Functional Maps#

Background#

We model a 3D shape \(\mathcal{X}_1\) as a compact two-dimensional manifold embedded in \(\mathbb{R}^3\). The space of square-integrable real-valued functions on the surface, \(\mathcal{L}^2(\mathcal{X}_1)\), is defined as:

\[\mathcal{L}^2(\mathcal{X}_1) := \{ f : \mathcal{X}_1 \rightarrow \mathbb{R} \mid \int_{\mathcal{X}_1} |f(x)|^2 dx < \infty \}\]

This is a Hilbert space with inner product:

\[\langle f, g \rangle_{\mathcal{L}^2(\mathcal{X}_1)} = \int_{\mathcal{X}_1} f(x) g(x) dx\]

In practice, shapes are discretized as point clouds or meshes, with \(n_1\) points \(\{x_i\}_{i=1}^{n_1}\). Functions are represented as vectors \(f \in \mathbb{R}^{n_1}\) with entries \(f_i = f(x_i)\). The mass matrix \(M_1 \in \mathbb{R}^{n_1 \times n_1}\) (diagonal, with entries \(m_i\)) allows discretization of the inner product:

\[\langle f, g \rangle_{\mathcal{L}^2(\mathcal{X}_1)} \approx f^\top M_1 g\]

The Laplace-Beltrami operator \(\Delta_1\) is discretized as a matrix. Its eigendecomposition yields eigenvalues \(\{\lambda_1^i\}\) and eigenfunctions \(\{\phi_1^i\}\) forming an orthonormal basis (LBO basis). For efficiency, we use a truncated basis \(\Phi_1^k = [\phi_1^1, \ldots, \phi_1^k] \in \mathbb{R}^{n_1 \times k}\).

Functional Maps#

Given two shapes \(\mathcal{X}_1\) and \(\mathcal{X}_2\), a pointwise correspondence \(T_{12}\) induces a pull-back operator (the functional map):

\[T^F_{21} : \mathcal{L}^2(\mathcal{X}_2) \rightarrow \mathcal{L}^2(\mathcal{X}_1), \quad T^F_{21}(g) = g \circ T_{12}\]

In a chosen basis, this operator is represented as a matrix \(C_{21}\) mapping coefficients. If \(\Pi_{12}\) is the pointwise correspondence matrix, then:

\[C_{21} = \Phi_1^\dagger \Pi_{12} \Phi_2\]

where \(\dagger\) denotes the Moore–Penrose pseudoinverse.

Pointwise Map Recovery#

To recover pointwise correspondences from functional maps, we use the nearest search in the embedding space.

\[T_{12} = \mathrm{NS}(\Phi_1, \Phi_2 C_{21}^\top)\]

Here, \(\mathrm{NS}\) denotes nearest search in the embedding space.

Truncated Basis and Approximations#

Using a truncated basis (\(k \ll n_1, n_2\)) enables efficient computation but introduces approximations in delta function representation and pointwise recovery. The row \(\Phi_1^k(x)\) is the spectral embedding of \(x\) in dimension \(k\). Linear operators cannot perfectly align these embeddings without additional priors.

Key Papers#

  1. Functional Maps: A Flexible Representation of Maps Between Shapes

  2. ZoomOut: Spectral Upsampling for Efficient Shape Correspondence

  3. Deep Geometric Functional Maps: Robust Feature Learning for Shape Correspondence

  4. Fast Sinkhorn Filters: Using Matrix Scaling for Non-Rigid Shape Correspondence

  5. Elastic Functional Maps

For more detailed examples and tutorials, see the ../tutorials/ section.