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

Abstract

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 $\mathcal {M}$ to the surface of an ellipsoid $\mathcal {E}^3(a,b,c)$ and from the 3-manifold $\mathcal {M}$ to an ellipsoid $\mathcal {E}^3(a,b,c)$, 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 $\partial \mathcal {M}$ and $\partial \mathcal {E}^3(a,b,c)$ and the volume-preserving OMT map between $\mathcal {M}$ and $\mathcal {E}^3(a,b,c)$, 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.

Appendix: Discrete 3-Manifold and Associated Ellipsoid

Let $\mathcal {M}$ be a simplicial 3-complex with a genus-zero boundary. Suppose the volume of $\mathcal {M}$ is one and the volume center lies at the origin. First, we need to determine the size of the ellipsoid to which $\mathcal {M}$ is mapped, i.e., we need to determine three principal semiaxes $a,b,c\,$ of the desired ellipsoid. This is the ellipsoid fitting problem, which is commonly seen in various fields. To this end, in view of probability theory and statistics, we adopt the concept of central moments [17] in statistics. Suppose that the cardinality of $\mathbb {V}(\mathcal {M})$ is N and $v_i=(x_i,y_i,z_i)\in \mathbb {V}(\mathcal {M})$ for $i=1,2,\ldots ,N$. The covariance matrix is

$$\begin{aligned} K = \frac{1}{N} \begin{bmatrix} \sum _{i=1}^N(x_i-\bar{x})(x_i-\bar{x}) &{} \sum _{i=1}^N(x_i-\bar{x})(y_i-\bar{y}) &{} \sum _{i=1}^N(x_i-\bar{x})(z_i-\bar{z}) \\ \sum _{i=1}^N(y_i-\bar{y})(x_i-\bar{x}) &{} \sum _{i=1}^N(y_i-\bar{y})(y_i-\bar{y}) &{} \sum _{i=1}^N(y_i-\bar{y})(z_i-\bar{z}) \\ \sum _{i=1}^N(z_i-\bar{z})(x_i-\bar{x}) &{} \sum _{i=1}^N(z_i-\bar{z})(y_i-\bar{y}) &{} \sum _{i=1}^N(z_i-\bar{z})(z_i-\bar{z}) \end{bmatrix} , \end{aligned}$$

(32)

where $\bar{x}=\sum _{i=1}^N x_i/N, \bar{y}=\sum _{i=1}^N y_i/N$ and $\bar{z}=\sum _{i=1}^N z_i/N$. The matrix K is positive definite because all vertices are not in the same plane. In addition, all eigenvalues of K are positive, with corresponding eigenvectors orthogonal to each other. We develop a method, specified in Algorithm 7, to estimate three semiaxes and the orientation of the ellipsoid using the eigenvalues and the eigenvectors, respectively [20]. Additionally, using this method, we obtain a closed curve $\gamma _A$, which corresponds to the ellipse $\{(x,y,0)\in \mathbb {R}^3~|~\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\}$, and two vertices $v_1$ and $v_2$, which correspond to $(-a,0,0)$ and (a, 0, 0) of $\mathcal {E}^3(a,b,c)$, respectively. These objects are necessary for the ellipsoidal CEM algorithm.

Ellipsoid fitting.

Fig. 8

Illustration of Algorithm 7 by fitting a liver-shaped 3-complex $\mathcal {M}$ to an ellipsoid