Preamble

Alternating Source-Detector Array Stationary CT System and Its Reconstruction Jia-bing Xiang, Yan-xin Wang, Yu-hang Yang, Wei Zhao, Bao-lei Li, and Bao-hua Sun 1 School of Physics, Beihang University, Beijing, 100191, China Hangzhou International Innovation Institute of Beihang, Beihang University, Hangzhou, 311115, China Tianmushan Laboratory, Hangzhou, 311115, China Beijing Hangxing Machinery Co.,Ltd., Beijing, 100013, China In this paper, we propose a novel design for a stationary CT system, termed the lternating ource- etector rray stationary CT (ASDA-sCT). The ASDA-sCT system comprises an array of miniature carbon nanotube X-ray sources and a detector array strategically positioned in the gaps between sources. To minimize projection loss caused by ray path obstruction, the X-ray sources are distributed within a short-scan trajectory that takes advantage of the fan-beam symmetry. After interpolation-based restoration of the discontinuities, CT images can be directly reconstructed using the filtered backprojection (FBP) algorithm with Parker’s weighting func- tion. We further investigate the influence of the number of X-ray sources on the reconstruction quality of the ASDA-sCT system and determine the optimal source number for different X-ray exit window sizes. However, the limited number of sources and the interpolation errors introduced during sinogram restoration remain critical barriers to achieving high-quality image reconstruction. To tackle these issues, we propose a tailored triple-stage three-stage dual-domain cascade neural network (TSDDC-Net), which incorporates prior knowledge to correct interpolation errors in the sinogram and compensate for the missing projection views. In the projection do- main, we introduce a novel multi-scale deformable convolution module (DFInception) that enhances feature extraction and improves the accuracy of sinogram refinement. In the image domain, a dual-encoder architecture is employed to independently extract features from the initial CT image reconstructed from raw interpolated projections and from the refined CT image reconstructed using the corrected sinogram. Ultimately, the well- designed deep learning model significantly enhances the quality of the reconstructed images. Experiments conducted on the Shepp-Logan phantom and abdominal CT datasets demonstrate the promising potential of the ASDA-sCT system for practical applications.

Keywords

Computed Tomography, Stationary CT, Sparse-view CT, Deep learning

INTRODUCTION

Computed tomography (CT) is an imaging technique that 2

reconstructs cross-sectional images of the interior of a body or object by measuring X-ray projections from multiple views with computational algorithms.

Due to its ability to reconstruct high-resolution three-dimensional images, CT is widely used in medical diagnostics and industrial non- destructive testing. Currently, all modern CT scanners adopt the third-generation rotate-rotate geometry where a single or double X-ray sources with detector modules rotate around the object for a full 360-degree scan[ ]. The development of CT technology focuses on three key advancements: en- hancing temporal resolution, reducing X-ray radiation expo-

sure, and minimizing manufacturing costs[ 3 ]. However, the 14

rotating mechanical structure of modern CT scanners con- 15

strains the further evolution of CT technology. Due to the enormously increasing centrifugal forces, the rotational scan time of CT’s gantry has reached its limit of 0.25 seconds[ Meanwhile, vibrations in the CT acquisition system caused

by mechanical rotation increase system noise and deteriorate 20

the spatial resolution of CT images. Last but not least, the cumbersome gantry and complex dynamic acquisition mech-

anism not only hinder portability but also significantly in- 23

Supported by the National Natural Science Foundation of China (No. 12325506). crease manufacturing costs.

In the 1980s, electron beam CT (EBCT) scanners were de-

veloped with a stationary design, eliminating mechanical mo- 26

tion to achieve shorter scan times of 33 to 100 ms for cardiac imaging[ ]. Nevertheless, the considerable expense associ- ated with its production has confined EBCT exclusively to laboratory research. With continuous development in X-ray source technology, the size of X-ray sources has gradually

reduced to the millimeter scale. Currently, miniature X-ray 32

sources can be classified based on electron generation meth- ods into thermal cathode X-ray sources[ ] and field emis-

sion cold cathode X-ray sources[ 8 , 9 , 12 ]. Miniature X-ray 35

sources with thermal cathode still rely on high-temperature heating of cathode materials to emit electrons, resulting in a long switching response time and limiting their performance

in high-speed dynamical applications. In contrast, miniature 39

X-ray sources with field emission cold cathode constructed from nanomaterials such as carbon nanotubes (CNTs) of-

fer significant advantages: they enable further miniaturiza- 42

tion, operate with lower power consumption, and achieve substantially shorter switching response times.

Recent stud- ies have demonstrated that active-current control schemes can accelerate the response of CNT emitters to the tens-of- nanoseconds range, thereby enabling blur-free, high-frame- rate imaging[ ]. In practical medical implementations, such as the stationary digital breast tomosynthesis system[ sealed 31-source CNT X-ray tube has achieved tube currents up to 38 mA and anode voltages up to 50 kVp. Long-term evaluations have confirmed stable performance, with a typi-

cal clinical lifetime exceeding three years under routine duty 53

cycles. Collectively, these advantages make field emission X-ray sources highly promising for applications in medical imaging and industrial non-destructive testing.

With the advancement of field emission cold cathodes, the practical application of stationary CT systems composed of multiple sources has become feasible. A stationary CT scan- ner arranges multiple fixed X-ray sources around the scanned object and achieves X-ray projection acquisition from differ- ent angles by rapidly switching X-ray sources through elec-

tronic control. Therefore, stationary CT scanners overcome 63

the constraints of centrifugal force and enable a faster scan-

ning speed without the dynamic rotation of the X-ray source 65

and detector. The CNT X-ray source array was first applied to stationary chest tomosynthesis and successfully reduced mo- tion artifacts in conventional tomosynthesis[ ]. Com- pared to stationary chest tomosynthesis, stationary CT imag- ing is far more complex than simply replacing the rotating X-ray tube with a stationary CNT X-ray source array. Since CT imaging requires nearly full-angle (360°) projection data, the fixed geometry of both the X-ray sources and the detec- tor module inevitably leads to X-ray path obstruction issues where the ray path may be obstructed by either the source ar- ray or the detector array. Enzhuo et al.[ ] designed rotation- free square and hexagonal stationary micro-CT systems. Two or three contiguous linear source arrays, combined with two or three linear detector arrays, form a square or hexagonal geometry. In this geometry design, the scan angle range was limited to approximately 180°, and the CT image was recon- structed using an iterative reconstruction algorithm. Chen et ] proposed a helical interlaced source-detector array CT architecture utilizing cylindrically distributed sources and a detector array. The missing data caused by source obstruction was compensated using interpolation, and a Katsevich-type reconstruction was implemented for fast approximate image reconstruction.

Bossuyt et al.[ ] developed a rectangular

multi-source X-ray cargo scanning system, where multiple 89

fixed X-ray sources arranged in a rectangular geometry pro-

vide multi-view projections without mechanical motion. A 91

model-based iterative reconstruction algorithm was employed to suppress aliasing artifacts for material discrimination in cargo inspection.

Gonzales et al.[ ] introduced a multi-plane

scanning scheme for baggage screening, in which two pairs of 95

linear source arrays and detector arrays are positioned in two separate transverse planes while the screened object moves along the z-axis. Due to the noncircular and limited-angle

scanning trajectory, CT reconstruction was performed using 99

an accelerated compressed sensing algorithm, and an analyti- cal reconstruction algorithm for this geometric configuration was provided in [ ]. Similarly, a three-plane stationary CT prototype was developed and experimentally demonstrated to meet the requirements for head CT imaging[ ]. To obtain complete and full-angle projection data without missing in- formation, Thompson et al.[ ] developed a source-detector double-ring configuration. The source-detector double-ring configuration consists of a source ring and a detector ring with a slight offset in the z-axial direction to avoid ray path obstruction. Although this geometric configuration enables the acquisition of projections from full angles, the geometric

deviations caused by tilted scanning can result in artifacts in 112

the reconstructed images. Due to incomplete projection data caused by non-ideal

scanning trajectories (e.g., sparse-view or limited-angle ac- 115

quisitions) or ray path obstructions in stationary CT sys- tems, conventional analytical reconstruction methods (filtered back-projection algorithm[ ], FBP) are often inapplicable.

Although iterative algorithms[ ] can reconstruct images from such incomplete data, their high computational com- plexity makes them impractical for real-time imaging ap-

plications, such as security screening and industrial non- 122

destructive testing. Recently, the advent of the data-driven

deep learning paradigm[ 26 , 27 ] has revolutionized CT recon- 124

struction by enabling faster reconstruction, enhancing image quality, and demonstrating great potential in mitigating ar-

tifacts caused by incomplete projection data. Deep learning 127

can compensate for missing information caused by undersam-

pling and limited angles by learning prior knowledge of the 129

scanned object from a vast amount of data. Deep learning- 130

based reconstruction methods can be categorized into three types based on the target domain: direct sinogram restora- tion in the projection domain[ ], artifact correction in the image domain[ ], and dual-domain joint optimiza- tion models that integrate information from both the pro-

jection and image domains[ 38 – 44 ]. Deep learning demon- 136

strates strong potential for mitigating projection incomplete- ness in stationary CT systems through the integration of prior

knowledge. Developing deep learning reconstruction frame- 139

works customized for the geometry and sampling character- istics of stationary CT systems remains a valuable and critical research direction.

In this work, we propose a new design of a station- ary CT system ( lternating ource- etector rray station- ary CT, ASDA-sCT) where the X-ray sources and detec- tors are alternately arranged on a circular ring.

Since detectors embedded between X-ray sources, projection occluded sources approximately recovered interpolation adjacent detector signals. restored sinogram directly reconstruct images using filtered backprojection algorithm. ensure projection angular coverage while

minimizing projection loss due to source occlusion, the 153

system adopts short-scan acquisition mode. Through simulation experiments, investigate optimal number X-ray sources different window sizes.

Beyond reconstructions using analytical iterative methods,

also introduce a tailored deep learning framework designed to 158

restore projection discontinuities correct undersampling artifacts inherent ASDA-sCT system.

We explore the optimal configuration of the X-ray source number and con- duct a comprehensive comparison among analytical, itera-

tive, and deep learning reconstruction methods. These exper- 163

iments validate the practical feasibility and superior recon- struction capability of the proposed system.

Our contributions are outlined as follows: We propose a novel stationary CT design (ASDA-sCT) with alternating X-ray sources and detectors on a cir- cular ring with a short scan mode.

The design of the ASDA-sCT system enables conve-

nient application of the analytical reconstruction algo- 171

rithm. Through simulation experiments, we investigate the effects of X-ray source number and X-ray exit win- dow size on reconstruction quality and determine the optimal source configuration.

We propose a three-stage dual-domain cascade network model tailored for the ASDA-sCT system. The model incorporates a novel DFInception module to enhance feature extraction in the projection domain. In the im- age domain, a dual-encoder architecture is used to bet-

ter fuse information from both initial and refined re- 181

constructions. These innovations effectively improve the quality of the reconstructed images.

The remaining content is organized as follows. In Sec- 184

, we provide a detailed description of the system design

and present analytical, iterative, and deep learning-based re- 186

construction methods. Section presents the experimental setup, results, and discussion. The conclusion are drawn in Section MATERIALS AND METHODS System Description In stationary CT systems, the fixed geometric configura- tion of X-ray sources and detectors inherently results in in- complete projection sampling. Specifically, full-angle scans suffer from unavoidable obstruction of ray paths by source or detector components, leading to missing projections and de- graded reconstruction quality. To overcome these limitations, we propose the ASDA-sCT system employing a short-scan acquisition mode. The geometric configuration of ASDA-

sCT is illustrated in Fig. 1 [FIGURE:1] . X-ray sources are uniformly dis- 200

tributed along a circular arc spanning from 0 to π + 2 γ m , 201

where denotes the fan angle of sources. A complete set of short-scan projections is acquired by rapidly switch- ing between the sources. The detector array is divided into two segments: a continuous detector array covers the angular range from while an interleaved detector ar- ray is embedded in the gaps between the X-ray sources. short-scan acquisition ASDA-sCT reduces projection decreases required number X-ray sources increases acquisition efficiency.

In the proposed ASDA-sCT system, a short-scan acquisition mode is adopted based on the geometric symmetry of fan- beam projections, which ensures sufficient angular sampling

for complete data acquisition with a scanning range of only 214

. Compared with a full 360° source arrangement, this configuration reduces the required number of X-ray sources for the same source density along the ring. Additionally, the detector segment without source placement can continuously

measure projection data without interruption, thereby mini- 219

mizing projection data loss. For regions with missing projec- tion data, interpolation from adjacent detector signals can be used to estimate the absent measurements, thereby preventing . Detectors are inserted into the gaps between the X-ray sources, forming an alternating arrangement of sources and detectors. the reconstruction task from degenerating into a limited-angle CT problem. illustrates the 3D model of the proposed ASDA-sCT system, where the detector arrays are interleaved

between adjacent X-ray sources. To minimize projection data 226

loss at the source positions, each detector module is designed to cover as much of the gap between two neighboring sources as possible. Along the source array, only the X-ray exit win-

dows are left uncovered, while all the remaining spaces be- 230

tween them are occupied by the detector arrays. Therefore, the size of each exit window corresponds to the spacing be- tween adjacent detector arrays, which directly defines the re- gions of missing projections in the sinogram.

In summary, the ASDA-sCT geometry effectively miti- gates the inherent ray-path obstruction problem in the design of stationary CT systems. However, projection discontinu- ities caused by source obstruction and artifacts from sparse angular sampling are still protogenetic issues for practical ap- plication. Hence, we focus on advancing the reconstruction algorithm to improve image quality under these constraints.

System Parameters The geometric configuration of the ASDA-sCT system is

summarized in Table 1 [TABLE:1] . The system utilizes miniature CNT 244

X-ray sources with a fan angle of 60 ◦ (half-angle γ m = 30 ◦ ), 245

uniformly distributed along a circular detector ring with a ra- 246

dius of 512 mm. This setup provides an effective field of view (FOV) diameter of 512 mm, adequately satisfying the imaging requirements of both medical diagnostics and secu-

rity screening applications. 250

In the ASDA-sCT system, imaging quality is strongly in-

3D schematic diagram of the ASDA-sCT. Parameters Value Distance between the source and isocenter Distance between the source and detector center (mm) Fan angle Dimension of reconstruction grids (pixels) Pixel size (mm) Size of detector elements (mm) Detector type Curved detector fluenced by the size of the X-ray exit window and the number of X-ray sources. Due to ray path obstruction, detectors can- not be positioned within the exit window region, resulting in missing projection data where the sources occupy space on the detector ring. As the X-ray exit window size increases, the available detector area within the source-detector alter- nating segment decreases, causing a larger fraction of projec- Detector array size as a function of source number under dif- ferent X-ray exit window sizes (5mm, 10mm, and 20mm). Smaller window sizes allow a greater number of sources to be accommo- dated at the same size of detector array. tions to be lost. proportion missing projections quantified

deleted ( ϵ = Ns

L = Ns (2 γ m + π ) R ) , (1) 261

where number X-ray sources, X-ray window, radius detector ring, length source-detector alternating segment.

When the sources fully occupy this segment ( ϵ = 100% ), all 265

projection region lost, reconstruction deteriorates limited-angle problem.

Moreover, window constrains maximum number deployable sources, which estimated

deleted ( N max = � (2 γ m + π ) R s · ϵ max

where denotes maximum acceptable proportion missing projections.

Since the X-ray sources and detectors are alternately arranged, the number of X-ray sources and de- tectors is identical within each alternating segment:

N = L d + s = (2 γ m + π ) R d + s , (3) 275

where is the number of X-ray sources, is the size of the detector array between adjacent sources, is the size of the X-ray exit window, is the radius of the detector ring, and is the arc length of the source-detector alternating segment.

In this study, we consider three configurations with X-ray exit window sizes of 5 mm, 10 mm, and 20 illustrated proportion missing projections increases linearly number sources.

Taking account non-negligible detectors embedded between sources, maximum tolerable projection proportion As illustrated in Fig. , as the number of X-ray sources increases, the size of the detector array correspondingly decreases. For a fixed window size, the maximum number of deployable sources is

determined by the minimum acceptable detector array size 290

that can be placed between adjacent sources:

N max = � (2 γ m + π ) R d min + s

Since the detector width cannot be infinitely small, we set the 293

minimum detector array size to 1 mm, based on the current 294

detector fabrication capabilities. Under this constraint, the maximum number of deployable sources is 357, 194, and 102 for window sizes of 5 mm, 10 mm, and 20 mm, respectively.

The number of X-ray sources plays a pivotal role in deter-

mining image reconstruction quality. A smaller number of 299

sources reduces projection loss but may lead to insufficient angular sampling, resulting in pronounced aliasing artifacts.

Conversely, increasing the number of sources enhances the projection number but exacerbates projection loss, potentially degrading reconstruction quality due to increased interpola- tion errors. Therefore, achieving an optimal balance between source number and projection completeness is essential in ASDA-sCT system design and will be further investigated in

Band-shaped discontinuities appear in the redundant region due to projection data loss.

Analytic Reconstruction using Parker’s Smooth Weighting Function Based on the geometric symmetry of the fan-beam, sym- metric projection paths satisfy the following relationship:

p ( γ, β ) = p ( − γ, π + β + 2 γ ) , (5) 313

where denotes the line integral projection, represents the position in the detector coordinate, and is the projection angle. In a full-angle circular trajectory fan-beam CT scan, each projection path is sampled twice. To eliminate redun- dant projections, the short-scan acquisition mode reduces the total scan angle from without compromising the completeness of the projection data. The redundant pro- jection region in a fan-beam short-scan sinogram is illustrated in Fig.

To avoid artifacts in the reconstructed image, sinogram re- dundancy must be reweighted to ensure that each ray con- tributes equally to the reconstructed image. This is achieved by applying a weighting function that equalizes the contri- bution of singly- and doubly-sampled projections.

In this work, we adopt the smooth weighting function proposed by Parker[

� , 0 ≤ β < 2 γ m − 2 γ

1 , 2 γ m − 2 γ ≤ β ≤ π − 2 γ

ω β ( γ ) =

� , π − 2 γ < β ≤ π + 2 γ m

The projections are then reweighted using this function, followed by image reconstruction via the filtered backprojec- tion algorithm. sCT system. Due to the absence of detectors at source po- sitions, periodically spaced band-like gaps appear in the re- dundant region of the sinogram. Geometric ray path analysis reveals that X-rays emitted from a given source are lost when projected onto the position of an opposing X-ray source, where no detectors are present. Due to the reversibility of light paths, the conjugate rays traveling in the opposite di- rection are also missing. Consequently, the conjugate projec- tion rays are missing in pairs, making it impossible to com- pensate for the loss solely through the arrangement of X-ray sources. To mitigate this discontinuity, linear interpolation using neighboring projection data can be applied to estimate the missing values.

Iterative Reconstruction Algorithms Compared to the analytical reconstruction algorithm, iter- ative reconstruction methods offer greater flexibility in han-

dling incomplete or non-uniform projection data. They en- 351

able stable image reconstruction under conditions such as sparse sampling or missing projections.

Moreover, prior

knowledge, such as non-negativity, total variation minimiza- 354

tion, and sparsity, can be incorporated as regularization terms during the iterative process to improve reconstruction quality.

Currently, the Simultaneous Algebraic Reconstruction

Technique (SART)[ 51 , 52 ] is one of the most widely used 358

iterative algorithms. Unlike the Algebraic Reconstruction

Technique (ART)[ 53 ], which updates pixel values sequen- 360

tially for each ray, SART employs a block-wise update strat- egy. In each iteration, it computes corrections from all pro- jection paths and averages them to update the image. The iterative update formula is given by:

x ( k +1) j = x ( k ) j + λ

� M i =1 a ij , (7) 365

where denotes the value of pixel at iteration is the measured projection for the -th ray, is the -th row of the system matrix, is its -th entry, and is the relaxation parameter. By averaging over all projections, SART reduces the impact of local noise, thereby improving reconstruction stability.

To mitigate the undersampling issue in the ASDA-sCT sys- tem, a compressed sensing (CS)-based reconstruction method with total variation (TV) regularization can be employed.

The reconstruction method is based on the Adaptive Steep- est Descent – Projection onto Convex Sets (ASD-POCS)

algorithm[ 54 ]. It minimizes the following objective function 377

where is the system matrix, is the measured projec- tion data, and is the regularization weight. The TV term,

TV( x ) , minimizes the ℓ 1 -norm of the image gradient, en- 381

couraging piecewise-smooth structures and preserving sharp edges.

ASD-POCS integrates gradient-based optimization with projection onto convex sets, ensuring convergence while enhancing image quality under sparse-view acquisition.

E. Deep Learning Reconstruction Method: A Three-Stage 386

Dual-Domain Cascade Neural Network The interpolation errors and the limited number of X-ray

sources significantly degrade CT image reconstruction qual- 389

ity. Recently, data-driven deep learning methods have demon- 390

strated strong potential in learning the underlying structure 391

of scanned objects to compensate for the missing informa- tion caused by sinogram discontinuities and undersampling.

Therefore, we propose a hree- omain ascade neural network (TSDDC-Net) that incorporates prior knowl- edge to improve reconstruction quality. An overview of the

proposed deep learning–based reconstruction framework is 397

presented in Fig.

1. Sinogram

Restoration Upsampling In the first stage, linear interpolation is used to coarsely re- store discontinuities in the measured sinogram. To alleviate undersampling artifacts, interpolation is also applied along the angular direction to enhance projection density. However, coarse restoration upsampling process inevitably intro- duces substantial interpolation errors, which limit the quality of the reconstructed images. Thus, a subsequent correction stage is necessary to correct interpolation errors and improve image quality.

2. Sinogram-Domain Network

In the second stage, a UNet-based Sinogram-Domain Net- work (SD-Net) is employed to correct interpolation errors in the sinogram. The network adopts the classical encoder- decoder structure, where multi-scale features are extracted and hierarchically upsampled to produce a refined sinogram.

To enhance the network’s feature extraction capability, we propose a Deformable Inception (DFInception) module, which integrates the deformable convolution[ ] with the In- ceptionNext module[ ]. Deformable convolution intro- duces learnable offset parameters to the standard convolution operation, enabling the network to adjust the sampling loca-

tions of the convolutional kernels adaptively. This mechanism 421

allows the receptive field to dynamically change its shape and position in response to the geometric structure of the input features. The deformable convolution operation can be ex- pressed as

Y ( p 0 ) = �

where denote the input and output feature maps, re- spectively, is the position of the current output location, represents the receptive field of the convolution kernel, the predefined offset of the -th sampling point in

is kernel weight at position p n , and ∆ p n ∈ R 2 is the learn- 431

able offset for sampling position . The core idea of the In- ception module is to apply multiple convolutional filters with different kernel sizes in parallel within the same layer, en- abling multi-scale feature extraction. The operation can be expressed as:

, X = Concat ( X i ) ,

where the input feature map is split into channel groups . Then, each is processed by a deformable convolution with kernel size . The outputs are then concatenated and added to the original input via a residual connection.

The design of the DFInception module is motivated by two key observations.

First, sinogram data consist of overlap- ping sinusoidal trajectories, which inherently are global pat- terns across the entire image. Standard convolutional layers are limited to local receptive fields and thus struggle to cap- ture such long-range dependencies. In contrast, our proposed DFInception module combines deformable convolutions and multi-scale kernels to effectively extend the receptive field

and capture long-range interactions with significantly fewer 450

parameters than Transformer-based models. Second, due to the large missing regions in sinograms, using elongated con- volutional kernels (e.g., ) is essential for enabling the network to directly connect the central pixels of the miss- ing regions with the surrounding measured areas in the high- resolution layers of the UNet architecture. This facilitates

early-stage compensation and significantly improves the per- 457

formance of error correction. When supervised by ground-truth well-sampled sinograms, SD-Net effectively learns to correct interpolation errors. The refined sinogram is subsequently reconstructed via the fil- tered backprojection algorithm to produce a refined CT im- age, which is then fed into the image-domain refinement stage. The network is trained using a mean squared error (MSE) loss:

L SD = ∥ ˆ S − S gt ∥ 2 2 , (11) 466

where is the refined sinogram, and represents the ground truth.

3. Image-Domain Network

Although sinogram-domain correction reduces major arti- facts, even small residual errors in the sinogram can lead to noticeable artifacts in the reconstructed CT images, such as blurring, structural distortion, and loss of edge contrast. This observation highlights that prior knowledge in the sinogram domain alone is insufficient for high-fidelity reconstruction.

To further enhance image quality, especially in terms of struc- tural details and edge preservation, we introduce an Image- Domain refinement Network (ID-Net) designed to correct in- terpolation errors that remain unaddressed or are newly intro- duced in the sinogram domain.

The network is fed with two CT images: one recon- structed from the SD-Net-corrected sinogram and the other from the raw interpolated sinogram. In this way, prior knowl-

Stage 1: Linear interpolation is applied to the measured sinogram to eliminate discontinuities and increase projection views.

Stage 2: A sinogram-domain network (SD-Net), equipped with the proposed DFInception module, refines the interpolated sinogram by correcting interpolation-induced errors.

Stage 3: Two separate encoders are employed to independently extract features from the initial CT image and the SD-Net-enhanced CT image, respectively. These features are then fused through a decoder to reconstruct the final image in the image domain.

edge introduced from the sinogram domain is effectively

utilized while retaining the original measured information. 485

Two separate encoders are employed to extract multi-scale features from both inputs, leveraging complementary infor-

mation from sinogram correction and initial reconstruction. 488

These features are then fused through a decoder to gener- ate the final high-quality CT image.

The network is su- pervised using CT images reconstructed from fully sampled sinograms, and trained with an MSE loss as well:

L ID = ∥ ˆ I − I gt ∥ 2 2 , (12) 493

where denotes the predicted CT image, and represents the ground truth.

Experiment Dataset We evaluate the design of the ASDA-sCT and its recon- struction method using a Shepp-Logan phantom and a pub- licly available abdominal CT dataset, authorized for the 2016

NIH-AAPM-Mayo Clinic Low Dose CT Grand Challenge1 501

by Mayo Clinic . In this dataset, 6146 full dose abdominal CT 2D slices from 42 anonymous patients are divided into a

training set of 4138 CT images (27 patients), a validating set 504

of 625 CT images (5 patients), and a testing set of 1356 CT

images (10 patients) with 70% allocated for training, 10% 506

allocating for validating, and 20% for testing. Before train- ing, the CT images and projections are normalized into [0,1].

The photon number was set to in the simulation. The number of iterations was set to 200 for both SART and POCS-TV. The regularization weight in POCS-TV was set to 1. SD-Net and ID-Net were trained for 200 epochs with

an initial learning rate of 0.0005. The system configuration is 513

listed in the Tab. . All experiments were run on a computer with an Intel(R) Xeon(R) Silver 4210R CPU and an NVIDIA GeForce GTX3090.

RESULTS AND DISCUSSION Find The Optimal Source Number In this study, we conducted simulated reconstruction exper- iments using the training dataset CT images under different numbers of X-ray sources, followed by a quantitative evalu- ation of the reconstruction results shown in Fig. . The ex- periments compared the reconstructed images from ASDA- sCT systems with different exit window sizes to those from sparse-view CT systems with the same number of sources but without sinogram discontinuities. The results demonstrate that both the number of sources and the size of the exit win- dow are critical factors influencing reconstruction quality. the ASDA-sCT system, the number of X-ray sources is lim- ited by the physical size of the CNT X-ray sources. For the smallest exit window size in our simulation (5 mm), the total number of X-ray sources can reach only about 357. There- fore, for medical diagnostic applications, the number of pro- jection views in ASDA-sCT falls below the Nyquist sampling requirement, which may lead to angular aliasing and degrada- tion of fine structural and lesion details in the reconstructed images. This undersampling issue must be addressed by ad-

vanced reconstruction algorithms to enable practical clinical 538

implementation. number sources When the number of sources is under 50 , the reconstruction quality of the station- ary CT system is comparable to that of the sparse-view CT system.

In this case, image degradation is mainly due to aliasing artifacts caused by undersampling. As the number of sources increases, the reconstruction quality gradually im- proves. However, the improvement in the stationary CT sys- tem is less than that in the sparse-view CT system, primarily because interpolation errors become more pronounced due to increased projection loss.

The study further reveals that the size of the source window has a substantial impact on reconstruction quality.

Larger window sizes necessitate interpolation over broader regions, which increases interpolation errors.

Consequently, for a fixed number of sources, systems with smaller window sizes yield lower interpolation errors and achieve better reconstruc- tion quality. Remarkably, when the window size is reduced to 5mm, the reconstruction quality of the stationary CT system becomes closely comparable to that of the sparse-view CT system with the same number of sources. This finding sug- gests that further reduction in source size could effectively suppress interpolation artifacts, enabling the stationary CT system to achieve image quality comparable to that of rotat- ing CT systems, while avoiding projection loss caused by ray path obstruction.

Overall, although increasing the number of sources exac- erbates interpolation errors, it more effectively mitigates the dominant aliasing artifacts caused by undersampling. There- fore, the optimal number of sources in the ASDA-sCT system should be maximized within the physical constraints of X-ray source placement. Specifically, the experimental results indi- cate that when the maximum allowed proportion missing

projections is set to 90% minimum detector array size is set 572

to 1mm , the optimal number of sources for exit window sizes of 5mm, 10mm and 20mm is 357, 194 and 102, respectively.

Experiments on the Shepp-Logan Phantom system under different X-ray exit window sizes. In the upper and lower triangular regions of the sinogram, known as ray- redundant regions, strip-like gaps appear due to the lack of detector coverage at source positions. For all three systems, with optimal numbers of X-ray sources corresponding to win- dow sizes of 5mm, 10mm, and 20mm, the proportion of miss- ing projections in the redundant ray regions is approximately

Index Measure (SSIM). Sinograms under different X-ray window sizes using Shepp-Logan phantom, along complete sinograms (without discontinuities), interpolated results, interpolation errors (display window:[-10, 10]). bottom, corresponds window 102), 194), 357), respectively. display window sinogram images 150].

Sinograms obtained under different X-ray exit window sizes using the Shepp–Logan phantom. From left to right in each row: projections of the rotational CT with the same scanning geometry, projections acquired by the ASDA-sCT system, interpolated projections of ASDA-sCT, and the corresponding interpolation error images. From top to bottom: exit window size of 20 mm (N = 102), 10 mm (N = 194), and 5 mm (N = 357), respectively. The display window settings are indicated below the column titles. 90%, as previously specified. Larger exit window sizes result in wider projection gaps and a greater number of missing pix- els requiring interpolation, which in turn leads to increased interpolation errors. Conversely, smaller window sizes allow for the deployment of more X-ray sources within the same

scanning trajectory. This not only reduces aliasing artifacts 589

but also decreases interpolation errors as well. roughly recover the missing projection data, linear in-

Reconstruction

results

Shepp-Logan phantom using different reconstruction methods:

Ground Truth, (Sparse-View (ASDA-sCT), (ASDA-sCT), POCS-TV (ASDA-sCT).

Subfigures (a)–(c) correspond window 10mm, 20mm, respectively.

Images marked asterisk denote residual respect ground truth. display window reconstructed images residual images [-0.2, 0.2].

Reconstruction results of the Shepp–Logan phantom using the reconstruction methods listed above. Here, sparse-view CT denotes the reconstruction from a rotational CT system using the same geometry and projection number as ASDA-sCT. Subfigures (a)–(c) correspond to exit window sizes of 20 mm, 10 mm, and 5 mm, respectively.

Images marked with an asterisk (*) denote the residuals with respect to the ground truth. The distinct errors and artifacts (red arrows) in the reconstructed images are caused by large interpolation errors at the corresponding locations in the sinogram. terpolation is applied based on the available neighboring signals within each source gap.

This approach performs well in regions where the projection values change grad- ually.

However, areas large projection gradients, typically structural edges image,

linear interpolation introduces significant errors.

Such errors 597

contribute visible artifacts degrade accuracy representation structural detail final reconstructed images.

However, in regions with large projection gradients—such as those indicated by the red ar- rows in the interpolation error images of Fig. —linear in- terpolation introduces substantial errors. These areas corre-

spond to the edges of the outer elliptical structure in the re- constructed images, as highlighted by the red arrows in the residual images of Fig. (third column). Consequently, the introduced interpolation errors primarily degrade the accu- racy of edge reconstruction and increase visible artifacts in the reconstructed images.

results

Shepp-Logan phantom using analytical iterative methods.

Compared

results

sparse-view ASDA-sCT reconstructions exhibit aliasing artifacts pronounced errors structures interpolation inaccuracies. window decreases, system accommodates sources, thereby reducing aliasing interpolation-related artifacts.

Although iterative algorithms POCS-TV effectively suppress aliasing artifacts,

methods

entail substantial increase computational complexity reconstruction time. presents a comparison between the reconstruction results of the Shepp–Logan phantom ob- tained from the ASDA-sCT system and those reconstructed from a conventional rotational CT (sparse-view CT) with the

same scanning configuration. For the sparse-view rotational 625

CT, the differences from the ground truth primarily arise from streaking aliasing artifacts caused by insufficient projection sampling. In contrast, the ASDA-sCT reconstructions exhibit not only aliasing artifacts but also pronounced edge errors at the edge of the reconstructed object, which are attributed to interpolation inaccuracies in the sinogram.

Among the three reconstruction methods, the FBP algorithm are inher- ently less stable and tends to be sensitive to local interpo- lation errors in the sinogram and the undersampling issue.

The SART method is able to partially suppress aliasing ar- tifacts by ensuring projection consistency.

The POCS-TV algorithm yields the best performance, as its total variation prior—assuming gradient sparsity—matches the piecewise- smooth characteristics of the Shepp–Logan phantom, result-

ing in minimal aliasing artifacts and edge errors. As the exit 640

window size decreases, more X-ray sources can be accommo-

dated within the same scanning trajectory, thereby increasing 642

the number of available projections and reducing aliasing ar- tifacts. Meanwhile, a smaller exit window size also narrows the missing projection regions, which enhances interpolation accuracy in the sinogram and consequently improves the fi- delity of edge reconstruction. Table summarizes the quanti- tative evaluation results for the Shepp–Logan phantom exper- iments. Consistent with the observations in Fig. , the POCS- TV algorithm achieves the best quantitative results due to its effective use of prior information. All three reconstruction methods show improved performance as the exit window size decreases, benefiting from increased sampling and reduced interpolation errors.

Experiments on the Abdominal CT Dataset present abdominal reconstruction

results

obtained using analytical algorithms, iterative

methods, and deep learning approaches under X-ray 658

window sizes 20mm, respectively. Among traditional methods, POCS-TV algorithm, which incorporates regularization, achieves effective suppression aliasing artifacts compared analytical algorithms.

However, strong regularization constraint leads excessive smoothing, resulting

diminished edge contrast, loss of fine structural details, and 665

appearance plastic-like artifacts. , Fig. and Fig. present abdominal CT reconstruction results ob-

tained using FBP, iterative reconstruction, and deep learning 668

methods under X-ray exit window sizes of 5 mm, 10 mm, and 20 mm, respectively. Although the FBP algorithm pro- vides the fastest reconstruction speed, it yields the worst im- age quality. As shown in the enlarged view of the red-box region in Fig. (c,2) (yellow arrows), the anatomical details of the soft tissue are almost indistinguishable due to the mix- ture of severe aliasing artifacts and interpolation errors. As shown by the quantitative results in Table , the reconstruc- tion performance of SART is only slightly better than that

of FBP. Visually, the differences between the two reconstruc- 678

tions are minimal, with almost no noticeable improvement. 679

Among traditional methods, the POCS-TV algorithm, which incorporates total variation regularization, achieves more ef- fective suppression of aliasing artifacts. However, the strong regularization constraint leads to the over-smoothing prob- lem. As observed in Fig. (c,4), the internal cavities and fine structural features of the soft tissue are oversmoothed.

The gray-level transitions become overly uniform, and most 686

texture details are lost, producing a plastic- or wax-like ap- pearance, commonly named as plastic-like artifacts. Such ex- cessive smoothing often results in a lack of fine texture de- tails, which may reduce the diagnostic reliability of the re- constructed image.

In contrast, data-driven deep learning

methods

exhibit 692

incomparable advantages artifact suppression structural detail preservation.

As illustrated in Figs. (b,), (b,) and (b,*), the reconstruction residuals of deep

learning–based methods are significantly smaller than those 696

of traditional algorithms. Table also demonstrates that when facing severe artifacts (at exit window sizes of 10

mm and 20 mm), deep learning methods achieve substan- 699

tial improvements in reconstruction accuracy. This advan- tage stems from their ability to exploit learned statistical priors of the scanned objects, thereby achieving more ef- fective artifact suppression and enhanced preservation of structural details. compared three representative

learning models—FBPConvNet [ 32 ], HD-Net [ 43 ], and 705

DuDoTrans For comparison, three representative

deep learning models—FBPConvNet [ 32 ], HD-Net [ 43 ], and 707

DuDoTrans [ ]—were evaluated alongside our proposed TSDD-Net model under the same experimental conditions.

All models were trained on the same dataset with their respec-

tive optimal training configurations as reported in the original 711

publications. image-domain-only model FBPConvNet still struggles limited sharpness incomplete recovery structures. comparison, models HD-Net, DuDoTrans, proposed TSDDC-Net, which incorporate prior knowledge projection

Reconstruction results on abdominal CT data using different methods under the ASDA-sCT system configuration with an X-ray exit window size of 5mm with e.g., methods listed above. Since the sampling is sufficiently dense, traditional methods can already achieve reasonably good reconstruction results. The deep learning–based methods provide more detailed reconstructions and show better suppression of reconstruction noise. As indicated by the yellow arrows, TSDDC-Net exhibits the most accurate recovery of fine details, producing results that are closest to the Ground Truth and demonstrating superior overall reconstruction quality.

Reconstruction

results

abdominal using different

methods

under ASDA-sCT system configuration X-ray window 10mm. display window reconstructed images [-200, 200], while residual images [-50, comparison includes:

Ground Truth, SART, POCS-TV, FBPConvNet, HD-Net, DuDoTrans, TSDDC-Net (proposed).

Subfigures: Reconstructed images, Residual images respect ground truth, Enlarged views red-boxed regions highlighting structural details.

Reconstruction results on abdominal CT data using different methods under the ASDA-sCT system configuration with an X-ray exit window size of 10mm with e.g., methods listed above. As indicated by the yellow arrows, the soft-tissue structures reconstructed by deep learning–based methods exhibit superior artifact suppression and more complete detail recovery compared with traditional approaches. Among them, the proposed TSDDC-Net achieves the highest reconstruction quality, particularly in restoring the fine streak-like soft-tissue structures highlighted by the lower yellow arrows.

Method

Exit Window Size = 5 Exit Window Size = 10 Exit Window Size = 20 FBP(Sparse-View CT) FBP(ASDA-sCT) SART(ASDA-sCT) POCS-TV(ASDA-sCT) 0.03030 Reconstruction

results

abdominal using different

methods

under ASDA-sCT system configuration X-ray window 20mm. display window reconstructed images [-200, 200], while residual images [-50, comparison includes:

Ground Truth, SART, POCS-TV, FBPConvNet, HD-Net, DuDoTrans, TSDDC-Net (proposed).

Subfigures: Reconstructed images, Residual images respect ground truth, Enlarged views red-boxed regions highlighting structural details.

Reconstruction results on abdominal CT data using different methods under the ASDA-sCT system configuration with an X-ray exit window size of 20mm with e.g., methods listed above. All reconstruction methods suffer from substantial loss of fine structural details. However, the proposed TSDDC-Net exhibits the most effective detail preservation and achieves the most accurate recovery of anatomical structures. image domains, demonstrate substantially improved reconstruction performance.

Among them, TSDDC-Net

consistently achieves the best

results

in minimizing artifacts 719

while preserving anatomical details. As shown in of the soft-tissue details. However, the internal cavity bound- aries are not clearly reconstructed, and the streak-like struc- tures indicated by the bottom yellow arrow appear blurred.

HD-Net, which integrates domain priors from both the pro- jection and image domains, delivers notably superior recon- struction results compared with FBPConvNet.

The recon- struction obtained by HD-Net exhibits enhanced clarity in soft-tissue contours. In comparison, DuDoTrans produces an overly smoothed reconstructed image, leading to a percepti- ble loss of high-frequency structural information. The pro- posed TSDDC-Net achieves the highest reconstruction accu- racy, accurately recovering the soft-tissue morphology, delin- eating the boundaries of internal cavities, and preserving the streak-like textural features indicated by the yellow arrows.

Overall, TSDDC-Net consistently achieves the best perfor-

mance in minimizing artifacts while preserving fine anatom- 737

ical details. This observation is further confirmed by the re- sults shown in Fig. (c,*) for all compared methods. system’s X-ray window increases 20mm, reduced number deployable X-ray sources leads severe interpolation errors aliasing

artifacts in the initial reconstruction image. As a result, 743

performance reconstruction

methods

noticeably degrades. Analytical iterative methods, particular,

suffer from significant blurring and detail loss. For instance, 746

anatomical structures shown become

indistinct and barely recognizable. Among the deep learning 748

methods, comparison models exhibit noticeable structural degradation, struggling recover details well. contrast, proposed TSDDC-Net preserve tissue boundaries structures, demonstrating superior stability robustness under challenging ASDA-sCT system configurations larger window sizes.

These

results

demonstrate superiority proposed TSDDC-Net, specifically designed ASDA-sCT systems.

For the sys- tem with an exit window size of 20 mm (Fig. ), the sparse angular sampling and large interpolation errors lead to se-

vere detail loss in the reconstruction results of all three tra-

ditional methods. Although deep learning–based approaches 760

can reconstruct the overall soft-tissue contours, the fine struc- tural textures and the morphology of internal cavities are largely missing.

When the exit window size decreases to 10 mm (Fig. ), the number of available projections in- creases. Therefore, all reconstruction methods show substan- tial improvement in image quality. Among them, the pro- posed TSDDC-Net achieves the best performance, with only minor detail loss observed in a few localized regions. As the exit window further narrows to 5 mm (Fig. ), the projec- tion sampling becomes sufficiently dense and is no longer considered sparse. Under this condition, traditional methods can reconstruct almost complete structural details with min-

imal artifacts. Deep learning–based approaches further en- 773

hance image quality, producing reconstructions that closely match the ground truth (the fully sampled rotational CT im-

age with N = 960 ) and exhibiting fidelity sufficient for clin- 776

ical diagnostic applications. In summary, the exit window size critically affects the performance of reconstruction meth- ods for the ASDA-sCT system. Smaller window sizes en- able denser projection sampling, leading to higher reconstruc-

tion fidelity. Deep learning methods, by leveraging learned 781

prior knowledge, can effectively compensate for missing in- formation caused by undersampling and further enhance im- age quality.

To further evaluate the effectiveness of the proposed dual-domain architecture, particularly the DFInception mod- ule in the projection domain and the dual-encoder design in the image domain, we conducted a series of ablation experiments, as presented in Table quantitative

results

Table confirm proposed TSDDC-Net, incorporating novel architectural components, achieves performance across evaluation metrics.

In the ablation experiments, we removed individual modules and compared the reconstruction performance before and after re- moval to verify their effectiveness in the network.

results

reveal models leveraging single-domain priors, either sinogram domain image domain, yield lower performance compared dual-domain model.

The results in the first three rows of Table indicate that utilizing priors from only a single domain, whether the sinogram or the image domain, leads to inferior reconstruction performance com- pared to the dual-domain model.

This demonstrates the ef- fectiveness of jointly exploiting complementary information from both domains to achieve superior reconstruction qual- ity. Specifically, the projection-domain-only model outper- forms the image-domain-only model. The relatively smooth variations and simpler structural patterns in projection data make it easier for the model to learn and generalize. contrast, image-domain data often exhibits complex aliasing, interpolation-induced artifacts, and noise, which substantially

increase learning difficulty. 811

Furthermore, the incorporation of the DFInception module enhances the sinogram refinement capability in the projec- tion domain, improving the quality of the refined CT image.

Meanwhile, the dual-encoder design in the image domain ef-

fectively integrates complementary features from the initial 816

reconstructed image and the SD-Net-refined image, leading to enhanced edge preservation and finer structural detail in the final reconstructed CT image.

Computational Efficiency

The deep learning–based methods offer another signifi- 821

cant advantage—computational efficiency and parallelizabil- ity. Table presents a summary of the reconstruction time re- quired by different methods for the ASDA-sCT system with

an exit window size of 10 mm (N=194) under identical ex- 825

perimental settings (see Section ). Each result is averaged over 500 random reconstruction runs. Among all methods, the FBP algorithm achieves the fastest speed but yields the lowest reconstruction quality.

Both SART and POCS-TV algorithms require repeated iterations of forward and back- ward projecting, resulting in substantially increased compu- tational costs. Consequently, these methods exhibit the slow- est reconstruction speed. FBPConvNet achieves the fastest

computational speed among the deep learning–based meth- 834

ods but yields the lowest reconstruction accuracy.

DuDo- Trans has a slower inference time because its Transformer- based architecture involves higher computational complexity and a larger number of parameters compared with convolu- tional networks. Between the two dual-domain convolutional networks, TSDDC-Net is slightly slower than HD-Net, but it achieves better reconstruction quality. Overall, TSDDC-Net enables near real-time reconstruction at sub-second speeds and demonstrates strong potential for real-time dynamic CT imaging applications.

SUMMARY

This study proposes an Alternating Source-Detector Array Stationary CT (ASDA-sCT) system capable of tomographic imaging without gantry rotation. The system utilizes a com- putationally efficient interpolation strategy to complete miss- ing projections, enabling direct reconstruction using the fil- tered backprojection algorithm. In the proposed design, max- imizing the number of X-ray sources is crucial for achieving optimal reconstruction quality. Furthermore, we introduce

TSDDC-Net, a deep learning reconstruction model specifi- 854

cally tailored for this system. By incorporating the novel DFInception feature extraction module and an advanced dual- domain architecture, TSDDC-Net effectively corrects inter- polation errors and suppresses aliasing artifacts. Experimen- tal results highlight the ASDA-sCT system’s strong potential

in clinical imaging and industrial nondestructive testing. 860

The ASDA-sCT system offers diverse potential applica- tions. By incorporating multi-row detectors and adopting a cone-beam CT (CBCT) geometry, the system can be con- figured into a stationary CBCT (sCBCT) system, effectively

addressing the slow scanning speed of conventional CBCT. 865

More importantly, sCBCT enables a nearly motion artifact- free real-time 4D dynamic reconstruction. To address under- sampling challenge inherent in the ASDA-sCT system, we

Method

Exit Window Size = 5 Exit Window Size = 10 Exit Window Size = 20 POCS-TV FBPConvNet HD-Net DuDoTrans TSDDC-Net

Method

Sinogram Domain Image Domain Sinogram Domain Image Domain SD-Net SD-Net + DFInception - ID-Net SD-Net ID-Net SD-Net + DFInception ID-Net SD-Net ID-Net + DEecoder 0.25670 SD-Net + DFInception ID-Net + DEecoder 0.24631

Method

POCS-TV Time (ms)

Method

FBPConvNet HD-Net DuDoTrans TSDDC-Net Time (ms) can increase the number of projections by employing a multi-

plane scanning strategy. Each scanning plane is equipped 870

with an ASDA-sCT scanner, capturing projections from dif- ferent view angles. This design offers a promising solution for applying the ASDA-sCT system in industrial CT inspec-

tion and security screening. 874

Due to the large amount of CT data of anatomical struc-

tures available in clinical practice and the excellent per- 876

formance of deep learning reconstruction, deep learning- 877

based reconstruction methods are expected to facilitate the

widespread clinical adoption of ASDA-sCT. The ASDA-sCT 879

system acquires fewer projections, reducing radiation dose

while maintaining diagnostic quality. In addition, its ex- 881

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