Data-Proximal Complementary ℓ¹-TV Reconstruction for Limited Data Computed Tomography

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The authors present a novel variational regularization framework that combines the advantages of different regularizers. The basic idea of this framework is to perform reconstruction in two stages.  The first step is a ℓ1-reconstruction, and the second step minimizes the TV norm for artifact reduction.  The simulation results demonstrated the effectiveness of the framework, but I still have the following concerns:

1. Why can the use of the data-proximity coupling avoid the disadvantages of the two regularization terms and leverage their respective advantages? The authors may not provide a detailed theoretical explanation, but a more comprehensive explaination should be offered.

2. The authors only conduct numerical simulation experiments. It is essential to perform at least one image reconstruction experiment using real CT data to verify the performance of your method in practical CT reconstruction.

3. The authors should consider some noteworthy limited-angle CT reconstruction methods, and make a comparison with one of them, such as "An Image Reconstruction Model Regularized by Edge-preserving Diffusion and Smoothing for Limited-angle Computed Tomography", "Directional-TV algorithm for image reconstruction from limited-angular-range data," or "Structure-guided computed tomography reconstruction from limited-angle projections."

4. The ℓ2-error and PSNR are correlated, and the authors could attempt other image quality assessment metrics, such as Full-Structure Similarity Index and Pearson-correlation-coefficient.

5. Line 228. "that in in this approach" should be "that in this approach".

Author Response

Dear Reviewer 1,

Thank you very much for your insightful review of our article. Your thoughtful comments and suggestions have been extremely helpful in strengthening our work. We appreciate the time and effort you dedicated to providing valuable feedback.

Attached, please find our detailed response to your comments.

Sincerely,

Simon Göppel, Jürgen Frikel, and Markus Haltmeier

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

In this study, the authors propose a new variational regularization framework specifically applied to the problem of image reconstruction in finite data CT. This framework combines the advantages of different regularization techniques and overcomes the shortcomings of traditional methods in dealing with the dual challenges of missing data and noise. This work solves the problem of limited data reconstruction in the field of computational tomography. In some cases, due to physical limitations or experimental conditions, the complete data cannot be obtained, resulting in a seriously under-determined image reconstruction problem, resulting in a large number of reconstruction artifacts. To solve this problem, a complementary ℓ1-TV reconstruction algorithm based on data proximity is proposed, which aims to solve both the limited data problem and the noise reduction problem. The paper is well organized and legible, but there are still the following problems that need to be revised.

 

1. The specific mathematical analysis of the proposed complementary ℓ1-TV algorithm is rather complicated, and its theoretical properties are not discussed in depth in this paper, especially the properties of algorithm convergence and optimal solutions.

2. Although the numerical experiment part shows the effectiveness of the proposed algorithm, in order to enhance the persuasion of the conclusion, more experimental samples can be added, more existing regularization methods can be compared, and the influence of parameter selection on the reconstruction quality can be explored.

3. The conclusions section could further emphasize the direct link between the available numerical experimental results and the theoretical advantages of the proposed algorithm, by explicitly stating how the experimental results concretely validate the dual role of data approximation coupling in terms of noise suppression and data complementation, and how this role is preferable to a single regularization method.

4. While the authors enumerate a wide range of possible directions for future research, one or two of the most promising or urgent research themes could be more explicitly identified in the conclusion, highlighting the direction of future research in the field.

5. In the introduction, although the background of the problem of finite-data CT and the limitations of the existing regularization methods are mentioned, the innovation and unique contribution of this study compared with the existing literature are not sufficiently highlighted. It is recommended that the unique features and contributions of the complementary ℓ1-TV algorithm be explicitly stated in the introduction in comparison to the prior art.

Author Response

Dear Reviewer 2,

Thank you very much for your insightful review of our article. Your thoughtful comments and suggestions have been extremely helpful in strengthening our work. We appreciate the time and effort you dedicated to providing valuable feedback.

Attached, please find our detailed response to your comments.

Sincerely,

Simon Göppel, Jürgen Frikel, and Markus Haltmeier

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

1. Although the quantitative results in Table 2 demonstrate the superiority of the proposed method, visually it can be seen that for such a simple Forbild phantom, the image quality of the image reconstructed by the proposed method is  close to those of the images reconstructed by TV and l1-TV methods. Why is the visual effect not significant?

2. The authors conducted experiments on the walnut data, but did not provide reconstructed images due to the lack of a high-quality reference image. I believe that observing whether the artifacts are significant may not necessarily require a perfect reference image.

Author Response

Thanks again for your valuable suggestions. We made the following updates to the article:

General remarks:

• Almost all figures were updated:
• In Figure 1 we removed the presentation of the NCAT phantom. 
• We included Figure 2, which now presents all phantoms that are used for reconstruction experiments.
• We rearranged Figure 3, since we believe that it now presents the results in a more clear way
• In Figure 4: We removed the original image, since it is now presented in Figure 2
• Figure 5 is new. It shows now the results of reconstructions from real CT data of a lotus phantom
• We rearranged Figure 6, since we believe that it now presents the results in a more clear way
• Since the article now includes more experiments using different data, we had to rewrite most of the text of section 4.

 

1. Although the quantitative results in Table 2 demonstrate the superiority of the proposed method, visually it can be seen that for such a simple Forbild phantom, the image quality of the image reconstructed by the proposed method is  close to those of the images reconstructed by TV and l1-TV methods. Why is the visual effect not significant?

Response: 

Upon closer evaluation of the proposed reconstruction with the TV or hybrid ell1-TV reconstructions one can observe that a little bit fewer limited-view artifacts are generated. Nevertheless, the TV  and hybrid ell1-TV methods appears to reconstruct the FORBILD phantom quite effectively, which we think is due to the piecewise constant nature of the phantom, which aligns well with TV regularization. 

Furthermore, for all reconstructions, we employed a moderate number of iterations. Increasing the number of iterations might result in additional improvements in artifact reduction performance, especially in the case of TV and $\ell^1$-TV reconstructions, as well as in the case of the proposed method. However, simultaneously, it may facilitate the generation of blocky artifacts. We did not delve deeper into how the number of iterations influences the performance of all algorithms. The number of iterations used in all computations was determined experimentally, because using a larger number of iterations did not significantly enhance the reconstruction quality.

 

2. The authors conducted experiments on the walnut data, but did not provide reconstructed images due to the lack of a high-quality reference image. I believe that observing whether the artifacts are significant may not necessarily require a perfect reference image.

Response:

We now included the reconstructions of real CT data of a lotus root, presented in Figure 5. This experiments shows that our method (along with all other methods) also performs effectively on real data, allowing for similar conclusions as those drawn in the case of synthetic data. However, it’s worth noting that in this instance, no extensive parameter search was conducted, leaving potential for further improvement.

Reviewer 2 Report

Comments and Suggestions for Authors

The author carefully replied to and addressed my questions; I believe this manuscript is acceptable for publication.

Author Response

Thanks again for taking the time to review our article.