Ellipsoidal conformal and area-/volume-preserving parameterizations and associated optimal mass transportations

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.