I Introduction
Xray Computed Tomography (CT) is one of the most powerful clinical imaging imaging tools, delivering highquality images in a fast and cost effective manner. However, the Xray is harmful to the human body, so many studies has been conducted to develop methods that reduce the Xray dose. Specifically, Xray doses can be reduced by reducing the number of photons, projection views or the size of the fieldofview of Xrays. Among these, the CT technique for reducing the fieldofview of Xray is called interior tomography. Interior tomography is useful when the regionofinterest (ROI) within a patient’s body is small (such as heart), because interior tomography aims to obtain an ROI image by irradiating only the ROI with xrays. Interior tomography not only can dramatically reduce the Xray dose, but also has cost benefits by using a smallsized detector. However, the use of an analytic CT reconstruction algorithm generally produces images with severe artifacts due to the transverse directional projection truncation.
Sinogram extrapolation is a simple approximation method to reduce the artifacts. However, sinogram extrapolation method still generates biased CT number in the reconstructed image. Recently, Katsevich et al [1]
proved the general uniqueness results for the interior problem and provided stability estimates. Using the total variation (TV) penalty, the authors in
[2] showed that a unique reconstruction is possible if the images are piecewise smooth. In a series of papers [3, 4], our group has shown that a generalized Lspline along a collection of chord lines passing through the ROI can be uniquely recovered [3]; and we further substantiated that the high frequency signal can be recovered analytically thanks to the Bedrosian identify, whereas the computationally expensive iterative reconstruction need only be performed to reconstruct the low frequency part of the signal after downsampling [4]. While this approach significantly reduces the computational complexity of the interior reconstruction, the computational complexity of existing iterative reconstruction algorithms prohibits their routine clinical use.In recent years, deep learning algorithms using convolutional neural network (CNN) have been successfully used for lowdose CT
[5, 6], sparse view CT [7, 8], etc. However, the more we have observed impressive empirical results in CT problems, the more unanswered questions we encounter. In particular, one of the most critical questions for biomedical applications is whether a deep learningbased CT does create any artificial structures that may mislead radiologists in their clinical decision. Fortunately, in a recent theory of deep convolutional framelets [9], we showed that the success of deep learning is not from a magical power of a blackbox, but rather comes from the power of a novel signal representation using nonlocal basis combined with datadriven local basis. Thus, the deep network is indeed a natural extension of classical signal representation theory such as wavelets, frames, etc; so rather than creating new informations, it attempts to extract the most information out of the the input data using the optimal signal representation.Inspired these findings, here we propose a deep learning framework for interior tomography problem. Specifically, we demonstrate that the interior tomography problem can be formulated as a reconstruction problem in an endtoend manner under the constraints that remove the null space signal components of the truncated Radon transform. Numerical results confirmed the proposed deep learning architecture outperforms the existing interior tomography methods in image quality and reconstruction time.
Ii Theory
Iia Problem Formulation
Here, we consider 2D interior tomography problem and follow the notation in [3]. The variable
denotes a vector on the unit sphere
. The collection of vectors that are orthogonal to is denoted asWe refer to realvalued functions in the spatial domain as images and denote them as for . We denote the Radon transform of an image as
(1) 
where and . The local Radon transform for the truncated fieldofview is the restriction of to the region which is denoted as . Then, the interior reconstruction is to find the unknown within the ROI from .
IiB Null Space of Truncated Radon Transform
The main technical difficulty of the interior reconstruction is the existence of the null space [3, 10]. To analyze the null space, we follow the mathematical analysis in [3]. Specifically, the analytic inversion of can be equivalently represented using the differentiated backprojection followed by the truncated Hilbert transform along the chord lines, so we analyze the interior reconstruction problem to take advantages of this. More specifically, if the unit vector along the chord line is set as a coordinate axis, then we can find the unit vector such that consists of the basis for the local coordinate system and denotes its coordinate value (see Fig. 1). We further define 1D index set parameterized by the :
Then, the null space of the is given by [3, 4]:
for some functions . A typical example of the null space image is illustrated in Fig. 2. This is often called as the cupping artifact. The cupping artifacts reduce contrast and interfere with clinical diagnosis.
Note that the null space signal is differentiable in any order due to the removal of the origin in the integrand. Accordingly, an interior reconstruction algorithm needs an appropriate regularization term that suppresses by exploiting this. Specifically, one could find an analysis transform such that its null space is composed of entire function, and use it for an analysisbased regularization term. For example, the regularization using TV [2] and Lspline model [3, 4] correspond to this. The main result on the perfect reconstruction in [3] is then stated as follows. If the null space component is equivalent to a signal within the ROI, then is identically zero due to the characterization of Hilbert transform pairs as boundary values of analytic functions on the upper half of the complex plane [3]; so TV or Lspline regularization provides the unique solution.
IiC CNNbased Null Space Removal
Instead of designing a linear operator such that the common null space of and to be zero, we can design a frame and its dual such that and for all and the groundtruth image . This framebased regularization is also an active field of research for image denoising, inpainting, etc [11].
One of the most important contributions of the deep convolutional framelet theory [9] is that and correspond to the encoder and decoder structure of a CNN, respectively, and the shrinkage operator emerges by controlling the number of filter channels and nonlinearities. Accordingly, a convolutional neural network represented by can be designed such that
(2) 
Then, our interior tomography algorithm is formulated to find the solution for the following problem:
(3) 
where denotes the groundtruth data available for training data, and denotes the CNN satisfying (2). Now, by defining as a rightinverse of , i.e. , we have
for some , since the right inverse is not unique due to the existence of the null space. See Fig. 2 for the decomposition of . Thus, is a feasible solution for (3), since
(4) 
and the data fidelity constraint is automatically satisfied due to the definition of the right inverse. Therefore, the neural network training problem to satisfy (4) can be equivalently represented by
(5) 
where denotes the training data set composed of groundtruth image an its truncated projection. A typical example of the right inverse for the truncated Radon transform is the inverse Radon transform, which can be implemented by the filtered backprojection (FBP) algorithm. Thus, in (5) can be implemented using the FBP.
After the neural network is learned, the inference can be done simply by processing FBP reconstruction image from a truncated radon data using the neural network , i.e. . The details of the network and the training procedure will be discussed in the following section.
Iii Method
Iiia Data Set
Ten subject data sets from AAPM LowDose CT Grand Challenge were used in this paper. Out of ten sets, eight sets were used for network training. The other two sets were used for validation and test, respectively. The provided data sets were originally acquired in helical CT, and were rebinned from the helical CT to angular scan fanbeam CT. The size artifact free CT images are reconstructed from the rebinned fanbeam CT data using FBP algorithm. From the CT image, sinogram is numerically obtained using a forward projection operator. The number of detector in numerical experiment is 736. Only 350 detectors in the middle of 736 detectors are used to simulate the truncated projection data. Using this, we reconstruct ROI images.
IiiB Network Architecture
The proposed network is shown in Fig. 3. The first layer is the FBP layer that reconstructs the cuppingartifact corrupted images from the truncated projection data, which is followed by a modified architecture of UNet [12]. A yellow arrow in Fig. 3 is the basic operator and consists of
convolutions followed by a rectified linear unit and batch normalization. The yellow arrows between the seperate blocks at every stage are omitted. A red arrow is a
average pooling operator and is located between the stages. The average pooling operator doubles the number of channels and reduces the size of the layers by four. Conversely, a blue arrow is average unpooling operator, reducing the number of channels by half and increasing the size of the layer by four. A violet arrow is the skip and concatenation operator. A green arrow is the simple convolution operator generating the final reconstruction image.IiiC Network training
The proposed network was implemented using MatConvNet toolbox in MATLAB R2015a environment. Processing units used in this research are Intel Core i77700 central processing unit and GTX 1080Ti graphics processing unit. Stochastic gradient reduction was used to train the network. As shown in Fig. 3, the inputs of the network are the truncated projection data, i.e. . The target data corresponds to the 256
256 size center ROI image cropped from the groundtruth data. The number of epochs was 300. The initial learning rate was
, which gradually dropped to . The regularization parameter was . Training time lasted about 24 hours.Iv Results
We compared the proposed method with existing iterative methods such as the TV penalized reconstruction [2] and the Lspline based multiscale regularization method by Lee et al [4]. Fig. 4 shows the groundtruth images and reconstruction results by FBP, TV, Lee method [4] and the proposed method. The graphs in the bottom row in Fig. 4 are the crosssection view along the white lines on the each images. Fig. 5 shows the magnitude of difference images between the ground truth image and reconstruction results of each method. The reconstructed images and the cutview graphs in Fig. 4 show that the proposed method results have more fine details than the other methods. The error images in Fig. 5 confirm that the high frequency components such as edges and textures are better restored in the proposed method than other method.
We also calculated the average values of the peak signaltonoise ratio (PSNR) and the normalized mean square error (NMSE) in Table I. The proposed method achieved the highest value in PSNR and the lowest value in NMSE with about dB improvement. The computational times for TV, Lee method [4] and the proposed method were 1.8272s, 0.3438s, and 0.0532s, respectively, for each slice reconstruction. The processing speed of the proposed method is about 34 times faster than the TV method and 6 times faster than Lee method [4].
FBP  TV  Lee method [4]  Proposed  

PSNR [dB]  9.4099  30.2004  27.0344  37.4600 
NMSE  8.2941e1  6.9137e3  1.4332e2  1.2994e3 
V Conclusion
In this paper, we proposed a deep learning network for interior tomography problem. The reconstruction problem was formulated as a constraint optimization problem under data fidelity and null space constraints. Based on the theory of deep convolutional framelet, the null space constraint was implemented using the convolutional neural network with encoder and decoder architecture. Numerical results showed that the proposed method has the highest value in PSNR and the lowest value in NMSE and the fastest computational time.
Acknowledgment
The authors would like to thanks Dr. Cynthia McCollough, the Mayo Clinic, the American Association of Physicists in Medicine (AAPM), and grant EB01705 and EB01785 from the National Institute of Biomedical Imaging and Bioengineering for providing the LowDose CT Grand Challenge data set. This work is supported by National Research Foundation of Korea, Grant number NRF2016R1A2B3008104. Yoseob Han and Jawook Gu contributed equally to this work.
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