## 1. Introduction

The Interior problem reconstructs a region of interest (ROI) inside a patient or an object from data along X-ray paths through the ROI [

1]. This ROI reconstruction problem does not have a unique solution in an unconstrained image space, even if we know the object support (OS) exactly [

2]. As a result, reconstructions with filtered backprojection (FBP) suffer from a Direct Component(DC) shift and low-frequency artifacts, which make it difficult to meet the clinical diagnostic needs of physicians.

An X-ray is harmful to the human body. Hence, it is highly desirable to perform interior reconstruction for radiation dose reduction. Over the last few decades, a variety of ROI image reconstruction algorithms, which can improve the quality of reconstructed ROIs by incorporating smart prior knowledge into the image reconstruction process, have been developed. In particular, a theoretically exact local reconstruction methodology, which is referred to as interior tomography, has been proved and developed. Kudo H. and Ye YB have proven that if a sub-region in an internal ROI is known, the ROI can be reconstructed using differentiated backprojection (DBP) projection on a convex set (POCS) technique [

3,

4,

5]. Subsequently, a singular value decomposition (SVD)-based DBP method has been developed for interior tomography [

6,

7]. Jin X et al. improved a novel continuous SVD method for interior reconstruction, assuming a known sub-region [

8]. However, in many important applications, such as perfusion cardiac CT and micro-CT, a sub-region is not always available.

Inspired by compressed sensing theory, researchers have proposed an accurate reconstruction algorithm which incorporates sparse prior information of ROI into an iterative reconstruction model. For example, Yu et al. showed that if ROI can be expressed as a segmented constant function, the algebraic iterative reconstruction algorithm with total variation (TV) sparse constraints can accurately reconstruct ROI images [

9,

10]. Yang et al. proposed that if ROI is a piecewise polynomial function, then ROI can be accurately reconstructed by minimizing the high-order TV sparsity constraint [

11]. With the development of compressed sensing theory, many new sparse priors have been introduced to improve the quality of reconstructed ROI. Esther Klann et al. developed an algorithm for interior tomography of a piecewise constant function using a Haar wavelet basis [

12]. John Paul Ward et al. proved that the exact reconstruction of ROI is guaranteed using a one-dimensional generalized total variation seminorm penalty [

13,

14]. Subsequently, Liu et al. employed the curvelet transform coefficients to regularize the interior problem and obtained a curvelet frame-based regularization method for interior tomography [

15]. Zhao et al. applied the Mumford–Shah-TV regularization method to the interior tomography and developed an algorithm based on split Bregman and OS-SART iterations [

16]. Tatiana A. Bubba et al. proposed a nonsmooth regularization approach based on shearlets for ROI tomography [

17]. Nevertheless, all of the above algorithms exploit the sparse property of the image but ignore the fact that the projection data usually contain noise during a real CT scan. Considering the noise characteristics of the projection data, Xu et al. proposed a statistical interior tomography method with a TV sparsity constraint [

18]. However, the ROI images reconstructed with TV regularization may lose some detailed features and contain blocky artifacts in the case of noisy projections.

One crucial issue in interior tomography is that intensities drop around the peripheral region of a ROI. When a priori information does not exist or is not sufficiently strong, image intensity could deviate dramatically from the truth near the boundary of the ROI. There are a number of ways to suppress this problem by introducing additional information about the object to be reconstructed. Currently, an effective means is to enforce object support (OS). However, this is not always available in practical scenarios. Hence, we hypothesize that the zeroth-order image moment is a surrogate, which can be easily measured in the projection domain.

Sparse representation and dictionary learning (DL) have achieved great success in the medical imaging community, such as in low-dose image reconstruction [

19,

20], limited-angle CT reconstruction [

21], spectral CT [

22], etc. The extensive experimental results have shown that the DL-based regularization term is superior to the TV-based candidate in terms of preserving image details and removing artifacts. Encouraged by the above findings, here, we introduce the DL and zeroth-order image moment into the statistical iterative reconstruction (SIR) framework for interior tomography to further lower the radiation dose. The zeroth-order image moment can be estimated in the projection domain using the Helgason–Ludwig consistency condition (HLCC). An alternating minimization algorithm is developed to optimize the objective function.

The novelties of the proposed method are as follows. Firstly, an l1-normdictionary learning penalty is introduced into the SIR framework for the ROI image reconstruction to obtain a reconstruction performance that is superior to the existing TV penalty. Secondly, most CS-based interior tomography methods assume that the object support is known before the reconstruction of the proposed SIRDL + HL method using direct current priors which can be estimated using projection data and the Helgason–Ludwig consistency condition. Thirdly, an alternating minimization algorithm is developed to minimize the associated objective function, transforming the image reconstruction problem into a sparse coding sub-problem and an image updating sub-problem. To accelerate the convergence speed of the proposed alternating minimization algorithm, an order subset strategy is applied during the iteration process.

The rest of this paper is organized as follows. In

Section 2, we first briefly review related mathematical theories and then establish the proposed reconstruction framework. In

Section 3, both the numerical simulation and Sheep lung real CT projection experiments are performed to evaluate the proposed method. In

Section 4, we discuss the results and conclude the paper.

## 4. Discussion and Conclusions

Mathematically, there is no unique solution to the interior problem in CT reconstruction, and the reconstructed ROI image exhibits the DC shifts. Based on the assumption of a constant in the ROI region, TV-based interior tomography can eliminate the above artifacts well; however, in practical applications, the above assumptions do not necessarily hold, so TV constraint-based interior tomography will still show a slight DC artifact under the tight object support constraint. As early as 1992, the mathematician Maass had investigated the singular value decomposition of the Radon transform under the 2D interior problem and pointed out that the ROI reconstruction was almost exact except for a smooth unknown bias function, which is an approximately constant low-frequency component. Therefore, the difference between the DC value of the reconstructed image during the iterative process and the DC value calculated beforehand as a new constraint can eliminate the DC offset artifacts well.

Since dictionary sparse representation based on image blocks can effectively preserve the local structure information of images while suppressing image noise, introducing dictionary-based sparse representation into the statistical iterative reconstruction framework can effectively solve the interior problem, especially in a low-dose case. Extensive comparison of the experimental results shows that the performance of the dictionary learning-based interior tomography is better than that of the TV-based interior tomography.

The DC value of the cross-section where the ROI region is located needed to be calculated. In this paper, we used the untruncated projection data to calculate the value.

Figure 10 and

Figure 11 illustrate that the DC value of the cross-section image has little effect on the reconstruction results. In fact, we can use the truncated data to estimate the DC value. For example, we can estimate the ratio of the area of a cross-section of the human body to the area of the organ of interest; then, we can estimate the DC value of the whole cross-section using the DC value of the projection through the ROI region and the above ratio. The next step is to use the truncated projection data to estimate the DC component of the image of the cross-section where the ROI region is located.

In conclusion, we have incorporated sparse representation in terms of a learned dictionary and the constraint in terms of an image DC value into the SIR framework for interior tomography. Experimental results have demonstrated that our proposed approach reduces the DC artifact effectively and the DL-based algorithm outperforms the TV-based method in preserving fine structures, especially in a low-dose situation.