**7. Conclusions**

In this work, we developed and implemented a polar harmonic multilevel system on the circular-polar grid based on the multiscale theory, and testified to the performance of the *PHMS* in four different quantitative measures, which suggests that the *PHMS* transform is suitable for the circular-shape Fourier transform. Another advantage of the *PHMS* is that the circular low frequency region in the frequency domain is consistent with the image spectrum distribution, which can process the multilevel structure more effectively.

**Acknowledgments:** The first author Guomin Sun is very grateful to the China Scholarship Council for funding the first author visiting the University of Tuscia. This work was supported by the National Natural Science Foundation of China under Grant 11271001, Grant 61370147, and Grant 61573085.

**Author Contributions:** Guomin Sun and Carlo Cattani designed the circular multilevel system and performed the experiments, Jinsong Leng analyzed the results. Guomin Sun and Carlo Cattani wrote the paper. All authors have read and approved the final manuscript.

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