Introduction

A harmonic map $H:M \rightarrow D$ can be viewed as an embedding from a manifold $\textbf{M}$ with disk topology to a planar domain $\textbf{D}$. A harmonic map is a critical point for the harmonic energy functional,

$$E(H)=\int_M \vert\vert\nabla H\vert\vert^2 d\mu_M,$$

and can be calculated by minimizing $E(H)$. The norm of the differential $\vert\vert\nabla H\vert\vert$ is given by the metric on $M$ and $D$, and $d\mu M$ is the measure on $M$ [7,5,1,2]. Since our source manifold $M$ is in the form of a discrete triangular mesh, we approximate the harmonic energy as [1,11,3],

$$E(H)=\sum_{[v_0,v_1]} k_{[v_0,v_1]} \vert\vert H(v_0)-H(v_1)\vert\vert^2,$$ (1)

where $[v_0,v_1]$ is an edge connecting two neighboring vertices $v_0$ and $v_1$, and $k_{[v_0,v_1]}$ is defined as

$$\frac{1}{2}(\frac{(v_0-v_2)\cdot(v_1-v_2)}{\vert\vert(v_0-v... ...dot(v_1-v_3)}{\vert\vert(v_0-v_3)\times(v_1-v_3)\vert\vert}),$$

where $\{v_0,v_1,v_2\}$ and $\{v_0,v_1,v_3\}$ are two conjoined triangular faces.

By minimizing the harmonic energy, a harmonic map can be computed using the Euler-Lagrange differential equation for the energy functional, i.e.

$$\Delta E = 0,$$ (2)

where $\Delta$ is the Laplace-Beltrami operator [7,5,1,2]. This will lead to solving a sparse linear least-square system for the mapping $H$ of each vertex $v_i$ [1,11,3]. If the boundary condition is given,

$$H\lfloor_{\partial M} : \partial M \rightarrow \partial D,$$ (3)

then the solution exists and is unique.

The theory of harmonic maps is based on conformal geometry theory [3,8]; the harmonic map between two topological disks is a diffeomorphism with minimal stretching energy and bounded angle distortion. Harmonic maps are invariant for the same source surface with different poses, thus making it possible to account for global rigid motion. Harmonic maps are highly continuous, stable and robust to noise. They depend on the geometry in a continuous manner.

Harmonic maps have many merits which are valuable for computer vision applications:

• First, the harmonic map is computed through global optimization, and takes into account the overall surface topology. Thus it does not suffer from local minima, folding, clustering, which are common problems due to local optimization.

• Second, the harmonic map is not sensitive to the resolution of the surface, and to the noise on the surface. Even if the data for the input surface is noisy, the result won't be affected significantly.

• Third, the harmonic map doesn't require the surface to be smooth. It can be accurately computed even when the surface includes sharp features.

• Forth, since the range of the map of an open surface is a unit disk which is convex, the harmonic map exists, and is a diffeomorphism, namely, the map is one to one and on-to. So it can allow us to establish correspondences on 2D and recover 3D registration from the same mapping.

• Fifth, the harmonic map is determined by the metric, not the embedding. This implies that the harmonic map is invariant for the same surface with different poses. Furthermore, if there is not too much stretching between two surfaces undergoing non-rigid deformations, they will induce similar harmonic maps.

Furthermore, harmonic maps are easy to compute and robust to numerical errors. By using a traditional finite element method [4], they are easy to implement.