# Finsler-Laplace-Beltrami Operators with Application to Shape Analysis

Simon Weber1,2*
<EMAIL_ADDRESS>Thomas Dagès3*
<EMAIL_ADDRESS>Maolin Gao1,2
<EMAIL_ADDRESS>Daniel Cremers1,2
<EMAIL_ADDRESS>
###### Abstract

††* equal contribution.11footnotetext: Technical University of
Munich22footnotetext: Munich Center for Machine Learning33footnotetext:
Technion - Israel Institute of Technology

The Laplace-Beltrami operator (LBO) emerges from studying manifolds equipped
with a Riemannian metric. It is often called the swiss army knife of geometry
processing as it allows to capture intrinsic shape information and gives rise
to heat diffusion, geodesic distances, and a multitude of shape descriptors.
It also plays a central role in geometric deep learning. In this work, we
explore Finsler manifolds as a generalization of Riemannian manifolds. We
revisit the Finsler heat equation and derive a Finsler heat kernel and a
Finsler-Laplace-Beltrami Operator (FLBO): a novel theoretically justified
anisotropic Laplace-Beltrami operator (ALBO). In experimental evaluations we
demonstrate that the proposed FLBO is a valuable alternative to the
traditional Riemannian-based LBO and ALBOs for spatial filtering and shape
correspondence estimation. We hope that the proposed Finsler heat kernel and
the FLBO will inspire further exploration of Finsler geometry in the computer
vision community.

Figure 1: Simplified overview of the derivation of the Finsler-Laplace-
Beltrami operator (FLBO), built from a Finsler metric. It generalizes the
traditional heuristic anisotropic Laplace-Beltrami operators (ALBO) [6, 29]
(top). The FLBO can directly replace the ALBOs in surface processing tasks
like shape correspondence, notably by constructing shape-dependent anisotropic
convolutions in their spectral domain.

##  1 Introduction

###  1.1 Finsler versus Riemann

In his PhD thesis of 1918, Paul Finsler introduced the notion of a
differentiable manifold equipped with a Minkowski norm on each tangent space
[22]. It constitutes a generalization of Riemannian manifolds [39] that Elie
Cartan later referred to as Finsler manifolds [12]. While Riemannian manifolds
are omni-present in computer vision and machine learning, Finsler manifolds
have remained largely unexplored.

Essentially, a Finsler metric is the same as a Riemannian one without the
quadratic assumption. The consequence is that the metric not only depends on
the location, but also on the tangential direction of motion. This implies
that the length of a curve between two points $A$ and $B$ may change whether
we are traversing it from $A$ to $B$ or the other way round, thus geodesic
curves from $A$ to $B$ may differ from those from $B$ to $A$.

Figure 2: Due to non-uniform wind currents, the geodesic curve from $A$ to $B$
differs from the one from $B$ to $A$. While not possible in Riemannian
geometry, such asymmetries are characteristic of Finsler manifolds.

This means that on a Finsler manifold, there is a built-in asymmetric
anisotropy, which contrasts with the symmetric Riemannian case. In the
Riemannian world, moving from an isotropic to a non isotropic metric allows
more flexibility to capture interesting features on the manifolds. For
instance, on a human hand shape, we may want a point on a finger to be close
in some sense to all the points along the ring of the finger more than those
along the length of the finger. Shifting from a Riemannian world to a Finsler
world allows to increase the flexibility of anisotropy by also allowing non-
symmetric distances. For many physical systems the Finsler metric is actually
the natural choice of distance. For instance, a boat on a river or a plane in
the wind might find that a path along the maximum current provides the
shortest travel time whereas the return voyage might require moving away from
the maximum current – see Fig. 2.

###  1.2 Contribution

The aim of this paper is to explore Finsler manifolds, study the Finsler heat
diffusion, introduce a Finsler-Laplace-Beltrami Operator (FLBO), and
demonstrate its applicability to shape analysis and geometric deep learning
via the shape matching problem. In particular:

  * •

We present a theoretical exploration of Finsler manifolds in a concise and
self-contained form that is accessible to the computer vision and machine
learning community.

  * •

We revisit the heat diffusion on Finsler manifolds and derive a tractable
equation.

  * •

We generalize the traditional (anisotropic) Laplace-Beltrami operator to a
Finsler-Laplace-Beltrami operator (FLBO).

  * •

By leveraging the relationship between the diffusivity coefficient and the
Riemannian metric, we propose an easy-to-implement discretization of the FLBO.

  * •

We perform extensive evaluation of the proposed approach for shape
correspondence based on the FLBO. We highlight the benefits of this new
operator in terms of accuracy on both full and partial shape datasets.

  * •

We release our code as open source to facilitate further research:
https://github.com/tum-vision/flbo.

##  2 Related work

The research domains most relevant to our work are geometric deep learning for
shape correspondence and the use of the Finsler metric in computer vision.

### Geometric deep learning for shape correspondence

Finding correspondences between shapes is a fundamental task in computer
vision and a key component of many geometry processing applications, such as
object recognition, shape reconstruction, texture transfer, or shape morphing.
Initially solved with handcrafted approaches [44, 16, 38], the shape
correspondence problem has recently benefited from the success of neural
networks [28, 27, 42, 43, 25, 23] to learn features achieving tremendous
results.

Initially seen in the Euclidean space, shapes are nowadays mostly considered
as embedded manifolds. Intrinsic information, such as geodesic distances, are
products of the heat diffusion paradigm, which has mostly been studied on
manifolds equipped with a Riemannian metric. For such manifolds, heat
diffusion is governed by the Poisson equation, and thus naturally the Laplace-
Beltrami operator (LBO) plays a central role [10]. The LBO is traditionally
isotropic on the manifold, but recent approaches have heuristically injected
anisotropy into it, producing anisotropic LBOs (ALBO) [1]. Coupled with
spatial filtering [29] ALBOs have shown impressive accuracy in shape
correspondence.

We here focus on methods centred around anisotropic ALBOs. Nevertheless, many
other types of approaches exist for full and partial shape correspondence. In
particular, we point out the vast literature based on the functional map
framework [35, 36, 30, 24, 8, 20, 41, 3, 11, 9].

In [31], the well-known heat kernel signature and wave kernel signature are
generalized by leveraging the spectral properties of deformable shapes and
learning task-specific spectral filters. This work is extended in [7] to
direction-sensitive feature descriptors by using the eigendecomposition of
anisotropic Laplace-Beltrami Operators (ALBO) [1]. Complementary to such
spectral approaches, a line of works builds on spatial filtering methods. The
idea is to construct patch operators directly on manifolds or graphs in the
spatial domain. In [33], a convolution operator acting on vector fields
without constraints on the filters themselves is defined. In [6], a local
intrinsic representation of a function defined on a manifold is extracted with
anisotropic heat kernels as spatial weighting functions. This idea is extended
in [29] by introducing an anisotropic convolution operator in the spectral
domain. In this work, the eigendecomposition of multiple ALBOs proposed in [1]
is filtered via Chebychev polynomials with learnable coefficients. This method
achieves impressive results for shape correspondence in terms of accuracy.

### Finsler computer vision

Computer vision on Finsler manifolds, a natural generalization of Riemannian
manifolds, is a largely uncharted research area. In [37], robotics
applications are pushed beyond Riemannian geometry and the Euler-Lagrange
equation is reviewed from a Finsler point of view. A few works in image
segmentation use a Finsler metric, either to add the orientation of an image
as an extra spatial dimension [14], or to reformulate geodesic evolutions for
region-based active contours [15, 13]. The heat equation on Finsler manifolds
has been defined in mathematics [34]. In [46], a geodesic distance based on
Finsler metrics is applied to contour detection. Nevertheless the chosen heat
equation is theoretically unmotivated and is solely addressed from a heuristic
point of view.

##  3 Background

We traditionally model a 3D shape as a two-dimensional compact Riemannian
manifold $(X,M)$, with $X$ a smooth manifold equipped with an inner product
$\langle u,v\rangle_{M}=u^{\top}M(x)v$ for vectors $u,v$ on the tangent space
$T_{x}X$ at each point $x\in X$. We alternatively model the shape as a Finsler
manifold $(X,\mathcal{F}_{x})$ equipped with a Randers metric
$\mathcal{F}_{x}:T_{x}X\rightarrow\mathbb{R}$ at point $x$. This manifold can
be viewed as a generalization of the Riemannian model, as the metric does not
necessarily induce an inner product. We point out that we recover a Riemannian
manifold if $\mathcal{F}_{x}$ is symmetric. In this section we briefly
introduce the Riemannian heat diffusion, the Randers metric and the Finsler
heat diffusion.

###  3.1 Riemannian heat diffusion

#### Laplace-Beltrami operator.

The Laplace-Beltrami Operator (LBO) on the Riemannian shape $X$ is defined as:

$\Delta_{X}f(x)=-\operatorname{div}_{X}(\nabla_{X}f(x)),$ (1)

with $\nabla_{X}f$ and $\operatorname{div}_{X}f$ being the intrinsic gradient
and divergence of $f(x)\in L^{2}(X)$. The LBO is a positive semi-definite
operator with a real eigen-decomposition $(\lambda_{k},\phi_{k})$:

$\Delta_{X}\phi_{k}(x)=\lambda_{k}\phi_{k}(x),$ (2)

with countable eigenvalues $0=\lambda_{0}\leq\lambda_{1}\leq\ldots$ growing to
infinity. The corresponding eigenfunctions $\phi_{0},\phi_{1}\ldots$ are
orthonormal for the standard inner product on $L^{2}(X)$:

$\langle\phi_{i},\phi_{j}\rangle=\delta_{ij},$ (3)

and form an orthonormal basis for $L^{2}(X)$. It follows that any function
$f\in L^{2}(x)$ can be expressed as:

$f(x)=\sum_{k\geq 0}\langle\phi_{k},f\rangle\phi_{k}(x)=\sum_{k\geq
0}\hat{f}(\lambda_{k})\phi_{k}(x),$ (4)

and $\hat{f}(\lambda_{k})=\langle\phi_{k},f\rangle$ are the coefficients of
the manifold Fourier transform of $f$. The Convolution Theorem on manifolds is
given by:

$(f\ast g)(x)=\sum_{k\geq
0}\hat{f}(\lambda_{k})\hat{g}(\lambda_{k})\phi_{k}(x),$ (5)

where $f*g$ is the convolution of signal $f$ with filter kernel $g$ and
$\hat{f}(\lambda_{k})\hat{g}(\lambda_{k})$ is the spectral filtering of $f$ by
$g$.

#### Heat diffusion.

The LBO governs the diffusion equation:

$\frac{\partial f(x,t)}{\partial t}=-\Delta_{X}f(x,t),$ (6)

where $f(x,t)$ is the temperature at point $x$ at time $t$. Given some initial
heat distribution $f_{0}(x)=f(x,0)$ the solution of Eq. 6 at time $t$ is given
by the convolution:

$f(x,t)=\int_{X}f_{0}(\xi)h_{t}(x,\xi)d\xi,$ (7)

with the heat kernel $h_{t}(x,\xi)$. In the spectral domain it is expressed
as:

$h_{t}(x,\xi)=\sum_{k\geq 0}e^{-t\lambda_{k}}\phi_{k}(x)\phi_{k}(\xi).$ (8)

#### Anistropic LBO.

Extending the traditional LBO presented above, an anistropic LBO (ALBO) is
created by changing the diffusion speed along chosen directions on the
surface, typically the principal directions of curvature [1]:

$\Delta_{\alpha}f(x)=-\operatorname{div}_{X}(D_{\alpha}(x)\nabla_{X}f(x)),$
(9)

where $D_{\alpha}(x)$ is a thermal conductivity tensor acting on the intrinsic
gradient direction in the tangent plane and $\alpha$ is a scalar expressing
the level of anisotropy. The ALBO can be extended to multiple anisotropic
angles $\theta$ [6]:

$\Delta_{\alpha\theta}f(x)=-\operatorname{div}_{X}(D_{\alpha\theta}(x)\nabla_{X}f(x)),$
(10)

by defining the thermal conductivity tensor:

$D_{\alpha\theta}(x)=R_{\theta}D_{\alpha}(x)R_{\theta}^{\top},$ (11)

where $R_{\theta}$ is a rotation in the tangent plane (around surface normals)
with angle $\theta$.

#### Anistropic convolution operator

Similarly, Eq. 5 is extended by introducing the anisotropic convolution
operator [29]:

$(f\ast g)_{\alpha\theta}(x)=\int_{y\in X}f(y)g_{\alpha\theta,x}(y)dy,$ (12)

where $g_{\alpha\theta,x}(y)$ is an oriented kernel function centered at point
$x$ from the spectral domain

$g_{\alpha\theta,x}(y)=\sum_{k\geq
0}\hat{g}(\lambda_{\alpha\theta,k})\phi_{\alpha\theta,k}(x)\phi_{\alpha\theta,k}(y).$
(13)

A straightforward derivation shows that:

$(f\ast g)_{\alpha\theta}(x)=\sum_{k\geq
0}\hat{g}(\lambda_{\alpha\theta,k})\hat{f}(\lambda_{\alpha\theta,k})\phi_{\alpha\theta,k}(x),$
(14)

where
$\hat{f}(\lambda_{\alpha\theta,k})=\langle\phi_{\alpha\theta,k},f\rangle$ are
the coefficients of the anisotropic Fourier transform of $f$. To extract
features along all directions, the anisotropic convolution operators are
summed and then:

$(f\ast g)_{\alpha}(x)=\int_{0}^{2\pi}(f\ast g)_{\alpha\theta}(x)d\theta.$
(15)

The design of $\hat{g}(\lambda)$ is crucial when defining the anisotropic
convolution operators. Following [18], Chebychev polynomials are adopted to
define polynomial filters $\hat{g}(\lambda)$ in [29].

###  3.2 Randers metric

Riemannian manifolds are equipped with a quadratic metric
$\mathcal{R}_{x}(v)=\lVert v\rVert_{M(x)}$ that naturally induces a positive-
definite inner product on the tangent space $T_{x}M$ at each point $x$.
Finsler geometry is often said to be the same as Riemannian geometry without
the quadratic constraint. A Finsler metric thus no longer induces an inner-
product and is not required to be symmetric. Formally, a Finsler metric is
given by the following properties.

######  Definition 1 (Finsler metric).

A Finsler metric $\mathcal{F}_{x}:T_{x}X\to\mathbb{R}_{+}$ at point $x$ is a
smooth metric satisfying the triangle inequality, positive homogeneity, and
positive definiteness axioms:

$\displaystyle\mathcal{F}_{x}(v+v^{\prime})\leq\mathcal{F}_{x}(v)+\mathcal{F}_{x}(v^{\prime}),\;$
$\forall v,v^{\prime}\in T_{x}X$ (16) $\displaystyle\mathcal{F}_{x}(\gamma
v)=\gamma\mathcal{F}_{x}(v),\;$
$\forall(\gamma,v)\\!\in\\!\mathbb{R}_{+}\\!\times\\!T_{x}X$ (17)
$\displaystyle\mathcal{F}_{x}(v)=0\iff v=0.$ (18)

We consider Randers metrics, one of the simplest type of Finsler metrics
generalizing the Riemannian one.

######  Definition 2 (Randers metric).

Let $M:X\to S_{d}^{++}$ be a tensor field of symmetric positive definite
matrices and a vector field $\omega:X\to\mathbb{R}^{d}$ such that
$\lVert\omega(x)\rVert_{M(x)^{-1}}<1$ for all $x\in X$. The Randers metric
associated to $(M,\omega)$ is given by

$\mathcal{F}_{x}(v)=\lVert
v\rVert_{M(x)}+\langle\omega(x),v\rangle,\quad\forall x\in X,v\in T_{x}X.$
(19)

The Randers metric $\mathcal{F}$ is the sum of the Riemannian metric
associated to $M$ and an asymmetric linear part governed by $\omega$. Notably
when $\omega\equiv 0$, $\mathcal{F}$ is a traditional Riemannian metric. The
condition $\lVert\omega(x)\rVert_{M(x)^{-1}}<1$ ensures the positivity of the
metric (see supplementary material).

From any metric we can construct a new metric associated to it, its dual,
which plays a central role in differential geometry.

######  Definition 3 (Dual Finsler metric).

The dual metric of a Finsler metric $\mathcal{F}_{x}$ is the metric
$\mathcal{F}_{x}^{*}:T_{x}X\to\mathbb{R}_{+}$ defined by

$\mathcal{F}_{x}^{*}(v)=\max\\{\langle
v,b\rangle;b\in\mathbb{R}^{d},\mathcal{F}_{x}(b)\leq 1\\}.$ (20)

When considering only Randers metrics, a subtype of Finsler metrics, the dual
metric becomes explicit.

######  Lemma 1 (Dual Randers metric).

Let $\mathcal{F}_{x}$ be a Randers metric given by $(M,\omega)$. The dual
metric of $\mathcal{F}_{x}$ is also a Randers metric $\mathcal{F}_{x}^{*}$
associated to $(M^{*},\omega^{*})$ that satisfies

$\begin{pmatrix}\alpha M^{*}&\omega^{*}\\\
{\omega^{*}}^{\top}&\frac{1}{\alpha}\\\
\end{pmatrix}=\begin{pmatrix}M&\omega\\\ \omega^{\top}&1\\\
\end{pmatrix}^{-1},$ (21)

where $\alpha=1-\langle\omega,M^{-1}\omega\rangle>0$.

A proof is provided in the supplementary material.

From Eq. 21 we can derive:

$M^{*}=\frac{1}{\alpha^{2}}\Big{(}\alpha
M^{-1}+(M^{-1}\omega)(M^{-1}\omega)^{\top}\Big{)},$ (22)

and

$\omega^{*}=-\frac{1}{\alpha}M^{-1}\omega,.$ (23)

It is simple to show that $\lVert\omega^{*}\rVert_{{M^{*}(x)}^{-1}}<1$ (see
supplementary material). The dual Randers metric $\mathcal{F}_{x}^{*}$ is the
Randers metric on the manifold $X$ associated to the new Riemannian and linear
components $(M^{*},\omega^{*})$:

$\mathcal{F}_{x}^{*}(v)=\lVert v\rVert_{M^{*}(x)}+\langle w^{*}(x),v\rangle.$
(24)

###  3.3 Finsler heat equation

The classical heat equation in the Riemannian case is the gradient flow of the
Dirichlet energy, and by analogy we can derive a Finsler heat equation as the
gradient descent of the dual energy $\tfrac{1}{2}\mathcal{F}_{x}^{*}(v)^{2}$,
in the following way [5, 34].

######  Definition 4 (Finsler heat equation).

Given a manifold $X$ equipped with a Finsler metric $\mathcal{F}_{x}$, the
heat equation on this Finsler manifold for smooth vector fields $u(x,t)$ with
$(x,t)\in X\times\mathbb{R}_{+}$ is given by the gradient descent of the dual
energy

$\partial_{t}u=\operatorname{div}(\mathcal{F}_{x}^{*}(\nabla
u)\nabla\mathcal{F}_{x}^{*}(\nabla u)).$ (25)

##  4 Finsler heat kernel

We now analyse the Finsler heat equation to derive a novel Laplacian operator
for shape matching. Our plan is the following:

  * •

Analyse the heat equation on a Finsler manifold equipped with a Randers
metric,

  * •

From a specific case where the solution is explicit, we exhibit a simplified
equation behaving like a regular heat equation with an external heat source,

  * •

The solution to this new problem is explicit and governed by the homogeneous
term, which is an anisotropic heat diffusion with diffusivity involving the
Randers metric,

  * •

The homogeneous solution is given by a convolution with an anisotropic kernel
depending on the Randers metric,

  * •

This heat kernel allows the construction of a new Finsler-Laplace-Beltrami
operator (FLBO) that benefits from all the properties of ALBOs.

###  4.1 Finsler heat diffusion

The heat equation is well-known to be related to distances and geodesics in
short time, and thus it is often analyzed when $t\to 0$. We look at the
following specific case in short time. To provide a tractable solution, we
also here assume that we slightly depart from Riemannian geometry by taking
small $\lVert\omega\rVert$.

######  Proposition 1 (Simplified Randers heat equation - distance trick).

If the initial solution $u_{0}=u(x,0)$ of the heat equation satisfies
$\mathcal{F}_{x}^{*}(\nabla u_{0})=1$, then in short time the Finsler heat
equation on a Randers metric associated to $(M,\omega)$ with small
$\lVert\omega\rVert$ simplifies to

$\partial_{t}u=\operatorname{div}\left((M^{*}-\omega^{*}{\omega^{*}}^{\top})\nabla
u\right)+\operatorname{div}(\omega^{*})\,.$ (26)

###### Proof.

We provide the proof in the supplementary material. ∎

The initial condition $\mathcal{F}_{x}^{*}(\nabla u_{0})=1$ holds when
$-u_{0}$ is a distance function, as the Finsler Eikonal equation is given by
$\mathcal{F}_{x}^{*}(-\nabla u)=1$ [32]. In practice, the solution to Eq. 26
is not our primary concern. We only use this particular initialisation to
exhibit a tractable solution that allows us to define a family of Finsler-
based LBOs. We name the coefficient
$D_{\mathcal{F}_{x}^{*}}=M^{*}-\omega^{*}{\omega^{*}}^{\top}$ as the Finlser
diffusivity. We can interpret Eq. 26 as a homogeneous heat equation:

$\partial_{t}u(x,t)=\operatorname{div}(D_{\mathcal{F}_{x}^{*}}\nabla u(x,t)),$
(27)

to which we add an external heat source $\operatorname{div}(\omega^{*}(x))$.
Given some initial heat condition $u_{0}(x)=u(x,0)$, the solution of Eq. 26 is
the sum of the solution of Eq. 27,

$u(x,t)=\int_{X}u_{0}(\xi)h_{t}(x,\xi)d\xi,$ (28)

where $h_{t}$ is the heat kernel for Eq. 27, and of any particular solution,
such as

$u^{d}(x,t)=\int_{X}\operatorname{div}(\omega^{*}(\xi))\hat{h}_{t}(x,\xi)d\xi\,,$
(29)

where $\hat{h}$ is the average over time of the heat kernel $h_{t}$:

$\hat{h}_{t}(x,\xi)=\int_{s=0}^{t}h_{(t-s)}(x,\xi)ds\,.$ (30)

###  4.2 Finsler-Laplace-Beltrami operator

In Riemannian geometry, the heat kernel is by definition given by the
convolution kernel of the solution of the homogeneous equation. An external
source introduces a convolution with the time average of the heat kernel. By
analogy, from the homogeneous Eq. 27 we define the Finsler-Laplace-Beltrami
operator (FLBO) as:

$\Delta_{\mathit{FLBO}}u(x)=-\operatorname{div}_{X}(D_{\mathcal{F}_{x}^{*}}\nabla_{X}u(x))\,.$
(31)

The FLBO is a special case of an ALBO on a Riemannian manifold, that was
derived after analyzing the heat equation when the manifold is equipped with a
Finsler metric instead. Unlike previous ALBOs, it is theoretically justified
rather than empirically designed based solely on heuristics. Since the FLBO in
an ALBO, it inherits all of its properties, such as the spectral decomposition
of a compact positive semi-definite operator

$\Delta_{\mathit{FLBO}}\phi_{k}^{\mathit{FLBO}}(x)=\lambda_{k}^{\mathit{FLBO}}\phi_{k}^{\mathit{FLBO}}(x).$
(32)

The FLBO also differs from the Laplace-Finsler operator (LF) deriving from Eq.
26:

$\Delta_{F}u(x)=-\operatorname{div}(\mathcal{F}_{x}^{*}(\nabla
u)\nabla\mathcal{F}_{x}^{*}(\nabla u)),$ (33)

In particular, the LF operator acts on a Finsler manifold and not a Riemannian
one like the FLBO. Thus, it cannot be used as a direct replacement of LBOs in
most applications.

In the previous section we have shown that the solution of the simplified
Randers heat diffusion equation exhibits the Finsler heat kernel
$h_{t}(x,\xi)$ associated with the homogeneous Finsler heat diffusion
equation.

######  Proposition 2.

In the spectral domain, the Finsler heat kernel can be expressed as

$h_{t}^{FLBO}(x,\xi)=\sum_{k\geq
0}e^{-t\lambda_{k}^{FLBO}}\phi_{k}^{FLBO}(x)\phi_{k}^{FLBO}(\xi).$ (34)

We have unified the Finsler heat diffusion equation and the anisotropic
Laplace-Beltrami operator formulation. The link comes from the Finsler heat
kernel that was exhibited in the explicit solution of the simplified equation
Eq. 26 in a special case. We point out that the asymmetric component $\omega$
of the Randers metric influences the diffusivity coefficient and then the
eigendecomposition of the Finsler heat kernel.

##  5 Experiments

###  5.1 Discretization

We present a possible discretization of our FLBO given a manifold discretized
by sampling $n$ vertices. As the FLBO is a positive semi-definite operator,
and by analogy with the LBO it can be approximated by an $n\times n$ sparse
matrix $L_{\mathit{FLBO}}=-S_{\mathit{FLBO}}^{-1}W_{\mathit{FLBO}}$ with the
mass matrix $S_{\mathit{FLBO}}$ and the stiffness matrix $W_{\mathit{FLBO}}$.
Our FLBO is not restricted to a particular discrete surface representation.
Following [6, 29], we focus on triangular meshes to discretize the Finsler
metrics, but representations for other discretizations can be easily deduced.
As in [6, 29], we introduce various angles $\theta$, each giving a Finsler
metric $(M_{\theta},\omega_{\theta})$, to produce convolution operators from
FLBOs sensitive to $\theta$, and then sum them up to get a final Finsler-based
convolution operator incorporating features from all directions. We use the
same notations as in [6, 29].

Our manifold is discretized as a triangular mesh with vertices $V$, edges $E$,
and faces $F$. For each triangle $ijk\in F$, we define
$U_{ijk}=(\hat{u}_{M},\hat{u}_{m},\hat{n})$ as an orthonormal reference frame
associated to the face unit normal $\hat{n}$ and the directions of principal
curvature $\hat{u}_{M},\hat{u}_{m}\in\mathbb{R}^{3}$. We denote
$\hat{e}_{ij}\in\mathbb{R}^{3}$ the unit vector along the embedding of the
edge $(i,j)\in E$ pointing from $i$ to $j$, and $\alpha_{ij}$ and $\beta_{ij}$
the angles at $k$ and $h$ of the adjacent triangles $ijk\in F$ and $ijh\in F$
respectively. We write $R_{\theta}\in\mathbb{R}^{3}$ the rotation matrix of
angle $\theta$ around the third basis vector (to be identified as $\hat{n}$).
The notations match those of Fig. 3.

Figure 3: Notations for unit vectors and angles on the triangular mesh. The
figure mimics the illustration in [6].

In all our experiments, the mass matrix $S_{\mathit{FLBO}}$ is computed as a
diagonal matrix with diagonal element $i$ equal to one third of the areas of
faces having $i$ as a vertex [19].

Our chosen Riemannian metric is also anisotropic using ideas from the ALBO
community. When $\omega\equiv 0$ we obtain the traditional ALBO [29]. The
diffusivity $D_{\alpha}(x)$ of ALBOs is usually created by introducing a
scaling factor $\alpha>0$ into one of its eigenvalues by various heuristic
formulae to break the isotropy along eigendirections [6, 29]. We firstly build
on the strategy of [29] to create the uniform anisotropic Riemannian
diffusivity

$D_{\alpha}=\begin{pmatrix}\frac{1}{1+\alpha}&\\\ &1\end{pmatrix},$ (35)

with $\alpha=10$ in our experiments as in [29]. For each angle $\theta$, we
rotate this diffusivity to create the oriented diffusivity matrix
$H_{\alpha\theta}$, also called shear matrix,

$H_{\alpha\theta}=R_{\theta}U_{ijk}D_{\alpha}U_{ijk}^{\top}R_{\theta}^{\top}.$
(36)

We can now define our anisotropic Riemannian metric at angle $\theta$. As we
want our construction to generalise the Riemannian case $\omega\equiv 0$, and
since $D_{\mathcal{F}_{x}^{*}}$ equals $M^{-1}(x)$ when $\omega\equiv 0$, we
choose

$M_{\alpha\theta}=H_{\alpha\theta}^{-1}.$ (37)

Note that our scheme recovers the isotropic Riemannian metric when $\alpha=0$
as the shear matrix is then the identity. For each angle $\theta$, we
introduce the Randers anisotropic component rotated by $\theta$ from the
direction of maximal curvature

$\omega_{\theta}=\tau R_{\theta}\hat{u}_{M},$ (38)

where $\tau$ is a hyperparameter to ensure
$\lVert\omega\rVert_{M_{\alpha\theta}^{-1}}<1$. We have thus constructed the
Randers metric $(M_{\alpha\theta},\omega_{\theta})$, from which we can easily
compute its dual $(M_{\alpha\theta}^{*},\omega_{\theta}^{*})$ and its Finsler
diffusivity $D_{\mathcal{F}_{x}^{*}}^{\alpha\theta}$. We then use the
generalisation of the classical cotangent weight scheme for anisotropic
diffusivities [6] to compute the weights of
$W_{\mathit{FLBO}}^{\alpha\theta}$:

$w_{ij}=\begin{cases}\frac{1}{2}\left(\frac{\langle\hat{e}_{kj},\hat{e}_{ki}\rangle_{D_{\mathcal{F}_{x}^{*}}^{\alpha\theta}}}{\sin\alpha_{ij}}+\frac{\langle\hat{e}_{hj},\hat{e}_{hi}\rangle_{D_{\mathcal{F}_{x}^{*}}^{\alpha\theta}}}{\sin\beta_{ij}}\right)&(i,j)\in
E,\\\ -\sum_{k\neq i}w_{ik}&i=j,\\\ 0&\text{else}.\end{cases}$ (39)

The FLBO at angle $\theta$ is finally given by
$\Delta_{\mathit{FLBO}}^{\alpha\theta}=-S_{\mathit{FLBO}}^{-1}W_{\mathit{FLBO}}^{\alpha\theta}$.
Note that if we had taken $\omega_{\theta}\equiv 0$, then we would have
recovered the ALBO from [29]. See Fig. 1 for a illustrative summary of how we
derive the FLBO.

#### Finsler-based anisotropic convolution operator

Since our FLBOs are ALBOs, we can use the method of [29] to create
$\theta$-oriented convolution kernels $g_{\alpha\theta,x}$ according to Eq. 13
by working in the spectral domain of $\Delta_{\mathit{FLBO}}^{\alpha\theta}$
and taking $\phi_{\alpha\theta,0},\phi_{\alpha\theta,1}\ldots$ to be the
eigenvectors of $\Delta_{\mathit{FLBO}}^{\alpha\theta}$. Convolution
$(f*g)_{\alpha\theta}$ of any function $f$ with this kernel is then computed
in the spectral domain (Eq. 14) and then oriented convolutions are summed to
get the final convolution operator $(f*g)_{\alpha}$ (Eq. 15). In our
experiments, we use eight uniformly spaced samples of $\theta\in[0,\pi)$ as in
[29]. Following [29], we encode the spectral coefficients
$\hat{g}(\lambda_{\alpha\theta,k})$ of $g_{\alpha\theta,x}$ using Chebychev
polynomials

$\hat{g}(\lambda_{\alpha\theta,k})=\sum\limits_{s=0}^{S-1}c_{\alpha\theta,s}T_{s}(\lambda_{\alpha\theta,k}),$
(40)

where $S=16$ in our tests as in [29], $T_{s}$ is the Chebychev polynomial of
the first kind of order $s$, i.e. $T_{s}(\cos\gamma)=\cos(s\gamma)$ for any
$\gamma$, and $c_{\alpha\theta,s}\in\mathbb{R}$ are coefficients to be
learned.

For visualisation purposes, we provide in the supplementary a plot of
convolution filters $g_{\alpha\theta,x}$ for various $\theta$ when choosing
the spectral coefficients of the kernel to be equal to one of the Chebychev
polynomials, similarly to [29].

###  5.2 An Application: Shape matching

As a concrete example, we use our FLBO for shape matching, which is a common
shape analysis application where LBOs play a central role. We compare the
results to those based on traditional Riemannian (anisotropic) LBOs, namely
FieldConv [33] and ACSCNN [29]. Note that ACSCNN coincides with our method
when $\tau=0$.

#### Datasets and evaluataion.

We work on four publicly available datasets: FAUST Remeshed [4, 38], SCAPE
Remeshed [2, 38], and SHREC’16 Partial Cuts [17] and Holes [17]. We follow the
Princeton benchmark protocol [26]. We provide the percentage of matches that
are at most $r$-geodesically distant from the groundtruth correspondence on
the reference shape, with $r\in[0,1]$.

#### Setting.

Following [29], our chosen architecture is SplineCNN [21]. We train each
dataset for $100$ epochs with Adam optimizer and and we set the learning rate
to $0.001$. We take as input SHOT descriptors [40]. In particular, we use the
open-source implementation of ACSCNN111https://github.com/GCVGroup/ACSCNN. The
preprocessing step to get the Finsler-based filters is done with our custom
implementation.

###  5.3 Full mesh correspondence.

#### FAUST (Remeshed)

dataset contains 100 human shapes corresponding to 10 different human beings
remeshed with the LRVD [45] to generate more challenging shapes with different
mesh connectivity [4, 38]. It does not have one-to-one correspondence between
shapes and they do not share the same number of vertices. As suggested in
[20], we use the first 80 shapes as the training set and we test on the
remaining 20 ones.

#### SCAPE (Remeshed)

dataset contains 71 human shapes corresponding to the same human being in
different positions. Each shape is remeshed as in FAUST Remeshed [2, 38]. The
first 51 shapes form the training set and we test on the other 20 shapes.

Method | Accuracy (r = 0) | Accuracy (r = 0.01)
---|---|---
FieldConv [33] | 53.39 | 75.01
ACSCNN [29] | 61.95 | 82.46
ours | $\tau=0.1$ | ${62.61}$ | ${82.90}$
$\tau=0.2$ | ${62.58}$ | ${82.90}$
$\tau=0.3$ | ${62.67}$ | ${82.94}$
$\tau=0.4$ | ${62.27}$ | ${82.66}$
| $\tau=0.5$ | ${62.38}$ | ${82.68}$

Table 1: Accuracy on FAUST Remeshed.

Method | Accuracy (r = 0) | Accuracy (r = 0.01)
---|---|---
FieldConv [33] | 37.41 | 61.37
ACSCNN [29] | 46.13 | 71.70
ours | $\tau=0.1$ | ${46.94}$ | ${72.44}$
$\tau=0.2$ | ${46.27}$ | ${71.91}$
$\tau=0.3$ | 45.91 | ${71.92}$
$\tau=0.4$ | 45.82 | 71.44
| $\tau=0.5$ | 45.26 | 71.27

Table 2: Accuracy on SCAPE Remeshed.

###  5.4 Partial correspondence.

SHREC’16 Partial Cuts and SHREC’16 Partial Holes datasets contain respectively
135 and 90 shapes of human and animals divided in 8 categories. The shapes are
deformed following two kinds of partiality, respectively cuts and holes. For
each category in Cuts, we use the first 10 shapes as a training dataset and we
test on the last 5 shapes. For each category in Holes, we use the first 6
shapes as a training dataset and we test on the last 4 shapes.

###  5.5 Results

We present quantitative results in Tabs. 1, 2, 3 and 4 for FAUST Remeshed,
SCAPE Remeshed, SHREC’16 Partial Cuts, and SHREC’16 Partial Holes
respectively. For the first two datasets we provide the accuracy within
$r\in\\{0,0.01\\}$ geodesic radius of the groundtruth, whereas we only display
the performance for $r=0$ in the last two datasets due to space constraints as
each of these datasets are decomposed into 8 subdatasets. Clearly, FieldConv
struggles compared to the more advanced ACSCNN. Our method systematically
outperforms, or is on par with, ACSCNN for a default choice of $\tau=0.1$.
This demonstrates that the FLBO can be used in complement to advanced
approaches to boost performance, requiring only a small additional
preprocessing overhead for computing the metric and the FLBO, which is
negligible compared to neural network training times. We test our method with
other values of $\tau$ and find that the results remain consistent, proving
that our approach is not highly sensitive to finely tuning this
hyperparameter. In the literature, providing the results as a plot with
respect to all $r\in[0,1]$ is sometimes performed. However, since our results
are close to those of ACSCNN, the curves tend to superimpose and do not
provide any additional insight, so we do not provide them here.

Method | Cat | Centaur | David | Dog | Horse | Michael | Victoria | Wolf
---|---|---|---|---|---|---|---|---
FieldConv [33] | 0.054 | 1.18 | 0 | 0.094 | 1.93 | 0.33 | 0.010 | 45.81
ACSCNN [29] | 38.10 | 74.99 | ${23.78}$ | 59.87 | 44.37 | 12.82 | ${31.23}$ | 93.75
ours | $\tau=0.1$ | ${40.13}$ | ${75.41}$ | 22.34 | 59.31 | 42.72 | ${16.04}$ | 27.95 | ${94.02}$
$\tau=0.2$ | $37.72$ | 72.83 | 22.48 | ${60.02}$ | 43.17 | ${14.30}$ | 29.72 | ${94.14}$
$\tau=0.3$ | ${40.26}$ | 73.40 | 21.59 | 57.61 | 43.49 | ${13.15}$ | 31.10 | ${94.17}$
$\tau=0.4$ | ${39.63}$ | 72.29 | 21.51 | 57.12 | ${44.83}$ | ${13.55}$ | 29.35 | 93.33
| $\tau=0.5$ | ${40.94}$ | 73.16 | 20.49 | ${60.17}$ | 43.85 | ${15.04}$ | 28.46 | ${94.06}$

Table 3: Accuracy on SHREC’16 Partial Cuts.

Method | Cat | Centaur | David | Dog | Horse | Michael | Victoria | Wolf
---|---|---|---|---|---|---|---|---
FieldConv [33] | N.A. | 0.038 | N.A. | 0.11 | N.A. | 0.11 | 0.049 | 0.39
ACSCNN [29] | 19.98 | 32.80 | 16.09 | 41.70 | 46.85 | 12.29 | 11.32 | 84.53
ours | $\tau=0.1$ | 19.95 | ${35.85}$ | ${17.96}$ | ${42.68}$ | ${47.87}$ | 12.00 | ${11.70}$ | ${84.64}$
$\tau=0.2$ | 17.99 | ${33.10}$ | ${16.40}$ | ${42.84}$ | ${49.77}$ | ${13.90}$ | 10.02 | ${85.23}$
$\tau=0.3$ | ${21.27}$ | ${35.48}$ | ${18.43}$ | ${43.29}$ | 45.07 | 11.31 | ${11.52}$ | ${85.74}$
$\tau=0.4$ | ${20.08}$ | 32.54 | ${16.41}$ | ${42.98}$ | 45.71 | ${13.16}$ | ${11.41}$ | ${85.72}$
| $\tau=0.5$ | 18.62 | ${35.04}$ | ${16.28}$ | ${42.68}$ | 45.38 | 11.89 | 10.19 | 84.23

Table 4: Accuracy on SHREC16’ Partial Holes. N.A. indicates a failure to
process the dataset.

We plot some qualitative correspondence results in Fig. 4. We find that our
method is able to provide a convincing mapping between shapes undergoing
various highly non-rigid deformations. It is also able to handle missing parts
such as cuts and holes, although performance naturally deteriorates as the
challenge is significantly harder.

Figure 4:  Visual shape correspondence results on the FAUST Remeshed (row 1),
SCAPE remeshed (row 2), and SHREC’16 Partial Cuts (columns 1 and 2 in rows 3
to 5), and SHREC’16 Partial Holes (columns 1 and 3 in rows 3 to 5) datasets.
The source shape is on the left, on which we perform dense correspondence
estimation on the shapes to the right. See the supplementary material for more
visual results on other shapes.

##  6 Conclusion

We revisited Finsler manifolds, a generalization of Riemannian manifolds that
naturally allow for anisotropies. By exploring their heat diffusion equation
we derived a novel Finsler heat kernel and a Finsler-Laplace-Beltrami operator
(FLBO). This FLBO generalizes traditional heuristic anisotropic LBOs while
being theoretically motivated. It can be used in place of LBOs or ALBOs and is
compatible with various advanced techniques such as spatial filtering or
geometric deep learning. We tested the pertinence of the FLBO in shape
correspondence experiments and found that our approach can easily complement
state-of-the-art geometric deep learning methods using ALBOs. We hope that the
concise and self-contained review of Finsler geometry and the derivation of a
tractable Finsler heat equation and FLBO will encourage further exploration of
the Finsler geometry in the field of computer vision.

#### Limitations and future works.

We have used some strong assumptions to derive the Finsler heat equation.
While we acknowledge this limitation, it is necessary, in our work, to exhibit
some tractable operators. However, we believe that a further theoretical focus
on this stage is an exciting and promising research direction. Our experiments
are also built on a straightforward discretization. Although it is not a major
drawback, as our goal is to show that traditional LBOs can be generalized to
Finsler-based LBOs, an extension of our example to other choices of
$(M,\omega)$ would also be crucial. Finally, beyond our simple application to
shape matching, we do believe that the importance of Finsler manifolds in
computer vision is largely uncharted and requires further stimulating works.

#### Acknowledgement.

This work was supported by the ERC Advanced Grant SIMULACRON.

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Supplementary Material

This supplementary material is organized as follows:
Sec. 1 provides some proofs concerning properties of Randers metrics mentioned
in the main paper: positivity, dual formula, and bounded norm of the dual
Randers vector field.
Sec. 2 details the proof of Proposition 1.
Sec. 3 gives additional details about runtime in our experiments.
Sec. 4 shows additional visual results of anisotropic kernels and shape
correspondence.

##  1 Classical proofs for the Randers metric

The proofs in this section of the supplementary material are not novel in the
Randers metric community. The purpose of putting them here is to present a
self-contained theory that is fully and simply explained, avoiding the need
for readers to search through other challenging mathematical papers. For prior
existing proofs, see for example [32]. Note that errors exist in various
reference materials, which further justifies rederiving the proofs. For
instance, in [32] the formulae for the dual Randers metric in Prop. 5.1 are
incorrect.

###  1.1 Proof of Randers metric positivity

Any vector in the tangent space $T_{x}X$ can be given in the form
$M(x)^{-1}v$. We can then write

$\displaystyle\mathcal{F}_{x}(M(x)^{-1}v)=\sqrt{v^{\top}M(x)^{-1}v}+\omega(x)^{\top}M(x)^{-1}v$
(41) $\displaystyle\quad=\lVert M(x)^{-\tfrac{1}{2}}v\rVert+\langle
M(x)^{-\tfrac{1}{2}}\omega(x),M(x)^{-\tfrac{1}{2}}v\rangle.$ (42)

Using the Cauchy-Schwartz inequality, we get
$\lvert\omega(x)^{\top}M(x)^{-1}v\rvert\leq\lVert
M(x)^{-\tfrac{1}{2}}\omega(x)\rVert\lVert M(x)^{-\tfrac{1}{2}}v\rVert$. The
assumption on $\omega$ provides $\lVert
M(x)^{-\tfrac{1}{2}}\omega(x)\rVert<1$, thus as soon as $v\neq 0$, we get
$\mathcal{F}_{x}(M(x)^{-1}v)>0$. ∎

###  1.2 Proof of dual Randers metric formulae

For conciseness, we drop the explicit dependence on $x$. The dual Finsler
metric $\mathcal{F}^{*}$ of the Randers metric $\mathcal{F}$ is given by
Definition 3. By positive homogeneity of $\mathcal{F}$, it is also given by

$\mathcal{F}^{*}(v)=\max\left\\{\frac{\langle
v,b\rangle}{\mathcal{F}(b)};b\neq 0\right\\}.$ (43)

As such,
$\tfrac{1}{\mathcal{F}^{*}(v)}=\min\left\\{\tfrac{\mathcal{F}(b)}{\langle
v,b\rangle};b\neq 0\right\\}$, and once again by homogeneity we have

$\frac{1}{\mathcal{F}^{*}(v)}=\min\\{\mathcal{F}(b);\langle v,b\rangle=1\\}.$
(44)

The Lagrangian for this constrained optimization problem is given by
$L(b,\mu)=\mathcal{F}(b)+\mu\langle v,b\rangle$. Computing the gradient
$\tfrac{\partial L}{\partial b}$ and setting it to $0$ to satisfy the KKT
conditions, the optimal $b$, denoted $b^{*}$, satisfies

$M\frac{b^{*}}{\lVert b^{*}\rVert_{M}}+\omega+\mu v=0.$ (45)

On one hand, by computing the scalar product with $b^{*}$, and recalling that
the constraint enforces $\langle v,b^{*}\rangle=1$, and recalling the formula
for the Randers metric (Eq. 19) we get $\mu=-\mathcal{F}(b^{*})$. Since
$b^{*}$ solves the optimization problem in Eq. 44, we have

$\mu=-\frac{1}{\mathcal{F}^{*}(v)}.$ (46)

On the other hand, we can compute $\lVert\omega+\mu v\rVert_{M^{-1}}$ from Eq.
45

$\lVert\omega+\mu v\rVert_{M^{-1}}=\frac{\lVert Mb^{*}\rVert_{M^{-1}}}{\lVert
b^{*}\rVert_{M}}=1.$ (47)

Taking the square, we get a degree two polynomial for which $\mu$ is a root

$\mu^{2}\lVert
v\rVert_{M^{-1}}^{2}+2\mu\langle\omega,v\rangle_{M^{-1}}+\lVert\omega\rVert_{M^{-1}}^{2}-1=0.$
(48)

Writing $\alpha=1-\lVert\omega\rVert_{M^{-1}}^{2}>0$, the roots are given by

$\mu_{\pm}=\frac{-\langle\omega,v\rangle_{M^{-1}}\pm\sqrt{\langle\omega,v\rangle_{M^{-1}}^{2}+\lVert
v\rVert_{M^{-1}}^{2}\alpha}}{\lVert v\rVert_{M^{-1}}^{2}}.$ (49)

Clearly, $\mu_{+}>0$ and $\mu_{-}<0$ for $v\neq 0$, yet $\mu<0$ following
$\mu=-\mathcal{F}(b^{*})$ as the metric is always positive. Therefore
$\mu=\mu_{-}$, and then by inverting Eq. 46 we get

$\displaystyle\mathcal{F}^{*}(v)$ $\displaystyle=\frac{\lVert
v\rVert_{M^{-1}}^{2}}{\langle\omega,v\rangle_{M^{-1}}+\sqrt{\langle\omega,v\rangle_{M^{-1}}^{2}+\lVert
v\rVert_{M^{-1}}^{2}\alpha}}$ (50)
$\displaystyle=\sqrt{v^{\top}\frac{1}{\alpha^{2}}\left(\alpha
M^{-1}+M^{-1}\omega\omega^{\top}M^{-1}\right)v}$ $\displaystyle\hskip
43.20012pt-\frac{1}{\alpha}\langle\omega,v\rangle_{M^{-1}},$ (51)

where we used the classical trick
$\tfrac{1}{x+\sqrt{y}}=\tfrac{x-\sqrt{y}}{x^{2}-y}$ to remove the square root
from the denominator. We now recognise the dual metric $\mathcal{F}^{*}$ of
the Randers metric $\mathcal{F}$ as a Randers metric associated to
$(M^{*},\omega^{*})$ with

$\begin{cases}M^{*}=\frac{1}{\alpha^{2}}(\alpha
M^{-1}+(M^{-1}\omega)(M^{-1}\omega)^{\top}),\\\
\omega^{*}=-\frac{1}{\alpha}M^{-1}\omega.\end{cases}\qed$ (52)

###  1.3 Proof of $\lVert\omega^{*}\rVert_{{M*}^{-1}(x)}<1$

For conciseness, we drop the explicit dependence on $x$ and we denote
$\tilde{\omega}=M^{-\tfrac{1}{2}}\omega$. Recalling that $M$ is symmetric and
that $\alpha>0$, and plugging $M^{-1}\omega=M^{-\tfrac{1}{2}}\tilde{\omega}$
into Eqs. 22 and 23, we get

$\lVert\omega^{*}\rVert_{{M^{*}}^{-1}}^{2}=\tilde{\omega}^{\top}(\alpha
I+\tilde{\omega}\tilde{\omega}^{\top})^{-1}\tilde{\omega}.$ (53)

We then invoke the Sherman-Morrison formula
$\left(A+uv^{\top}\right)^{-1}=A^{-1}-\frac{1}{1+v^{\top}A^{-1}u}A^{-1}uv^{\top}A^{-1}$
for matrix $A=I$ and vectors $u=v=\alpha^{-\frac{1}{2}}\tilde{\omega}$.
Writing
$q_{M,\omega}=\alpha^{-1}\tilde{\omega}^{\top}\tilde{\omega}=\alpha^{-1}\omega^{\top}M^{-1}\omega$
for conciseness, we get

$\displaystyle\lVert\omega^{*}\rVert_{{M^{*}(x)}^{-1}}^{2}$
$\displaystyle=q_{M,\omega}\left(1-\frac{q_{M,\omega}}{1+q_{M,\omega}}\right)$
(54) $\displaystyle=\frac{q_{M,\omega}}{1+q_{M,\omega}}$ (55)
$\displaystyle=1-\frac{1}{1+\alpha^{-1}\omega^{\top}M^{-1}\omega}<1.\qed$ (56)

Figure 5:  Finsler-based anisotropic kernels $g_{\alpha\theta,x}$ centered at
the same point $x$ with different rotation angles $\theta$ equal to $0$ (left)
and $\tfrac{3\pi}{8}$ (right). Here the filter is chosen to be the Chebychev
polynomial $\hat{g}(\lambda)=T_{5}(\lambda)$.

##  2 Proof of Proposition 1

For conciseness, we drop the explicit dependence on $x$. Denoting
$\mathcal{P}(u)=\mathcal{F}^{*}(\nabla u)\nabla\mathcal{F}^{*}(\nabla u)$, the
Finsler heat equation is given by

$\partial_{t}u=\operatorname{div}(\mathcal{P}(u)).$ (57)

Differentiating the Finsler dual metric Eq. 24, we have

$\nabla\mathcal{F}^{*}(\nabla u)=M^{*}\frac{\nabla u}{\lVert\nabla
u\rVert_{M^{*}}}+\omega^{*}.$ (58)

Given the initial condition, the solution $u$ will approximately satisfy
$\mathcal{F}^{*}(\nabla u)=1$ in short time, i.e. for small time steps $t$. As
such, in short time we have

$\mathcal{P}(u)=M^{*}\frac{\nabla u}{\lVert\nabla
u\rVert_{M^{*}}}+\omega^{*}.$ (59)

Following [5] (Corrolary 2.10), if $u$ follows this condition, then there
exists a positive scalar $\lambda$ giving the proportionality between the
following vectors

$M^{*}\frac{\nabla u}{\lVert\nabla
u\rVert_{M^{*}}}+\omega^{*}=\lambda\big{(}(M^{*}-\omega^{*}{\omega^{*}}^{\top})\nabla
u+\omega^{*}\big{)}.$ (60)

This is due to the fact that $\nabla u$ belongs to the shared 0-level set of
functions of $v$ on the tangent space $f_{1}(v)=\lVert
v\rVert_{M^{*}}+{\omega^{*}}^{\top}v-1$ and $f_{2}(v)=\lVert
v\rVert_{M^{*}-\omega^{*}{\omega^{*}}^{\top}}^{2}+2{\omega^{*}}^{\top}v-1$
which share the same sign for all $v$, and with gradients computed at
$v=\nabla u$ equal to $\nabla f_{1}(\nabla u)=M^{*}\frac{\nabla
u}{\lVert\nabla u\rVert_{M^{*}}}+\omega^{*}$ and $\nabla f_{2}(\nabla
u)=2((M^{*}-\omega^{*}{\omega^{*}}^{\top})\nabla u+\omega^{*})$. By computing
the scalar product of Eq. 60, i.e. of the equation $\nabla f_{1}(\nabla
u)=\tfrac{\lambda}{2}\nabla f_{2}(\nabla u)$, with $\nabla u$, and recalling
that $\mathcal{F}^{*}(\nabla u)=1$, we have that

$\nabla u^{\top}\nabla f_{1}(\nabla u)=\mathcal{F}^{*}(\nabla u)=1$ (61)

and that

$\displaystyle\nabla u^{\top}\frac{1}{2}\nabla f_{2}(\nabla u)$
$\displaystyle=\lVert\nabla
u\rVert_{M^{*}-\omega^{*}{\omega^{*}}^{\top}}^{2}+(1-\lVert\nabla
u\rVert_{M^{*}})$ (62) $\displaystyle=\lVert\nabla
u\rVert_{M^{*}}^{2}-(1-\lVert\nabla u\rVert_{M^{*}})^{2}$ $\displaystyle\hskip
28.80008pt+(1-\lVert\nabla u\rVert_{M^{*}})$ (63) $\displaystyle=\lVert\nabla
u\rVert_{M^{*}}.$ (64)

Another calculation gives us that

$\displaystyle 1+\lVert\nabla
u\rVert_{M^{*}-\omega^{*}{\omega^{*}}^{\top}}^{2}$
$\displaystyle=1+\lVert\nabla u\rVert_{M^{*}}^{2}-(1-\lVert\nabla
u\rVert_{M^{*}})^{2}$ (65) $\displaystyle=2\lVert\nabla u\rVert_{M^{*}},$ (66)

thus $\nabla u^{\top}\frac{1}{2}\nabla f_{2}(\nabla
u)=\tfrac{1}{2}(1+\lVert\nabla
u\rVert_{M^{*}-\omega^{*}{\omega^{*}}^{\top}}^{2})$. As $\nabla f_{1}(\nabla
u)=\tfrac{\lambda}{2}\nabla f_{2}(\nabla u)$, we can deduce that

$\lambda=\frac{2}{1+\nabla u^{\top}(M^{*}-\omega^{*}{\omega^{*}}^{\top})\nabla
u}.$ (67)

Under the assumption that the metric is close to Riemannian, i.e.
$\lVert\omega\rVert$ is small, then we can approximate $\lVert\nabla
u\rVert_{M^{*}}=\mathcal{F}^{*}(\nabla u)=1$, and thus approximate the
proportionality coefficient $\lambda$ by

$\lambda=\frac{2}{1+\lVert\nabla u\rVert_{M^{*}}}=1.$ (68)

Using Eqs. 59 and 60, we conclude that for this particular initialisation the
heat equation becomes in short time

$\partial_{t}u=\operatorname{div}\left((M^{*}-\omega^{*}{\omega^{*}}^{\top})\nabla
u\right)+\operatorname{div}(\omega^{*}).\qed$ (69)

##  3 Implementation and runtime

In terms of runtime and memory, the difference with [29] comes from a small
overhead due to computing the metric as preprocessing. Our additional actions
involve cheap basic operations on $3\times 3$ matrices, e.g. inversion (Eq.
37) and multiplication (Eqs. 22, 23, 36 and 38). Note that further
preprocessing operations, e.g. Eq. 39, are the same as those in [29] and thus
have the same cost. For example, our implementation on a CPU ‘2 GHz Intel Core
i5’ takes on average 1.70s per shape in the FAUST (Remeshed) dataset (1.82s
per shape in the SCAPE (Remeshed) dataset), compared to 1.55s (resp. 1.76s)
for [29]. Once this preprocessing is performed, we obtain an anisotropic LBO
discretised as a matrix (line 432) of the same size as in [29]. Therefore,
runtime of the core shape-matching learning algorithm is the same and is
significantly longer, around 30s per epoch for FAUST (Remeshed) (resp. 10s for
SCAPE (Remeshed)) on 1 GPU ‘Quadro P6000’.

##  4 Further visual results

###  4.1 Example kernels

We show in Fig. 5 two of our anisotropic kernels $g_{\alpha\theta,x}$ at
different angles, when choosing a predetermined Chebychev polynomial for its
spectral decomposition.

###  4.2 More shape correspondence results

We display further visual shape correspondence results in Figs. 6, 7, 8 and 9.
Our method clearly outperforms FieldConv while providing on par visual results
to ACSCNN. The superiority of our method is revealed in the quantitative
results of the main paper.

FieldConv ACSCNN Ours

Figure 6:  Visual shape correspondence results on the FAUST Remeshed dataset.
The source shape is on the left, on which we perform dense correspondence
estimation on the shapes to the right, using either FieldConv, ACSCNN, or our
approach.

FieldConv ACSCNN Ours

Figure 7:  Visual shape correspondence results on the SCAPE Remeshed dataset.
The source shape is on the left, on which we perform dense correspondence
estimation on the shapes to the right, using either FieldConv, ACSCNN, or our
approach.

ACSCNN Ours

Figure 8:  Visual shape correspondence results on the SHREC’16 Cuts dataset.
The source shape is on the left, on which we perform dense correspondence
estimation on the shapes to the right, using either ACSCNN or our approach.
FieldConv fails to provide meaningful correspondence results on partial shapes
so we do not display its qualitative performance.

ACSCNN Ours

Figure 9:  Visual shape correspondence results on the SHREC’16 Holes dataset.
The source shape is on the left, on which we perform dense correspondence
estimation on the shapes to the right, using either ACSCNN or our approach.
FieldConv fails to provide meaningful correspondence results on partial shapes
so we do not display its qualitative performance.