Optimal mass transport for geometric modeling based on variational principles in convex geometry

Abstract

In geometric modeling, surface parameterization plays an important role for converting triangle meshes to spline surfaces. Parameterization will introduce distortions. Conventional parameterization methods emphasize on angle-preservation, which may induce huge area distortions and cause large spline fitting errors and trigger numerical instabilities.To overcome this difficulty, this work proposes a novel area-preserving parameterization method, which is based on an optimal mass transport theory and convex geometry. Optimal mass transport mapping is measure-preserving and minimizes the transportation cost. According to Brenier’s theorem, for quadratic distance transportation costs, the optimal mass transport map is the gradient of a convex function. The graph of the convex function is a convex polyhedron with prescribed normal and areas. The existence and the uniqueness of such a polyhedron have been proved by the Minkowski-Alexandrov theorem in convex geometry. This work gives an explicit method to construct such a polyhedron based on the variational principle, and formulates the solution to the optimal transport map as the unique optimum of a convex energy. In practice, the energy optimization can be carried out using Newton’s method, and each iteration constructs a power Voronoi diagram dynamically. We tested the proposal algorithms on 3D surfaces scanned from real life. Experimental results demonstrate the efficiency and efficacy of the proposed variational approach for the optimal transport map.

Appendix

We give a brief proof for the main theorem 3. Details can be found in [19]:

The cell voronoi decomposition has a dual Delaunay triangulation. Suppose, without loss of generality, two cells W _i , W _j are adjacent and they share a common edge e _ij . The edge e _ij has a dual Delaunay edge $\bar{e}_{ij}.$ The norm with respect to ρ is defined as

$$|e|_\rho = \int\limits_{e} \rho(x) {\rm d}x,$$

(5)

and |e| is just the traditional Euclidean length.

By direct computation, we can show

$$\frac{\partial w_{i}}{\partial h_{j}} = \frac{\partial w_{j}}{\partial h_{i}} = \frac{|e_{ij}|_\rho}{|\bar{e}_{ij}|}.$$

Proof As shown in Fig. 6, suppose two power diagram cells W _i , W _j are adjacent to edge e _ij  = W _i ∩ W _j . W _i has site p _i and power h _i , and W _j has site p _j and power h _j . Then e _ij can be represented by

$$\|x-p_{i}\|_{2} - h_{i} = \|x-p_{j}\|_{2} - h_{j},$$

which induced

$$2\langle p_{j}-p_{i}, x\rangle = h_{i} - h_{j} + \|p_{j}\|_{2} - \|p_{i}\|_{2}.$$

(6)

If we change h _i by δ h _i , e _ij will shift by $\epsilon$ along p _j  − p _i . Then we get

$$2 \left \langle p_{j}-p_{i}, x + \epsilon\frac{p_{j}-p_{i}}{\|p_{j}-p_{i}\|} \right \rangle = h_{i} + \delta h_{i} - h_{j} + \|p_{j}\|_{2} - \|p_{i}\|_{2}.$$

(7)

By subtracting Eqs. (7) from (6), we can easily get

$$\epsilon = \frac{\delta h_{i}}{2\|p_{j}-p_{i}\|}.$$

Therefore,

$$\delta w_{j} = |e_{ij}|\epsilon + o(\epsilon_{2}) = |e_{ij}|\frac{\delta h_{i}}{2\|p_{j}-p_{i}\|} + o(\epsilon_{2}).$$

And finally,

$$\frac{\delta w_{j}}{\delta h_{i}} = \frac{|e_{ij}|}{2\|p_{j}-p_{i}\|} = \frac{|e_{ij}|}{2|\bar{e}_{ij}|}.$$

The proof for $\frac{\delta w_{i}}{\delta h_{j}} = \frac{|e_{ij}|}{2|\bar{e}_{ij}|}$ is similar.□

Fig. 6

Illustration of the power diagram cell intersections. It shows that when the power of W _i changes, the cell W _i is enlarged along p _j  − p _i

Therefore the differential 1-form ω = ∑ ^k _i = 1 w _i dh _i is a closed 1-form, dω = 0. By Brunn-Minkowski inequality [40], the admissible space

$${{\mathcal{H}}} := \left \{{\bf h}| \forall i, w_{i}({\bf h}) > 0, \sum_{i} h_{i} = 0\right \}$$

is non-empty and convex. Therefore, $E({\bf h}) = \int^{{\bf h}}\omega$ is well defined. The gradient of E is $(w_{1},\ldots, w_{k}),$ the Hessian matrix of E is as follows. The off diagonal element is given by

$$\frac{\partial_{2} E}{\partial h_{i} \partial h_{j}} = \frac{\partial w_{i}}{\partial h_{j}} = \frac{|e_{ij}|_\rho}{|\bar{e}_{ij}|}.$$

(8)

Because ∑ _i w _i (h) = const, therefore the diagonal element is given by

$$\frac{\partial w_{i}}{\partial h_{i}} = -\sum_{j\neq i} \frac{\partial w_{i}}{\partial h_{j}},$$

(9)

the negative Hessian matrix is diagonal dominant. So the energy is concave on ${{\mathcal{H}}}.$ Therefore, the gradient map ∇ E: h → w is injective. This shows the uniqueness.

For a given ${\varvec{\mu}},$ we can define the energy

$$E_\mu({\bf h}):= \sum_{i} \mu_{i} h_{i} - \int\limits^{{\bf h}} \omega$$

(10)

which is positive definite. Furthermore, the desired solution is the unique global minimum. The gradients along the boundary of ${{\mathcal{H}}}$ point to the interior, so the minimum is in the interior. This shows the existence.

Note that, in the above theorem, the domain $\Upomega$ could be infinite, even the entire of ${{\mathbb{R}}^{n}},$ as long as the total volume is finite, such as spherical area-preserving mapping.