CADRE: A novel unsupervised reconstruction algorithm for limited-angle CT of ancient wooden structures

Abstract

Background

Non-destructive testing (NDT) is crucial for the preservation and restoration of ancient wooden structures, with Computed Tomography (CT) increasingly utilized in this field. However, practical CT examinations of these structures—often characterized by complex configurations, large dimensions, and on-site constraints—frequently encounter difficulties in acquiring full-angle projection data. Consequently, images reconstructed under limited-angle conditions suffer from poor quality and severe artifacts, hindering accurate assessment of critical internal features such as mortise-tenon joints and incipient damage.

Objective

This study aims to develop a novel algorithm capable of achieving high-quality image reconstruction from incomplete, limited-angle projection data.

Methods

We propose CADRE (Contour-guided Alternating Direction Method of Multipliers-optimized Deep Radon Enhancement), an unsupervised deep learning reconstruction framework. CADRE innovatively integrates the ADMM optimization strategy, the learning paradigm of Deep Radon Prior (DRP) networks, and a geometric contour-guidance mechanism. This approach synergistically enhances reconstruction performance by iteratively optimizing network parameters and input images, without requiring large-scale paired training data, rendering it particularly suitable for cultural heritage applications.

Results

Systematic validation using both a digital dougong simulation model of the Yingxian Wooden Pagoda and a physical wooden dougong model from Foguang Temple demonstrates that, under typical 90° and 120° limited-angle conditions, the CADRE algorithm significantly outperforms traditional FBP, iterative reconstruction algorithms SART and ADMM-TV, and other representative unsupervised deep learning methods (Deep Image Prior, DIP; Residual Back-Projection with DIP, RBP-DIP; DRP). This superiority is evident in quantitative metrics such as PSNR and SSIM, as well as in visual quality, including artifact suppression and preservation of structural details. CADRE exhibits exceptional capability in accurately reproducing internal mortise-tenon configurations and fine features within ancient timber.

Conclusion

The CADRE algorithm provides a robust and efficient solution for limited-angle CT image reconstruction of ancient wooden structures. It effectively overcomes the limitations of existing methods in handling incomplete data, significantly enhances the quality of reconstructed images and the characterization of internal fine structures, and offers strong technical support for the scientific understanding, condition assessment, and precise conservation of cultural heritage, thereby holding substantial academic value and promising application prospects.

Keywords

ancient wooden structures limited-angle CT reconstruction algorithm unsupervised deep learning non-destructive testing (NDT)

Introduction

Ancient Chinese wooden structures, as invaluable cultural heritage, embody a wealth of historical and cultural information. However, these edifices, through prolonged exposure to natural ageing, environmental erosion, sustained loads, and other external factors, commonly face issues such as structural degradation, insect infestation, and cracking.¹ Potential internal defects within timber components, including cracks, decay, and boreholes, severely threaten the structural integrity and safety of these ancient buildings.^2–4 Consequently, the precise investigation of the internal condition of timber components and the assessment of their structural safety performance have become critical aspects of ancient building preservation, urgently requiring the development of advanced detection technologies. Accurately acquiring detailed structural information, such as internal defect distribution and the configuration of mortise-tenon joints, is of paramount theoretical and practical significance for guiding subsequent heritage restoration, reinforcement, and in-depth academic research.³

Traditional inspection methods for ancient timber structures primarily rely on visual examination and percussion (tapping) techniques; these methods are highly subjective and struggle to detect deep internal defects.⁵ With technological advancements, timber structure inspection has evolved from qualitative assessment to quantitative analysis using scientific instruments, with detection ranges extending from the surface to the interior. Commonly used modern non-destructive testing (NDT) techniques for timber components include the micro-drilling resistance drill (Resistograph), stress wave tomography (Impulse Tomograph), ultrasonic testing, and infrared thermography.^6,7 Despite significant progress, each of these techniques has inherent limitations: the micro-drilling resistance drill is mainly suitable for assessing superficial damage and provides only linear probing; stress wave tomography faces uncertainties in distinguishing between voids and decayed regions; and ultrasonic and infrared thermography have limited capability in detecting deep-seated defects. Furthermore, some minimally invasive or intrusive methods, such as core drilling, can cause irreversible damage to the precious heritage fabric.

In contrast, radiation-based detection techniques, particularly Computed Tomography (CT), offer a noninvasive means to acquire detailed internal three-dimensional (3D) structural information, including the location and size of defects, without damaging the object under investigation.⁸ CT has seen increasingly widespread application in the field of cultural heritage preservation in recent years. For on-site inspection of large-scale ancient wooden structures, CT systems based on Cobalt-60 (⁶⁰Co) gamma-ray sources have demonstrated unique suitability due to their strong penetrating power, relatively lenient requirements for the scanning environment, compact source size, lack of need for complex high-voltage power supplies, and ease of maintenance (Figure 1).^9,10

However, the application of CT technology for on-site inspection of ancient wooden structures still faces significant challenges. The considerable size and complex geometry of timber components, coupled with spatial constraints at heritage sites, often preclude the deployment of conventional full-angle CT scanning systems (which require 360° rotation).¹¹ For instance, when examining models like the dougong (bracket sets) of the Yingxian Wooden Pagoda, it may only be possible to acquire projection data within a limited angular range of 90° to 150°. Such incomplete projection data lead to a significant degradation in the quality of reconstructed images, characterized by severe streak artifacts and structural distortions, thereby hindering the accurate identification of internal defects.¹² Traditional analytical reconstruction algorithms, such as Filtered Back-Projection (FBP),¹³ are particularly prone to pronounced artifacts under limited-angle data conditions. Although algebraic iterative reconstruction techniques like the Simultaneous Algebraic Reconstruction Technique (SART),¹⁴ and iterative algorithms based on Total Variation (TV)¹⁵ regularization, such as the Alternating Direction Method of Multipliers-TV (ADMM-TV),¹⁶ can suppress artifacts and improve image quality to some extent, their reconstruction performance remains unsatisfactory when data incompleteness is severe, often failing to completely eliminate artifacts and accurately recover fine structural details.¹⁷

In recent years, deep learning-based CT image reconstruction methods have achieved remarkable progress. However, most successful applications rely on a supervised learning paradigm, which necessitates a large number of high-quality full-angle CT images paired with corresponding limited-angle CT images for training.¹⁸ Acquiring such extensive, high-quality paired datasets is extremely difficult, if not infeasible, in the context of precious ancient wooden structures.¹⁹ Consequently, the development of unsupervised deep learning reconstruction methods, which do not depend on paired data, has become a critical research direction for addressing the challenges of limited-angle CT imaging in architectural heritage.

To tackle these challenges, this paper proposes a novel unsupervised deep learning algorithm named CADRE (Contour-guided ADMM-optimized Deep Radon Enhancement), specifically designed for limited-angle CT image reconstruction of ancient wooden structures. Building upon the Deep Radon Prior (DRP) network framework, this algorithm innovatively integrates the optimization strategy of Alternating Direction Method of Multipliers (ADMM)²⁰ with a geometric contour-guidance mechanism. The CADRE algorithm enhances reconstruction performance through three core designs: (1) the introduction of the ADMM framework, which decomposes the complex optimization problem into multiple more tractable sub-problems, ensuring stable convergence and reconstruction robustness; (2) the utilization of the DRP network's projection loss consistency in the Radon domain to improve reconstruction quality; and (3) the incorporation of geometric contour information from the scanned object as a prior to guide the reconstruction process, effectively preserving structural geometric fidelity and suppressing artifacts. Systematic experiments conducted on a digital dougong model of the Yingxian Wooden Pagoda and a physical wooden dougong model from Foguang Temple have validated that the CADRE algorithm achieves reconstruction quality significantly superior to existing mainstream methods under typical 90° and 120° limited-angle conditions. This research provides an effective technical pathway for the precise NDT of ancient wooden structures, holding considerable theoretical value and application prospects for advancing the digital preservation and scientific study of cultural heritage.

The remainder of this paper is organized as follows: Section 2 reviews related NDT techniques and research progress in limited-angle CT reconstruction algorithms. Section 3 elaborates on the proposed CADRE algorithm framework, its mathematical model, and network architecture. Section 4 presents and analyzes the reconstruction results from both simulated and real timber component experimental data. Section 5 provides a discussion of the findings and outlines future research directions. Finally, Section 6 concludes the paper.

Related work

Non-destructive testing techniques for ancient wooden structures

The field of NDT for ancient wooden structures has progressively evolved from traditional, experience-dependent manual practices to quantitative assessment based on precision scientific instruments. Traditional inspection methods, such as visual examination and percussion sounding, while simple and convenient, are highly reliant on the inspector's experience, leading to strong subjectivity, and they are ineffective at detecting early-stage or concealed defects within timber components.⁷ With technological advancements, various modern NDT techniques have been applied to assess the health status of ancient timber structures.

The Pilodyn wood tester and the micro-drilling resistance drill (Resistograph) indirectly assess the surface density and mechanical strength of wood along specific paths by measuring physical parameters, such as pin penetration depth or drilling resistance.^21,22 However, the Pilodyn is limited to shallow-depth inspections, and the resistance drill primarily provides linear probing data, which is insufficient for comprehensively mapping the two- or three-dimensional distribution of internal defects. Stress wave tomography (Impulse Tomograph) constructs a two-dimensional image of the internal condition by analyzing the propagation velocity of stress waves, allowing for intuitive visualization of defective areas.²³ Nevertheless, it faces limitations in accurately distinguishing between internal voids and decayed regions and in quantifying the extent of damage. Ultrasonic testing identifies internal discontinuities by leveraging the different propagation characteristics of ultrasonic waves in various media; however, as wood is a naturally anisotropic and heterogeneous material, its complex texture (e.g. annual rings, knots) and variable moisture content can easily interfere with signal propagation, thus affecting detection accuracy.²⁴ Infrared thermography infers internal defects or moisture content variations by capturing anomalies in the surface temperature distribution of timber components, but its detection depth is limited, and it is susceptible to changes in ambient temperature.²⁵ Although minimally invasive methods like core drilling can directly provide information about the wood's interior, they cause a degree of damage to the heritage object, which contradicts the principle of minimal intervention in heritage conservation.²⁶

In comparison, radiation-based Computed Tomography (CT), particularly systems utilizing X-rays or γ-rays, can noninvasively acquire detailed three-dimensional images of the internal structure of timber components.²⁷ This allows for the accurate identification and localization of internal defects (e.g. cracks, insect damage, decay) and constructional features (e.g. mortise-tenon joints).^11,28 Industrial CT has seen mature application in modern wood science and quality control, where it is used to effectively detect variations in wood density, growth patterns, and internal flaws.²⁹ However, the direct application of conventional industrial or medical CT equipment for on-site inspection of large-scale ancient wooden structures presents numerous challenges. These include the bulky nature of the equipment, difficulties in transportation and installation, stringent requirements for scanning angles (typically a full 360° rotation), and the complexity of on-site operation.^30,31 These practical constraints underscore the urgent need to develop CT technologies suitable for the specific environmental and inspection demands of architectural heritage, especially techniques capable of achieving high-quality imaging from limited scanning angles.

Limited-angle CT reconstruction algorithms

Limited-angle CT refers to a data acquisition process where, due to physical constraints, the detector can only acquire projection data from an angular range smaller than 180° (for parallel-beam geometry) or 180°+ fan angle (for fan-beam/cone-beam geometries). This data incompleteness leads to the so-called “missing wedge” or “missing region” problem in the frequency domain, rendering image reconstruction a challenging ill-posed inverse problem.³² The resulting images are often plagued by severe streak artifacts, structural distortions, and a loss of resolution. Limited-angle CT image reconstruction algorithms can be broadly categorized into analytical, iterative, and deep learning-based methods according to their mathematical principles and implementation.

Analytical reconstruction algorithms, exemplified by FBP, are based on the Fourier-slice theorem and the inverse Radon transform. They are computationally efficient and straightforward to implement. However, under limited-angle conditions, the inherent mathematical limitations of the FBP algorithm lead to very significant streak artifacts and image distortion, severely degrading image quality and potentially obscuring critical structural details.¹³ Iterative reconstruction algorithms approach the problem by establishing an objective function composed of a data fidelity term and a regularization prior term, then progressively approximating the true solution through an iterative optimization strategy. Typical iterative algorithms include the Algebraic Reconstruction Technique (ART) and its improved versions, such as the SART. By utilizing all projection data simultaneously in each iteration to update the image, SART can effectively suppress noise and some artifacts, exhibiting greater robustness than ART.¹⁴ Furthermore, iterative methods incorporating TV regularization, such as the ADMM²⁰ -based ADMM-TV algorithm, perform well in preserving edges and reducing noise, making them particularly suitable for low-dose and sparse-angle CT reconstruction.¹⁶ Although iterative algorithms outperform FBP in limited-angle scenarios, they still struggle to completely eliminate artifacts when the angular range is severely restricted or noise levels are high. Moreover, the regularization can introduce over-smoothing, leading to a loss of fine details, and their computational cost is relatively high.

In recent years, Deep Learning (DL) has shown enormous potential in the field of CT image reconstruction due to its powerful non-linear feature learning capabilities. Supervised deep learning methods typically train a deep neural network to learn a mapping from low-quality (e.g. limited-angle, sparse-angle, or noisy) reconstructed images to high-quality reference images. Recent state-of-the-art approaches have further pushed the boundaries of this paradigm, employing sophisticated techniques like score-based generative models, diffusion models, and advanced iterative networks with deep neural priors to achieve remarkable fidelity.^33–35 However, these powerful methods still highly dependent on large-scale, high-quality paired training datasets, which requires a vast number of limited-angle CT images and their corresponding full-angle counterparts. In the domain of ancient wooden structures, acquiring such datasets is extremely difficult, if not entirely impractical.¹⁹ To overcome the data bottleneck of supervised learning, unsupervised deep learning methods have emerged. For instance, the Deep Image Prior (DIP) utilizes the structure of a Convolutional Neural Network (CNN) itself as an implicit image prior, optimizing the network parameters to fit the observed data,³⁶ thereby enabling image reconstruction from a single incomplete data instance. The DRP further leverages projection consistency in the Radon domain, optimizing the network via a cross-domain loss and iteratively updating the network's input to avoid overfitting.³⁷ This process ensures that the Radon transform of the reconstructed image aligns with the measured projection data, thus enhancing reconstruction quality. The Residual Back-Projection with Deep Image Prior (RBP-DIP) attempts to combine traditional reconstruction concepts with the DIP framework and has achieved some success in limited-angle CT.³⁸ Nevertheless, existing unsupervised deep learning methods still face several common challenges when processing severely ill-posed limited-angle CT data. These include slow or unstable optimization convergence, significant fluctuations in reconstruction quality during iterations, sensitivity to network architecture and hyperparameters, and an insufficient ability to preserve fine structures and complex textures.^18,37 These shortcomings limit their application in the analysis of ancient timber components, which demands high precision and fidelity.

Innovation of this study

Although existing NDT techniques and limited-angle CT reconstruction algorithms for ancient wooden structures have made certain progress, they still exhibit clear limitations when faced with the practical constraints of on-site inspections and the demand for high-fidelity imaging of fine internal structures. Specifically, in limited-angle CT reconstruction, traditional analytical algorithms produce severe artifacts, iterative algorithms are deficient in detail recovery and computational efficiency, supervised deep learning methods are constrained by data availability, and current unsupervised deep learning methods have yet to achieve an optimal balance between convergence stability, artifact suppression, and detail preservation.

To address these research gaps and technical challenges, the CADRE algorithm proposed in this study aims to provide new solutions in several key aspects:

Fusion of Optimization Framework and Deep Prior: The CADRE algorithm combines the stable convergence properties of the ADMM framework with the powerful learning capabilities of DRP-inspired networks, effectively overcoming the training instability and slow convergence commonly found in existing unsupervised methods.

Introduction of Geometric Contour Guidance: By explicitly incorporating the geometric contour of the target object as a prior to guide the reconstruction process, CADRE is designed to more accurately recover structural edges and suppress the spread of artifacts outside the region of interest. This enhances the fidelity of critical geometric features like mortise-tenon joints, a crucial aspect often overlooked by existing unsupervised methods.

Unsupervised Learning Paradigm: By adhering to an unsupervised learning path, the method avoids dependance on large-scale paired training data, making it inherently more suitable for the data-scarce field of cultural heritage preservation.

Therefore, this research focuses on enhancing the imaging quality and practical utility of limited-angle CT for the inspection of ancient wooden structures. Through the CADRE algorithm, this work aims to provide a more robust, accurate, and adaptable NDT analysis tool for the field.

Methodology: the CADRE framework

The central challenge in limited-angle CT image reconstruction is to recover a high-quality tomographic image from an incomplete set of projection data. Deep learning, particularly unsupervised learning methods, offers a promising avenue to address this ill-posed problem.

Deep radon prior (DRP) and its limitations

The DRP is a representative unsupervised CT reconstruction method previously proposed by our research team. It employs an untrained deep neural network, typically an encoder-decoder architecture, as an image generator. The core idea is to optimize the network parameters $θ$ and an input vector z (usually initialized with random noise or a fixed initial image, such as an FBP reconstruction) to minimize the discrepancy between the forward projection of the generated image and the measured projection data p. The optimization objective of DRP can be expressed as:

{θ *, z *} = \underset{θ, z}{\arg min} L (A φ_{D R P} (z; θ), p)

(1)

Where $x = φ_{D R P} (z *; θ *)$ is the final reconstructed image, $φ_{D R P} (\cdot; θ)$ denotes the neural network with parameters $θ$ , A is the CT forward projection operator, and $L (\cdot, \cdot)$ is a loss function (e.g. L₁ or L₂ norm) that measures the difference in the projection domain. In this work, z is initialized with the FBP reconstruction from the corresponding limited-angle data.

Although DRP provides a novel approach to unsupervised CT reconstruction, its practical application faces several challenges. These include potentially slow or unstable convergence, especially with severely incomplete projection data; significant fluctuations in image quality during the iterative process; and a limited ability to achieve high-fidelity reconstruction of fine structures, such as sharp edges and complex textures, which are crucial for analyzing the internal details of ancient wooden structures.

Proposed CADRE algorithm

To address the limitations of DRP and other unsupervised methods, we propose a novel unsupervised deep learning framework named CADRE (Contour-guided ADMM-optimized Deep Radon Enhancement). CADRE is designed to significantly improve limited-angle CT image quality by enhancing the optimization strategy and integrating multiple priors.

Objective function

Let $O (z; θ)$ represent a deep neural network with parameters $θ$ and input z. The general objective of CADRE is to find the optimal network parameters $θ *$ and an iteratively optimized input $z *$ that minimize a composite objective function. This function consists of a data fidelity term in the projection domain and a regularization term in the image domain, expressed as:

{θ *, z *} = \underset{θ, z}{\arg min} \frac{λ}{2} L (A O (z; θ), p) + R (O (z; θ))

(2)

Here, $λ$ is a hyperparameter that balances the data fidelity term and the regularization term $R (O (z; ϑ))$ . The fidelity term $L (A O (z; θ), p)$ ensures consistency with the measured projection data p, while the regularization term introduces prior knowledge to stabilize the solution and enhance image quality. In this study, we employ the image-domain TV as the regularizer, i.e. $R (O (z; θ)) = ‖ \nabla O (z; θ) ‖_{1}$ , to promote gradient sparsity and preserve details. While we focus on TV regularization, the CADRE framework is versatile and allows for the use of other regularizers in the future.

ADMM-based optimization framework

To effectively solve the optimization problem in equation (2), we utilize the ADMM framework. We introduce an auxiliary splitting variable d to decouple the terms. For TV regularization, we set the constraint $d = \nabla O (z; θ)$ , where d approximates the image gradient. Consequently, equation (2) can be reformulated as the following constrained optimization problem:

{θ *, d *} = \underset{θ, d}{\arg min} \frac{λ}{2} L (A O (z; θ), p) + ‖ d ‖_{1}, s . t . d = \nabla O (z; θ)

(3)

where

‖ d ‖_{1}

penalizes the L₁ norm of the gradient d (anisotropic TV). Note that the input z is also optimized iteratively alongside

θ

and d. This constrained problem is solved by minimizing its augmented Lagrangian function:

L_{β} (θ, d, b; z) = \frac{λ}{2} L (A O (z; θ), p) + ‖ d ‖_{1} + \frac{β}{2} ‖ \nabla O (z; θ) - d + b ‖_{2}^{2} - \frac{β}{2} ‖ b ‖_{2}^{2}

(4)

where b is the scaled dual variable (Lagrange multiplier) and

β

> 0 is a penalty parameter. The ADMM algorithm decomposes this into three sub-problems, which are solved iteratively for the (k + 1)-th iteration with a fixed input

z^{k}

Step 1: $θ$ -Subproblem Update (Network Parameter Optimization):

With $d^{k}$ and $b^{k}$ fixed, the minimization problem with respect to the network parameters $θ$ is:

θ^{k + 1} = \underset{θ}{\arg min} \frac{λ}{2} L (A O (z^{k}; θ), p) + \frac{β}{2} ‖ \nabla O (z^{k}; θ) - d^{k} + b^{k} ‖_{2}^{2}

(5)

This step is typically solved using a gradient-based optimizer, such as Adam, for several iterations. In CADRE, the Huber loss is chosen for the projection loss function $L (\cdot, \cdot)$ . The Huber loss combines the advantages of L₁ and L₂ losses and is defined as:

L_{δ} (r) = {\begin{matrix} \frac{1}{2} r^{2} & for | r | \leq δ \\ δ (| r | - \frac{1}{2} δ) & otherwise \end{matrix}

(6)

where r is the residual (in our case,

r = A O (z^{k}; θ) - p

) and δ is a threshold parameter, which is fixed to 1.0 in this work. This choice enhances the model's robustness to potential outliers in the projection data.

Step 2: d -Subproblem Update (Regularization Prior Enforcement):

With $θ^{k + 1}$ and $b^{k}$ fixed, the sub-problem for the auxiliary variable d is:

d^{k + 1} = \underset{d}{\arg min} ‖ d ‖_{1} + \frac{β}{2} ‖ \nabla O (z^{k}; θ^{k + 1}) - d + b^{k} ‖_{2}^{2}

(7)

This problem is equivalent to solving for a proximal operator, i.e. $d^{k + 1} = {prox}_{{‖ \cdot ‖}_{1} / β} (\nabla O (z^{k}; θ^{k + 1}) + b^{k})$ . For the L₁ norm, this sub-problem has a closed-form solution based on a soft-thresholding shrinkage operator.

Step 3: $b$ -Subproblem Update (Dual Variable Update):

With $θ^{k + 1}$ and $d^{k + 1}$ fixed, the dual variable b is updated as:

b^{k + 1} = b^{k} + \nabla O (z^{k}; θ^{k + 1}) - d^{k + 1}

(8)

Iterative process and integrated mechanisms

The complete optimization process of CADRE involves iterating through the ADMM updates while also refining the network input z and applying a contour constraint.

Note on the Contour Constraint $C$ : The contour information C plays two primary roles: (1) it refines the network output to conform to known object boundaries during the iterative update of the network input z (see Step 5), and (2) it is used to remove potential artifacts outside the object region in the final reconstructed image. It does not appear directly in the core objective function (equation (2)) or the ADMM sub-problems (equations (5)–(8)) but acts as a guiding and post-processing step within the iterative framework.

Step 4: Contour-Guided Mechanism:

To enhance geometric fidelity and suppress artifacts, we introduce a contour guidance step. A binary contour mask C (where the target object region is 1 and the background is 0) is first obtained. A high-precision contour can be generated from external measurement systems, such as 3D laser scanners, which provide accurate geometric boundaries independent of the CT data quality. A more direct alternative is to apply simple thresholding to an initial FBP reconstruction; the efficacy of this approach is dependent on the initial image quality and the contrast between the object and its background. In each full iteration, after the ADMM steps yield the updated network parameters $θ^{k + 1}$ , the network output for the current input $z^{k}$ is multiplied element-wise with the contour mask $C$ :

O_{C} (z^{k}; θ^{k + 1}) = O (z^{k}; θ^{k + 1}) ⊙ C

(9)

where ⊙ denotes element-wise multiplication. This operation focuses the reconstruction effort on the object's interior, mitigating the impact of external artifacts on subsequent iterations.

Step 5: Network Input z Update (Input Image Optimization):

Unlike DIP-type methods that often use a fixed random input, CADRE treats z as an optimizable variable. The update rule is designed to bring the Radon transform of the contour-guided output closer to the measured projections:

z^{k + 1} = O_{C} (z^{k}; θ^{k + 1}) + η \cdot A^{T} (p - A O_{C} (z^{k}; θ^{k + 1}))

(10)

where

A^{T}

is the back-projection operator and

η

is the update step size (set to 0.8 in this study). This update allows the input z to dynamically adapt to the reconstruction task.

Step 6: Overall Iterative Process:

The updated input $z^{k + 1}$ from Step 5 is then used for the next full iteration (returning to Step 1 for new ADMM updates). This entire process (Steps 1–5) is repeated until a predefined maximum number of iterations K (set to 150 epochs in this study) is reached. The final reconstructed image $x_{f i n a l}$ is the contour-guided network output from the last iteration: $x_{f i n a l} = O_{C} (z^{K}; θ^{K})$ .

The complete optimization procedure of the CADRE algorithm is summarized in Algorithm 1.

Network architecture

The overall framework of the CADRE unsupervised CT reconstruction algorithm is illustrated in Figure 2. The neural network $O (\cdot; \cdot)$ employed within CADRE is an improved encoder-decoder architecture, similar to the U-Net structure (as shown in Figure 3). It features symmetric down-sampling (encoder) and up-sampling (decoder) paths, with skip connections to fuze multi-level features for preserving image details. Both the network input and output are CT images. To enhance the network's ability to capture key features, a Convolutional Block Attention Module (CBAM)³⁹ is integrated after each convolutional block in the encoder. The design of the network architecture was also inspired by experiences from related neural architecture search studies, such as the Neural Architecture Search with Deep Radon Prior (NAS-DRP),¹⁸ with optimizations made to the up-sampling methods and skip connections for the specific task of CT reconstruction.

Figure 1.

Schematic diagram of the structural complexity of ancient timber architecture.

Figure 2.

Overall framework of the CADRE unsupervised CT reconstruction algorithm.

Figure 3.

Architecture of the network used in the CADRE algorithm.

By integrating a deep neural network as a powerful learned prior within a principled iterative optimization framework, our method facilitates a synergistic optimization process guided by feedback from both the image and Radon domains. This hybrid approach is designed to overcome the limitations of methods that rely on either classical regularization or a single deep learning prior alone. Consequently, the proposed framework aims to achieve a superior balance between artifact suppression and detail preservation for challenging limited-angle CT tasks.

Experimental validation and results

This section details the series of experiments conducted to validate the performance of the proposed CADRE algorithm. It includes the experimental setup, datasets used, evaluation metrics, and an analysis of the reconstruction results on both simulated and real-world timber component CT data.

Experimental setup

Benchmarks and implementation details

To comprehensively evaluate the performance of CADRE, we selected several representative algorithms for comparison. These benchmarks include a traditional analytical method, FBP; commonly used iterative algorithms, namely the SART and the ADMM-TV; and state-of-the-art unsupervised deep learning methods, specifically the DIP, RBP-DIP, and the DRP.

The hyperparameters for all benchmark algorithms were meticulously tuned through experiments to achieve their respective optimal performances. All deep learning-based methods were implemented in the PyTorch framework, utilizing an encoder-decoder network architecture with 5 down-sampling and 5 up-sampling layers. Each model was trained for 150 epochs using the Adam optimizer with a learning rate of 5 × 10⁻⁴. All computational experiments were executed on a computer equipped with an Intel Core i9-13900 CPU, 96 GB of system memory, and an NVIDIA GeForce RTX 4090 GPU. For the proposed CADRE algorithm, the reconstructed image size was set to 512 × 512 pixels. The Huber loss was used for the data fidelity term, and the input update step size was set to η = 0.8. The ADMM parameters were set to λ = 2 and β = 10.

Datasets and evaluation scenarios

We employed two datasets to validate the effectiveness and generalization capability of the CADRE framework. The first is a digital model of a dougong (bracket set) from the Yingxian Wooden Pagoda, which is a 1:1 scale simulated model with precise geometry and a known ground truth, as shown in Figure 4(a). The second is a physical wooden model of a dougong from the Foguang Temple, a 1:7.5 scale model of a seven-purlin bracket set, shown in Figure 4(b), which was used to evaluate the algorithm's performance under real-world scanning conditions. For quantitative analysis and fair comparison, all image data in the experiments were normalized to a value range of [0, 1]. Reflecting the practical constraints of on-site inspections of ancient wooden structures, our study focuses on CT reconstruction performance under two typical and challenging limited-angle scenarios: 90° and 120° angular ranges.

Figure 4.

Timber component models used in the experiments. (a) Exterior view of the simulated dougong model from the Yingxian Wooden Pagoda. (b) Exterior view of the physical wooden dougong model from the Foguang Temple.

Evaluation metrics

To objectively and quantitatively evaluate the performance of the different reconstruction algorithms, this study uses two widely applied image quality metrics in the image processing field: the Peak Signal-to-Noise Ratio (PSNR) and the Structural Similarity Index Measure (SSIM).

The PSNR is defined based on the Mean Squared Error (MSE) between the reconstructed image x and a reference image $x_{g t}$ (the ground truth in simulations or the full-angle FBP reconstruction in real-data experiments). Its formula is as follows:

P S N R (x, x_{g t}) = 10 \cdot \log_{10} (\frac{M A X_{I}^{2}}{M S E}) = 10 \cdot \log_{10} (\frac{M A X_{I}^{2}}{\frac{1}{N} \sum_{i = 1}^{N} {(x_{i} - {(x_{g t})}_{i})}^{2}})

(11)

Here, $M A X_{I}$ is the maximum possible pixel value of the image (1 for a normalized image), and N is the total number of pixels. The unit for PSNR is decibels (dB). A higher PSNR value indicates a smaller pixel-level error between the reconstructed and reference images, signifying lower distortion.

The SSIM comprehensively evaluates the similarity between the reconstructed and reference images from three aspects: luminance (l), contrast (c), and structure (s). Its formula is:

S S I M (x, x_{g t}) = \frac{(2 μ_{x} μ_{x_{g t}} + c_{1}) (2 σ_{x, x_{g t}} + c_{2})}{(μ_{x}^{2} + μ_{x_{g t}}^{2} + c_{1}) (σ_{x}^{2} + σ_{x_{g t}}^{2} + c_{2})}

(12)

Here, $μ$ and $σ^{2}$ are the mean and variance of the image, respectively; $σ_{x_{g t}}$ is the covariance; and $c_{1}$ and $c_{2}$ are stabilization constants to avoid division by zero. The value of SSIM ranges from 0 to 1. A value closer to 1 indicates that the two images are more similar in terms of structural information, corresponding to higher visual fidelity in the reconstruction. In general, higher PSNR and SSIM values both indicate superior reconstruction quality, reflecting lower levels of noise and artifacts and better preservation of structural details.

Simulation study: Yingxian Pagoda Dougong model

This section presents a simulation experiment conducted on the 1:1 scale digital dougong model from the Yingxian Wooden Pagoda, as depicted in Figure 4(a). We utilized the MCGPU_CT numerical simulation platform to generate limited-angle CT projection data at 90° and 120° to evaluate the capability of the CADRE algorithm in reconstructing fine internal details, such as complex mortise-tenon joints.

Simulated data acquisition

The numerical simulation was performed using the MCGPU_CT Monte Carlo simulation platform,⁴⁰ a GPU-accelerated tool previously developed by our team (a schematic is shown in Figure 5). This platform significantly reduces simulation time while maintaining the high accuracy and scalability of Monte Carlo methods. The primary parameters for the CT scan simulation of the dougong model are detailed in Table 1.

Figure 5.

Schematic diagram of the MCGPU_CT Monte Carlo simulation system.

Table 1.

Parameters for the CT scan simulation of the Yingxian Pagoda dougong model.

Parameter	Value
Radiation Source	⁶⁰Co
Source Activity	100 Ci
Detector Type	GAGG(Ce)
Detector Dimensions	500 × 1
Pixel Size	10 mm × 10 mm
Distance of Source-to-Detector	5000 mm
Distance of Source-to-Object-Center	2500 mm
Projection Angular Interval	0.5°
Reconstructed Image Size	512 × 512

After performing a full-angle (360°, 720 projections) scan of the digital model with the MCGPU_CT platform, we extracted subsets corresponding to 0°–90° (180 projections) and 0°–120° (240 projections) to serve as the limited-angle projection data. The sinograms for these two limited-angle conditions are shown in Figure 6(a) and (b), respectively. Figure 6(c) displays the reference image (Ground Truth) reconstructed from the full-angle data using the FBP algorithm.

Figure 6.

Simulated data from the Yingxian Pagoda dougong model. (a) Sinogram from the 0–90° limited-angle scan. (b) Sinogram from the 0–120° limited-angle scan. (c) Reference image reconstructed from the full-angle 0–360° scan via FBP.

Analysis of reconstruction results

Figure 7 provides a visual comparison of the results from different reconstruction algorithms on a selected CT slice of the dougong model under 90° and 120° limited-angle conditions. The corresponding quantitative evaluation metrics (PSNR and SSIM) are summarized in Table 2. To facilitate detailed inspection, a Region of Interest (ROI) is highlighted with a yellow box and magnified in each reconstructed image.

Figure 7.

Comparison of reconstruction results for the Yingxian Pagoda digital model under 90° (top row) and 120° (bottom row) limited-angle conditions.

Table 2.

Quantitative evaluation of limited-angle reconstruction results for the Yingxian Pagoda dougong model.

MetricsMethods	Limited angle 90°		Limited angle 120°
MetricsMethods	PSNR (dB)	SSIM	PSNR (dB)	SSIM
FBP	21.23	0.469	22.27	0.441
SART	22.86	0.711	24.54	0.795
ADMM-TV	23.32	0.712	25.91	0.810
DIP	21.83 ± 0.03	0.481 ± 0.031	23.57 ± 0.06	0.721 ± 0.025
RBP-DIP	24.85 ± 0.40	0.798 ± 0.020	27.73 ± 0.10	0.854 ± 0.010
DRP	28.45 ± 0.09	0.867 ± 0.006	32.23 ± 0.34	0.909 ± 0.004
CADRE	32.35 ± 0.10	0.916 ± 0.001	34.11 ± 0.13	0.940 ± 0.010

A visual analysis of Figure 7 reveals the following: The conventional FBP algorithm produced severe streak artifacts and significant image distortion under both 90° and 120° limited-angle conditions, resulting in the poorest reconstruction quality with obscured structural details. Iterative algorithms like SART and ADMM-TV suppressed artifacts to some extent compared to FBP but yielded blurry images with a noticeable loss of detail, particularly in structurally complex regions. In contrast, the unsupervised deep learning methods (DIP, RBP-DIP, DRP) generally outperformed the traditional algorithms. Specifically, the DIP reconstructions still exhibited some noise amplification and texture distortion. While RBP-DIP and DRP generated clearer images, with DRP showing relatively good detail preservation, all three methods still suffered from a loss of fine features to varying degrees. The proposed CADRE algorithm, however, demonstrated the best reconstruction performance under both limited-angle scenarios. Its reconstructions effectively eliminated streak artifacts while clearly rendering the structural boundaries and internal textures of the timber components. The magnified ROIs explicitly show CADRE's superior ability to reconstruct the fine contours of the internal mortise-tenon joints and maintain sharp edges.

The quantitative metrics in Table 2 further corroborate these visual observations. Under both 90° and 120° limited-angle conditions, the CADRE algorithm significantly outperformed all other benchmark methods in both PSNR and SSIM. For instance, under the more challenging 90° scenario, CADRE achieved an average PSNR of 32.35 dB and an average SSIM of 0.916. Under the 120° scenario, its average PSNR and SSIM were 34.11 dB and 0.940, respectively, showing an improvement over the next-best method, DRP (PSNR 32.23 dB, SSIM 0.909), and a much larger margin over traditional and other unsupervised methods (with PSNR gains ranging from 1.88 dB to 11.84 dB). These data provide strong evidence of CADRE's exceptional performance in artifact suppression, noise reduction, and preservation of structural details and overall image fidelity in limited-angle CT reconstruction.

It is noteworthy that as the scanning angle increased from 90° to 120°, the reconstruction quality (PSNR and SSIM values) improved for all algorithms, which is expected as more projection information becomes available. This confirms the importance of increasing the scanning angle to enhance reconstruction quality. However, even with the more severely incomplete data from the 90° scan, the CADRE algorithm still achieved results far superior to the other methods, demonstrating its excellent robustness and adaptability to data incompleteness. This is highly valuable for practical inspections where severe spatial constraints may be encountered.

To further assess the ability of each algorithm to preserve structural details, Figure 8 displays horizontal intensity profiles extracted from the same location (indicated by a yellow dashed line) across a different CT slice for each reconstruction method.

Figure 8.

Comparison of horizontal intensity profiles from the reconstruction results for the Yingxian Pagoda digital model under 90° (top) and 120° (bottom) limited-angle conditions.

As shown in Figure 8, the profile generated by the CADRE algorithm most closely matches the ground truth. It accurately reflects the pixel intensity variations across both structural edges and relatively uniform internal regions. In contrast, the profiles from the other methods exhibit varying degrees of oscillation, distortion, or over-smoothing, failing to accurately reproduce the original structural information. This result again highlights the superior capability of the CADRE algorithm in maintaining structural integrity and detailed accuracy during the reconstruction process, which is crucial for subsequent analysis of the internal details and structure of timber heritage.

Real-data validation: Foguang Temple Dougong model

To further validate the performance and generalization capability of the CADRE algorithm on a physical model and under real-world scanning conditions, this section presents experiments conducted on a 1:7.5 scale wooden model of a dougong from the Foguang Temple (see Figure 4(b)). The model's maximum dimension of approximately 700 mm exceeds the single-scan range of most small-scale XCT systems, making it a more representative proxy for large-scale ancient timber components. The experiment was performed using the Co-60 CT system at the Institute of Nuclear and New Energy Technology, Tsinghua University. Limited-angle projection data at 90° and 120° were extracted from a full-angle (360°) scan for reconstruction.

Experimental data acquisition

The core experimental platform for this study was the Co-60 CT system at Tsinghua University (see Figure 9). This system uses a Co-60 isotope as a gamma-ray source. The high-energy gamma rays it emits have strong penetrating power, uniform spatial distribution, and stable energy and radiation levels, making it highly suitable for inspecting large and dense timber components. The system is equipped with a CsI(Tl) scintillator detector array, providing precise radiation measurements and projection data. The main technical parameters of this CT system are listed in Table 3. By acquiring data with this system and then subsampling the projections by angle, we can simulate limited-angle scanning scenarios to evaluate the performance of different reconstruction algorithms under the influence of real-world noise and physical effects.

Figure 9.

The Co-60 CT scanning system at the Institute of Nuclear and New Energy Technology, Tsinghua University.

Table 3.

Main technical parameters of the Co-60 CT scanning system.

Parameter	Value
Radiation Source	⁶⁰Co
Source Activity	~30 Ci
Detector Type	CsI (Tl)
Detector Dimensions	460 × 8
Pixel Size	3.2 mm × 5 mm
Maximum Detectable Object Size	700 mm
Geometric Magnification	1.86
Distance of Source-to-Detector	1300 mm
Distance of Source-to-Object-Center	700 mm
Projection Angular Interval	0.365°
Reconstructed Image Size	512 × 512

A full-angle scan of the physical dougong model was performed using the Co-60 CT system, from which limited-angle projection data corresponding to 90° and 120° scanning ranges were extracted. The sinograms for these two limited-angle conditions are shown in Figure 10(a) and (b), respectively. Figure 10(c) displays the image reconstructed from the full-angle data using the FBP algorithm, which serves as the reference image for this real-data experiment.

Figure 10.

Experimental data from the Foguang Temple wooden dougong model. (a) Sinogram from the 0–90° limited-angle scan. (b) Sinogram from the 0–120° limited-angle scan. (c) Reference image reconstructed from the full-angle 0–360° scan via FBP.

Analysis of reconstruction results

Figure 11 displays the limited-angle reconstruction results from different algorithms for a representative CT slice of the physical dougong model, with the full-angle FBP reconstruction serving as a visual reference. The corresponding quantitative evaluation metrics (PSNR and SSIM) are compiled in Table 4.

Figure 11.

Comparison of reconstruction results for the Foguang Temple wooden dougong model under 90° (top row) and 120° (bottom row) limited-angle conditions.

Table 4.

Quantitative evaluation of limited-angle reconstruction results for the Foguang Temple wooden dougong model.

MetricsMethods	Limited angle 90°		Limited angle 120°
MetricsMethods	PSNR (dB)	SSIM	PSNR (dB)	SSIM
FBP	13.69	0.441	16.66	0.552
SART	15.13	0.587	20.70	0.796
ADMM-TV	15.51	0.591	22.48	0.798
DIP	13.88 ± 0.02	0.344 ± 0.003	17.31 ± 0.04	0.496 ± 0.031
RBP-DIP	24.44 ± 0.64	0.671 ± 0.008	28.57 ± 0.99	0.821 ± 0.004
DRP	28.81 ± 0.86	0.817 ± 0.006	30.89 ± 0.08	0.862 ± 0.008
CADRE	31.19 ± 0.25	0.876 ± 0.014	36.28 ± 0.84	0.961 ± 0.004

As seen in the visual results in Figure 11, the reconstruction quality for all algorithms degraded to some extent compared to the simulation study, due to complex physical factors in real-world scans such as noise, scatter, and non-ideal detector response. The streak artifacts in the FBP reconstructions are more severe with real data, nearly obscuring most structural details under the 90° limited-angle condition (Figure 11, top row), making internal structures difficult to identify. While the SART and ADMM-TV algorithms still show some artifact suppression, the images are blurry, and their ability to recover fine details is limited, failing to meet the needs for analyzing features like mortise-tenon joints. Consistent with the simulation results, the unsupervised deep learning methods (DIP, RBP-DIP, DRP) also demonstrated an advantage over traditional methods with real data, with the DRP algorithm managing to delineate the main contours and some internal structures of the dougong. However, the superiority of the CADRE algorithm was even more pronounced in the reconstruction of the physical model. Its reconstructions not only significantly reduced artifacts but were also clearly superior to the other methods in preserving structural integrity, edge sharpness, and fine internal details, such as micro-cracks or the intricate forms of joint connections. By observing the magnified ROIs (highlighted by yellow boxes), one can clearly see CADRE's capability in rendering detail under complex, real-world data conditions.

The quantitative results in Table 4 further support this visual assessment. Under both limited-angle conditions, the average PSNR and SSIM values for the CADRE algorithm were the highest. For example, at 120°, CADRE achieved a PSNR of 36.28 dB and an SSIM of 0.961, significantly higher than the other algorithms. Even in the highly challenging 90° limited-angle scenario with real-world noise, CADRE still achieved a PSNR of 31.19 dB and an SSIM of 0.876, demonstrating its robustness and stability in practical applications. These results indicate that the proposed CADRE algorithm can effectively handle real-world limited-angle CT data, providing a high-quality imaging basis for the NDT of ancient wooden structures.

Ablation study

To thoroughly analyze the contribution of each key component within the CADRE algorithm, we conducted an ablation study based on the data acquired from the physical dougong model under both 90° and 120° limited-angle conditions. The experiment evaluated the performance change by selectively removing specific modules from the complete CADRE framework. The results are presented in Table 5.

Table 5.

Results of the ablation study on key components of the CADRE algorithm using the Foguang Temple dougong model limited-angle dataset.

MetricsVariants	Limited angle 90°		Limited angle 120°
MetricsVariants	PSNR (dB)	SSIM	PSNR (dB)	SSIM
CADRE	31.19 ± 0.25	0.876 ± 0.014	36.28 ± 0.84	0.961 ± 0.004
w/o CG	30.08 ± 0.05	0.848 ± 0.005	35.17 ± 0.13	0.935 ± 0.011
w/o CBAM	30.86 ± 0.08	0.859 ± 0.011	35.91 ± 0.41	0.958 ± 0.006
w/o ADMM	30.53 ± 0.16	0.852 ± 0.004	35.40 ± 0.22	0.941 ± 0.002
w/o CG&CBAM&ADMM	28.81 ± 0.86	0.817 ± 0.006	30.89 ± 0.08	0.862 ± 0.008

Based on the experimental data in Table 5, the removal of the contour-guidance mechanism (w/o CG) had the most significant impact on reconstruction quality, causing the PSNR to drop by approximately 1.11 dB under both tested angular ranges, with a corresponding clear decrease in SSIM. This strongly confirms the critical role of the contour-guidance strategy in preserving structural details and suppressing artifacts outside the region of interest. Similarly, disabling the CBAM attention module (w/o CBAM) or removing the ADMM optimization framework (w/o ADMM) also led to a clear degradation in both PSNR and SSIM values (e.g. under the 90° condition, removing CBAM and ADMM resulted in PSNR drops of 0.33 dB and 0.66 dB, respectively; under the 120° condition, the drops were 0.37 dB and 0.88 dB, respectively). These results validate the positive contributions of the CBAM in helping the network focus on key image features and of the ADMM framework in ensuring stable and efficient algorithm convergence. Therefore, the results of the ablation study powerfully demonstrate the critical synergistic contribution of the integrated contour guidance, CBAM attention module, and ADMM optimization framework to achieving high-quality limited-angle CT reconstruction. The complete CADRE framework is able to achieve an optimal balance between effectively suppressing artifacts and maximally preserving structural details.

Discussion

The CADRE algorithm's superior performance in limited-angle CT of ancient wooden structures stems from its innovative synergistic design. Unlike unsupervised methods such as DIP that rely on a single, implicit network prior and can suffer from unstable convergence, CADRE's ADMM framework provides a stable, explicit iterative path that effectively balances data fidelity and regularization. A key innovation is the geometric contour-guidance mechanism—an explicit structural prior absent in other methods—which significantly enhances interior detail preservation by incorporating macroscopic object information. This component is crucial for resolving complex features like mortise-tenon joints, and its effectiveness has been validated by our ablation study. The framework's ability to learn from incomplete data is further strengthened by integrating powerful concepts from the Deep Radon Prior (DRP) to ensure consistency in the projection domain, while an embedded CBAM attention module improves the recovery of fine structural details. It is this organic combination of three complementary components—a mathematical optimization framework, an explicit geometric prior, and a deep Radon-domain prior—that enables CADRE to more effectively address the severe ill-posedness of the limited-angle problem, yielding superior results compared to techniques that rely solely on a single image- or Radon-domain prior.

The success of the CADRE algorithm not only represents a significant advancement in limited-angle CT reconstruction technology but also brings substantial application value to the field of cultural heritage preservation. High-quality reconstruction of internal structures is fundamental to understanding and assessing the health of ancient buildings. The clear images provided by this algorithm allow for the accurate visualization of critical details that were previously difficult to observe directly, such as the precise morphology of mortise-tenon joints. This will provide essential data to support conservation engineers and researchers in formulating precise restoration plans, conducting structural safety assessments, and carrying out scientific studies on material degradation mechanisms. Furthermore, CADRE's robust performance even under conditions of severe data incompleteness (e.g. a 90° limited angle) enhances the feasibility and effectiveness of using CT for in-situ NDT of ancient wooden structures in constrained environments, which is significant for advancing the field from traditional, experience-based conservation to precise, science-based preservation.

Despite the promising performance of the CADRE algorithm, several factors require consideration for its practical application and future development. First, the computational complexity is increased compared to traditional methods, and optimizing computational efficiency will be necessary when processing very large datasets or aiming for near-real-time reconstruction. Second, the reconstruction results may be influenced by factors such as the ADMM penalty parameter and the accuracy of the contour information, necessitating further research into the algorithm's robustness and adaptive parameter strategies. Furthermore, the applicability and performance of the algorithm need additional validation for extremely large components that exceed the single-scan field of view of the CT system, or for cases involving highly complex material properties (e.g. the presence of metallic components).

To address these challenges and expand the application prospects of CADRE, future research can focus on several directions. These include algorithm lightweighting through more efficient network architectures and optimizers; deeper integration of physical imaging models and material-specific prior knowledge into the learning framework; the development of intelligent fusion techniques for multi-modal NDT data (e.g. CT with ultrasound or terahertz imaging) to obtain more comprehensive structural information; and extending the current two-dimensional framework to three-dimensional or even four-dimensional (to account for temporal changes) reconstruction, coupled with uncertainty quantification, to provide more dynamic and reliable decision support for heritage conservation.

Conclusion

In this paper, we successfully developed a novel unsupervised deep learning algorithm, CADRE, to address the critical issues of poor image quality and severe artifacts in the limited-angle CT inspection of ancient wooden structures. This algorithm innovatively integrates the Alternating Direction Method of Multipliers (ADMM) optimization framework, a Deep Radon Prior-inspired network learning approach, and a geometric contour-guidance mechanism, effectively enhancing the ability to reconstruct high-quality CT images from incomplete projection data.

Through systematic experimental validation on both a simulated digital model and a physical timber model, the CADRE algorithm demonstrated consistently and significantly superior performance under typical 90° and 120° limited-angle conditions. Its quantitative metrics (PSNR, SSIM) and visual quality (artifact suppression, detail clarity) surpassed those of traditional filtered back-projection, iterative reconstruction algorithms, and other representative unsupervised deep learning methods. It exhibited exceptional capability in accurately resolving critical internal details, such as mortise-tenon joints.

The main contribution of this research is the introduction of a robust and efficient unsupervised learning framework that effectively solves a practical challenge in limited-angle CT reconstruction within the field of cultural heritage. The success of the CADRE algorithm not only provides a powerful technical tool for the precise non-destructive inspection and health assessment of ancient wooden structures but also offers valuable insights for addressing ill-posed inverse problems in related fields. Future work will focus on further improving the algorithm's efficiency and robustness, as well as expanding its application to three-dimensional reconstruction and multi-modal data fusion.

Footnotes

Acknowledgments

This work was supported by the National Key R&D Program of China (Grant No. 2023YFF0906300). We gratefully acknowledge the support of the Beijing Key Laboratory of Nuclear Detection Technology.

ORCID iDs

Jintao Fu

Tianchen Zeng

Ximing Liu

Funding

This work was supported by the National Key R&D Program of China (Grant No. 2023YFF0906300).

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

References

Jirouš-Rajković

Miklečić

. Enhancing weathering resistance of wood—a review. Polymers (Basel) 2021; 13: 1980.

Liu

Bai

. Thermodynamic modelling for the prediction and optimisation of long-term performance in wooden structures. Int J Heat Technol 2023; 41: 1341–1348.

Wang

Macchioni

, et al. Comprehensive evaluation method of historical timber structural building taking Fujiu Zhou house as an example. Forests 2021; 12: 1172.

Shabani

Kioumarsi

Plevris

, et al. Structural vulnerability assessment of heritage timber buildings: a methodological proposal. Forests 2020; 11: 881.

Guan

Zhang

, et al. Application of nondestructive testing technologies in preserving historic trees and ancient timber structures in China. Forests 2021; 12: 318.

Deuse

. Non-destructive diagnostics for timber structures in historic buildings : investigation methods and testing tools. Diagnóstico no destructivo de estructuras de madera en edificios históricos: métodos de investigación y herramientas de prueba, https://repositorio.unican.es/xmlui/handle/10902/12378 (September 2017, accessed 4 June 2025).

Zhao

Huo

, et al. Application progress and prospect of defect detection technology for timber structure members. Russ J Nondestruct Test 2024; 60: 455–469.

Zabler

Maisl

Hornberger

, et al. X-ray imaging and computed tomography for engineering applications. tm - Technisches Messen 2021; 88: 211–226.

Yanmin

Liye

Jigang

, et al. Development of single-vehicle mobile 60Co container inspection system. At Energy Sci Technol 2003; 37: 503–507.

10.

Tingrui

. 60Co γ-ray source and radiation technology. Univ Chem 2019; 34: 13–20.

11.

Björngrim

Myronycheva

Fjellström

. The use of large-scale X-ray computed tomography for the evaluation of damaged structural elements from an old timber bridge. Wood Mat Sci Eng 2022; 17: 1028–1029.

12.

Singh

Muralidhar

Munshi

. Image reconstruction from incomplete projection data using combined ART- CBP algorithm. ProQuest. DOI: 10.14429/dsj.52.2185.

13.

Schofield

King

Tayal

, et al. Image reconstruction: part 1 – understanding filtered back projection, noise and image acquisition. J Cardiovasc Comput Tomogr 2020; 14: 219–225.

14.

Andersen

Kak

. Simultaneous Algebraic Reconstruction Technique (SART): a superior implementation of the art algorithm. Ultrason Imaging 1984; 6: 81–94.

15.

Jin

Chen

, et al. Anisotropic total variation for limited-angle CT reconstruction. In: IEEE nuclear science symposium & medical imaging conference, 2010, pp.2232–2238. DOI: 10.1109/NSSMIC.2010.5874180.

16.

Zhang

Wang

Yan

, et al. Image reconstruction based on total-variation minimization and alternating direction method in linear scan computed tomography. Chin Phys B 2013; 22: 078701.

17.

Greffier

Frandon

Larbi

, et al. CT Iterative reconstruction algorithms: a task-based image quality assessment. Eur Radiol 2020; 30: 487–500.

18.

Cong

, et al. Neural architecture search with Deep Radon Prior for sparse-view CT image reconstruction. Med Phys 2025; 52: 3044–3058.

19.

Meng

Yan

, et al. Addressing data scarcity in crack detection via CrackModel: a novel dataset synthesis approach. Buildings 2025; 15: 1053.

20.

Wang

Zeng

Wang

, et al. ADMM-based deep reconstruction for limited-angle CT. Phys Med Biol 2019; 64: 115011.

21.

Fundova

Funda

. Non-destructive wood density assessment of Scots pine (Pinus sylvestris L.) using Resistograph and Pilodyn. PLoS One 2018; 13: e0204518.

22.

Gendvilas

Downes

Lausberg

, et al. Predicting wood density using resistance drilling: the effect of varying feed speed and RPM. Forests 2024; 15: 579.

23.

Feng

, et al. Image reconstruction of internal defects in wood based on segmented propagation rays of stress waves. Appl Sci 2018; 8: 1778.

24.

Paniagua

Corrales

Torres

, et al. Influence of timber moisture content on the ultrasonic wave velocity measurement of Tectona grandis L.F. and Cupressus lusitanica M. from Costa Rica. Forests 2022; 13: 1296.

25.

López

Basterra

Ramón-Cueto

, et al. Detection of singularities and subsurface defects in wood by infrared thermography. Int J Archit Herit 2014; 8: 517–536.

26.

Zheng

Xing

, et al. Research on tree ring micro-destructive detection technology based on digital micro-drilling resistance method. Forests 2022; 13: 1139.

27.

Morigi

Casali

Bettuzzi

, et al. Application of X-ray computed tomography to cultural heritage diagnostics. Appl Phys A 2010; 100: 653–661.

28.

Gejdoš

Gergeľ

Michajlová

, et al. The accuracy of CT scanning in the assessment of the internal and external qualitative features of wood logs. Sensors 2023; 23: 8505.

29.

Kobayashi

Hwang

Okochi

, et al. Non-destructive method for wood identification using conventional X-ray computed tomography data. J Cult Herit 2019; 38: 88–93.

30.

Reims

Schoen

Boehnel

, et al. Strategies for efficient scanning and reconstruction methods on very large objects with high-energy x-ray computed tomography. In: Developments in X-Ray tomography IX, Vol. 9212, 2014, pp.72–80: SPIE. DOI: 10.1117/12.2062002.

31.

Jintao

Yuewen

, et al. Experimental research on the use of cobalt-60 computed tomography for heritage timber buildings. J Archit Hist 2023; 4: 122–131.

32.

Gong

Zeng

. Self-guided limited-angle computed tomography reconstruction based on anisotropic relative total variation. IEEE Access 2020; 8: 70465–70476.

33.

Huang

Chen

, et al. RED: residual estimation diffusion for low-dose PET sinogram reconstruction. Med Image Anal 2025; 102: 103558.

34.

Zhang

Wang

, et al. Score-based generative model with conditional null-space learning for limited-angle tomographic reconstruction in medical imaging. IEEE Trans Comput Imaging 2025; 11: 556–569.

35.

Pan

Gao

, et al. Iterative residual optimization network for limited-angle tomographic reconstruction. IEEE Trans Image Process 2024; 33: 910–925.

36.

Lempitsky

Vedaldi

Ulyanov

. Deep image prior. In: 2018 IEEE/CVF conference on computer vision and pattern recognition, 2018, pp.9446–9454. DOI: 10.1109/CVPR.2018.00984.

37.

Sun

, et al. Deep radon prior: a fully unsupervised framework for sparse-view CT reconstruction. Comput Biol Med 2025; 189: 109853.

38.

Shu

Entezari

. RBP-DIP: residual back projection with deep image prior for ill-posed CT reconstruction. Neural Netw 2024; 180: 106740.

39.

Woo

Park

Lee

, et al. CBAM: convolutional block attention module. Published online July 18, 2018. DOI: 10.48550/arXiv.1807.06521.

40.

Zeng

Sun

, et al. An MCGPU-based CT simulation study for wooden architectural structures. In: Proceedings of the 2025 American nuclear society annual meeting. DOI: 10.13182/T140-48830.