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Convolutional Sparse Coding for Compressed Sensing CT Reconstruction
Peng Bao, Wenjun Xia, Kang Yang, Weiyan Chen, Mianyi Chen, Yan Xi, Shanzhou Niu, Jiliu Zhou, He Zhang, Huaiqiang Sun, Zhangyang Wang, Yi Zhang
TL;DR
Sparse-view CT reconstruction must recover high-quality images from undersampled projections, a challenge associated with reducing radiation dose. The paper proposes PWLS-based CSC models, and experiments on simulated and real data show competitive performance against several state-of-the-art methods.
Problem
Reducing CT views can lower radiation dose but leaves insufficient projection data, making high-quality reconstruction challenging.
Source: §I · p. 1Method
The paper proposes PWLS-CSC with CSC regularization and a variant, PWLS-CSCGR, that adds gradient regularization on feature maps.
Source: §I · p. 2Results
The proposed methods were competitive with several state-of-the-art methods in experiments on simulated and real data.
Source: §V · p. 12Takeaways & Limitations
PWLS-CSCGR was reported to work with four filters and to be robust to the training set, including without one.
Source: §V · p. 12
Abstract
from arXiv · showhide
Over the past few years, dictionary learning (DL)-based methods have been successfully used in various image reconstruction problems. However, traditional DL-based computed tomography (CT) reconstruction methods are patch-based and ignore the consistency of pixels in overlapped patches. In addition, the features learned by these methods always contain shifted versions of the same features. In recent years, convolutional sparse coding (CSC) has been developed to address these problems. In this paper, inspired by several successful applications of CSC in the field of signal processing, we explore the potential of CSC in sparse-view CT reconstruction. By directly working on the whole image, without the necessity of dividing the image into overlapped patches in DL-based methods, the proposed methods can maintain more details and avoid artifacts caused by patch aggregation. With predetermined filters, an alternating scheme is developed to optimize the objective function. Extensive experiments with simulated and real CT data were performed to validate the effectiveness of the proposed methods. Qualitative and quantitative results demonstrate that the proposed methods achieve better performance than several existing state-of-the-art methods.
I. INTRODUCTION
Sparse-view CT can reduce radiation dose, but undersampled projections make high-quality reconstruction challenging. The paper introduces CSC-based models for this setting and describes a PWLS formulation for CT reconstruction.
I. INTRODUCTION
- Reducing the number of CT views can lower radiation dose but leaves insufficient projection data, making high-quality reconstruction challenging.The paper focuses on the sparse-view problem.Source: §I · p. 1
- With highly sparse views, iterative reconstruction can remain unsatisfactory without priors, while TV-based methods can produce patchy effects.Source: §I · p. 1
- Patch-based dictionary learning can create shifted redundant features and boundary aggregation artifacts, motivating CSC's direct modeling of shift invariance.Source: §I · p. 2
- PWLS-CSC uses CSC regularization with externally trained filters; PWLS-CSCGR adds feature-map gradient regularization to suppress ringing linked to inaccurate filters.Source: §I · p. 2
A. Penalized Weighted Least-Squares for CT Image Reconstruction
- The PWLS formulation assumes calibrated, log-transformed projection data are approximately Gaussian and balances data fidelity against regularization.Its variance model uses bin-adaptive r_p and scaling parameter ε.Source: A. Penalized Weighted Least-Squares… · p. 2
B. Convolutional Sparse Coding
Convolutional sparse coding represents an image as summed filter–feature-map convolutions, avoiding patch-based aggregation. PWLS-CSC combines this representation with a PWLS model, trains filters on external full-sampled CT data, and alternates image and feature-map updates.
B. Convolutional Sparse Coding
- CSC represents an image as summed convolutions of filters and corresponding feature maps, with feature maps the same size as the image.This whole-image representation avoids patch aggregation artifacts and can obtain rotation-invariant features.Source: B. Convolutional Sparse Coding · p. 3
A. PWLS-CSC Model
- PWLS-CSC combines CSC regularization with the PWLS model and trains its filters using an external full-sampled CT dataset.The filters are trained by the method cited as.Source: A. PWLS-CSC Model · p. 3
- The reconstruction alternates between estimating an intermediate image with fixed feature maps and representing that image with fixed filters.Because CSC does not represent low-frequency image components well, CSC processes only the high-frequency component, which is then summed with the low-frequency component.Source: A. PWLS-CSC Model · p. 3
- ADMM is used to solve the feature-map optimization after introducing a dual variable constrained to equal the feature maps.The image update uses the separable paraboloid surrogate method, and the feature-map step can be solved in the Fourier domain.Source: A. PWLS-CSC Model · p. 3
B. PWLS-CSCGR
PWLS-CSCGR adds gradient regularization on feature maps to address artifacts associated with inaccurate filters. Its solution alternates between the PWLS-CSC image update and a gradient-constrained feature-map update.
B. PWLS-CSCGR
- PWLS-CSCGR adds gradient regularization on feature maps to PWLS-CSC, addressing artifacts that can arise from inaccurate filters.The paper states that applying the gradient constraint to feature maps is superior to applying it in the image domain.Source: B. PWLS-CSCGR · p. 4
- The gradient term uses filters for gradients along the x- and y-axes, with τ as a hyperparameter.The feature-map optimization is set up using the gradient operators and solved with ADMM.Source: B. PWLS-CSCGR · p. 4
- PWLS-CSCGR retains the PWLS-CSC image update and changes the feature-map solution to include the gradient constraint.The paper says the main difference from solving the earlier formulation is solving for M; the algorithm alternates updates and outputs the reconstructed image.Source: B. PWLS-CSCGR · pp. 4–5
IV. EXPERIMENTAL RESULTS
The experiments use simulated and real clinical CT data, with fixed default filters and regularization settings unless specified otherwise. Performance is quantitatively evaluated with PSNR, RMSE, and SSIM.
IV. EXPERIMENTAL RESULTS
- Experiments used simulated and real clinical CT data, with CSC and CSCGR phases accelerated on a GTX 1080 Ti GPU.The stated setup was MATLAB 2017a on a PC with an AMD Ryzen 5 1600 CPU and 16 GB RAM.Source: §IV · p. 5
- The default setup used 32 filters of size 10 × 10, β = 0.005, λ = 0.005, ρ = 100 × λ + 1, and τ = 0.06.All parameters were constant except ρ, which was self-adaptive according to.Source: §IV · p. 5
- PSNR, RMSE, and SSIM were used to quantitatively evaluate the compared methods.The convolution computation was transformed into the frequency domain and treated as a dot product.Source: §IV · p. 5
A. Simulated Results
For the simulated CT experiments, filters were trained using high-frequency components from ten randomly selected Mayo dataset images. The setup simulated fan-beam projections with 64 views, and compared the proposed methods with five listed alternatives.
A. Simulated Results
- Ten randomly selected full-dose Mayo CT slices supplied high-frequency components for training the predetermined filters.The dataset contains 5,936 full-dose CT slices, each 1 mm thick, from 10 patients.Source: A. Simulated Results · p. 5, Figure 1 · p. 5
- The fan-beam simulation used 64 evenly spaced projection views over 360° and a flat detector with 512 bins.The simulated image arrays were 20 × 20 cm2; the source-to-center and detector-to-center distances were each 40 cm.Source: A. Simulated Results · p. 5
- The simulated-results comparison included FBP, PWLS-TGV, PWLS-DL, PWLS-ULTRA, and LEARN.The reported iteration counts were 2,000 outer iterations and 20 and 100 inner iterations for PWLS and CSCGR, respectively; PWLS-CSC used the same iteration counts as PWLS-CSCGR.Source: A. Simulated Results · p. 5
1) Abdominal Case:
In the abdominal case, PWLS-CSC and PWLS-CSCGR remove artifacts while preserving more detail, especially contrast-enhanced blood vessels in the liver. The enlarged ROI shows these vessels nearly lost in several comparison methods but retained by the proposed methods.
1) Abdominal Case:
- PWLS-CSC and PWLS-CSCGR remove artifacts while preserving more abdominal detail, especially contrast-enhanced blood vessels in the liver.Their difference images are nearly indistinguishable visually, although the absolute difference images show better performance for PWLS-CSCGR.Source: 1) Abdominal Case · p. 5
- PWLS-DL and PWLS-ULTRA smooth some organ details during artifact suppression, while LEARN also fails to recover small details.The passage attributes smoothing by patch-based DL to averaging overlapped patches, which suppresses noise but can smooth small details.Source: 1) Abdominal Case · p. 5
- In the enlarged liver ROI, PWLS-CSC and PWLS-CSCGR preserve most details, including contrast-enhanced vessels that are almost lost in PWLS-DL, PWLS-ULTRA, and LEARN.The proposed methods also retain a coherent visual effect on the reference image, including mottle-like liver structures.Source: 1) Abdominal Case · p. 5
2) Thoracic Case:
In the thoracic images, PWLS-CSC and PWLS-CSCGR preserve more detail while removing artifacts. Enlarged regions show that they recover tiny vessels and retain curve-like structures that other methods do not adequately maintain.
2) Thoracic Case:
- PWLS-CSC and PWLS-CSCGR preserve more thoracic detail in some regions while effectively removing artifacts.The comparison methods suppress most artifacts but smooth some organs and lose important details to varying degrees.Source: 2) Thoracic Case · p. 5
- Only PWLS-CSC and PWLS-CSCGR satisfactorily recover the tiny vessels, while PWLS-CSCGR suppresses ringing artifacts seen with PWLS-CSC.The ringing is attributed probably to inaccurate filters; gradient regularization aids its suppression.Source: 2) Thoracic Case · p. 6
- The curve-like structures are clearly visible with PWLS-CSC and PWLS-CSCGR, whereas the other methods cannot adequately maintain them.The cited comparison concerns two small regions enlarged from the blue and red boxes in the thoracic image.Source: 2) Thoracic Case · pp. 5–6
3) Quantitative Evaluation:
Across sparse-view settings, PWLS-CSCGR achieves the best quantitative performance, while PWLS-CSC becomes stronger than PWLS-DL and PWLS-ULTRA at higher view counts. In real dental CT data, the proposed methods suppress noise and artifacts, with PWLS-CSCGR preserving tooth detail.
3) Quantitative Evaluation:
- Across experiments using 48, 64, and 80 projection views, the paper concludes that CSC and PWLS-CSCGR are more robust to different sampling rates than other methods.The abdominal image is separately described as reconstructed at 48, 60, and 80 projection views.Source: 3) Quantitative Evaluation · p. 6
B. Real Data
- The real-data experiment used projection data from a UEG Dental CT scanner acquired at 110 kVp and 13 mA/10 ms per projection.It used 500 evenly distributed views for the reference and 125 downsampled views for method comparisons; the reconstructed image was 256 × 256 pixels.Source: B. Real Data · p. 6
- The full-sampled reference still contains noticeable noise and artifacts because of hardware limitations.The reference was reconstructed from 500 projection views.Source: B. Real Data · p. 6
3) Quantitative Evaluation:
- PWLS-CSCGR achieves the best abdominal quantitative performance at all reported view counts; PWLS-CSC trails PWLS-DL and PWLS-ULTRA at 48 views but outperforms them at 64 and 80.The passage reports a similar thoracic trend: PWLS-CSCGR has the highest scores on all three metrics, and PWLS-CSC outperforms PWLS-TGV and PWLS-DL.Source: 3) Quantitative Evaluation · p. 6, Table I · p. 7
B. Real Data
- In the 125-view dental reconstructions, PWLS-CSC and PWLS-CSCGR suppress noise and artifacts without PWLS-DL's patch-aggregation side effect, and PWLS-CSCGR clearly preserves tooth detail.PWLS-DL's tooth details are blurred; PWLS-ULTRA and LEARN also leave artifacts and blur details.Source: B. Real Data · pp. 6–7
C. Effect of Filter Parameters
The filter-parameter experiments show diminishing gains as the filter count grows, little sensitivity to filter size, and only small improvement as training samples increase.
1) Number of Filters:
- With more filters, both images’ metrics rise then stabilize; 32 filters balance performance and computational time.The authors attribute diminishing gains at larger counts to reduced benefit after representation precision improves.Source: 1) Number of Filters · pp. 7–8
2) Filter Sizes:
- Performance is not quite sensitive to filter size across the tested sizes of 6, 8, 10, 12, and 14; the authors set size to 10.Size 10 is approximately the maximum in Fig. 9.Source: 2) Filter Sizes · p. 9
C. Effect of Filter Parameters
- As training samples increase, performance improves only slightly, indicating robustness to training-set size for the current number of filters.The evidence is reported for the current filter count.Source: 2) Filter Sizes · p. 9
4) Filter Trained with Different Training Sets:
The experiments find similar outcomes across filter-training sets and stable convergence, while the parameter sweeps identify settings for β, λ, and τ.
4) Filter Trained with Different Training Sets:
- Reconstructions from different filter-training sets look visually similar and have close quantitative results; adaptive filters score highly but take time to update each iteration.The tested sets include undersampled CT, natural images, and intermediate reconstruction images.Source: 3) Number of Training Samples · p. 9, Figure 13 · p. 11
1) Convergence analysis:
- As iteration count increases, PSNR and RMSE converge to stable values, supporting efficient optimization of the energy functions to a satisfactory solution..Source: 1) Convergence analysis · pp. 9–10
3) Number of Training Samples:
- Table IV reports quantitative results for different numbers of training samples.The supplied passage identifies the table’s scope but gives no cell values or comparison outcome.Source: Table IV · p. 10
2) Effect of
- The passage reports that PSNR and SSIM decrease as β increases, while smaller β slows algorithm convergence; it does not specify a chosen β value.Source: 2) Effect of · p. 10
- PSNR and SSIM peak at λ = 0.005 and then slowly decrease for larger λ values.The authors set λ to 0.005.Source: 2) Effect of · p. 11
- PSNR and SSIM change slowly with τ, indicating low sensitivity; the authors set τ to 0.06.Source: 2) Effect of · p. 11
V. DISCUSSION AND CONCLUSION
The discussion presents two PWLS-based CSC methods and reports competitive performance and robustness across experiments. It also notes reconstruction time as a practical concern.
V. DISCUSSION AND CONCLUSION
- The paper proposes PWLS-CSC and PWLS-CSCGR, adding gradient regularization on feature maps to the basic CSC model to preserve structural information.The authors report evaluation with simulated and real data and competitive results against several state-of-the-art methods.Source: §V · p. 12
- PWLS-CSCGR works with only four filters and is robust to training-set choice, including operation without a training set.The authors characterize it as an unsupervised learning method.Source: §V · p. 12
- Reconstruction is time-consuming and has a computational burden similar to PWLS-DL; training 32 filters with 10 images takes 85 seconds of GPU time.The authors note that training is offline and that parallel computing and advanced optimization could accelerate reconstruction.Source: §V · p. 12
- CSC represents a signal as summed convolutions of filters and feature maps, and the paper contrasts its unsupervised formulation with data-driven CNN methods.The discussion says CSC is robust to training-sample count and can work without training data.Source: §V · p. 12