**5. Conclusions**

In this paper, we proposed a new concept in nonlocal TV, in which the first and second order derivatives are combined in the regularization term. By combining the two terms, we were able to compensate for each other's weaknesses, i.e., staircase artifact and loss in smooth intensity changes in the first derivative and image blurring in the second derivative. Furthermore, we proposed a specially designed proximal splitting algorithm that is based on Passty's framework. The key idea is to split the original cost function to minimize as finely as possible to accelerate convergence and simplify necessary computations. This allows as to make the final iterative algorithm into a form of row-action type, which

is known to converge very quickly compared to other standards such as Chambolle–Pock algorithm and proximal gradient. In our experiments, we experimentally confirmed that the proposed algorithm converges within 20 iterations even for the case of brain CT imaging in which the requirement of image contrast is very severe. The simulation results with the brain CT image were performed for both the sparse-view CT and the low-dose CT. We showed that our proposed algorithm works well in practice.

As future work, our proposed nonlocal TV can be compared with the latest technology e.g., deep leaning [27,28] or other applied methods such as low-rank minimization [29,30].

Recently, image reconstruction methods using deep learning have been actively investigated. Our proposed method can be compared with existing deep leaning [27,28] as advanced compressed sensing. Additionally, our proposed method can be applied to low-rank TV, which can improve image quality by combing low-rank minimization and Total Variation [29,30].

**Author Contributions:** Conceptualization, Y.K. and H.K.; methodology, Y.K.; software, Y.K.; validation, Y.K. and H.K.; formal analysis, Y.K. and H.K.; investigation, Y.K. and H.K.; resources, Y.K. and H.K.; data curation, Y.K. and H.K.; writing—original draft preparation, Y.K.; writing—review and editing, Y.K. and H.K.; visualization, Y.K.; supervision, H.K.; project administration, H.K.; funding acquisition, H.K. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported by JST CREST Grant Number JPMJCR1765, Japan.

**Conflicts of Interest:** The authors declare no conflicts of interest.

### **References**


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