# An Overall Uniformity Optimization Method of the Spherical Icosahedral Grid Based on the Optimal Transformation Theory

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theory and Methods

**R**

^{n}, two subspaces with measures, (

**X**, ψ) ∈

**R**

^{n}and (

**Y**, φ) ∈

**R**

^{n}, having the same total measures, that is ${\int}_{\mathit{X}}^{}\psi dx}={\displaystyle {\int}_{\mathit{Y}}^{}\phi dy$, there is a measure-preserving mapping T:

**X**→

**Y**to make φ(B) = ψ(T

^{−1}(B)), where B ⊂

**Y**, T

^{−1}(B) ⊂

**X**. Then the transportation cost can be formalized as in Equation (1):

**X**×

**Y**→

**R**is a cost function of T; T

_{opt}= argmin{C(T)} is the optimal transportation mapping.

**φ**= {φ

_{1}, φ

_{2}, …, φ

_{nv}} corresponding to the point set {p

_{i}} ⊂ S

^{2}, which cannot be covered by any hemisphere, such that $\psi ({S}^{2})={\displaystyle {\sum}_{i=1}^{nv}{\phi}_{i}}$. A set of cells,

**W**= {w

_{i}}, decomposing the unit sphere, that is ${S}^{2}={\displaystyle {\cup}_{i=1}^{nv}{w}_{i}}$, can be found to make φ(w

_{i}) = ψ

_{i}. Then a mapping T: w

_{i}→ p

_{i}is the optimal transportation mapping minimizing the transportation cost. The existence and uniqueness of the complete solutions in the continuous space have been proved by Gu (2013). Surfaces can be discretized by triangular meshes. The discrete spherical optimal transportation mapping is an approximate preserved area mapping from a topological spherical triangular mesh to a spherical triangular mesh. A topological spherical triangular mesh is

**M**{

**V**,

**E**,

**F**}, with point set

**V**= {v

_{1}, v

_{2}, …, v

_{nv}}, edge set

**E**= {e

_{1}, e

_{2}, …, e

_{ne}} and face set

**F**= {f

_{1}, f

_{2}, …, f

_{nf}}. ∀v

_{i}∈

**V**, there is a set of first-order adjacent vertices

**v**= {${v}_{{i}_{1}}$, ${v}_{{i}_{2}}$, …, ${v}_{{i}_{Ns}}$}, Ns is the number of first-order adjacent vertices. ∀ψ

_{adj}_{i}∈

**ψ**, it can be defined as one-third of total area of triangles formed by v

_{i}and

**v**, its construction is illustrated in Figure 1 and its formulization is defined in Equation (2),

_{adj}_{i}v

_{j}v

_{j}

_{+1}.

**M**,

**ψ**) → (

**N**,

**φ**), such that φ

_{i}= ψ

_{i}and C(T) in Equation (3) is minimal, in which

**N**{

**V***,

**E**,

**F**} is the image of

**M**,

**φ**is a measure set of

**N**.

**V**, a cell decomposition set

**W**= {w

_{i}} can be constructed, where w

_{i}’s area is φ

_{i}. When φ

_{i}equals ψ

_{i}, the transportation cost is minimal equal to zero. Updating the center of cell w

_{i}as ${v}_{i}^{\ast}$ obtains the image of the approximate preserved area mapping.

**V**= {v

_{1}, v

_{2}, …, v

_{nv}} ∈ S

^{2}and its weight set

**r**= {r

_{1}, r

_{2}, …, r

_{nv}} ∈ R, c

_{i}(v

_{i}, r

_{i}) is a circle on a sphere with center v

_{i}and radius r

_{i}. Spherical power distance between any point p ∉

**V**and c

_{i}is defined in Equation (4):

_{i}) is a geodesic distance between p and tangent point, intersection of a line through p tangent to circle and the circle, as shown in Figure 2.

_{i}, r

_{i})} is a cell decomposition of sphere, that is ${S}^{2}={\displaystyle {\cup}_{i=1}^{nv}{w}_{i}}$, where w

_{i}= {v

_{i}∈

**V**| pow(p, v

_{i}) < pow(p, v

_{j}), ∀v

_{j}∈

**V**−{v

_{i}}}. A spherical power diagram of a random point set is shown in Figure 3.

_{i}and radius r

_{i}, the red spherical polygon is spherical power cell of each point, and the blue spherical triangle is cell’s dual triangle.

_{i}and v

_{j}, there is pow (q, v

_{i}) = pow (q, v

_{j}) (Figure 4), then:

_{l}and R

_{k}are triangle power radius of ∆v

_{i}v

_{j}v

_{k}and ∆v

_{i}v

_{j}v

_{l.}. d

_{l}and d

_{k}are vertical distances from the triangle center o

_{l}and o

_{k}to edge [v

_{i}, v

_{j}], respectively. γ

_{i}and γ

_{j}are the distances between v

_{i}and q, v

_{j}and q, respectively.

_{i}= d(q, v

_{i}), γ

_{j}= d(q, v

_{j}), and γ

_{i}+ γ

_{j}= γ

_{ij}, there is:

_{j}with respect to h

_{i}, there is:

_{i}∩ ∆v

_{i}v

_{j}v

_{l}), ${\phi}_{i}^{jk}$ = area(w

_{i}∩ ∆v

_{i}v

_{j}v

_{k}), φ is the power cell area. There is:

_{1}) −

**ψ**

_{1}, φ(w

_{2}) −

**ψ**

_{2}, …, φ(w

_{nv}) −

**ψ**

_{nv}), and the Hessian matrix can be expressed as follows:

**ψ**, where ψ

_{virtual}_{i_virtual}= 4πR

^{2}/nv for any spherical icosahedral grid point; R is the earth radius. There is a mapping T: (Ico-Grid,

**ψ**) → (AREA-Ico-Grid,

_{virtual}**φ**), where Ico-Grid is composed of a grid point set

**V**= {v

_{i}}, a grid edge set

**E**, a grid set

**F**and a grid point set of AREA-Ico-Grid,

**V*** = {${v}_{i}^{\ast}$} is the image of

**V**under the mapping T, its grid edge set and grid set are the same as Ico-Grid’s. When the areas of the grids by cell decomposition are equal to each other, that is

**φ**(v*) =

**ψ**(T

_{virtual}^{−1}(v*)), v* ⸦ AREA-Ico-Grid, T

^{−1}(v*) ⸦ Ico-Grid, T is the approximate area-preserved mapping. The Algorithm 1 of area quasi-uniformity optimization for the spherical icosahedral grid is described as follows.

Algorithm 1: Area Uniformity Optimization for the Spherical Icosahedral Grid |

Input: a spherical icosahedral grid Ico-Grid {V, E, F}, step λ, initial height vector h_{0}, threshold δC |

Output: a spherical icosahedral grid with area quasi-uniformity AREA-Ico-Grid {V*, E, F} |

(1) Compute the virtual measure ψ = {ψ_{virtual}_{i_virtual}}; |

(2) According to h_{0} and V, decompose the sphere into a cell set W = {w_{i}}, and calculate each power cell area to get the original measure φ of Ico-Grid by, φ ← _{0}φ; |

(3) Calculate the transportation cost C between φ and _{0}ψ by Equation (3). If C < δC, _{virtual}V* ← V and move to step 5. If not, move to the next step; |

(4) Update cell decomposition of S^{2} |

(4.1) Calculate gradient ∇C = (φ(w_{1}) − ψ_{1}, φ(w_{2}) − ψ_{2}, …, φ(w_{nv}) − ψ_{nv})^{T}; |

(4.2) Compute the transportation cost C via ∇C. If C<δC, move to step 5. If not, move to step 4.3; |

(4.3) According to Equations (10)–(13), calculate a Hessian matrix H; |

(4.4) Establish the relationship between hessian matrix and gradient, that is Hδh = ∇C; |

(4.5) Update height vector, h = h + λδh; |

(4.6) Decompose the sphere into a new cell set W = {w_{i}} based on h; compute cells’ area φ = {φ_{W}_{wi}}; |

(5) Compute the centers of cells in W, and obtain CW = {c_{wi}}; |

(6) V* ← CW, and output result; |

(7) End. |

## 3. Results and Discussions

#### 3.1. The Grid Quality Evaluation

#### 3.1.1. The Grid Area Uniformity

_{area}was used to measure the grid area uniformity, which can be calculated with Equation (14):

_{avg}is the average area of the spherical icosahedral grid; earth radius was set to 6, 371.007 km.

_{max}and minimum grid area A

_{min}and average area A

_{avg}of NOPT grid and OURS grid at certain resolutions are listed in Table 1.

_{A}= A

_{min}/A

_{max}, of different spherical icosahedral grids at different resolutions are presented in Figure 6.

_{area}| are depicted in Table 5, and Figure 7 (|D

_{area}| < 0.08%).

_{area}of all grids are in [−38.0%, 22.0%] (Table 3). The ones of HR grid and OURS grid are mainly in (−2.0%, 2.0%), which is smaller than other four grids. There are more than 90% grids with D

_{area}in (−0.06%, 0.06%) in OURS grid. The cumulative proportions have increased logarithmically as |D

_{area}| increases, in which the increasing rate of OURS grid is the fastest and that of NOPT grid is the slowest (Figure 7). The cumulative proportions of OURS grid with |D

_{area}| of less than 0.044% is more than 90.00%, and the ones of other five grids are only 0.02% (NOPT grid), 0.62% (SPRG grid), 13.24% (HR grid), 0.37% (SCVT grid), and 1.25% (XU grid). Although the area ratio of HR grid is bigger than OURS grid, the proportion of grid with smaller is far less than that of OURS grid.

_{area}are symmetry. The distribution of NOPT grid has fractal characteristics and is not continuous. Because of the larger number of grids with larger D

_{area}, its distribution is shown by darker color. The distributions of the SPRG grid, SCVT grid and XU grid are similar. As the larger number of grids with smaller D

_{area}(−0.06%, 0.06%) in OURS grid, its distribution is shown by the lightest color and some grids with larger D

_{area}are mainly located along the triangular boundaries. Grids with larger D

_{area}are also located around these boundaries, however, these regions are far larger than that of OURS grid.

#### 3.1.2. The Grid Interval Uniformity

_{min}and the maximum distance d

_{max}(Table 6), the ratios between them (Table 7) and the grid length relative deviation (as Equation (16)) are calculated to describe the grid interval uniformity.

_{d}= d

_{min}/d

_{max}, of different grids at different resolutions are calculated in Table 7 and presented in Figure 10.

_{length}| are depicted in Table 10 and Figure 11. Because of higher cumulative proportion in OURS grid in a smaller interval, only a part of results with |D

_{length}| < 1.00% are demonstrated.

_{length}of all grids are in [−15%, 21%] (Table 8), and the ones of HR grid and OURS grid are mainly in (−1%, 1%). The proportion of grid in this range of OURS grid is about 7 times that of NOPT grid, although the interval ratio of NOPT grid is bigger than that of OURS grid. The cumulative proportions also increase logarithmically as |D

_{length}| increases, in which the increasing rate in OURS grid is the fastest and that of NOPT grid is the slowest. The cumulative proportions of OURS grid is more than 99.00% with |D

_{length}| less than 0.50%. Ones of other grids are only 3.86% (NOPT grid) 17.36% (SPRG grid), 75.17% (HR grid), 21.64% (SCVT grid), and 20.47% (XU grid) in the same range.

_{length}are these pentagons, and distributions of all grids are symmetry. Near these pentagons, the lengths of grids are less than the average length and shown by darker color in all grids. The distribution of NOPT grid has fractal characteristics and is not continuous, too. The D

_{length}in the remaining grids are reduced from the pentagons to triangular centers. The grids with larger D

_{length}are mainly located along the triangular boundaries in OURS grid, and those grids are spread like pentagon or star from these pentagons.

#### 3.2. The Numerical Accuracy Evaluation

_{2}-norm error (as Equation (19)), L

_{∞}-norm error (as Equation (20)) of discretization have been calculated (Table 11 and Table 12 and Figure 13 and Figure 14).

^{ana}and f

^{num}are the analytical and numerical solutions of Laplacian operator, respectively.

_{2}-norm errors and L

_{∞}-norm errors of all grids have been reduced as resolution increase, ones of OURS grid and HR grid are smaller than the others. The L

_{∞}-norm error (meaning the maximal error) of OURS grid has been reduced from 9.15 × 10

^{−2}(NOPT grid), 2.57 × 10

^{−2}(SPRG grid), 1.66 × 10

^{−2}(HR grid), 3.87 × 10

^{−2}(SCVT and XU grid) to 1.59 × 10

^{−2}at level 8. The L

_{2}-norm error (meaning the RMS error) of OURS grid has been reduced from 4.52 × 10

^{−4}(NOPT grid), 2.03 × 10

^{−4}(SPRG grid), 6.05 × 10

^{−5}(HR grid), 2.33 × 10

^{−4}(SCVT and XU grid) to 5.86 × 10

^{−4}at the same grid resolution, in which the enhancement of average accuracy is nearly 8 times that of NOPT grid and has more than 11.62% compared to the HR grid.

#### 3.3. Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Gassmann, A. A global hexagonal C-grid non-hydrostatic dynamical core (ICON-IAP) designed for energetic consistency. Q. J. R. Meteorol. Soc.
**2013**, 139, 152–175. [Google Scholar] [CrossRef] - Schubert, J.J.; Stevens, B.; Crueger, T. Madden-Julian oscillation as simulated by the MPI Earth System Model: Over the last and into the next millennium. J. Adv. Model. Earth Syst.
**2013**, 5, 71–84. [Google Scholar] [CrossRef] - Zängl, G.; Reinert, D.; Rípodas, P.; Baldauf, M. The ICON (ICOsahedral Non-hydrostatic) modelling framework of DWD and MPI-M: Description of the non-hydrostatic dynamical core. Q. J. R. Meteorol. Soc.
**2015**, 141, 563–579. [Google Scholar] [CrossRef] - Skamarock, W.C.; Klemp, J.B.; Duda, M.G.; Fowler, L.D.; Park, S.-H.; Ringler, T.D. A Multiscale Nonhydrostatic Atmospheric Model Using Centroidal Voronoi Tesselations and C-Grid Staggering. Mon. Weather. Rev.
**2012**, 140, 3090–3105. [Google Scholar] [CrossRef] [Green Version] - Satoh, M. Atmospheric Circulation Dynamics and General Circulation Models; Springer: Berlin/Heidelberg, Germany, 2014. [Google Scholar]
- Dubos, T.; Dubey, S.; Tort, M.; Mittal, R.; Meurdesoif, Y.; Hourdin, F. DYNAMICO-1.0, an icosahedral hydrostatic dynamical core designed for consistency and versatility. Geosci. Model Dev.
**2015**, 8, 3131–3150. [Google Scholar] [CrossRef] [Green Version] - Walko, R.L.; Avissar, R. The Ocean-Land-Atmosphere Model (OLAM). Part I: Shallow-Water Tests. Mon. Weather Rev.
**2008**, 136, 4033–4044. [Google Scholar] [CrossRef] - Walko, R.L.; Avissar, R. The Ocean-Land-Atmosphere Model (OLAM). Part II: Formulation and Tests of the Nonhydrostatic Dynamic Core. Mon. Weather Rev.
**2008**, 136, 4045–4062. [Google Scholar] [CrossRef] - Revokatova, A.; Nikitin, M.; Rivin, G.; Rozinkina, I.; Nikitin, A.; Tatarinovich, E. High-Resolution Simulation of Polar Lows over Norwegian and Barents Seas Using the COSMO-CLM and ICON Models for the 2019–2020 Cold Season. Atmosphere
**2021**, 12, 137. [Google Scholar] [CrossRef] - Hsu, L.-H.; Chen, D.-R.; Chiang, C.-C.; Chu, J.-L.; Yu, Y.-C.; Wu, C.-C. Simulations of the East Asian Winter Monsoon on Subseasonal to Seasonal Time Scales Using the Model for Prediction Across Scales. Atmosphere
**2021**, 12, 865. [Google Scholar] [CrossRef] - Hay, H.C.F.C.; Matsuyama, I. Nonlinear tidal dissipation in the subsurface oceans of Enceladus and other icy satellites. Icarus
**2018**, 319, 68–85. [Google Scholar] [CrossRef] - Suzuki, K.; Nakajima, T.; Satoh, M.; Tomita, H.; Takemura, T.; Nakajima, T.; Stephens, G.L. Global cloud-system-resolving simulation of aerosol effect on warm clouds. Geophys. Res. Lett.
**2008**, 35, 610–616. [Google Scholar] [CrossRef] - Cheng, Y.; Dai, T.; Zhang, H.; Xin, J.; Chen, S.; Shi, G.; Nakajima, T. Comparison and evaluation of the simulated annual aerosol characteristics over China with two global aerosol models. Sci. Total Environ.
**2021**, 763, 143003–143017. [Google Scholar] [CrossRef] [PubMed] - Goto, D.; Kikuchi, M.; Suzuki, K.; Hayasaki, M.; Yoshida, M.; Nagao, T.M.; Choi, M.; Kim, J.; Sugimoto, N.; Shimizu, A.; et al. Aerosol model evaluation using two geostationary satellites over East Asia in May 2016. Atmos. Res.
**2019**, 217, 93–113. [Google Scholar] [CrossRef] - Yamashita, Y.; Takigawa, M.; Goto, D.; Yashiro, H.; Satoh, M.; Kanaya, Y.; Taketani, F.; Miyakawa, T. Effect of Model Resolution on Black Carbon Transport from Siberia to the Arctic Associated with the Well-Developed Low-Pressure Systems in September. J. Meteorol. Soc. Jpn.
**2021**, 99, 287–308. [Google Scholar] [CrossRef] - Cheng, Y.; Dai, T.; Goto, D.; Schutgens, N.A.J.; Shi, G.; Nakajima, T. Investigating the assimilation of CALIPSO global aerosol vertical observations using a four-dimensional ensemble Kalman filter. Atmos. Chem. Phys.
**2019**, 19, 13445–13467. [Google Scholar] [CrossRef] [Green Version] - Korn, P. Formulation of an unstructured grid model for global ocean dynamics. J. Comput. Phys.
**2017**, 339, 525–552. [Google Scholar] [CrossRef] - Peixoto, P.S. Accuracy analysis of mimetic finite volume operators on geodesic grids and a consistent alternative. J. Comput. Phys.
**2016**, 310, 127–160. [Google Scholar] [CrossRef] - Peixoto, P.S.; Barros, S.R.M. Analysis of grid imprinting on geodesic spherical icosahedral grids. J. Comput. Phys.
**2013**, 237, 61–78. [Google Scholar] [CrossRef] - Wang, N.; Lee, J.L. Geometric Properties of the Icosahedral-Hexagonal Grid on the Two-Sphere. Soc. Ind. Appl. Math.
**2011**, 33, 2536–2559. [Google Scholar] [CrossRef] [Green Version] - Weller, H.; Thuburn, J.; Cotter, C.J. Computational modes and grid imprinting on five quasi-uniform spherical c-grids. Mon. Weather Rev.
**2012**, 140, 2734–2755. [Google Scholar] [CrossRef] [Green Version] - Cheong, H.; Kang, H. Eigensolutions of the spherical Laplacian for the cubed-sphere and icosahedral-hexagonal grids. Q. J. R. Meteorol. Soc.
**2015**, 141, 3383–3398. [Google Scholar] [CrossRef] - Miura, H.; Kimoto, M. A comparison of grid quality of optimized spherical hexagonal—pentagonal geodesic grids. Mon. Weather Rev.
**2005**, 133, 2817–2833. [Google Scholar] [CrossRef] - Subich, C.J. Higher-order finite volume differential operators with selective upwinding on the icosahedral spherical grid. J. Comput. Phys.
**2018**, 368, 21–46. [Google Scholar] [CrossRef] - Xu, G. Discrete Laplace-Beltrami Operator on Sphere and Optimal Spherical Triangulations. Int. J. Comput. Geom. Appl.
**2006**, 16, 75–93. [Google Scholar] [CrossRef] - Korn, P.; Linardakis, L. A conservative discretization of the shallow-water equations on triangular grids. J. Comput. Phys.
**2018**, 375, 871–900. [Google Scholar] [CrossRef] - Tomita, H.; Tsugawa, M.; Satoh, M.; Goto, K. Shallow Water Model on a Modified Icosahedral Geodesic Grid by Using Spring Dynamics. J. Comput. Phys.
**2001**, 174, 579–613. [Google Scholar] [CrossRef] - Tomita, H.; Satoh, M.; Goto, K. An Optimization of the Icosahedral Grid Modified by Spring Dynamics. J. Comput. Phys.
**2002**, 183, 307–331. [Google Scholar] [CrossRef] - Iga, S.; Tomita, H. Improved smoothness and homogeneity of icosahedral grids using the spring dynamics method. J. Comput. Phys.
**2014**, 258, 208–226. [Google Scholar] [CrossRef] - Iga, S. An equatorially enhanced grid with smooth resolution distribution generated by a spring dynamics method. J. Comput. Phys.
**2017**, 330, 794–809. [Google Scholar] [CrossRef] - Heikes, R.; Randall, D.A. Numerical Integration of the Shallow-Water Equations on a Twisted Icosahedral Grid. Part I: Basic Design and Results of Tests. Mon. Weather Rev.
**1995**, 123, 1862–1880. [Google Scholar] [CrossRef] - Heikes, R.; Randall, D.A. Numerical Integration of the Shallow-Water Equations on a Twisted Icosahedral Grid. Part II. A Detailed Description of the Grid and an Analysis of Numerical Accuracy. Mon. Weather Rev.
**1995**, 123, 1881–1887. [Google Scholar] [CrossRef] - Heikes, R.P.; Randall, D.A.; Konor, C.S. Optimized icosahedral grids: Performance of finite-difference operators and multigrid solver. Mon. Weather Rev.
**2013**, 141, 4450–4469. [Google Scholar] [CrossRef] - Du, Q.; Gunzburger, M.D.; Ju, L. Constrained centroidal Voronoi tessellations for surfaces. Siam J. Sci. Comput.
**2003**, 24, 1488–1506. [Google Scholar] [CrossRef] - Du, Q.; Gunzburger, M.D.; Ju, L. Voronoi-based finite volume methods, optimal Voronoi meshes, and PDEs on the sphere. Comput. Methods Appl. Mech. Eng.
**2003**, 192, 3933–3957. [Google Scholar] [CrossRef] - Ju, L.; Ringler, T.; Gunzburger, M. Voronoi Tessellations and Their Application to Climate and Global Modeling; Springer: Berlin/Heidelberg, Germany, 2011; Volume 80, pp. 313–342. [Google Scholar]
- Du, Q.; Vance, F.; Max, G. Centroidal Voronoi Tessellations: Applications and Algorithms. SIAM Rev.
**1999**, 41, 637–676. [Google Scholar] [CrossRef] [Green Version] - Miura, H. Application of the Synchronized B Grid Staggering for Solution of the Shallow-Water Equations on the Spherical Icosahedral Grid. Mon. Weather Rev.
**2019**, 147, 2485–2509. [Google Scholar] [CrossRef] - Wang, N.; Bao, J.W.; Lee, J.L.; Moeng, F.; Matsumoto, C. Wavelet Compression Technique for High-Resolution Global Model Data on an Icosahedral Grid. J. Atmos. Ocean. Technol.
**2015**, 32, 1650–1667. [Google Scholar] [CrossRef] - Jubair, M.; Alim, U.; Röber, N.; Clyne, J.; Mahdavi-Amiri, A. Icosahedral Maps for a Multiresolution Representation of Earth Data. In Proceedings of the VMV’16 Proceedings of the Conference on Vision, Modeling and Visualization, Bayreuth, Germany, 10–12 October 2016. [Google Scholar]
- Gu, X.; Luo, F.; Sun, J. Variational principles for Minkowski type problems, discrete optimal transport, and discrete Monge-Ampere equations. Math. Methods Solid State Superfluid Theory
**2013**, 20, 383–398. [Google Scholar] [CrossRef] [Green Version] - Cui, L.; Qi, X.; Wen, C.; Lei, N.; Li, X.; Zhang, M.; Gu, X. Spherical optimal transportation. Comput. -Aided Des.
**2019**, 115, 181–193. [Google Scholar] [CrossRef] - Tu, Y.; Wen, C.; Wen, Z.; Wu, J.F. Isometry Invariant Shape Descriptors for Abnormality Detection on Brain Surfaces Affected by Alzheimer’s Disease. In Proceedings of the 2018 40th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC), Honolulu, HI, USA, 17–21 July 2018. [Google Scholar]
- Giri, A.; Choi, G.; Kumar, L. Open and closed anatomical surface description via hemispherical area-preserving map. Signal Process.
**2020**, 180, 107867–107880. [Google Scholar] [CrossRef] - Su, Z.; Zeng, W.; Wang, Y.; Lu, L.Z.; Gu, X.F. Shape Classification Using Wasserstein Distance for Brain Morphometry Analysis. In Information Processing in Medical Imaging (IPMI); Springer: Berlin/Heidelberg, Germany, 2015. [Google Scholar]
- Su, Z.; Wang, Y.; Shi, R. Optimal mass transport for shape matching and comparison. IEEE Trans. Pattern Anal. Mach. Intell.
**2015**, 37, 2246–2259. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Lei, N.; Su, K.; Cui, L.; Yau, S.-T.; Gu, X.D. A Geometric View of Optimal Transportation and Generative Model. Comput. Aided Geom. Des.
**2019**, 68, 1–21. [Google Scholar] [CrossRef]

**Figure 8.**Spherical distributions of D

_{area}of different grids at level 9. (

**a**) NOPT grid. (

**b**) SPRG grid. (

**c**) HR grid. (

**d**) SCVT grid. (

**e**) XU grid. (

**f**) OURS grid.

**Figure 11.**Curves of cumulative proportion with |D

_{length}| < 1.00% for different grids at level 9.

**Figure 12.**Spherical distributions of D

_{length}at level 9. (

**a**) NOPT grid. (

**b**) SPRG grid. (

**c**) HR grid. (

**d**) SCVT grid. (

**e**) XU grid. (

**f**) OURS grid.

**Figure 13.**L

_{2}-norm error of Laplacian operator. (

**a**) L

_{2}-norm error of all grids (

**b**) L

_{2}-norm error of OURS and HR grid from level 5 to 8.

**Figure 14.**L

_{∞}-norm error of Laplacian operator. (

**a**) L

_{∞}-norm error of all grids (

**b**) L

_{∞}-norm error of OURS and HR grid from level 5 to 8.

**Figure 15.**Spherical distributions of Laplacian operator error of different grids. (

**a**) NOPT grid. (

**b**) SPRG grid. (

**c**) HR grid. (

**d**) SCVT grid. (

**e**) XU grid. (

**f**) OURS grid.

Level | nv | A_{avg} | NOPT | OURS | ||
---|---|---|---|---|---|---|

A_{min} | A_{max} | A_{min} | A_{max} | |||

0 | 12 | 42, 505, 466.06 | 42, 505, 466.06 | 42, 505, 466.06 | 42, 505, 466.06 | 42, 505, 466.06 |

1 | 42 | 12, 144, 418.88 | 11, 115, 261.91 | 12, 556, 071.10 | 11, 115, 254.48 | 12, 556, 081.85 |

2 | 162 | 3, 148, 553.04 | 2, 812, 532.18 | 3, 339, 347.67 | 2, 885, 170.41 | 3, 216, 806.64 |

3 | 642 | 794, 494.69 | 705, 296.93 | 923, 852.78 | 735, 897.42 | 799, 720.75 |

4 | 2, 562 | 199, 088.83 | 176, 460.40 | 237, 913.42 | 185, 750.43 | 199, 596.50 |

5 | 10, 242 | 49, 801.37 | 44, 123.63 | 59, 942.43 | 46, 638.43 | 49, 880.62 |

6 | 40, 962 | 12, 452.17 | 11, 031.44 | 15, 015.28 | 11, 700.00 | 12, 500.00 |

7 | 163, 842 | 3, 113.16 | 2, 757.89 | 3, 755.66 | 2, 919.71 | 3, 114.73 |

8 | 655, 362 | 778.30 | 689.48 | 939.35 | 729.62 | 778.67 |

9 | 2, 621, 442 | 194.57 | 172.37 | 234.84 | 182.34 | 194.60 |

Level | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |
---|---|---|---|---|---|---|---|---|

${\tilde{A}}_{\mathrm{avg}}$ | 1.047 | 1.047 | 1.047 | 1.047 | 1.047 | 1.047 | 1.047 | |

NOPT | ${\tilde{A}}_{\mathrm{min}}$ | 0.930 | 0.928 | 0.930 | 0.928 | 0.928 | 0.928 | 0.928 |

${\tilde{A}}_{\mathrm{max}}$ | 1.218 | 1.251 | 1.260 | 1.263 | 1.263 | 1.264 | 1.264 | |

r_{A} | 0.764 | 0.742 | 0.740 | 0.735 | 0.735 | 0.734 | 0.734 | |

SPRG | ${\tilde{A}}_{\mathrm{min}}$ | 0.879 | 0.848 | 0.820 | 0.803 | 0.791 | 0.781 | 0.776 |

${\tilde{A}}_{\mathrm{max}}$ | 1.080 | 1.079 | 1.080 | 1.081 | 1.083 | 1.087 | 1.092 | |

r_{A} | 0.814 | 0.786 | 0.760 | 0.743 | 0.730 | 0.718 | 0.711 | |

HR | ${\tilde{A}}_{\mathrm{min}}$ | 0.998 | 1.008 | 1.012 | 1.014 | 1.013 | 1.016 | 1.016 |

${\tilde{A}}_{\mathrm{max}}$ | 1.062 | 1.066 | 1.068 | 1.068 | 1.069 | 1.070 | 1.070 | |

r_{A} | 0.940 | 0.946 | 0.950 | 0.949 | 0.948 | 0.950 | 0.950 | |

SCVT | ${\tilde{A}}_{\mathrm{min}}$ | 0.861 | 0.812 | 0.760 | 0.719 | 0.677 | 0.639 | 0.599 |

${\tilde{A}}_{\mathrm{max}}$ | 1.082 | 1.081 | 1.080 | 1.081 | 1.081 | 1.081 | 1.081 | |

r_{A} | 0.796 | 0.751 | 0.710 | 0.665 | 0.626 | 0.591 | 0.554 | |

XU | ${\tilde{A}}_{\mathrm{min}}$ | 0.861 | 0.812 | 0.760 | 0.719 | 0.676 | 0.638 | 0.600 |

${\tilde{A}}_{\mathrm{max}}$ | 1.082 | 1.081 | 1.080 | 1.081 | 1.082 | 1.082 | 1.082 | |

r_{A} | 0.796 | 0.751 | 0.710 | 0.665 | 0.625 | 0.590 | 0.555 | |

OURS | ${\tilde{A}}_{\mathrm{min}}$ | 0.970 | 0.977 | 0.981 | 0.984 | 0.982 | 0.982 | 0.981 |

${\tilde{A}}_{\mathrm{max}}$ | 1.054 | 1.050 | 1.049 | 1.051 | 1.048 | 1.048 | 1.047 | |

r_{A} | 0.920 | 0.930 | 0.940 | 0.936 | 0.937 | 0.937 | 0.937 |

Interval (%) | NOPT | SPRG | HR | SCVT | XU | OURS |
---|---|---|---|---|---|---|

[−100, −38.0] | 0 | 0 | 0 | 0 | 0 | 0 |

(−38.0, −34.0) | 0 | 0 | 0 | 162 | 162 | 0 |

(−34.0, −30.0) | 0 | 0 | 0 | 0 | 0 | 0 |

(−30.0, −26.0) | 0 | 0 | 0 | 960 | 960 | 0 |

(−26.0, −22.0) | 0 | 162 | 0 | 1, 920 | 1, 920 | 0 |

(−22.0, −18.0) | 0 | 960 | 0 | 2, 880 | 3, 840 | 0 |

(−18.0, −14.0) | 0 | 4, 800 | 0 | 14, 400 | 13, 440 | 0 |

(−14.0, −10.0) | 192 | 34, 560 | 0 | 42, 240 | 42, 240 | 0 |

(−10.0, −6.0) | 436, 770 | 180, 480 | 0 | 112, 320 | 115, 200 | 12 |

(−6.0, −2.0) | 1, 132, 800 | 630, 720 | 22, 786 | 314, 880 | 317, 760 | 0 |

(−2.0, 2.0) | 366, 240 | 887, 520 | 2598, 480 | 1, 403, 040 | 1, 341, 584 | 2, 621, 430 |

(2.0, 6.0) | 60, 480 | 647, 040 | 176 | 728, 640 | 784, 336 | 0 |

(6.0, 10.0) | 0 | 235, 200 | 0 | 0 | 0 | 0 |

(10.0, 14.0) | 380, 160 | 0 | 0 | 0 | 0 | 0 |

(14.0, 18.0) | 96, 000 | 0 | 0 | 0 | 0 | 0 |

(18.0, 22.0) | 148, 800 | 0 | 0 | 0 | 0 | 0 |

(22.0, 100.0) | 0 | 0 | 0 | 0 | 0 | 0 |

total | 2, 621, 442 | 2, 621, 442 | 2, 621, 442 | 2, 621, 442 | 2, 621, 442 | 2, 621, 442 |

Interval (%) | N | p (%) |
---|---|---|

[−100, −0.18) | 72 | 0.00 |

[−0.18, −0.14) | 60 | 0.00 |

[−0.14, −0.10) | 17, 695 | 0.68 |

[−0.10, −0.06) | 44, 733 | 1.71 |

[−0.06, −0.02) | 495, 834 | 18.91 |

[−0.02, 0.02) | 1, 342, 129 | 51.20 |

(−0.02, 0.60) | 720, 919 | 27.50 |

(0.06, 100] | 0 | 0 |

total | 2, 621, 442 | 100.00 |

Interval (%) | NOPT | SPRG | HR | SCVT | XU | OURS |
---|---|---|---|---|---|---|

[0, 0.004) | 0 | 0 | 1.07 | 0 | 0.26 | 12.57 |

[0.004, 0.012) | 0.02 | 0.07 | 3.27 | 0.07 | 0.59 | 33.35 |

[0.012, 0.02) | 0.02 | 0.22 | 5.87 | 0.07 | 0.73 | 51.20 |

[0.02, 0.028) | 0.02 | 0.22 | 8.63 | 0.22 | 0.93 | 66.98 |

[0.028, 0.036) | 0.02 | 0.51 | 11.06 | 0.22 | 1.25 | 81.04 |

[0.036, 0.044) | 0.02 | 0.62 | 13.24 | 0.37 | 1.25 | 92.56 |

[0.044, 0.052) | 0.02 | 0.77 | 15.10 | 0.66 | 1.39 | 96.17 |

[0.052, 0.06) | 0.02 | 0.99 | 16.73 | 0.92 | 1.39 | 97.61 |

[0.06, 0.068) | 0.02 | 1.65 | 22.95 | 2.38 | 1.76 | 99.32 |

[0.068, 0.076) | 0.02 | 2.23 | 28.11 | 3.04 | 2.16 | 99.99 |

[0.076, 0.3) | 0.02 | 4.69 | 43.54 | 6.23 | 5.35 | 99.99 |

[0.3, 0.5) | 0.02 | 7.87 | 56.64 | 10.25 | 9.87 | 99.99 |

[0.5, 0.7) | 0.02 | 11.17 | 66.66 | 14.39 | 14.25 | 99.99 |

[0.7, 0.9) | 0.02 | 14.39 | 74.64 | 18.82 | 18.53 | 99.99 |

[0.9, 2) | 13.97 | 33.86 | 99.12 | 53.52 | 51.18 | 99.99 |

[6, 26) | 59.49 | 82.6 | 100 | 93.33 | 93.32 | 100 |

[26, 100] | 100 | 100 | 100 | 100 | 100 | 100 |

Level | NOPT | OURS | ||||
---|---|---|---|---|---|---|

d_{avg} | d_{min} | d_{max} | d_{avg} | d_{min} | d_{max} | |

0 | 7529.85 | 7529.85 | 7529.85 | 7529.85 | 7529.85 | 7529.85 |

1 | 3764.92 | 3526.83 | 4003.02 | 3764.92 | 3526.83 | 4003.02 |

2 | 1914.33 | 1763.41 | 2079.28 | 1915.30 | 1739.23 | 2087.62 |

3 | 961.22 | 881.71 | 1050.16 | 961.85 | 866.34 | 1059.12 |

4 | 481.12 | 440.85 | 526.42 | 481.47 | 432.15 | 532.56 |

5 | 240.62 | 220.43 | 263.38 | 240.81 | 215.84 | 267.18 |

6 | 120.32 | 110.21 | 131.71 | 120.42 | 107.83 | 134.40 |

7 | 60.16 | 55.11 | 65.86 | 60.20 | 54.13 | 67.12 |

8 | 30.08 | 27.55 | 32.93 | 30.10 | 27.06 | 33.56 |

9 | 15.04 | 13.78 | 16.47 | 15.05 | 13.53 | 16.78 |

Level | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |
---|---|---|---|---|---|---|---|---|

${\tilde{d}}_{\mathrm{avg}}$ | 1.182 | 1.182 | 1.182 | 1.182 | 1.182 | 1.182 | 1.182 | |

NOPT | ${\tilde{d}}_{\mathrm{min}}$ | 1.107 | 1.107 | 1.107 | 1.107 | 1.107 | 1.107 | 1.107 |

${\tilde{d}}_{\mathrm{max}}$ | 1.319 | 1.322 | 1.323 | 1.323 | 1.323 | 1.323 | 1.323 | |

r_{d} | 0.839 | 0.837 | 0.837 | 0.837 | 0.837 | 0.837 | 0.837 | |

SPRG | ${\tilde{d}}_{\mathrm{min}}$ | 1.077 | 1.059 | 1.043 | 1.017 | 1.017 | 0.999 | 0.983 |

${\tilde{d}}_{\mathrm{max}}$ | 1.276 | 1.279 | 1.279 | 1.284 | 1.285 | 1.279 | 1.259 | |

r_{d} | 0.844 | 0.828 | 0.815 | 0.802 | 0.791 | 0.781 | 0.781 | |

HR | ${\tilde{d}}_{\mathrm{min}}$ | 1.076 | 1.074 | 1.074 | 1.074 | 1.074 | 1.074 | 1.075 |

${\tilde{d}}_{\mathrm{max}}$ | 1.351 | 1.361 | 1.364 | 1.364 | 1.365 | 1.364 | 1.382 | |

r_{d} | 0.796 | 0.789 | 0.787 | 0.787 | 0.787 | 0.787 | 0.778 | |

SCVT | ${\tilde{d}}_{\mathrm{min}}$ | 1.065 | 1.035 | 1.005 | 0.975 | 0.946 | 0.917 | 0.890 |

${\tilde{d}}_{\mathrm{max}}$ | 1.275 | 1.277 | 1.277 | 1.277 | 1.277 | 1.277 | 1.280 | |

r_{d} | 0.835 | 0.810 | 0.787 | 0.763 | 0.741 | 0.718 | 0.695 | |

XU | ${\tilde{d}}_{\mathrm{min}}$ | 1.065 | 1.035 | 1.005 | 0.975 | 0.945 | 0.916 | 0.888 |

${\tilde{d}}_{\mathrm{max}}$ | 1.275 | 1.277 | 1.277 | 1.277 | 1.277 | 1.277 | 1.276 | |

r_{d} | 0.835 | 0.810 | 0.787 | 0.763 | 0.740 | 0.717 | 0.696 | |

OURS | ${\tilde{d}}_{\mathrm{min}}$ | 1.088 | 1.085 | 1.084 | 1.083 | 1.087 | 1.091 | 1.091 |

${\tilde{d}}_{\mathrm{max}}$ | 1.330 | 1.337 | 1.342 | 1.350 | 1.349 | 1.354 | 1.353 | |

r_{d} | 0.818 | 0.812 | 0.808 | 0.802 | 0.806 | 0.806 | 0.806 |

NOPT | SPRG | HR | SCVT | XU | OURS | |
---|---|---|---|---|---|---|

[−100, −15] | 0 | 0 | 0 | 0 | 0 | 0 |

(−15, −13] | 0 | 0 | 0 | 60 | 60 | 0 |

(−13, −11] | 0 | 0 | 0 | 120 | 120 | 0 |

(−11, −9] | 0 | 0 | 0 | 360 | 420 | 0 |

(−9, −7] | 0 | 5, 730 | 0 | 720 | 1, 020 | 0 |

(−7, −5] | 0 | 28, 800 | 0 | 2, 640 | 2, 280 | 0 |

(−5, −3] | 0 | 151, 680 | 0 | 100, 800 | 106, 030 | 0 |

(−3, −1] | 1, 488, 960 | 620, 160 | 0 | 301, 632 | 301, 632 | 0 |

(−1, 1) | 446, 880 | 966, 240 | 2, 447, 056 | 1, 536, 160 | 1, 471, 680 | 2, 621, 430 |

[1, 3) | 60, 480 | 679, 680 | 174, 374 | 678, 950 | 738, 200 | 0 |

[3, 5) | 0 | 168, 960 | 0 | 0 | 0 | 0 |

[5, 7) | 446, 400 | 192 | 0 | 0 | 0 | 0 |

[7, 9) | 144, 960 | 0 | 0 | 0 | 0 | 0 |

[9, 11) | 33, 750 | 0 | 0 | 0 | 0 | 0 |

[11, 13) | 0 | 0 | 0 | 0 | 0 | 0 |

[13, 15) | 0 | 0 | 0 | 0 | 0 | 0 |

[15, 17) | 12 | 0 | 0 | 0 | 0 | 0 |

[17, 19) | 0 | 0 | 0 | 0 | 0 | 12 |

[19, 21) | 0 | 0 | 12 | 0 | 0 | 0 |

[21, 100] | 0 | 0 | 0 | 0 | 0 | 0 |

total | 2, 621, 442 | 2, 621, 442 | 2, 621, 442 | 2, 621, 442 | 2, 621, 442 | 2, 621, 442 |

Interval (%) | HR | OURS | ||
---|---|---|---|---|

N | p (%) | N | p (%) | |

[−100, −1.1] | 16 | 0.00 | 0 | 0 |

(−1.1, −0.9] | 46, 592 | 1.78 | 0 | 0 |

(−0.9, −0.7] | 137, 776 | 5.26 | 0 | 0 |

(−0.7, −0.5] | 835, 680 | 31.88 | 0 | 0 |

(−0.5, −0.3] | 468, 576 | 17.87 | 325, 328 | 12.41 |

(−0.3, −0.1] | 288, 704 | 11.01 | 659, 536 | 25.16 |

(−0.1,0.1) | 211, 232 | 8.06 | 676, 512 | 25.81 |

[0.1, 0.3) | 166, 432 | 6.35 | 705, 728 | 26.92 |

[0.3, 0.5) | 134, 192 | 5.12 | 234, 528 | 8.95 |

[0.5, 0.7) | 110, 118 | 4.20 | 19, 798 | 0.76 |

[0.7, 0.9) | 91, 328 | 3.48 | 0 | 0 |

[0.9, 1.0) | 73, 328 | 2.80 | 0 | 0 |

[1.1, 1.3) | 50, 640 | 1.93 | 0 | 0 |

[1.3, 1.5) | 6, 816 | 0.26 | 0 | 0 |

[1.5, 1.7) | 0 | 0 | 0 | 0 |

[1.7, 2.3) | 0 | 0 | 12 | 0 |

[2.3, 100] | 12 | 0 | 0 | 0 |

total | 2, 621, 442 | 100 | 2, 621, 442 | 100 |

Interval (%) | NOPT | SPRG | HR | SCVT | XU | OURS |
---|---|---|---|---|---|---|

[0, 0.02) | 0 | 0.51 | 2.20 | 0.73 | 1.32 | 5.02 |

[0.02, 0.10) | 0.02 | 3.41 | 11.01 | 4.06 | 4.32 | 25.81 |

[0.10, 0.18) | 0.79 | 5.93 | 20.37 | 7.54 | 7.47 | 45.63 |

[0.18, 0.26) | 1.78 | 9.01 | 30.87 | 10.77 | 10.33 | 66.58 |

[0.26, 0.34) | 3.86 | 11.65 | 44.40 | 13.84 | 13.40 | 86.02 |

[0.34, 0.42) | 3.86 | 14.10 | 65.49 | 17.80 | 16.96 | 97.28 |

[0.42, 0.50) | 3.86 | 17.36 | 75.17 | 21.64 | 20.47 | 99.24 |

[0.50, 0.58) | 5.51 | 20.43 | 80.03 | 25.20 | 24.45 | 99.96 |

[0.58, 0.66) | 5.58 | 23.80 | 83.90 | 29.66 | 28.66 | 99.99 |

[0.66, 0.74) | 7.82 | 26.97 | 87.00 | 34.42 | 32.75 | 99.99 |

[0.74, 0.82) | 12.54 | 30.34 | 89.58 | 40.17 | 37.39 | 99.99 |

[0.82, 2.00) | 22.72 | 66.59 | 99.99 | 90.00 | 89.23 | 99.99 |

[2.00, 5.00) | 76.15 | 98.67 | 99.99 | 97.62 | 97.62 | 99.99 |

[5.00, 7.00) | 92.30 | 99.63 | 99.99 | 98.97 | 98.83 | 99.99 |

[7.00, 100] | 100 | 100 | 100 | 100 | 100 | 100 |

2 | 3 | 4 | 5 | 6 | 7 | 8 | |
---|---|---|---|---|---|---|---|

NOPT | 1.31 × 10^{−1} | 3.78 × 10^{−2} | 1.20 × 10^{−2} | 4.49 × 10^{−3} | 1.96 × 10^{−3} | 9.26 × 10^{−4} | 4.52 × 10^{−4} |

SPRG | 1.33 × 10^{−1} | 3.56 × 10^{−2} | 9.47 × 10^{−3} | 2.74 × 10^{−3} | 9.30 × 10^{−4} | 3.84 × 10^{−4} | 1.47 × 10^{−4} |

HR | 1.41 × 10^{−1} | 3.79 × 10^{−2} | 9.85 × 10^{−3} | 2.58 × 10^{−3} | 7.23 × 10^{−4} | 2.05 × 10^{−4} | 6.05 × 10^{−5} |

SCVT | 1.33 × 10^{−1} | 3.55 × 10^{−2} | 9.47 × 10^{−3} | 2.81 × 10^{−3} | 1.04 × 10^{−3} | 4.61 × 10^{−4} | 2.23 × 10^{−4} |

XU | 1.33 × 10^{−1} | 3.55 × 10^{−2} | 9.47 × 10^{−3} | 2.81 × 10^{−3} | 1.04 × 10^{−3} | 4.61 × 10^{−4} | 2.23 × 10^{−4} |

OURS | 1.38 × 10^{−1} | 3.75 × 10^{−2} | 9.78 × 10^{−3} | 2.57 × 10^{−3} | 6.98 × 10^{−4} | 1.99 × 10^{−4} | 5.42 × 10^{−5} |

2 | 3 | 4 | 5 | 6 | 7 | 8 | |
---|---|---|---|---|---|---|---|

NOPT | 3.52 × 10^{−1} | 1.28 × 10^{−1} | 8.08 × 10^{−2} | 8.89 × 10^{−2} | 9.10 × 10^{−2} | 9.10 × 10^{−2} | 9.15 × 10^{−2} |

SPRG | 2.81 × 10^{−1} | 7.62 × 10^{−2} | 3.56 × 10^{−2} | 3.23 × 10^{−2} | 3.00 × 10^{−2} | 2.70 × 10^{−2} | 2.57 × 10^{−2} |

HR | 2.84 × 10^{−1} | 8.67 × 10^{−2} | 3.23 × 10^{−2} | 1.41 × 10^{−2} | 6.75 × 10^{−3} | 3.31 × 10^{−3} | 1.66 × 10^{−3} |

SCVT | 2.81 × 10^{−1} | 7.61 × 10^{−2} | 3.92 × 10^{−2} | 3.88 × 10^{−2} | 3.87 × 10^{−2} | 3.87 × 10^{−2} | 3.87 × 10^{−2} |

XU | 2.81 × 10^{−1} | 7.61 × 10^{−2} | 3.92 × 10^{−2} | 3.88 × 10^{−2} | 3.87 × 10^{−2} | 3.87 × 10^{−2} | 3.87 × 10^{−2} |

OURS | 2.85 × 10^{−1} | 7.81 × 10^{−2} | 2.93 × 10^{−2} | 1.35 × 10^{−2} | 6.59 × 10^{−3} | 3.27 × 10^{−3} | 1.59 × 10^{−3} |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Luo, F.; Zhao, X.; Sun, W.; Li, Y.; Duan, Y.
An Overall Uniformity Optimization Method of the Spherical Icosahedral Grid Based on the Optimal Transformation Theory. *Atmosphere* **2021**, *12*, 1516.
https://doi.org/10.3390/atmos12111516

**AMA Style**

Luo F, Zhao X, Sun W, Li Y, Duan Y.
An Overall Uniformity Optimization Method of the Spherical Icosahedral Grid Based on the Optimal Transformation Theory. *Atmosphere*. 2021; 12(11):1516.
https://doi.org/10.3390/atmos12111516

**Chicago/Turabian Style**

Luo, Fuli, Xuesheng Zhao, Wenbin Sun, Yalu Li, and Yuanzheng Duan.
2021. "An Overall Uniformity Optimization Method of the Spherical Icosahedral Grid Based on the Optimal Transformation Theory" *Atmosphere* 12, no. 11: 1516.
https://doi.org/10.3390/atmos12111516