Figure 1.  Neural Geometry Processing. We encode input genus-0 surfaces as
overfitted neural networks and propose operators on them. Specifically, we
describe how to compute the geometric Jacobian and the First and Second
Fundamental Forms, and hence compute curvatures. We also define a Laplace
Beltrami operator directly using the neural representation, thus enabling
processing of scalar (or vector) fields on the underlying surface as well as
performing spectral analysis. We avoid any unnecessary discretization, as
commonly encountered while using a traditional surface representation (e.g., a
polygonal mesh).

# Neural Geometry Processing via Spherical Neural Surfaces

Romy Williamson  University College London<EMAIL_ADDRESS>and
Niloy J. Mitra  University College London<EMAIL_ADDRESS>
###### Abstract.

Neural surfaces (e.g., neural map encoding, deep implicits and neural radiance
fields) have recently gained popularity because of their generic structure
(e.g., multi-layer perceptron) and easy integration with modern learning-based
setups. Traditionally, we have a rich toolbox of geometry processing
algorithms designed for polygonal meshes to analyze and operate on surface
geometry. However, neural representations are typically discretized and
converted into a mesh, before applying any geometry processing algorithm. This
is unsatisfactory and, as we demonstrate, unnecessary. In this work, we
propose a spherical neural surface representation (a spherical
parametrization) for genus-0 surfaces and demonstrate how to compute core
geometric operators directly on this representation. Namely, we show how to
construct the normals and the first and second fundamental forms of the
surface, and how to compute the surface gradient, surface divergence and
Laplace Beltrami operator on scalar/vector fields defined on the surface.
These operators, in turn, enable us to create geometry processing tools that
act directly on the neural representations without any unnecessary meshing. We
demonstrate illustrative applications in (neural) spectral analysis, heat flow
and mean curvature flow, and our method shows robustness to isometric shape
variations. We both propose theoretical formulations and validate their
numerical estimates. By systematically linking neural surface representations
with classical geometry processing algorithms, we believe this work can become
a key ingredient in enabling neural geometry processing.

neural representation, continuous surface representation, spectral analysis,
meshfree representation, neural geometry processing

††submissionid: 0461††journal: TOG††ccs: Computing methodologies Shape
analysis††ccs: Computing methodologies Mesh geometry models††ccs: Computing
methodologies Shape representations††ccs: Computing methodologies Machine
learning approaches

##  1\. Introduction

The emergence of neural networks to represent surfaces has rapidly gained
popularity as a generic surface representation. Many variants (Park et al.,
2019; Mescheder et al., 2018; Groueix et al., 2018; Qi et al., 2016) have been
proposed to directly represent a single surface, as an overfitted network, or
a collection of surfaces, as encoder-decoder networks. The ‘simplicity’ of the
approach lies in the generic network architectures (e.g., MLPs, UNets) being
universal function approximators. Such deep representations can either be
overfitted to given meshes (Davies et al., 2020; Morreale et al., 2021) or
directly optimized from a set of posed images (i.e., images with camera
information) via an optimization, as popularized by the Nerf formulation
(Mildenhall et al., 2021). Although there are some isolated and specialized
attempts (Novello et al., 2022; Chetan et al., 2023; Yang et al., 2021), no
established set of operators directly works on such neural objects, encoding
an explicit surface, to facilitate easy integration with existing geometry
processing tasks.

Traditionally, triangle meshes, more generally polygonal meshes, are used to
represent surfaces. They are simple to define, easy to store, and benefit from
having a wide range of available algorithms supporting them. Most geometry
processing algorithms rely on the following core operations: (i) estimating
differential quantities (i.e., surface normals, and first and second
fundamental forms) at any surface location; (ii) computing surface gradients
and surface divergence on scalar/vector fields; and (iii) defining a Laplace
Beltrami operator (i.e., the generalization of the Laplace operator on a
curved surface) for analyzing the shape itself or functions on the shape.

Meshes, however, are discretized surface representations, and accurately
carrying over definitions from continuous to discretized surfaces is
surprisingly difficult. For example, the area of discrete differential
geometry (DDG) (Desbrun et al., 2006) was entirely invented to meaningfully
carry over continuous differential geometry concepts to the discrete setting.
However, many commonly used estimates vary with the mesh quality (i.e.,
distribution of vertices and their connections), even when encoding the same
underlying surface. In the case of the Laplace Beltrami operator, the ‘no free
lunch’ result (Wardetzky et al., 2007) tells us that we cannot hope for any
particular discretization of the Laplace Beltrami operator to simultaneously
satisfy all of the properties of its continuous counterpart. In practice, this
means that the most suitable choice of discretization (see (Bunge et al.,
2023) for a recent survey) may change depending on the specifics of the target
application.

In the absence of native geometric operators for neural representations, the
common workaround for applying a geometry processing algorithm to neural
representations is discretizing these representations into an explicit mesh
(e.g., using Marching cubes) and then relying on traditional mesh-based
operators. Unfortunately, this approach inherits the disadvantages of having
an explicit mesh representation, as listed above. A similar motivation
prompted researchers to pursue a grid-free approach for volumetric analysis
using Monte Carlo estimation for geometry processing (Sawhney and Crane, 2020)
or neural field convolution (Nsampi et al., 2023) to perform general
continuous convolutions with continuous signals such as (overfitted) neural
fields.

We avoid discretization altogether. First, we create a smooth and
differentiable representation — spherical neural surfaces — where we encode
individual genus-0 surfaces using dedicated neural networks. Then, we exploit
autograd to compute surface normals, tangent planes, and Fundamental Forms
directly with continuous differential geometry. Knowing the First Fundamental
Form allows us to compute attributes of the parametrization, such as local
area-distortion. Knowing the Second Fundamental form allows us to compute
surface attributes such as mean curvature, Gauss curvature, and principal
curvature directions — without any unnecessary discretization error.
Essentially, encoding surfaces as seamless neural maps, almost trivially,
allows us to access definitions from differential geometry.

Other estimates require more thought. We use the computed surface normals and
mean curvatures to apply the continuous Laplace Beltrami operator to any
differentiable scalar/vector field. Additionally, to enable spectral analysis
on the resultant Laplace Beltrami operator, we propose a novel optimization
scheme and a neural representation of scalar fields that allows us to extract
the lowest $k$ spectral modes, using the variational formulation of the
Dirichlet eigenvalue problem. We use Monte Carlo estimates to approximate the
continuous inner products and integration of functions on the (neural)
surface. Our framework of spherical neural surfaces makes optimizing many
expressions involving surface gradients and surface integrals of smooth
functions possible. We hope that our setup opens the door to solving other
variational problems posed on smooth (neural) surfaces.

We evaluate our proposed framework on standard geometry processing tasks. In
the context of mesh input, we assess the robustness of our estimates to
different meshing of the same underlying surfaces (Figure 11) and
isometrically related shapes (Figure 10). We compare the results against
ground truth values when they are computable (e.g., analytical functions - see
Figure 3 and Figure 4) and contrast them against alternatives, where
applicable (Figure 11).

Our main contribution is introducing a novel representation and associated
operators to enable neural geometry processing.
We:

  1. (i)

introduce spherical neural surfaces as a representation to encode genus-0
surfaces seamlessly as overfitted neural networks;

  2. (ii)

compute surface normals, First and Second Fundamental Forms, curvature,
continuous Laplace Beltrami operators, and their smallest spectral modes
without using any unnecessary discretization;

  3. (iii)

show illustrative applications towards scalar field manipulation and neural
mean curvature flow; and

  4. (iv)

demonstrate the robustness and accuracy of the setup on different forms of
input.

##  2\. Related Works

#### Neural representations

Implicit neural representations, such as signed distance functions (SDFs) and
occupancy networks, have recently been popularized to model 3D shapes.
Notably, DeepSDF (Park et al., 2019) leverages a neural network to represent
the SDF of a shape, enabling continuous and differentiable shape
representations that can be used for shape completion and reconstruction
tasks; (Davies et al., 2020) use neural networks to overfit to individual
SDFs; Occupancy Networks (Mescheder et al., 2018) learn a continuous function
to represent the occupancy status of points in space, providing a powerful
tool for 3D shape modeling and reconstruction from sparse input data. Explicit
neural representations, on the other hand, use neural networks to directly
predict structured 3D data such as meshes or point clouds. Mesh R-CNN
(Gkioxari et al., 2019) extends Mask-RCNN to predict 3D meshes from images by
utilizing a voxel-based representation followed by mesh refinement with a
graph convolution network operating over the mesh’s vertices and edges. In the
context of point clouds, PointNet (Qi et al., 2016) and its variants are
widely used as backbones for shape understanding tasks like shape
classification and segmentation. Hybrid representations combine the strengths
of both implicit and explicit methods. For example, Pixel2Mesh (Wang et al.,
2018) generates 3D meshes from images by progressively deforming a template
mesh using a graph convolutional network, integrating the detailed geometric
structure typical of explicit methods with the continuous nature of implicit
representations. More relevant to our work is the AtlasNet (Groueix et al.,
2018) that models surfaces as a collection of parametric patches, balancing
flexibility and precision in shape representation, and Neural Surface Maps
(Morreale et al., 2021), which are overfitted to a flattened disc
parametrization of surfaces to enable surface-to-surface mapping.

#### Estimating differential quantities

Traditionally, researchers have investigated how to carry over differential
geometry concepts (do Carmo, 1976) to surface meshes, where differential
geometry does not directly apply because mesh faces are flat, with all the
‘curvature’ being at sharp face intersections. Taubin (1995) introduce several
signal processing operators on meshes. Meyer et al. (2002) used averaging
Voronoi cells and the mixed Finite-Element/Finite-Volume method; (Cazals and
Pouget, 2005) used osculating jets; while (de Goes et al., 2020) used discrete
differential geometry (Desbrun et al., 2006) to compute gradients, curvatures,
and Laplacians on meshes. Recently, researchers have used learning-based
approaches to ‘learn’ differential quantities like normals and curvatures on
point clouds (Guerrero et al., 2018; Ben-Shabat et al., 2019; Pistilli et al.,
2020).

More related to ours is the work by Novello et al. (2022), who represent
surfaces via implicit functions and analytically compute differential
quantities such as normals and curvatures. With a similar motivation, Chetan
et al. (2023) fit local polynomial patches to obtain more accurate derivatives
from pre-trained hybrid neural fields; they also use these as higher-order
constraints in the neural implicit fitting stage. A similar idea was explored
by Bednarik et al. (2020), as a regularizer for the AtlasNet setup, by using
implicit differential surface properties during training. The work by Yang and
colleagues (2021) is closest to ours in motivation - they also mainly focus on
normal and curvature estimates but they additionally demonstrate how to
perform feature smoothing and exaggeration using the local differential
quantities in the target loss functions and thus driving the modification of
the underlying neural fields. However, we go beyond normal and curvature
estimates and focus on operators such as surface gradient, surface divergence
and Laplace Beltrami operators for processing scalar and vector fields defined
on the surfaces.

#### Laplace Beltrami operators on meshes

The Laplace-Beltrami operator (LBO) is an indispensable tool in spectral
geometry processing, as it can be used analyze and manipulate the intrinsic
properties of shapes. Spectral mesh processing (Lévy and Zhang, 2010)
leverages the eigenvalues and eigenfunctions of the LBO for tasks such as mesh
smoothing and shape segmentation. Many subsequent efforts have demonstrated
the use of LBO towards shape analysis and understanding (e.g., shapeDNA
(Reuter et al., 2006), global point signatures (Rustamov, 2007), heat kernel
signature (Sun et al., 2009)), eventually culminating into the functional map
framework (Ovsjanikov et al., 2012). An earlier signal processing framework
(Taubin, 1995) has been revisited with the LBO operator (Lévy and Zhang, 2010)
for multi-scale processing applications (e.g., smoothing and feature
enhancement). Although the Laplace-Beltrami operator on triangular meshes or,
more generally, on polyhedral meshes, can be computed, these operators depend
on the underlying discretization of the surface (i.e., placement of vertices
and their connectivity). See (Bunge et al., 2023) for a discussion. In the
case of triangular meshes, the commonly used discretizations are the Uniform
LBO and the cotan LBO. In Section 6, we compare our neural LBO with the cotan
discretization and discuss their relative merits.

##  3\. Spherical Neural Surfaces

We introduce a new neural representation called a ‘Spherical Neural Surface’
(SNS) to represent continuous surfaces. An SNS is a Multi-Layer Perceptron
(MLP) $S_{\theta}:\,\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}$, which we train
so that the restriction to the unit sphere
($\mathbb{S}^{2}\subset\mathbb{R}^{3}$) is a continuous parametrization of a
given surface. In terms of notation, we will also use $S_{\theta}$ or $S$ as
shorthand for the set $S_{\theta}(\mathbb{S}^{2})$. In our current work, we
primarily focus on creating SNSs from input triangle-meshes and analytically-
defined surfaces, but there is potential to create SNSs from other
representations such as pointclouds, SDFs, etc. Figure 12 shows initial
results for generating an SNS from a neural SDF.

###  3.1. Overfitting an SNS to a Triangle-Mesh

Given a genus-0 manifold triangle-mesh, we first find an injective embedding
of the mesh vertices onto the unit sphere, using a spherical embedding
algorithm (Schmidt et al., 2023; Praun and Hoppe, 2003). We extend the
embedding to a continuous embedding by projecting points on the sphere to the
discrete sphere, and employing barycentric coordinates so that every point
$p_{i}$ on the sphere corresponds to a unique point $q_{i}$ on the target
surface. We overfit the network $S_{\theta}$ to this parametrization by
minimizing the MSE between the ground truth and predicted surface positions,
as:

(1)
$\mathcal{L}_{MSE}:=\frac{1}{N}\sum\limits_{i=1}^{N}\left\|S_{\theta}(p_{i})-q_{i}\right\|^{2}.$

Further, to improve the fitting, we encourage the normals derived from the
spherical map to align with the normals of points on the mesh, via a
regularization term,

(2)
$\mathcal{L}_{\textit{normal}}:=\frac{1}{N}\sum_{i=1}^{N}\left\|\mathbf{n}_{S_{\theta}}({p_{i}})-\mathbf{n}_{\textit{mesh}}({p_{i}})\right\|^{2}=\frac{2}{N}\sum_{i=1}^{N}(1-\cos{\alpha_{i}(p_{i})}).$

In this expression, $\mathbf{n}_{\textit{mesh}}({p_{i}})$ is the (outwards)
unit-normal on the mesh, and $\mathbf{n}_{S_{\theta}}({p_{i}})$ is the
corresponding (outwards) unit-normal of the smooth parametrization at $p_{i}$,
which is derived analytically from the Jacobian of $S_{\theta}$ at $p_{i}$
(see Section 3.2); the angle $\alpha(p_{i})$ is the angle between
$\mathbf{n}_{\textit{mesh}}({p_{i}})$ and $\mathbf{n}_{S_{\theta}}({p_{i}})$.
See Figure 2 for an example.

Figure 2.  Spherical Neural Surface. Given an input surface — a mesh in this
case — we progressively overfit an MLP to represent the surface as
$S_{\theta}$. The loss term is simply the MSE between the ground truth and
predicted surface positions, plus a (scaled) normals regularization term
(Equation 2).

###  3.2. Computing Differential Quantities using SNS

One of the advantages of Spherical Neural Surfaces as a geometry
representation is that it is extremely natural to compute many important
quantities from continuous differential geometry - without any need for
approximation or discretization. This is thanks to the automatic
differentiation functionality built into modern machine learning setups and
algebraically tracking some changes of variables.

For example, to compute the outwards-pointing normal $\mathbf{n}_{S}$, we
simply compute the $3\times 3$ Jacobian of $S$ — $\mathbf{J}({S})$ — and then
turn this into a Jacobian for a local 2D parametrization by composing it with
a $3\times 2$ orthonormal matrix ($R$):

(3)
$\mathbf{J}^{\textit{local}}_{S}=\mathbf{J}({S})\,R=\begin{pmatrix}\textbar&\textbar\\\
S_{u}&S_{v}\\\ \textbar&\textbar\\\ \end{pmatrix}.$

The columns of $\mathbf{J}^{\textit{local}}_{S}$ are the partial derivatives
$S_{u}$ and $S_{v}$. The matrix $R$ defines orthogonal unit vectors on the
tangent plane of the sphere at each point, and the functions $u$ and $v$ are
the corresponding orthonormal coordinates on the tangent plane of the sphere.
Then, the normalized cross product of $S_{u}$ and $S_{v}$ is the outward-
pointing unit normal to the surface:

(4) $\mathbf{n}_{S}=\frac{S_{u}\times S_{v}}{\left\|S_{u}\times
S_{v}\right\|}.$

In terms of standard spherical polar coordinates ($\theta$ and $\phi$), we
chose to use the following rotation matrix:

(5) $R=\begin{pmatrix}\cos\theta\cos\phi&-\sin\phi\\\
\cos\theta\sin\phi&\cos\phi\\\ -\sin\theta&0\\\ \end{pmatrix}.$

This matrix is an orthonormal version of the usual spherical polar coordinates
Jacobian. We avoid degeneracy at the poles by selectively re-labeling the $x$,
$y$, and $z$ coordinates so that $\theta$ is never close to zero or $\pi$.

Now, from $\mathbf{J}^{\textit{local}}_{S}$, we can simply write the First
Fundamental Form (FFF) of the sphere-to-surface map as:

(6) $\mathbf{I}_{S}=\begin{pmatrix}E&F\\\ F&G\\\
\end{pmatrix}=(\mathbf{J}^{\textit{local}}_{S})^{T}\,\mathbf{J}^{\textit{local}}_{S}.$

To compute the Second Fundamental Form (SFF), we first compute the second
partial derivatives of the parametrization with respect to the local co-
ordinates, and then find the dot products of the second derivatives with the
normal, $\mathbf{n}=\mathbf{n}_{S_{\theta}}$. The second derivatives are:

(7) $S_{uu}=\mathbf{J}(S_{u})S_{u},\quad S_{uv}=\mathbf{J}(S_{u})S_{v},\quad
S_{uu}=\mathbf{J}(S_{v})S_{v}.$

We can now express the SFF as:

(8) $\mathbf{II}_{S}=\begin{pmatrix}e&f\\\ f&g\\\
\end{pmatrix}=\begin{pmatrix}S_{uu}\cdot\mathbf{n}&S_{uv}\cdot\mathbf{n}\\\
S_{uv}\cdot\mathbf{n}&S_{vv}\cdot\mathbf{n}\\\ \end{pmatrix}.$

In terms of the FFF and SFF, the Gauss curvature is

(9) $K=\frac{eg-f^{2}}{EG-F^{2}},$

and the mean curvature is

(10) $H=\frac{Eg-2Ff+Ge}{2(EG-F^{2})}.$

Furthermore, we can algebraically solve the quadratic equation
$\text{det}(\mathbf{I}_{S}-\lambda\mathbf{II}_{S})=0$, to obtain explicit
expressions for the principal curvatures, that we then use to find the
principal curvature directions directly on the Spherical Neural Surface. In
Figure 3, we compare our differential estimates against analytical
computations for an analytic surface; in Figure 6 we also show curvature
estimates and principal curvature directions on other SNS approximations of
input triangle meshes (ground truth values are not available).

Figure 3.  First and Second Fundamental Forms. Here, we display the
differential quantities computed using our framework, on an SNS that has been
overfitted to the mesh of a surface with an analytical parametrization
(bottom) and we show that the results are extremely close to the differential
quantities that have been computed symbolically, using Matlab.

##  4\. Mathematical Background

This section briefly summarizes the relevant background for continuous
surfaces that we need to develop a computation framework for Laplace-Beltrami
and spectral analysis using SNS (Section 5).

###  4.1. The Laplace Beltrami Operator on Smooth Surfaces

The (Euclidean) Laplacian $\Delta$ of a scalar field
$f:\mathbb{R}^{n}\rightarrow\mathbb{R}$ is defined as the divergence of the
gradient of the scalar field. Equivalently, $\Delta f$ is the sum of the
unmixed second partial derivatives of the field $f$:

(11) $\Delta f:=\nabla\cdot\nabla
f=\sum_{i=1}^{n}\frac{\partial^{2}f}{\partial^{2}x_{i}}.$

The Laplace-Beltrami operator, or surface Laplacian, is the natural
generalization of the Laplacian to scalar fields defined on curved domains. If
$\Sigma\subset\mathbb{R}^{3}$ is a regular surface and
$f:\Sigma\rightarrow\mathbb{R}$ is a scalar field, then the Laplace-Beltrami
operator on $f$ is defined as the surface divergence of the surface gradient
of $f$, and we denote this by $\Delta_{\Sigma}f$ to emphasize the dependence
on the surface $\Sigma$. Thus,

(12) $\Delta_{\Sigma}f:=\nabla_{\Sigma}\cdot\nabla_{\Sigma}f.$

The surface gradient of the scalar field $f$ (written $\nabla_{\Sigma}f$), and
the surface divergence of a vector field
${\mathbf{F}}:\Sigma\rightarrow\mathbb{R}^{3}$ (written $\nabla_{\Sigma}\cdot
F$), can be computed by smoothly extending the fields to fields $\tilde{f}$
and $\tilde{\mathbf{F}}$ respectively, which are defined on a neighborhood of
$\Sigma$ in $\mathbb{R}^{3}$.

The surface gradient is defined as the projection of the gradient of
$\tilde{f}$ into the local tangent plane:

(13)
$\nabla_{\Sigma}f=\nabla\tilde{f}-(\nabla\tilde{f}\cdot\mathbf{n})\mathbf{n},$

where $\mathbf{n}$ is the unit surface normal at the point. The surface
divergence of the vector field $\mathbf{F}$ is defined as

(14)
$\nabla_{\Sigma}\cdot\mathbf{F}=\nabla\cdot\tilde{\mathbf{F}}-\mathbf{n}^{T}{\mathbf{J}}(\tilde{\mathbf{F}})\mathbf{n},$

where
$\mathbf{J}(\tilde{\mathbf{F}}):=\begin{bmatrix}\frac{\partial{\tilde{\mathbf{F}}}}{\partial{x_{1}}},&\frac{\partial{\tilde{\mathbf{F}}}}{\partial{x_{2}}},&\frac{\partial{\tilde{\mathbf{F}}}}{\partial{x_{3}}}\\\
\end{bmatrix}$ is the Jacobian of $\mathbf{F}$. Since
$\mathbf{J}(\tilde{\mathbf{F}})\mathbf{n}$ is the directional derivative of
$\tilde{\mathbf{F}}$ in the normal direction, then
$\mathbf{n}^{T}\mathbf{J}(\tilde{\mathbf{F}})\mathbf{n}=\mathbf{J}(\tilde{\mathbf{F}})\mathbf{n}\cdot\mathbf{n}$
can be thought of as the contribution to the three-dimensional divergence of
$\tilde{\mathbf{F}}$ that comes from the normal component of
$\tilde{\mathbf{F}}$, and by ignoring the contribution from the normal
component, we get the two-dimensional ‘surface divergence’. Although these
expressions depend a-priori on the particular choice of extension, the surface
gradient and surface divergence are in fact well-defined (i.e., any choice of
extension will give the same result).

By expanding out the definitions of surface gradient and surface divergence,
we can derive an alternative formula for the surface Laplacian
($\Delta_{\Sigma}f$) in terms of the (Euclidean) Laplacian
($\Delta\tilde{f}$), the gradient ($\nabla\tilde{f}$) and the Hessian
($\mathbf{H}(\tilde{f})=\mathbf{J}(\nabla\tilde{f})^{T}$) as,

(15)
$\Delta_{\Sigma}f=\Delta\tilde{f}-2H\nabla\tilde{f}\cdot\mathbf{n}-\mathbf{n}^{T}\mathbf{H}(\tilde{f})\mathbf{n}.$

The dependence on the surface is captured by the normal function $\mathbf{n}$,
and the mean curvature $H$. Please refer to (Reilly, 1982; Xu and Zhao, 2023)
for a derivation. In the case when $\tilde{f}$ is a ‘normal extension’ — i.e.,
it is locally constant in the normal directions close to the surface — then
the second and third terms disappear, meaning that $\Delta_{\Sigma}f$ is
consistent with $\Delta\tilde{f}$.

If we substitute in the coordinate function $\mathbf{x}$, then the first and
third terms disappear in the above equation, and we are left with the familiar
equation,

(16) $\Delta_{\Sigma}\mathbf{x}=-2H\mathbf{n}.$

This formula is often used in the discrete setting to compute the mean
curvature from the surface Laplacian.

###  4.2. Spectrum of the Laplace Beltrami Operator

Given two scalar functions, $f$ and $g$, defined on the same regular surface
$\Sigma$, we can define their inner product to be,

(17) $\left<f,g\right>_{L^{2}(\Sigma)}:=\int_{\Sigma}fg\,dA\,.$

Then, the $L^{2}$ norm of a scalar function can be expressed as,

(18)
$\left\|f\right\|_{L^{2}(\Sigma)}=\sqrt{\left<f,f\right>_{L^{2}(\Sigma)}}=\sqrt{\int_{\Sigma}|f|^{2}\,dA}\,.$

The Laplace Beltrami operator, $\Delta_{\Sigma}$, is a self-adjoint linear
operator with respect to this inner product. This means that there is a set of
eigenfunctions of $\Delta_{\Sigma}$ that form an orthonormal basis for the
space $L^{2}(\Sigma)$ \- the space of ‘well-behaved’ scalar functions on
$\Sigma$ that have a finite $L^{2}$-norm. This basis is analogous to the
Fourier basis for the space of periodic functions on an interval.

In the discrete mesh case, the eigenfunctions of the Laplace-Beltrami operator
are computed by diagonalizing a sparse matrix (Lévy and Zhang, 2010; Bunge et
al., 2023) based on the input mesh. In the continuous case, however, we no
longer have a matrix representation for $\Delta_{\Sigma}$. Instead, we exploit
a functional analysis result that describes the first $k$ eigenfunctions of
$\Delta_{\Sigma}$ as the solution to a minimization problem, as described
next.

First, we define the Rayleigh Quotient of a scalar function $f$ to be the
Dirichlet energy of the scalar function, divided by the squared $L^{2}$-norm
of the function:

(19)
$Q_{\Sigma}(f):=\frac{\left\|\nabla_{\Sigma}f\right\|_{L^{2}(\Sigma)}^{2}}{\left\|f\right\|^{2}_{L^{2}(\Sigma)}}.$

Then, the eigenfunctions of $\Delta_{\Sigma}$ are the minimizers of the
Rayleigh Quotient, as given by,

$\Psi_{0},\Psi_{1},...,\Psi_{k-1}=\text{argmin}_{\Psi_{0}^{*},...,\Psi_{k-1}^{*}}\sum_{i=0}^{k-1}Q_{\Sigma}(\Psi_{i}^{*})$
(20) $\textit{such
that}\quad\left<\Psi_{i},\Psi_{j}\right>_{L^{2}(\Sigma)}=0\quad\text{for
all}\quad i\neq j.$

We use $\Psi_{0}$ to represent the first eigenfunction, which is always a
constant function. The constraint states that the eigenfunctions are
orthogonal inside the inner-product space $L^{2}(\Sigma)$.

Figure 4.  Neural Spectral Basis against Analytical Solutions. The sphere is a
genus-0 surface for which the eigenspaces of the Laplace-Beltrami operator are
very well understood: the eigenspaces are the spaces spanned by linear
combinations of the spherical harmonics for each frequency. This means that
each of our optimized eigenfunctions should belong to one of these
eigenspaces, and therefore, the Rayleigh quotient should match the energy
level of the corresponding spherical harmonics. For evaluation, we compute the
Rayleigh quotient using a large number of samples for Monte Carlo integration,
since we cannot analytically compute the integral. For reference, we present
the ground truth Rayleigh quotients, in green.

In addition, the Rayleigh Quotient of the eigenfunction $\Psi_{i}$ is the
positive of the corresponding eigenvalue, i.e.,

(21)
$\Delta_{\Sigma}\Psi_{i}(x)=-Q_{\Sigma}(\Psi_{i})\Psi_{i}(x)\quad\forall\,x\in\Sigma.$

Physically, the eigenfunctions of $\Delta_{\Sigma}$ represent the fundamental
modes of vibration of the surface $\Sigma$. The Rayleigh Quotient of
$\Psi_{0}$ is zero, and as the Rayleigh quotient increases, the vibrational
modes increase in energy and the eigenfunctions produce more ‘intricate’ (high
frequency) patterns.

Figure 5.  Encoding Scalar Fields on Neural Surfaces. We encode scalar fields
on the sphere (and corresponding extensions into $\mathbb{R}^{3}$) as MLPs
$g_{\eta}:\mathbb{R}^{3}\rightarrow\mathbb{R}$. Because $S_{\theta}$,
considered as a parametrization, is bijective, then we can implicitly define
any smooth scalar field $h$ on the surface in terms of one of the functions
$g_{\eta}$ (restricted to $\mathbb{S}^{2}$). We visualize the one-to-one
correspondence between scalar fields defined on the surface, and scalar fields
defined on the sphere, by using vertex-colors to display the fields.  Figure
6.  First and Second Fundamental Forms. An SNS represents a smooth, seamless
parametrization of a genus-0 surface, and we can compute the partial
derivatives of this parametrization using automatic differentiation. We
compute the normals and the First Fundamental Form of the parametrization from
the first derivatives. Subsequently, computing the second derivatives enables
us to construct the Second Fundamental Form of the parametrization. Hence, we
compute Mean Curvature, Gaussian Curvature, and Principal Curvature Directions
at any point without any unnecessary discretization error.

##  5\. Spectral Analysis using SNS

To find the continuous eigenmodes of the Laplace-Beltrami operator on an SNS,
we require a continuous and differentiable representation for scalar functions
defined on the surface. In our framework, we represent such smooth scalar
fields on $\mathbb{S}^{2}$ by MLPs, $g_{\eta}$, from $\mathbb{R}^{3}$ to
$\mathbb{R}$. We only ever evaluate the MLP $g_{\eta}$ on
$\mathbb{S}^{2}\in\mathbb{R}^{3}$, but because the domain is $\mathbb{R}^{3}$
then it defines an extension of the scalar field, and this allows us to
compute $\nabla g_{\eta}$. Then, if
$S_{\theta}:\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}$ is an SNS, any smooth
scalar field $h$ defined on $S_{\theta}$ can be defined implicitly by the
equation,

(22) $h\circ S_{\theta}=g_{\eta}.$

Applying the chain rule to the implicit equation, we can compute the gradient
of $h$:

(23) $\nabla h=(\mathbf{J}(S_{\theta})^{T})^{-1}\nabla g_{\eta},$

which allows us to compute $\nabla_{S_{\theta}}h$ (Equation 13) and
$Q_{S_{\theta}}(h)$ (Equation 19) without explicitly computing $h$.

To optimize for a (smooth) scalar function $g_{\eta_{k}}$ so that $h$
approximates the $k$th non-constant eigenfunction of the Laplace-Beltrami
operator, we use gradient descent to optimize the weights, $\eta_{k}$. We use
a combination of two loss terms:

(24)
$\mathcal{L}_{Rayleigh}:=Q_{S}(h)\quad\text{and}\quad\mathcal{L}_{ortho}:=\sum_{i=0}^{k-1}\left<h,h_{i}\right>_{L^{2}(S)}^{2},$

where we sequentially compute the eigenfunctions, and $h_{i}$ is the $i$th
smallest eigenfunction to be found. Recall that we represent each such
eigenfunction using a dedicated MLP (except for $h_{0}$, which is the constant
eigenfunction). The loss term $\mathcal{L}_{ortho}$ encourages the scalar
field $h$ to be orthogonal to all previously-found eigenfunctions (including
the constant eigenfunction). The quantity $Q_{S_{\theta}}(h)$ depends on the
Jacobian of $S_{\theta}$, through $\nabla_{S_{\theta}}h$ and $\nabla h$.
However, since the surface does not change during optimization and $\theta$
stays fixed, we precompute $(\mathbf{J}(S_{\theta})^{T})^{-1}$. Note that, if
we were to optimize over the subspace of the unit-norm functions in
$L^{2}(\Sigma)$, then we could replace the Rayleigh Quotient with the
Dirichlet energy. However, it was a more natural design choice for us to
parametrize the entire space $L^{2}(\Sigma)$ and optimize the Rayleigh
Quotient, rather than to parametrize only the space of unit-norm functions in
$L^{2}(\Sigma)$ and minimize the Dirichlet Energy.

Additionally, for stability, we included a regularization term designed to
keep the $L^{2}$-norm of $h$ close to one:

(25) $\mathcal{L}_{unit}:=(\left\|h\right\|_{L^{2}(S)}-1)^{2}.$

The combined loss term is expressed as,

(26)
$\mathcal{L}=\mathcal{L}_{Rayleigh}+\lambda_{ortho}\mathcal{L}_{ortho}+\lambda_{unit}\mathcal{L}_{unit}.$

We choose large initial values for the coefficients because we find that this
improves stability, and then we reduce the coefficients so that the Rayleigh
Quotient is the dominant term during the later stages of optimization (see
Section 6 for details).

Finally, since we cannot exactly compute the inner products in these loss
terms, we approximate the inner products by Monte-Carlo sampling.
Specifically, we approximate,

(27)
$\left<f,g\right>_{L^{2}(S)}\approx\frac{Area(S)}{N}\sum_{i=1}^{N}f(p_{i})g(p_{i}),$

where $\\{p_{i}\\}_{1\leq i\leq N}$ is a set of uniformly distributed points
on the surface $S_{\theta}$; $f(p_{i})$ and $g(p_{i})$ denote calls to the
scalar field MLPs. (In the overfitting stage, we normalize all our Spherical
Neural Surfaces to have an area equal to $4\pi$.) For the Monte Carlo step, we
generate $N$ uniformly distributed points on the surface $S_{\theta}$ via the
following steps (rejection sampling):

  1. (i)

Generate $M\gg N_{target}$ samples ($p_{i}$) from the 3D normal distribution
$\mathcal{N}(\mathbf{\mu}=\mathbf{0},\mathbf{\Sigma}=\mathbf{I})$. Normalize,
to unit norm, to produce a dense uniform distribution of points on the sphere
$\mathbb{S}^{2}\subset\mathbb{R}^{3}$.

  2. (ii)

Compute the local area distortion ($d_{i}=\sqrt{E_{i}G_{i}-F_{i}^{2}}$), for
each point $p_{i}$, using the First Fundamental Form.

  3. (iii)

Perform rejection sampling, such that the probability of keeping point $p_{j}$
is equal to $\frac{d_{j}}{\sum_{i=1}^{M}d_{i}}N_{target}$.

Following this process, the expected number of points will equal $N_{target}$
— a specified parameter — however, the actual number of points ($N$) may vary.

Figure 7.  Spectral Modes on Neural Surfaces. We propose how to compute the
lowest few spectral modes of the neural Laplace Beltrami operator
$\Delta_{S_{\theta}}$ via minimization of the Rayleigh quotient, without
requiring any unnecessary discretization. Here we show the lowest few spectral
modes of different SNSs. The lowest spectral mode is constant, and not
included in this figure. Each of the spectral modes is represented by a
dedicated MLP (see Equation 22).  Figure 8.  Heat Flow (left) and Mean
Curvature Flow (right). (Left) We can evolve a given scalar field, $f$,
specified over the surface of an SNS, $S_{\theta}$, using the Partial
Differential Equation (PDE) $f_{t}=\Delta_{S_{\theta}}f$ (the heat equation).
We represent the evolving scalar field as a small MLP, whose weights are
‘finetuned’ for up to 100 epochs after every time step. Darker colors denote
low values (cold) with lighter colors being high (hot). (Right) Taking
advantage of the differentiable nature of our representation, we can also
compute a mesh-free Mean Curvature Flow (MCF) in which the Mean Curvature and
normals are updated at every iteration. We update the Spherical Neural Surface
using a small number of finetuning steps, after every iteration of the flow.
This formulation of MCF prevents singularities from forming, without any
special handling to prevent them (Kazhdan et al., 2012).  Figure 9.  Laplace
Beltrami on Neural Scalar Fields. We present two ways to compute Laplace
Beltrami operators on any smooth scalar field, $f$, defined on a neural
surface $S_{\theta}$ (top): first, using mean curvature estimates (top-middle)
(Equation 15) and second, using the divergence of gradient definition
(divgrad) (bottom-middle) (Equation 12). We represent scalar fields on
surfaces using dedicated MLPs. (Bottom) For comparison, we also show results
using the cotan LBO (with a lumped mass matrix) on a dense mesh, whose
vertices are the image of the vertices in a dense icosphere mesh under the
transformation $S_{\theta}$.  Figure 10.  Spectral SNS Modes across Isometric
Poses. If we compute the first few eigenfunctions on SNSs that represent near-
isometric surfaces, and order the functions according to their Rayleigh
quotients, we see a clear correspondence between corresponding eigenfunctions
on each surface. Although our computation of the LBO and its spectrum involves
extrinsic terms such as mean curvature and normals, the Laplace Beltrami
operator is intrinsic so its spectrum does not depend on the particular
embedding of a surface into $\mathbb{R}^{3}$, therefore the eigenmodes of
isometric SNSs should appear similar (up to a possible sign change).  Figure
11.  Robustness of Neural LBO. We demonstrate that our optimized
eigenfunctions are largely unaffected by differences in mesh density on the
starting mesh, because we work directly with a smooth surface. Here we show
that while the cotan discretization can be very inconsistent across changes in
mesh density, the neural LBO is more consistent when applied to a scalar field
defined on an SNS that is generated from different meshes of the same surface.

##  6\. Evaluation

#### Implementation Details

All of our networks in our experiments were trained on Nvidia GeForce P8 GPUs.
The SNS network is a Residual MLP with input and output of size three, and
eight hidden layers with 256 nodes each. The networks used to represent scalar
fields (in the eigenfunction and heat flow experiments) are very small
Residual MLPs with input size three, output size three and two hidden layers
with 10 nodes each. (To make the output value a scalar, we take the mean of
the three output values.) For both scalar fields and SNS, the activation
function is ‘Softplus’. We use the RMSProp optimizer with a learning rate of
$10^{-4}$ and a momentum of 0.9. We trained each SNS for up to 20,000 epochs
(eight-to-ten hours), and we fixed the normals regularization coefficient to
be $10^{-4}$. The time per epoch increases slightly with the number of
vertices in the mesh, but for simpler shapes (such as the sphere) the
optimization converges in fewer epochs.

#### Eigenfunction Optimization

In the eigenfunction experiment, we used the initial parameter values
$\lambda_{ortho}=10^{3}$ and $\lambda_{unit}=10^{4}$ and then we reduced the
coefficients linearly to $\lambda_{ortho}=1$ and $\lambda_{unit}=10^{2}$, over
10,000 epochs. These settings prevent the scalar field from collapsing to the
zero-function during the initial stages of optimization, whilst allowing the
Rayleigh Quotient to dominate the loss in the later stages. We optimized each
eigenfunction for up to 40,000 epochs (approximately six hours on our setup).

We generate $M=100,000$ points for the initial uniform distribution on the
sphere, and we use $N_{target}=10,000$ in the rejection sampling process. The
points stay fixed during each optimization stage. We believe that this is a
possible reason why the Rayleigh Quotient computed on the training points is
often slightly lower than the Rayleigh Quotient that is computed with a
different, larger set of uniform samples, and a better sampling strategy might
improve the accuracy of our optimization.

#### Comparison against Analytic Functions

For some of our SNS computations, there exist special cases for which an exact
analytic solution can be calculated. To test our estimates of curvature and
principal curvature directions, we overfitted an SNS to a mesh of the closed
surface with the parametrization

(28)
$\mathbf{r}(\theta,\phi)=(1+0.4\sin^{2}\theta\sin(6\theta)\sin(6\phi))\begin{pmatrix}\sin\theta\cos\phi;\sin\theta\sin\phi;\cos\theta\end{pmatrix}.$

We used the Matlab Symbolic Math Toolbox™ to compute the normals, mean
curvature, Gauss curvature and principal curvature directions in closed form.
In Figure 3, we show these differential quantities on the mesh of the analytic
surface and on the SNS. The computed curvatures align extremely closely,
therefore the overfitting quality is very high and the curvature computations
are accurate. There is a slight discrepancy at the poles, because we have used
a global analytic parametrization which not valid where
$\theta\in\\{0,\pi\\}$. On the other hand, in the SNS formulation we use
selective co-ordinate relabelling to avoid issues related to using spherical
polar co-ordinates close to the poles.

For evaluation of our optimized eigenfunctions, we overfitted an SNS to a
sphere-mesh with 10,242 vertices and optimized for the first five non-constant
eigenfunctions. The eigenspaces of the LBO on a sphere are the spaces spanned
by the spherical harmonics of each frequency (refer to (Jarosz, 2008)). The
first three spherical harmonics have eigenvalue -2 and the following five
spherical harmonics have eigenvalue -6. Although the basis for each eigenspace
is not unique, the first three eigenfunctions should also have an eigenvalue
of -2 and the following five eigenfunctions should have an eigenvalue of -6,
meaning that the Rayleigh Quotients should be equal to 2 and 6 (see Figure 4).
We computed the ground-truth eigenvalues by manually integrating the Cartesian
formulas (Jarosz, 2008) and we estimated the integral of the neural
eigenfunctions using 100k Monte Carlo samples. Estimates were within $1\%$ of
the ground-truth, and four out of five results were within $0.5\%$.

#### Neural LBO on Scalar Fields

To evaluate the accuracy of our Laplace-Beltrami operator on SNSs, we
generated random scalar fields using sine waves, and visually compared the
results of applying three different forms of the LBO, on five different
surfaces (see Figure 9). To compute $\Delta_{S_{\theta}}f$ using the mean
curvature formulation (Equation 15), we computed the gradient and Hessians of
the scalar field analytically and combined these with the SNS estimates of the
normals and mean curvature. We also computed $\Delta_{S_{\theta}}f$ using the
‘divergence of gradient’ definition (Equation 12) - in this version, all of
the differential calculations were carried out by autograd. We show the
results using vertex-colors, on a dense icosphere mesh.

Finally, we computed the result of applying the cotan LBO to the scalar field
(sampled at the vertices of the dense icosphere mesh). At this resolution, all
three versions of the LBO are extremely close, however, we notice that the
cotan estimate on the ‘spike ball’ (column 2) contains a small discontinuity,
not present in the SNS estimates. From a smooth shape and a smooth scalar
field, the resulting $\Delta_{S}$ should be smooth, which is the case for SNS
estimates.

#### Heat Flow and Mean Curvature Flow

We can approximate the evolution of any initial scalar field $f$ on an SNS
$S_{\theta}$, according to the process defined by the heat equation,
$f_{t}=\Delta_{S_{\theta}}f.$ In our implementation, we represent the field
$f$ implicitly, using Equation 22. In each time step, we compute
$\Delta_{S_{\theta}}f$ for the current scalar field $f$, using the divergence
of gradient formula in Equation 12. Then we select a small value ($d=10^{-3}$)
and we compute $f+d\Delta_{S_{\theta}}f$ at some fixed sample points (in
Figure 8-left, we used 10,242 points). We then update the scalar field, by
finetuning the network for up to 100 epochs to fit the new sample points,
using a simple MSE loss. Qualitatively, the experiment aligns with our
expectations. The scalar field gradually becomes more smooth over time, and
cold areas warm up to a similar heat level as the surroundings.

Similarly, we approximate a Mean Curvature Flow of an SNS by updating the SNS
to $S_{\theta}-dH\mathbf{n}$ at each time step, and finetuning for up to 100
epochs. The results in Figure 8-right show the spiky analytic shape becoming
closer to a sphere, over 150 iterations.

Figure 12.  Handling Neural Implicits. Spherical Neural Surfaces can be
generated from other neural representations, such as DeepSDF. Here, we have
produced an SNS from a DeepSDF representation for a camera, by overfitting the
SNS to a low-resolution marching-cubes mesh reconstruction and then finetuning
this SNS to better align with the SDF.

#### Neural Implicits

To demonstrate robustness to the input representation, we provide an
illustratory example of an SNS generated from a DeepSDF (Park et al., 2019),
and show the normals and principal curvature directions on its SNS. We took a
pretrained DeepSDF network and the optimized latent code for the camera shape
(Comi, 2023). Then, as initialization, we overfitted an SNS to a coarse
marching-cubes mesh. Then we took a set of samples on the SNS and projected
them to deepSDF representation defined by the signed distance function (SDF),
by moving them in the direction of the gradient of the SDF (which approximates
the surface normal) by the signed distance at that point, and use them to
finetune the SNS.

##  7\. Conclusion

We have presented spherical neural surfaces as a novel representation for
genus-0 surfaces along with supporting operators in the form of differential
geometry estimates (e.g., normals, First and Second Fundamental Forms), as
well as surface gradients and surface divergence operators. We also presented
how to compute a continuous Laplace Beltrami operator and its lowest spectral
modes. We demonstrated that ours, by avoiding any unnecessary discretization,
produces robust and consistent estimates even under different meshing
concerning vertex sampling as well as their connectivity.

### Limitations and Future Work

#### Beyond genus-0 surfaces

Since our SNSs rely on a sphere for the parametrization, it limits us to
genus-0 surfaces. If we could choose from a range of canonical surfaces (e.g.,
torus), this would allow us to seamlessly process higher genus shapes and,
possibly, to produce neural surfaces without such high distortions. Another
option would be to rely on local parametrizations, but then we would have to
consider blending across overlapping parametrizations.

#### Speed

A severe limitation of our current realization is its long running time. While
we expect that better and optimized implementations would increase efficiency
significantly, we also need to make changes to our formulation. Specifically,
our current spectral estimation is sequential, which is not only slow, but
leads to an accumulation of errors for later spectral modes. Hence, in the
future, we would like to jointly optimize for multiple spectral modes.

#### SNS from neural representations

We demonstrated our setup mainly on mesh input and also on neural input in the
form of deepSDF. In the future, we would like to extend our SNS to other
neural representations in the form of occupancy fields or radiance fields.
However, this will require locating and sampling points on the surfaces, which
are implicitly encoded – we need the representation to provide a projection
operator. We would also like to support neural surfaces that come with level-
of-detail. Finally, an interesting direction would be to explore end-to-end
formulation for dynamic surfaces encoding temporal surfaces with SNS, and
enabling optimization with loss terms involving first/second fundamental forms
as well as Laplace-Beltrami operators (e.g., neural deformation energy).

#### Acknowledgements

This project has received funding from the UCL AI Centre. RW wishes to thank
Johannes Wallner for his constructive comments regarding Section 4.

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