TY - JOUR
T1 - Ellipsoidal conformal and area-/volume-preserving parameterizations and associated optimal mass transportations
AU - Lin, Jia Wei
AU - Li, Tiexiang
AU - Lin, Wen Wei
AU - Huang, Tsung Ming
N1 - Publisher Copyright:
© 2023, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.
PY - 2023/8
Y1 - 2023/8
N2 - In this paper, we propose the conformal energy minimization (CEM), stretching energy minimization (SEM) and volume stretching energy minimization (VSEM) algorithms by using the Jacobi conformal projection to compute the ellipsoidal conformal, area- and volume-preserving parameterizations from the boundary of a simply connected closed 3-manifold M to the surface of an ellipsoid E3(a, b, c) and from the 3-manifold M to an ellipsoid E3(a, b, c) , respectively. At each correction step of SEM and VSEM, the coefficients of the Laplacian matrices are modified by imposing local area/volume stretch factors in the denominators. Furthermore, to find the area-preserving optimal mass transportation (OMT) map between ∂M and ∂E3(a, b, c) and the volume-preserving OMT map between M and E3(a, b, c) , in light of SEM and VSEM, we propose the ellipsoidal area-preserving OMT (AOMT) and volume-preserving OMT (VOMT) algorithms, which are combined with the project gradient method, while preserving the local area/volume ratios and minimizing the transport costs and distortions. The numerical results demonstrate that the transformation of a 3D irregular image into an appropriate ellipsoid or cuboid incurs a smaller transport cost and reduces the difference in the conversion compared with that into a ball or cube.
AB - In this paper, we propose the conformal energy minimization (CEM), stretching energy minimization (SEM) and volume stretching energy minimization (VSEM) algorithms by using the Jacobi conformal projection to compute the ellipsoidal conformal, area- and volume-preserving parameterizations from the boundary of a simply connected closed 3-manifold M to the surface of an ellipsoid E3(a, b, c) and from the 3-manifold M to an ellipsoid E3(a, b, c) , respectively. At each correction step of SEM and VSEM, the coefficients of the Laplacian matrices are modified by imposing local area/volume stretch factors in the denominators. Furthermore, to find the area-preserving optimal mass transportation (OMT) map between ∂M and ∂E3(a, b, c) and the volume-preserving OMT map between M and E3(a, b, c) , in light of SEM and VSEM, we propose the ellipsoidal area-preserving OMT (AOMT) and volume-preserving OMT (VOMT) algorithms, which are combined with the project gradient method, while preserving the local area/volume ratios and minimizing the transport costs and distortions. The numerical results demonstrate that the transformation of a 3D irregular image into an appropriate ellipsoid or cuboid incurs a smaller transport cost and reduces the difference in the conversion compared with that into a ball or cube.
KW - Area- and volume-preserving parameterizations
KW - Discrete optimal mass transportation
KW - Ellipsoidal conformal
KW - Jacobi conformal projection
KW - Simplicial 3-complex with a genus-zero boundary
UR - http://www.scopus.com/inward/record.url?scp=85163824889&partnerID=8YFLogxK
U2 - 10.1007/s10444-023-10048-w
DO - 10.1007/s10444-023-10048-w
M3 - Article
AN - SCOPUS:85163824889
SN - 1019-7168
VL - 49
JO - Advances in Computational Mathematics
JF - Advances in Computational Mathematics
IS - 4
M1 - 50
ER -