal of CT image reconstruction, such as to obtain the high resolution of reconstruction image or to preserve the particular edge, there are various choices for the regularization function. For example, in CT image reconstruction, the total variation (TV) norm which is generally used for preserving edges of reconstructed image [13,19].
Let f (i, j) be the pixel value of an image at (i, j), then, the TV norm of an image can be written as In [35], the authors considered the sparsity of image under the gradient transform, and an adaptive steepest descent-projection onto convex sets (ASD-POCS) algorithm was proposed. The model of ASD-POCS algorithm is where ε denotes the noise.
A high quality reconstructed image can be obtained using ASD-POCS algorithm for the sparse-view sampling over 360 0 . However, the limited-angle artifacts will be presented near edges of reconstructed image when the scanning angular range is seriously limited [45]. To overcome these problems, some authors have improved the CT reconstruction algorithm with TV. In [24] the authors proposed a new alternating optimization program for limited-angle reconstruction problem. In [25], the authors developed a novel iterative reconstruction algorithm using weighted TV as the objective function. In [10], the authors proposed an anisotropic TV reconstruction algorithm. Although these methods reduce the limited-angle artifacts near edges, the edges of object are still distorted [44].
In [20], the authors pointed out that TV regularization favors piecewise constant functions that may destroy relevant information. Recently, wavelet frames have been used in sparse-view CT image reconstruction [40,48,46,21,15,47,16], the basic idea is that the reconstructed image can be fine approximated by sparse coefficients under a proper wavelet tight frame. It is well-known that 1 norm or 0 quasi-norm prefers sparse solutions [48,46], then, the sparse regularization solution f λ is obtained from the following optimization problem where Ω is a convex set, λ > 0 is a regularization parameter, W is a wavelet frame, p = 0 or 1 denotes the 1 norm or 0 quasi-norm. A high quality reconstructed image can be obtained by this model for sparse-view CT reconstruction, however, the limited-angle artifacts will be presented near edges of reconstructed image for limited-angle problem that the scanning angular range is seriously limited.
To reconstruct a high quality image under the incomplete data or inconsistent data, a prior image has been incorporated into the CT image reconstruction by some scholars. In [8], a prior image constrained compressed sensing (PICCS) algorithm was proposed. The prior image of this algorithm, which was obtained from the union of interleaved dynamical data sets, was utilized to sparse-view image reconstruction for the individual time frames. In [38,9], the PICCS algorithm was utilized to cardiac CT to improve the temporal resolution. In [26], the PICCS algorithm was utilized to prospectively study CT dose reduction. The PICCS algorithm can effectively preserve the edges of the reconstructed image for the sparse-view CT image reconstruction when the parameter is chosen properly. The model of PICCS algorithm [8,38] is where W 1 and W 2 denote the discrete gradient transforms, f p denotes a prior image, and λ > 0 is a regularization parameter. The PICCS algorithm includes the algebraic reconstruction technique (ART) and the standard steepest descent method. A prior image, which is used to the follow-up detection or diagnosis, can be obtained from full-scan data using FBP algorithm [8,26,28] or ASD-POCS algorithm before the equipment is installed or in the first diagnosis. For example, imaging of a pipeline in service which is attached to wall or installed on the ground, the pipeline is only scanned in a limited angular range due to the restriction of scanning environment. To make the follow-up detection conveniently, we obtain a prior image from full-scan data before the pipeline is installed. The CT image of pipeline in service is close to the prior image except for few small features, such as crack. In medical fields, the prior image should be obtained from full-scan data in the first diagnosis, then, the prior image can be used for same patient in the follow-up diagnosis who is scanned in a limited angular range for reducing the dose of X-Ray [26] or for the restriction of C-arm CT [28]. The CT image of same patient is close to the prior image except for few small features, such as tumor.
In this paper, to better suppress the limited-angle artifacts near edges that occur in limited-angle CT image reconstruction and preserve the edges, a new model based on a wavelet tight frame and a prior image is proposed. The model is a constrained optimization model whose objective function includes a data fidelity term and two regularization terms. One of the regularization terms is based on the sparsity of the image under a wavelet tight frame, the other is the difference between the high-frequency of prior image and that of reconstructed image.
Unlike the common procedure used in [48,46,8,38,9,26], where the 1 norm or 0 quasi-norm was used to measure the sparsity of all wavelet coefficients, we utilize the 0 quasi-norm to promote the sparsity of wavelet coefficients of low-frequency, which can suppress the limited-angle artifacts as the 0 quasi-norm can cut off the small wavelet coefficients of low-frequency. And also different from [8,38,9,26], whose all information of prior image was used to the reconstruction procedure, we only utilize the high-frequency of prior image that only includes the image detail parts and make the difference between the high-frequency of prior image and that of reconstructed image to be minimized, which can preserve the edges of the reconstructed image and make the reconstructed image be closer to the desired solution.
More detail discussions of the reconstruction model will be found in Subsection 3.1. The objective function incorporating the 0 quasi-norm will make this problem into a non-convex and non-smooth minimization problem, and it is difficult to obtain a global minimization of the object function. To deal with this non-convex and non-smooth minimization problem, we use the Moreau envelope to approximate the regularization terms of original objective function [31] and use hard thresholding (HT) to deal with 0 quasi-norm [46,43]. An alternative minimizing iterative is therefore implemented to solve our model. Moreover, this paper provides the convergence analysis of the alternating iterative algorithm for solving our model. We show that each bounded sequence, which is generated by iteratively minimizing iterative, converges to a critical or a stationary point. Finally, the experimental results indicate that our algorithm can efficiently suppress the limited-angle artifacts and noise and preserve the edges of reconstructed image for limited-angle CT reconstruction. The remain of the paper is organized as follows. A brief description of mathematical principles and some definitions used in this work are provided in Section 2. In Section 3, we introduce the proposed minimization model and the corresponding numerical algorithm. In Section 4, we give the convergence analysis of the alternating iterative algorithm for solving our model under certain conditions. Finally, our numerical experimental results of limited-angle CT reconstruction are presented in Section 5 and conclusions are given in Section 6.

2.
Preliminaries. In this section, a brief description of mathematical principles and some definitions used in this work are provided.
2.1. Wavelet and tight frame. A brief introduction of the wavelet tight frame and its construction is given in this subsection. More details are available in [48,21,2,6,18,30,11,7]. Let H be a Hilbert space, if f 2 = n | f, e n | 2 , f or any f ∈ H , then the sequence {e n } ⊂ H is a tight frame for H .
There are two associated operators in a tight frame, one is the analysis operator W that is defined by the sequence { f, e n } is called the canonical frame coefficient sequence, and the other is its synthesis operator W T that is defined by Then, the sequence {e n } ⊂ H is a tight frame if and only if W T W = I, where I is an identical operator of H . It implies the following canonical expansion can be obtained under a given tight frame f = n f, e n e n , f or any f ∈ H .
Tight frame system is widely used in image processing, which is generated by the filters {h l } r l=0 of framelets. Let {V n } n∈N be multiresolution analysis (MRA) derived from the refinable function φ with refinement mask h 0 . In Fourier domain, the definition of refinable function φ isφ(2·) =ĥ 0φ . Hereφ is the Fourier transform of φ, andĥ 0 is the Fourier series of refinement mask h 0 . Let X ⊂ L 2 (R) be a countable set, Ψ = {ψ 1 , ψ 2 , ..., ψ r } ⊂ V 1 . The construction of a tight wavelet frames is to find Ψ that is the same as to find the filters {h l } such that The sequences {h l } r l=1 are called the high pass filters, the refinement mask h 0 is called the low pass filter. The equality above can be rewritten in the Fourier domain asψ l (2·) =ĥ lφ , l = 1, 2, ..., r, for some 2π-periodic functions {ĥ l } r l=1 . In [30], the Unitary Extension Principle (UEP) was proposed that can be used to construct a tight frame. One of the full UEP conditionŝ where ω ∈ {ω ∈ R : [φ,φ](ω) = 0}. The piecewise linear B-spline framelets are used in this work. The refinement mask isĥ 0 (ω) = cos 2 ( ω 2 ), whose corresponding filter , whose corresponding filters are

Some definitions.
In this subsection, we will introduce some definitions used in our work.
Definition 5. Let f : R N → (−∞, +∞) be a proper and lower semi-continuous function. The Fréchet subdifferential ∂f (x) is defined as [4] ∂f (x) = {u : lim inf for any x ∈ domf and ∂f (x) = ∅ if x ∈ domf . For each x ∈ domf , x is called the stationary or critical point of f if it satisfies 0 ∈ ∂f (x).
3. CT reconstruction model and numerical algorithm. In this section, we propose a new minimization model for limited-angle CT reconstruction problem, which is based on a wavelet tight frame and a prior image. An efficient numerical solver is also provided for the proposed minimization model.

3.1.
CT reconstruction model. In limited-angle reconstruction problem, the main difficult problem is that the reconstructed image will present some limitedangle artifacts near edges because the projection data are incomplete (see Figure  2). In order to solve this problem, a priori knowledge about the solution has to be incorporated into the reconstruction model. It is known that the reconstructed images can be well sparsely approximated using a proper wavelet tight frame. Thus, the wavelet transform of image under a tight frame is spare can be regarded as a prior knowledge and used to our reconstruction model.  , in this situation, the sparsity of the wavelet coefficients of lowfrequency of the reconstructed image with limited-angle artifacts is weaker than that of the desired image. To correct the limited-angle artifacts near edge, the 0 quasinorm of the wavelet coefficients of low-frequency of reconstructed image is utilized that penalizes smaller wavelet coefficients (i.e., the smaller wavelet coefficients of the low-frequency of reconstructed image will be set to zero by hard thresholding, which makes the wavelet coefficients of low-frequency more sparse).
In limited-angle CT reconstruction problem, the edges of reconstructed image are distorted (see Figure 3), which corresponding to the high-frequency of reconstructed image in the wavelet domain (see Figures 4 and 5). The quality of reconstructed image can be improved by an available prior image. As seen from Figure 3, the edges of (a) are well preserved, if we regard the high-frequency of the prior image  as a prior information (the prior image is a reconstructed image from full-scan data using FBP algorithm), and minimize the 2 norm of the difference between the high-frequency of prior image and that of reconstructed image, then, this method will make the high-frequency of reconstructed image be closer to that of prior image, and preserve the edges of reconstructed image for limited-angle reconstruction problem. In additional, because the noise is also included in the high-frequency of reconstructed image, the noise will be effectively suppressed by minimizing the 2 norm of the difference between the high-frequency of prior image (not include the noise or include the low level of noise) and that of reconstructed image.
Therefore, the high-frequency information of prior image and the sparsity of wavelet coefficients of low-frequency are utilized to suppress the limited-angle artifacts and to preserve the edges of reconstructed image. We propose to reconstruct a high quality image by solving the following minimization model denotes the system matrix of X-ray transform in a small scanning angular range, γ and λ are the positive regularization parameters, and g ∈ R M denotes the projection data.
3.2. Numerical algorithm. The non-convex and non-continuous of 0 quasi-norm and the nonseparable property of (W f ) L 0 inevitably make the numerical algorithm difficult in solving the minimization model (8) directly. To solve the problem (8), the Moreau envelope function (see, Definition 1) is introduced to approximate the regularization terms of original objective function (see Remark 1) [31], then the (8) becomes the following minimization problem where α = (α L , α G ) and β > 0. The minimization model (9) can be reformed as an unconstrained minimization model using an indicator function (see, Definition 4) as follows (10) arg min It is clear that the objective function in model (10) is close to that in model (8) when the parameter β is sufficiently large.
The model (10) is essentially consistent with the following model with two variables by the following proposition We use the following notes for convenience Let Ω be a convex set, Q(f, α) is noted in (12) and H(α G ) is noted in (13), W is a wavelet tight frame system. For any λ > 0, γ > 0, and β > 0, if a pair (f * , α * ) is a solution of (11), then . Moreover, f * is a solution of the model (10) with α * satisfying (14) and (15) if and only if a pair (f * , α * ) is a solution of (11).
Proof. The idea of this proof is similar to [31].
The Proposition 1 means that the solution of model (10) can be obtained by solving model (11). Thus, we will focus on solving the model (11).
To solve model (11), we can perform this minimization problem efficiently by iteratively minimizing with respect to f and α separately. This means that we will convert the model (11) into two sub-problems that are solved iteratively in an alternating fashion.
In [4], the authors indicated that minimizing the sum of a smooth function with a non-smooth function using the proximal forward-backward scheme can be regarded as a proximal regularization of the smooth function linearized at a given point. Motivated by this idea, we adopt this scheme to solve the sub-problem 1 and write it as an iterative scheme. Linearizing the θ(f ) := 1 2 Af − g 2 2 at a given point f n , we have that An equivalent form of (18) is that We can use the simultaneous algebraic reconstruction technique (SART) (the system matrix A need not be stored) [5,27] in this step if we do it like this. The (19) has a closed form solution and the solution is To solve the sub-problem 2, because β we solve the low-frequency part α L and the high-frequency part α G of wavelet coefficients α separately. (21) High-frequency: arg min Inverse Problems and Imaging Volume 11, No. 6 (2017), 917-948 We write (21) and (22) as an iterative scheme (23) The (23) has a closed form solution, which can be easily obtained by the first order optimality condition. The solution is of the objective function of (24) at a given point α n L to solve (24), a new iterative scheme can expressed as The (26) also has a closed form solution [46], which can be obtained using hard thresholding (see, Definition 3). The closed form solution is therefore, α n+1 = (α n+1 L , α n+1 G ). Next, the process of the alternating minimization with respect to f and α separately are summarized in the form of a pseudo-code. N ite denotes the maximum number of iteration, L θ and L ϕ are constants. The implementation steps of the alternating minimization for CT reconstruction are presented as follows Our algorithm: initialization: Given λ, β, and γ > 0, Step 1. updating f: Obtaining f n+1 using (20).
Step 2. updating α: Solving α n+1 by (25) and (27). end while 4. Convergence analysis. In this section, the convergence analysis of our algorithm is provided and we need a sequence of the lemmas to establish the convergence theorem.
Let η ∈ (0, +∞], and Φ η be the family of all continuous and concave functions F : [0, η) → R + which satisfy three conditions: be a proper and lower semi-continuous function and Ω 1 be a compact set. Suppose that Q is constant on Ω 1 and satisfies the KL property at each point of Ω 1 . Then, there exist ε 1 > 0, η > 0 and F ∈ Φ η such that for allz in Ω 1 and all z in the following intersection where dist(z, Ω 1 ) := inf { y − z 2 : y ∈ Ω 1 }. One has, We use the following notations for convenience: And two assumptions for the convergence analysis in the following are needed: 1. θ(f ) is a continuously differentiable function with θ being Lipschitz continuous with a constant L θ > 0; 2. ϕ(f, α L ) is a continuously differentiable function about the variable α L with ∂ α L ϕ being Lipschitz continuous with a constant L ϕ > 0.
Lemma 5. Suppose that assumptions 1 and 2 hold. Let {z n := (f n , α n )} be a sequence generated by our algorithm which is assumed to be bounded. For each positive integer n, define ) and there exists M > 0 and M > 0 such that whereM := 2 * max{β + ρβ, M }.
Proof. From (19), we have the following result by the optimality condition From (26), we have the following result by the optimality condition (23), we have the following result by the optimality condition It is clear that According to (40), (41), and (42), we obtain that (A n f , A n α L , 0) ∈ ∂Q(f n+1 , α n+1 ). Because {(f n , α n )} is bounded, and using the Mean Value Theorem and (30), there exists M, M > 0 such that On the other hand, using the (13), we have that
Next, we will establish a convergence theorem for our reconstruction algorithm.
Theorem 1. Suppose ρ > L ϕ and τ > L θ 2 . Let φ(f, α G ) and Q(f, α) be defined in (31) and (12), respectively. W is a wavelet tight frame system. For the sequence {z n := (f n , α n )} generated by our algorithm is assumed to be bounded, then, the following assertions hold: There exists a constant C, such that of Q(f, α).
Proof. We begin with proving the Item 1. By Lemma 3 and Lemma 4, we have that According to (23), we have that Because the sequence f n+1 and f n are generated by our algorithm, therefore, f n+1 ∈ Ω and f n ∈ Ω, adding (43), (44) and (45) together, and using the fact that we have that According to (46), we have that Inverse Problems and Imaging Volume 11, No. 6 (2017), 917-948 Because ρ ≥ L ϕ , we have that According to (25), we have that where the last equality follows that W is a wavelet tight frame system [32]. According to (47), we have that The (51) can be written as follows: According to (48) and (52), we have that Thus, we finish the proof of Item 1. Next, we will prove the Item 2. According to the definition of Q(f, α), we can obtain that Q(f, α) ≥ 0. Because τ ≥ L θ 2 and ρ ≥ L ϕ in our algorithm, we can obtain that Q(f n+1 , α n+1 ) ≤ Q(f n , α n ), therefore, we have that the sequence {Q(f n , α n )} converges, we have that This proves Item 2.
Next, we will prove the Item 3. Because {(f n , α n )} is bounded, if {(f * , α * )} is a limited point of {(f n , α n )}, it implies that there exists a sub-sequence {(f nq , α nq )} such that lim q→+∞ (f nq , α nq ) = (f * , α * ). We claim that the λ α L 0 term of Q(f, α) is lower semi-continuous. If lim n→+∞ α n L = α * L , we have that supp(α * L ) ⊆ supp(α n L ) where supp(x) = {i|x i = 0}, it implies that α n L 0 ≥ α * L 0 . Therefore, we have that lim inf n→+∞ λ α n L 0 ≥ lim inf n→+∞ λ α * L 0 = λ α * L 0 . It implies that the λ α L 0 term of Q(f, α) is lower semi-continuous by the definition of lower semi-continuous. Thus, we obtain that From (26), we obtain that ]. Let n = n q − 1 in the above inequality and let q → ∞, and using (55), we have that From (56) and (57), we have that Because the rest terms of objective function Q(f, α) are quadratic function, we have that According to Item 1, we have that the sequence {Q(f n , α n )} converges, then, there exists a constant C, such that This proves the Item 3. Next, we will prove the last Item. By Lemma 5, we have that Let n → +∞, using the Item 2 and the Remark 1 of [4], we have that This implies that (f * , α * ) is a critical or a stationary point of Q(f, α).
Next, we will prove that (f n , α n ) converges to a critical point or a stationary point (f * , α * ). According to the proof of Item 3, we know that there exists a subsequence {(f nq , α nq )} satifying lim q→+∞ (f nq , α nq ) = (f * , α * ). So we need to prove that {z n := (f n , α n )} is a convergent sequence.
According to the Item 1, we have that {Q(f n , α n )} is a non-increasing sequence, it implies that Q(f * , α * ) < Q(f n , α n ) for all n > 0 by (58). Again from (58), for any η > 0, there exists a n 0 ∈ N + such that Q(f n , α n ) < Q(f * , α * )+η for all n > n 0 . Let ω = {z * ∈ R N × R P : ∃ an increasing sequence of integers {n l }, such that z n l → z * as l → +∞}, and ω can be regarded as an intersection of compact sets ω = q∈N n>q z n , thus, ω is compact set. It is clear that lim n→∞ dist(z n , ω) = 0, it implies that for any ε 1 there exists a positive integer n 1 such that dist(z n , ω) < ε 1 for all n > n 1 . Summing up all these facts, for all n > l := max{n 0 , n 1 }, we have that Since ω is compact and nonempty, and since Q is finite and constant on ω, and Q is a proper and lower semi-continuous function (the proof of Item 3), Q satisfies the KL property at each point of ω (the examples 2 and 3 of [4]). According to the Lemma 2, for all n > l, we can obtain According to Lemma 5, we have that From the concavity of F we have that From Item 1, (60), and(61), we have that ρ . and it implies that From the fact that 2 √ ab ≤ a + b for all a, b ≥ 0, we can obtain that (64) 2 z n+1 − z n 2 ≤ C∆ n,n+1 + z n − z n−1 2 .
Summing up (64) for i = l + 1, ..., n, we have that Since F ≥ 0, for all n > l, we have that This easily shows that The (67) implies that ∞ n=l+1 z n+1 − z n 2 → 0 as l → ∞. For q > p > l, we have that It implies that {z n } is a Cauchy sequence, which is a convergent sequence, then, we have that (f n , α n ) → (f * , α * ). So we finish the proof of Item 4. 5. Numerical experimental. In this section, we conduct some numerical experiments, which include simulated and practical data, to test the performance of our algorithm for limited-angle CT reconstruction, and we compare our algorithm from limited-angle data with FBP algorithm from full-scan data due to the FBP algorithm has been widely used in commercially CT and can exactly reconstruct the object from full-scan data. In additional, our algorithm is used to compare with PICCS algorithm. All experiments are implemented on a 3.40 GHz intel(R) Core(TM) i3-4130 CPU processor with 8G memory.

Parameters and iteration number selections.
Influenced by some factors, such as noise level, reconstructed object and projection data, etc., optimizing the parameters and the iteration number are a difficult task in CT reconstruction, which empirically selected by visual inspection. In the experiments, the parameters and the iteration number of our algorithm are selected by trial and error.

5.2.
Quantitative characterization of the reconstructed image. In this paper, we quantitatively characterize the reconstruction quality by considering the root mean square error (RMSE) [46], peak signal-to-noise ratio (PSNR) [37], and mean structural similarity index (MSSIM) [41], which can be used for measuring the degree of similarity between the reconstructed image from limited-angle projection data and the reference image that is reconstructed from full-scan data using ASD-POCS or desired image. It should be noted that the different quantitatively characterization results will be obtained if we use the different reference images.

5.3.
Reconstruction from simulated data with Gaussian noise. In limitedangle CT image reconstruction, a digital NURBS based cardiac-torso (NCAT) phantom [36], which was used in nuclear medicine research, is used to test our algorithm. The scanning parameters of the simulated limited-angle CT are given in Table 1. The scanning ranges [0, 100 0 ], [0, 80 0 ] and [0, 60 0 ] are investigated, and the number of projection views are 100, 80, 60, respectively. The standard deviation and the average value of Gaussian noise added to the projection data are 0.5% g ∞ (1% g ∞ ) and zero, respectively. For the noise levels 0.5% g ∞ , the reconstruction parameters of our algorithm are ρ = 1, τ = 128, β = 0.8 * 128, λ = 8 * 128, and γ = 8 * 128. For the noise levels 1% g ∞ , the reconstruction parameters of our algorithm are ρ = 1, τ = 128, β = 0.8 * 128, λ = 8 * 128, and γ = 10 * 128. The maximum iteration number is N ite = 1500 for all situations. Figure 6 shows the NCAT phantom and prior image that are used for our algorithm and PICCS algorithm. Figure 6. NCAT phantom and prior image. The first column is NCAT phantom, the second column is the prior image and the third column is the absolute value of the difference between phantom and prior image. The prior image is the same with NCAT phantom. The reconstructed images for different scanning ranges using FBP algorithm, PICCS algorithm and our algorithm are shown in Figure 7. The noise levels are 0.5% g ∞ . The first row is the reconstructed results using FBP algorithm. The subsequent rows are the reconstructed results using PICCS algorithm and our algorithm. From left to right, Figure 7 shows the reconstructed results for different scanning ranges [0, 60 0 ], [0, 80 0 ], [0, 100 0 ] and [0, 360 0 ] in each columns. As seen from Figure 7, the limited-angle artifacts occur in the reconstructed images using FBP algorithm and the reconstructed images are distorted near edges of the object, however, the limited-angle artifacts are better suppressed and the edges of reconstructed image are better preserved using PICCS algorithm and our algorithm for limited-angle reconstruction problem.
We also consider the high noise levels 1% g ∞ . Figure 8 shows the reconstructed images for different scanning ranges using FBP algorithm, PICCS algorithm and our algorithm. The first row is the reconstructed results using FBP algorithm. The subsequent rows are the reconstructed results using PICCS algorithm and our algorithm. From left to right, Figure 8   In Table 2, we summarize the characterized quantitatively results of reconstruction quality. From Table 2, we can obtain that our algorithm, for scanning ranges [0, 100 0 ], [0, 80 0 ], and [0, 60 0 ], achieves the minimum RMSE and maximum PSNR (MMSIM) compared to the FBP algorithm for full-scan range and the PICCS algorithm. It implies that a higher image quality can be reconstructed using our algorithm because our model makes the 2 norm of the difference between the highfrequency of reconstructed image and that of prior image to be minimized, and uses the 0 quasi-norm to promote the sparsity of wavelet coefficients of low-frequency. Next, a modified chest phantom, which includes three circles but the prior image [49] does not includes them (see Figure 9), is used to test our algorithm. The scanning parameters of the simulated CT system are given in Table 3    From Figure 10, with the decrease of the scanning range, the quality of the reconstructed images becomes deteriorate. The limited-angle artifacts are better suppressed and the edges of reconstructed image are better preserved using our algorithm and PICCS algorithm for limited-angle reconstruction problem. And the three circles, which are not included in prior image, can also be better reconstructed for the scanning ranges [0, 120 0 ].
The reconstruction quality is characterized quantitatively in Table 4. From Table  4, we can obtain that our algorithm achieves the minimum RMSE and maximum PSNR (MMSIM) compared to FBP algorithm for full-scan range and PICCS algorithm, however, the quality of the three circles reconstructed using FBP algorithm for full-scan range is better than that using our algorithm and PICCS algorithm for limited-angle range by visual inspection. This experiment indicates that our algorithm and PICCS algorithm will not miss some important information that is not included in the prior image, and the reconstructed information is not included in the prior image (the circles) which becomes vestigial for the scanning angular range [0, 100 0 ] and [0, 80 0 ] using PICCS algorithm and our algorithm since the projection data are incomplete.
In additional, a simulated phantom, which includes three cracks with different directions but the prior image dose not include them (see Figure 11), is used to test our algorithm. The scanning parameters of the simulated limited-angle CT are given in Table 1. The parameters of simulated phantom are listed in Table 5, I denotes the ellipse intensity value, h denotes the horizontal semi-axis of the ellipse's length, v denotes the vertical semi-axis of the ellipse's length, x 0 denotes the x-axis center of the ellipse, y 0 denotes the y-axis center of the ellipse, and r denotes the angle between the horizontal semi-axis and the x-axis. The standard deviation and the average value of Gaussian noise added to the projection data are 0.1% g ∞ and zero, respectively. In the experiments, we consider the scanning ranges [0, 100 0 ] and [0, 120 0 ], and the number of projection views are 100 and 120, respectively.
The reconstructed results for the different scanning ranges using PICCS algorithm and our algorithm are shown in Figure 12. The first row is the reconstructed results for the scanning ranges [0, 120 0 ]. The subsequent row is the reconstructed results for the scanning ranges [0, 100 0 ]. From left to right, Figure 12 shows the reconstructed results using PICCS algorithm and our algorithm in each column. Figure 12 indicates that our algorithm and PICCS algorithm not only take advantage of the prior image but also not miss some important information that is Figure 11. Phantom and prior image. The first column is the phantom, the subsequent columns are the prior image and the absolute value of the difference between phantom and prior image.  not included in the prior image. And the cracks becomes vestigial for the scanning angular range [0, 100 0 ] using PICCS algorithm and our algorithm, which are not included in the prior image, since the projection data are incomplete. The reconstruction quality is characterized quantitatively in Table 6. From Table  6, we can obtain that our algorithm achieves the minimum RMSE and maximum MMSIM compared to PICCS algorithm, however, the PSNR using our algorithm is smaller than that using PICCS algorithm.  (8), then, the soft thresholding method is utilized to solve the 1 minimization sub-problem, which is named as " 1 − 0 algorithm" according to the regularization terms. Likewise, "our algorithm" is named as " 2 − 0 algorithm". In the following experiments, we will test the performance of 1 − 0 algorithm for limited-angle CT reconstruction, and the reconstructed results are compared with 2 − 0 algorithm. A modified chest phantom (see Figure 9) is used to test the performance of 1 − 0 algorithm, the scanning parameters of the simulated limited-angle CT are given in Table 3. The scanning ranges [0, 100 0 ] and [0, 120 0 ] are investigated, and the number of projection views are 143 and 171, respectively. The standard deviation and the average value of Gaussian noise added to the projection data are 0.1% g ∞ and zero, respectively. The reconstructed results for the different scanning ranges using 1 − 0 algorithm and 2 − 0 algorithm are shown in Figure 13. The first column is the reconstructed results for the scanning ranges [0, 100 0 ]. The subsequent column is the reconstructed results for the scanning ranges [0, 120 0 ]. From top to bottom, Figure 13 shows the reconstructed results using 1 − 0 algorithm and 2 − 0 algorithm in each rows. From Figure 13, the limited-angle artifacts are better suppressed and the edge of reconstructed image is better preserved using 1 − 0 algorithm and 2 − 0 algorithm for limited-angle reconstruction problem. And the reconstruction quality using 2 − 0 algorithm is as well as that using 1 − 0 algorithm.
The reconstruction quality is characterized quantitatively in Table 7. From Table  7, the RMSE, PSNR, and MSSIM value obtained using 2 − 0 algorithm are close to that using 1 − 0 algorithm. The reason of this result is because the information of reference image, which is close to the desire solution, is introduced into the reconstruction model. A simulated phantom (see Figure 11) is used to test the performance of 1 − 0 algorithm, the scanning parameters are given in Table 1. The scanning ranges [0, 100 0 ] and [0, 120 0 ] are investigated, and the number of projection views are 100 and 120, respectively. The standard deviation and the average value of Gaussian noise added to the projection data are 0.1% g ∞ and zero, respectively.
The reconstructed results for the different scanning ranges using 1 − 0 algorithm and the 2 − 0 algorithm are shown in Figure 14. The first row is the reconstructed results for the scanning ranges [0, 120 0 ]. The subsequent row is the reconstructed results for the scanning ranges [0, 100 0 ]. From left to right, Figure 14 shows the reconstructed results using 1 − 0 algorithm and 2 − 0 algorithm in each columns. From Figure 14, the reconstruction quality using 2 − 0 algorithm is as well as that using 1 − 0 algorithm.
The reconstruction quality is characterized quantitatively in Table 8. From Table  8, the RMSE, PSNR, and MSSIM value obtained using the 2 − 0 algorithm are close to that using 1 − 0 algorithm. The reason of this result is because the information of reference image, which is close to the desire solution, is introduced into the reconstruction model.  The maximum iteration number is N ite = 1400. Figure 15 shows the reconstructed images from practical data using different algorithms, region of interest (ROI) is labelled by red rectangle. The image on the left column is the prior image. The subsequent columns are the reconstructed results using FBP algorithm for scanning rangs [0, 80 0 ] and [0, 360 0 ], and the reconstructed result using our algorithm. As seen from Figure 15, the reconstructed images using our algorithm, the slope artifact can be better suppressed and the edges of the object can be better preserved for limited-angle problem. We can obtain a better quality of reconstruction image using our algorithm for limited-angle problem as that using FBP algorithm for full-scan range. As seen from Figure 16, the reconstructed result using our algorithm for limited-angle scanning range is better than that using FBP algorithm for the full-scan range (i.e. [0, 360 0 ]) in terms of suppressing the noise.
We regard the prior image as reference image, the reconstruction quality is characterized quantitatively in Table 9. From Table 9, we can obtain that our algorithm achieves the minimum RMSE and maximum PSNR (MMSIM) compared to the FBP algorithm for full-scan range.   6. Conclusions and discussion. To suppress the limited-angle artifacts of limitedangle CT reconstruction, we propose a minimization model that is based on wavelet tight frame and prior image. We perform this minimization problem efficiently by iteratively minimizing. Moreover, we show that each bounded sequence, which is generated by iteratively minimizing separately, converges to a critical or stationary point. The experimental results indicate that our method can suppress the artifacts in reconstructed images, can recover the edge structure information more effectively and can obtain a better quality of reconstructed image for limited-angle problem as that using FBP algorithm for full-scan range. But above all, the introduced prior image will not miss the important information that is not included in the prior image, and the information, using PICCS algorithm and our algorithm, which becomes vestigial for the scanning angular range is seriously limited because the projection data are incomplete. What's more, the chosen prior image is very important for our algorithm, the error solution will be obtained using our algorithm if the chosen prior image is inappropriate. For example, more boundary of a prior image does not match with the reconstructed object. Thus, an image, which is chosen as a prior image, should be close to the underline image except for few small features. A prior image, which is used to the follow-up detection or diagnosis, can be obtained from full-scan data using FBP algorithm or ASD-POCS algorithm before the equipment is installed or the first diagnosis. This research has investigated only in fan-beam limited-angle CT and the parameters and iteration number of our algorithm are selected by trial and error. In the future, we will investigate this model for other applications and consider how to optimize the parameters and iterations.