Geometric implicit neural representations for signed distance
functions
Luiz Schirmer∗Tiago Novello Vin´ ıcius da Silva Guilherme Schardong
Daniel Perazzo H´ elio Lopes Nuno Gon¸ calves Luiz Velho
November 11, 2025
Abstract
Implicit neural representations(INRs) have emerged
as a promising framework for representing signals in
low-dimensional spaces. This survey reviews the ex-
isting literature on the specialized INR problem of
approximatingsigned distance functions(SDFs) for
surface scenes, using either oriented point clouds or a
set of posed images. We refer to neural SDFs that in-
corporate differential geometry tools, such as normals
and curvatures, in their loss functions asgeometric
INRs. The key idea behind this 3D reconstruction ap-
proach is to include additionalregularizationterms in
the loss function, ensuring that the INR satisfies cer-
tain global properties that the function should hold
– such as having unit gradient in the case of SDFs.
We explore key methodological components, includ-
ing the definition of INR, the construction of geo-
metric loss functions, and sampling schemes from a
differential geometry perspective. Our review high-
lights the significant advancements enabled by geo-
metric INRs in surface reconstruction from oriented
point clouds and posed images.
Keywords:Neural Fields, Implicit Representa-
tions, Differential Geometry
1 Introduction
Animplicit neural representation(INR) is aneu-
ral networkthat parameterizes a signal in a low-
dimensional domain. This representation differs from
∗Corresponding author:luiz.schirmer@ufsm.brclassical methods, as it encodes the signal implicitly
in its parameters by mapping coordinates to target
signal values. For example, in the case of an implicit
surface, an INRftakes a 3D pointpand returns the
isosurface valuef(p). In this scenario, we aim for the
INR to approximate the input data as closely as pos-
sible, similar to the problem of approximating signals
usingradial basis functions. INRs provide a compact,
high-quality, and smooth approximation for discrete
data. Furthermore, INRs allow calculating higher-
order derivatives in closed form through automatic
differentiation, which is present in modern machine
learning frameworks.
INRs are smooth, compact networks that are fast
to evaluate and have a high representational capacity.
This has motivated their use in several contexts for
example: images [4, 38], face morphing [44, 73, 74],
signed/unsigned distance functions [37, 34, 46, 20, 49,
61, 45, 5, 8, 42], displacement fields [65], surface an-
imation [29, 35], multiresolution signals [39, 43, 20],
occupancy [30], constructive solid geometry [26], ra-
diance fields [31, 41], textures [40], 3D reconstruction
from images and videos [58, 59, 19], among others.
These works leverage the fact that INRs are compo-
sitions of smooth maps to explore their derivatives
during training. From these various applications, ge-
ometry processing is a noteworthy field with many
applications [58, 34, 20, 59].
The parametersθof an INRfareimplicitlydefined
as the solution to a non-linear equationL(θ) = 0,
whereLis aloss functionthat ensuresffits the sam-
ples{p i,f(p i)}of the ground-truth functionfand
1arXiv:2511.07206v1  [cs.CV]  10 Nov 2025
satisfies certain properties held byf. For instance,
when fittingfto thesigned distance function(SDF)
of a surface, a term is added to the loss function to
enforce the gradient∇fof the network to be unitary;
the Eikonal equation∥∇f∥= 1. This is a fundamen-
tal concept in INRs, as SDFs are solutions to this
partial differential equation. The primary benefit of
adding this constraint is that the sampling{p i,f(p i)}
is often sparse and concentrated near the ground-
truth surface. Consequently, trainingfonly on these
samples could introduce noise in regions with no data.
Imposing the Eikonal equation on additional points
helps regularizing the INR training.
In a neural SDFf, the outputf(p) is a distance
value that can be positive or negative, indicating
whether a point is inside or outside the underlying
compact surface. A distance value of zero indicates
that the point lies on the implicit surfaceS. The gra-
dientN=∇fprovides thenormalfield ofS, and its
Hessian Hess(f), theshape operator, gives the curva-
tures. In this work, we present a survey ongeomet-
ricapproaches that explore these differential objects
during the training and inference of INRs.
We define ageometricINR as a neural network
f:R3→Rapproximating a SDF of a regular surface
S, i.e.∥∇f∥ ≈1, such that its parametersθare im-
plicitly defined byL(θ) = 0, withLenforcing geomet-
rical properties ofSthrough∇fandHf. To enforce
the SDF property, an Eikonal termR
Ω(∥∇f∥−1)2dp
is added toL, where Ω is the training domain. An-
other important geometric term arises from forcing
the alignment of the normalsNofSwith the gradi-
ent∇f, i.e.R
S 
1− ⟨∇f, N⟩
dS.
To bring an in-depth discussion about geometric
INRs we consider the followingtraining pipeline. It
begins with theinput data, which could be either
an oriented point cloud consisting of points and nor-
mals sampled from the underlying surfaceS, or a set
of posed images taken from a scene havingSas a
surface. Next, aneural network(INR)f:R3→R
with parametersθis defined to fit the SDF ofS. This
fitting is achieved through optimization of ageomet-
ric loss functionLusing a variant of the gradient
descent algorithm. However, computing the gradient
∇Lmay be infeasible in practice due to the size of the
data set. Thus, it is common to consider mini-batches(samplingstep) which exploit geometric properties
of the underlying surfaceSto speed up the training.
Once the INRfis trained, the SDF properties of
fcan be leveraged for various applications, such as
geometryinferenceusingsphere tracingor surface
evolution usinglevel-set methods.
We present recent frameworks that enhance the
training performance of INRs by exploring geometri-
cal losses and curvature information to sample points
during training. Additionally, we discuss approaches
that utilize geometric INRs for 3D reconstruction
from posed images, where the neural SDF is used
to represent the scene geometry. Finally, we provide
examples of dynamic geometric INR approaches for
learning surface animation from oriented point clouds
and (time-dependent) posed images. To achieve this,
the network domain must be extended to space-time
R3×Rto encode the time variable [35, 29].
TerminologyIn the visual computing commu-
nity, implicit neural representations have also been
referred to as neural fields, neural implicits, and
coordinate-based neural networks. In this paper, we
focus on the terminology “implicit neural representa-
tions” despite some references using the other terms.
The paper is organized as follows. Section 2 dis-
cusses the main aspects of implicit surface reconstruc-
tion, focusing on the application of the Eikonal equa-
tion, oriented point-cloud-based reconstruction and
classical image-based approaches. Section 3 shows
a geometric framework to solve the geometric im-
plicit neural representation problem, where we de-
tail the input data, loss function details, and dataset
sampling. Section 4 presents applications considered
state-of-the-art for INRs, where we focus on neu-
ral implicit surface reconstruction from images, from
oriented point clouds, multi-resolution, and dynamic
INRs with applications for deforming objects and an-
imation. The final remarks are drawn at the conclu-
sion in Section 5.
2
2 Implicit surface reconstruc-
tion
Implicit representations are commonly used in com-
puter graphics to represent 3D shapes. Unlike ex-
plicit representations (e.g. using triangle meshes) im-
plicit representations encode a surfaceSas the zero-
level set of a functionf:R3→R. For the surfaceS
to be regular, the zero must be a regular value off,
that is,∇f̸= 0 onS=f−1(0). Again, SDFs are a
common example of an implicit representation, where
fis the solution of the Eikonal equation:
∥∇f∥= 1 subject tof= 0 onS.(1)
In this work, we present recent strategies to solve
(1) by parameterizingfwith an INRf:R3→R,
with parametersθ. To approximate a solution of this
equation, it is common to define a loss functionL
to enforcefto be a solution. Solving this equation
reveals that⟨∇f, N⟩= 1 onS, indicating that∇f
aligns with the normalsNofS. We refer to a solution
of the above problem as ageometric INR.
Before presenting examples of training pipelines for
geometric INRs, we recall some classic approaches.
2.1 Oriented point cloud based recon-
struction
Radial basis functions(RBFs) [3] is a classical
method that can be used to approximate the SDF of a
surfaceSfrom a sample{p i,fi}of this function. The
RBF is expressed ass(p) =Pλiϕ(∥p−p i∥),where
the coefficientsλ i∈Rare determined by imposing
s(pi) =f i. Theradial functionϕ:R+→Ris a real
function andp iare the centers of the RBF [34]. Note
that the RBF representation depends on the data
since its interpolantsdepends on the input points
pi.
Poisson surface reconstruction[16] is another
classical method widely used in computer graph-
ics to reconstruct a surface from an oriented point
cloud{p i, Ni}. It revolves around solving thePois-
son’s equation, using{p i, Ni}. The objective is to
reconstruct an implicit functionfof the underlyingsurface by asking it to be zero atp iand to have gra-
dients atp iequal toN i. The pairs{p i, Ni}are used
to define a vector fieldV. Then,fis computed by
optimizing min f∥∇f−V∥which results in a Poisson
problem: ∆f=∇ ·V.
2.2 Image-based reconstruction
There are many classical works that aim to recon-
struct the surfaceSof a 3D scene from a set of un-
ordered images{I j}[52]. Generally, these methods
focus on obtaining an oriented point cloud{p i, Ni}
usingstructure-from-motion[57]. The surfaceScan
then be reconstructed using Poisson surface recon-
struction [16]. COLMAP [47] is a standard example
of this approach. It extracts features from eachI j,
e.g., using SIFT [23], and searches for feature corre-
spondences between the images, using RANSAC [9].
Finally, using bundle adjustment, it computes camera
positions and the pointsp isuch that the correspond-
ing viewing rays intersect.
Recently, implicit neural representations initially
developed for the novel view synthesis problem such
as NeRFs [31] have been gaining popularity for rep-
resenting these systems. Recently, adaptations for
NeRFs have been created for the task of implicit sur-
face reconstruction [58, 59, 63, 21]. We will present
this problem in more detail in Subsection 4.2
3 Geometric INR framework
3.1 Overview of the problem
This section presents an overview of the framework
used to solve the geometric INR problem of train-
ing the parametersθof an INRf:R3→Rto
approximate the SDF of a desired surfaceS. This
pipeline describes the problem for both point-based
and image-based surface reconstruction. To present
this pipeline we follow the scheme in Figure 1.
Theinputcan be either a sample of oriented
points{p i, Ni}n
i=1from theground truthsurfaceS
or a set of posed images{I j}taken from a scene
havingSas its surface. Theoutputis an INRfap-
proximating the SDF ofS. To estimate an SDF with
3
its zero-level set ofSfromI j, it is assumed that
the existence of a projection invertible matrix trans-
formationW jfor each image, mapping from screen
to world coordinates. Although the problem seems
fundamentally different, we shall discuss their simi-
larities later. The framework explores the normals
Nto define a loss function and the curvatures of the
surface to sample the mini-batches.
Sample  of a surface {pi,Ni}STraining Find a minimum of  using gradient descent./uni2112  /uni2112(/uni03B8)=/uni2112data(/uni03B8)+/uni2112Eik(/uni03B8)
Sampling

is smooth
S/uni2248f−1(0)f:/uni211D3/uni2192/uni211D
Neural network
{/uni007C/uni2207f/uni007C=1 in /uni03A9,f=0 on SEikonal problemInput
OutputInferenceLoss function
DataPoint cloud {pi,Ni}Posed images {/uni2110j}or
Figure 1: Geometric INR pipeline: The input data
can be either an oriented point cloud{p i, Ni}or a
set of posed images{I j}. A neural networkfis then
defined to fit a solution to the Eikonal equation. To
trainf, we define a loss function consisting of two
terms: data constraint and Eikonal constraint. For
the point-based data, we simply enforcef(p i) = 0
and∇f(p i) =N i. For the image-based data, we rely
on volume rendering techniques. Finally, gradient
descent is used to optimize the resulting loss function.
Next, we define aneural network(INR)f, with
parametersθ, to fit the SDFfof the ground-truth
surfaceS. For this, we define aloss functionL=
Ldata+L Eikto enforcefto be a solution of Eikonal
equation (1). The data termL dataforcesfto fit
the input data.L Eikforcesfto be a solution of
the Eikonal equation; thus, it works like a (implicit)
regularization.
For image-based reconstruction, the data term
Ldatais modeled using volume rendering [58, 59].This involves defining a differentiable rendering func-
tionI(θ, W j) for the camera corresponding to each
posed image (I j, Wj). Thus, the data constraint
Ldatafor trainingθis defined by forcingI(θ, W j) =
Ij. Some techniques also incorporate geometric reg-
ularizations, such as curvature-based methods [19],
or different approaches to map the SDF to a density-
based operator [63].
Thetrainingstep consists of using a variant of the
gradient descent algorithm to find a minimum ofL.
However, in practice, computing the gradient∇L(θ)
may be unfeasible; thus, we considersamplingmini-
batches of the input data. Once we have the INRf
trained, we can infer its geometry torenderits zero-
level setf−1(0).
The following sections present each component of
geometric INR training in detail.
3.2 Input data
Here, we describe the two kinds of data options (input
of the pipeline in Figure 1) to reconstruct the surface
S.
Oriented point cloudLet{p i, Ni}n
i=1be an ori-
ented point cloud sampled fromS, wherep i∈S
andN iare the normals toSatp i. We aim to re-
construct the SDFfofSby enforcingf(p i) = 0
and∇f(p i) =N i. However, this approach may re-
sult in a neural SDF with spurious components on
its zero-level set. To mitigate such noise, we sample
additional points{q k}outside{p i}to regularize the
training. As a result, we obtain a set of points being
the union of{p i}and{q k}with their corresponding
SDF values.
Posed imagesFor the image-based setting, we as-
sume a set of imagesI jwith their corresponding
projection matricesW j. COLMAP [47] is commonly
used whenever the projection matrices are unavail-
able.W jis the productK j·[Rj|tj] of the intrinsic
matrixK j∈R4×4and the extrinsic matrix [R j|tj].
The intrinsic matrixK jincludes parameters of the
camera, such as focal length and center of projec-
tion. The extrinsic matrix [R j|tj] consists of the
4
orthogonal matrixR jfor camera orientation and the
camera position−t j. Note that in COLMAP, the
images are assumed to have significant overlap, oth-
erwise the feature matching step fails, resulting in
poor approximations ofW j.
3.3 Network architecture
We assume the INRf:R3→Rto be parametrized
by amultilayer perceptron(MLP) defined as follows.
f(p) =W n◦fn−1◦fn−2◦ ··· ◦f 0(p) +b n (2)
wheref i(pi) =φ(W ipi+bi) is theith layer, andp iis
the output off i−1, i.e.p i=fi−1◦···◦f 0(p). Here we
apply the smooth activation functionφ:R→Rto
each coordinate of the affine map, which is formed by
the linear mapW i:RNi→RNi+1and the biasb i∈
RNi+1. The operatorsW iare represented as matrices,
andb ias vectors, combining their coefficients to form
the parametersθof the functionf. In the following
section we define a loss function to trainθto fitfto
the input data.
The choice of the activation functionφhas a great
impact on the representation capacity off. For ex-
ample, using sines, that isφ= sin, results in a power-
ful INR architecture for surface reconstruction from
oriented point clouds [49, 34].
For image-based reconstruction, it is common to
use Fourier feature mapping [54] to represent the neu-
ral SDF [58]. In this approach, the activation func-
tion isφ=ReLU, and the first layer projects the
input onto a list of sines and cosines. However, this
method often results in slow training time. To speed
up training and rendering times, hashgrid-based rep-
resentation was proposed in [32, 19, 59].
3.4 Loss function
We now define a loss functionLto train the parame-
tersθof the INRf. Again, we start with the oriented
point-based case.
Oriented point-based renderingWe use the in-
put data{p i, Ni}n
i=1sampled from a surfaceSand
the Eikonal equation (1) to define the loss functionLas the composition ofL data, andL Eik. This loss
is used to optimizeθsuch thatf:R3→Rapproxi-
mates the SDFfofS.
L(θ)=1
nX
if(pi)2+
1−⟨∇f(p i), Ni⟩
| {z }
Ldata+Z
Ω 
1−∥∇f∥2dp
| {z }
LEik.
(3)
Here,L Eikencouragesfto be the SDF of a setX
by ensuring that∥∇f∥= 1,L dataencouragesXto
containS; i.e.f=fon{p i}. In addition, it asks for
the alignment between∇fand the normals ofSto
regularize the orientation near the zero-level set.
Typically, an additional term is added to (3) to pe-
nalize points outsideS, forcingfto be the SDF ofS
(i.e.,X=S) [49]. A common approach is to extend
Ldatato consider (off-surface) points outsideSby us-
ing an approximation of the SDF ofS[34]. The SDF
approximation at a pointpcan be computed using
|f(p)| ≈min i∥p−p i∥. The sign off(p) at a specific
pointpis negative ifplies inside the surfaceSand
positive otherwise. Note that for each pointp iwith a
normalN i, the sign of⟨p−p i, Ni⟩indicates the side
of the tangent plane thatpbelongs to [34]. There-
fore, we can estimate the sign off(p) by adopting
the dominant signs of the numbers⟨p−p j, Nj⟩, with
{pj}being a subset of the point clound.
Image-based renderingThe traditional image-
based pipeline for surface reconstruction converts the
SDFfto a volume density functionσby compos-
ingfwith a density distribution functionρ, that is,
σ=ρ◦f. Then, volume rendering is applied to create
an image given a camera view. Getting the intensity
for a pixelpalong a view directionvcan be com-
puted usingray marching: where we march along a
rayr(t) =o+tvand accumulate the colorsc 
r(t)
and densitiesσ 
r(t)
[15]. The integral is done in the
nearandfar raybounds:t nandt f. The volume
rendering equation is given by:
I(r) =Ztf
tnc 
r(t)
σ 
r(t)
expZt
tnσ 
r(s)
ds
dt.
(4)
5
To solve (4), most methods discretize the ray and
employ numerical methods [27]. This results in a
differentiable rendering which allows us to optimize
the model parametersθ[31].
After rendering each pixel corresponding to a input
imageI jwith projection matrixW j, we can compare
the output imageI(θ, W j) with its ground-truthI j
resulting in aPhotometric loss:
Ldata(θ) =1
NmX
j∥I(θ, W j)−I j∥2.(5)
3.5 Sampling
Point-based samplingLet{p i, Ni}n
i=1be a sam-
ple of the ground-truth surfaceS. During optimiza-
tion, we may not be able to compute the gradient
∇θLdata(θ) considering the whole dataset. Therefore,
it is common to divide it into minibatches. Novello
et al. [34] considered the principal curvature informa-
tion to prioritize regions with more geometric varia-
tions of the data during minibatch selection. Regions
of higher absolute principal curvatures encode more
detailed information than regions with lower absolute
curvatures.
Novello et al. [34] proposed splitting{p i}into
three sets based on their curvatures:V 1(low),V 2
(medium), andV 3(high). During training, they pri-
oritized the points with more geometrical information
inV 2andV 3while sampling fewer points fromV 1to
avoid redundancy. In Figure 2, we can see a compar-
ison between uniform sampling (first row) and this
curvature-based sampling (second row). Note how
this sampling strategy enhanced convergence during
training.
Image-based samplingFor the image-based re-
construction case, the sampling for theL dataterm
consists of choosing points{p i}along each rayr(t) =
p+tv, wherepis the pixel position andvis the
view direction. This approach is used to discretize
the volume rendering equation [31]. Most methods
follow NeRF, which splits the ray domain interval
[tn, tf] into evenly spaced times{t i}. Then, a time
t′
iis uniformly chosen within each interval [t i, ti+1].
Figure 2: Neural implicit surfaces approximating the
Armadillo model. The columns indicate the zero-
level sets after 29, 52, 76, and 100 epochs of train-
ing. Line 1 shows the results using minibatches sam-
pled uniformly. Line 2 presents the results using the
adapted sampling of minibatches with 10% / 70% /
20% of points with low/medium/high features. Im-
age from [34].
There are numerous methods focused on improving
the sampling strategy for NeRFs [18, 2].
3.6 Inference
Sphere tracingOne important advantage of neu-
ral SDFs is in rendering, as we can use thesphere
tracingalgorithm [12]. It approximates the intersec-
tion between a rayr(t) =p 0+tv, withp 0being
the starting point, and the zero-level set of a neu-
ral SDFfby iterating the systemp i+1=pi+f(p i)v.
Note that this requires multiple inferences during ren-
dering. The challenge of operating this algorithm in
real-time was addressed in [5]. Figure 3 provides the
sphere tracing of the zero-level set of neural SDFs
representing the Armadillo and Bunny. The algo-
rithm can accurately ray-cast the surface, avoiding
spurious components. Finally, for shading the sur-
face, we compute the normals by simply evaluating
the gradient∇f.
Curvature estimationAnother advantage of a
neural SDFfis that we can compute the curvatures
6
Figure 3: Sphere tracing of neural SDFs representing
the Armadillo and Bunny models. Both INRs have
the same architecture and were trained on the same
data during 500 epochs. Image adapted from [34]
of its level sets analytically, as automatic differentia-
tion gives its second partial derivatives. To illustrate
an application, consider the input data{p i, Ni}sam-
pled from a triangle mesh representing the underlying
surfaceS. Novello et al. [34] proposed transferring
the curvatures of the level sets offto the vertices of
the triangle mesh. In Figure 4, we trained a neural
SDF for the Dragon model and calculated the mean
curvature using ∆f; blue indicates higher curvatures,
and red indicates low curvatures. Additionally, in
Figure 5, we show the principal curvatures and di-
rections. The study of discrete curvatures of triangle
meshes is a significant topic in discrete differential
geometry.
4 Applications for INRs
This section reviews a list of methods and applica-
tions of geometric INRs. We divided it in oriented
point-cloud methods (Sec. 4.1), Image-based recon-
struction (Sec. 4.2), Multiresolution INRs (Sec. 4.3),
and Dynamic INRs (Sec. 4.4).
4.1 Surface reconstruction from ori-
ented point clouds
DeepSDF [37] introduced SDFs in the task of rep-
resenting surfaces as level sets of neural networks.
Figure 4: Visual comparison of the discrete and con-
tinuous mean curvatures of the Dragon model. The
top row shows the discrete mean curvature, while the
bottom row shows the mean curvatures computed
from the neural SDF usingPyTorchframework. Im-
age from [34]
Figure 5: Principal curvatures and directions of the
Dragon. Maximal curvatures are shown on the left,
while minimal curvatures are on the right. Notice
how their directions align nicely with the mesh’s
ridges and valleys.
However, it does not incorporate any geometric reg-
ularization in the loss function, such as enforcing the
Eikonal equation (1). Such regularizations were later
introduced by IGR [11] and SIREN [49] which repre-
sent the underlying surface using a neural INRfand
7
enforce the Eikonal equation during training. They
consider as input an oriented point cloud{p i, Ni},
thus, the data term in the loss function simply asks
forf(p i) = 0 and∇f(p i) =N i. Figure 6 shows some
level sets of MLPs trained using IGR.
Figure 6: Level sets of MLPs trained with IGR
method. Image adapted from [11].
The major difference between IGR and SIREN lies
in the architecture of the INR. SIREN proposes pa-
rameterizing the INR using a sinusoidal MLP (φ=
sin in Equation 2) with an initialization scheme that
ensures stability and good convergence during train-
ing. Due to their smoothness and large representa-
tion capacity, SIRENs have emerged as one of the
most popular architectures, influencing many other
works in surface representation [34, 51, 14, 8]. Fig-
ure 7 shows a surface reconstruction using SIREN to
fit an SDF from an oriented point cloud.
Figure 7: Siren Surface Reconstruction. Figure
adapted from [49]
.
Novello et al. [34] introduced a geometric INRwhich explores the curvature of the data during the
sampling stage to speed up training by selecting re-
gions with more geometric details. Section 3.5 pro-
vide additional details. This method also proposes
a loss function that incorporates the curvature of
the INR level sets. For this, it uses the closed-form
derivatives of the network to estimate differential
measures, such as normals and curvatures. This es-
timation is feasible because the oriented point cloud
lies in the neighborhood of the INR zero-level set.
Later, Jiao et al. [14] applied this approach to 3D
shape reconstruction and analysis of real medical
data. They leveraged geometric INRs to create a
shape atlas that captures the effects of age, sex, and
weight on 3D shapes, enabling shape reconstruction
and evolution.
Representing surfaces using SDFs has a common
limitation: they can only represent closed surfaces.
This is because SDFs inherently assume a clear dis-
tinction between the “inside” and “outside” of the
underlying surface, making them less suitable for
open surfaces. Fainstein et al. [8] addressed this limi-
tation by extending the approach in [34] to open sur-
faces usingunsigneddistance functions (UDFs).
4.2 Neural Implicit Surface Recon-
struction from Images
IDR [62] is one of the first geometric INR approaches
to learn a neural SDF from images. IDR simulta-
neously learns the neural SDF, camera parameters,
and a neural renderer that approximates the light
reflected from surfaces toward the camera. The ge-
ometry is represented as the zero-level set of an MLP,
while a neural renderer, based on the rendering equa-
tion, implicitly models lighting and materials. Vol-
ume SDF [63] adopts a different approach by using
differentiable volume renderingto optimize the INR
parameters. It maps the SDF to a volume density
function, assigning a value of 1 to points on the zero-
level set and 0 to points far from this set. The neu-
ral SDF is optimized throughvolume renderingfrom
the images and is regularized using the Eikonal con-
straint.
Exploring volume rendering in 3D reconstruction
has been motivated by the groundbreaking results of
8
Neural radiance fields(NeRF) [31, 66, 69, 2]. NeRF
employs differentiable volume rendering to optimize
the parameters of aReLUMLP combined with a
Fourier feature mapping[54]. These networks are
trained using volume rendering with supervision from
a set of posed images. In this approach, surface ge-
ometry is represented as a density function, where the
value is 1 for points on the surface and 0 for points
far from it. However, surfaces extracted from such
neural densities (using marching cubes [22]) tend to
be noisy (see Figure 8).
To address the noise issues in NeRF-based surface
extraction, Oechsle et al. [36] proposed UNISURF.
This method improves the reconstruction quality by
replacing NeRF’s density function with anoccupancy
network[30], offering a more robust representation
of surface geometry. Figure 8 shows their method
evaluated on the DTU dataset Jensen et al. [13].
Figure 8: Reconstruction of scan 24 of DTU dataset
evaluated with UNISURF.
However, UNISURF does not leverage Eikonal reg-
ularization because its underlying model is not an
SDF. To our knowledge, NeuS [58] is the first geo-
metric INR method to adapt the NeRF pipeline for
training neural SDFs from posed images. It employs
a density distribution function (see Sec 3.4) to map
SDF values to density values.
NeuS motivated several techniques, addressing as-
pects such as incorporating depth maps [68], han-
dling sparse views [21], baking the neural SDF [64],
enhancing gradient consistency [24], patch warping
(NeuralWarp) [6], and applying curvature regulariza-
tion (Neuralangelo) [19]. Figure 9 shows results from
NeuS on the room scene from the dataset presented
by Azinovi´ c et al. [1].
Neural RGB-D Surface Reconstruction [1] presents
Figure 9: Results from NeuS on the Room scene
from the dataset available in the paper Neural RGB-
D Surface Reconstruction[1]. The figure shows the
scene reconstruction as well as the normal map in
comparison to the input data. In these experiments,
we ran the NeuS method with its default settings for
20,000 steps.
a method for 3D scene reconstruction that effectively
combines RGB and depth data using a hybrid scene
representation based on a truncated signed distance
function (TSDF) and a volumetric radiance field.
The authors address limitations of traditional meth-
ods that struggle with noisy depth measurements and
incomplete geometry by leveraging color information
to fill in gaps where depth data is lacking. Their
approach not only improves the quality of geometry
reconstructions but also optimizes camera poses to
reduce misalignment artifacts, demonstrating supe-
rior performance compared to methods like NeRF,
NeuS and UNISURF with depth constraints, partic-
ularly in complex indoor environments.
Next, NeuS2 [59] and InstantNSR [72] used the
9
the hashgrid-based network architecture of instant-
NGP [32] to make training and rendering faster. In
the same vein, Neuralangelo [19] also uses hashgrids
mixed with finite gradients methods to make training
have a much better performance. Figure 10 presents
an overview of the NeuS2 architecture.
Figure 10: Given a pointp, NeuS2 combine its fea-
ture from a hashgrid with its coordinates as the in-
put for the neural SDF, which also outputs geometric
features. These are subsequently combined with the
viewing direction and fed into a RGB network to pro-
duce the color value. Figure adapted from [59].
Table 1, shows a comparison between some tech-
niques on the scene 122 of DTU dataset, both in the
view synthesis task and the geometry reconstruction.
Neuralangelo [19] is, currently, the state-of-the-art in
both tasks. It is a example of geometric INR since it
parameterizes the underlying surface as a neural SDF
and includes the mean curvature of the level sets as a
regularizer. The tests were conducted on a computer
with an AMD Ryzen 7 5700 CPU, 32 GB of RAM,
and an NVIDIA RTX 3080 graphics card. Figure 11
shows a qualitative evaluation with Neuralangelo us-
ing the Tanks and Temples dataset[17].
Figure 11: Qualitative comparison on Tanks and
Temples dataset[17].
Other than NeuS2, other techniques tried to im-
prove the speed of these representations. For ex-
ample, MARF [51] introduce the Medial Atom RayMethod Mean PSNR (↑) Chamfer dist.(↓)
NeuS 30.1 0.56
VolSDF 30.38 0.58
UNISURF 27.32 0.66
Neuralangelo34.91 0.45
Table 1: Performance of Neural SDF methods on
the scene 122 of DTU dataset. The PSNR measures
the quality of the synthesized view while the Cham-
fer distance measures the quality of the geometry of
the SDF by computing the Chamfer distance for the
synthesized mesh.
Fields, a neural object representation enabling dif-
ferentiable surface rendering with a single network
evaluation per camera ray. MARFs address chal-
lenges like multi-view consistency and surface dis-
continuities by using a medial shape representation,
offering cost-effective geometrically grounded surface
normals and analytical curvature computation. They
map camera rays to multiple medial intersection can-
didates and demonstrate applicability in sub-surface
scattering, part segmentation, and representing artic-
ulated shapes. With the ability to learn shape priors,
MARFs hold promise for tasks like shape retrieval
and completion.
Recent research on recovering scene properties
from images often employ neural SDFs and differ-
ential geometry, with deep neural networks demon-
strating inverse rendering of indoor scenes from a
single image. However, these methods typically yield
coarse lighting representations and lack fine environ-
mental details. Approaches like Lighthouse [50] and
NeRFFactor [71] train on natural illumination maps
to address environment estimation challenges. PAN-
DORA [7] and RefNeRF utilize multi-view reflections
but assume a distant environment modeled with a flat
2D map, while PhySG [70] models surfaces as SDFs
for multi-view image rendering. ORCa [56], building
on Novello et al. [34]’s concepts, uses reflections on
glossy objects to capture hidden environmental infor-
mation by converting these objects into radiance-field
cameras. This transforms object surfaces into virtual
sensors, capturing reflections as 2D projections of the
10
5D environment radiance field. This technique en-
ables depth and radiance estimation, novel-view syn-
thesis beyond the field of view, and imaging around
occluders, using differential geometry to estimate cur-
vature for neural implicit surfaces. Figure 12 shows
the results of this method and a diagram of its archi-
tecture.
To test these different types of SDF reconstruction
techniques, and inspired by nerfstudio [55], Yu et al.
[67] created SDFStudio, a platform that allows the
testing and prototyping of different SDF extraction
techniques from images. Table 4.2 show a comparison
between the methods, following the input, geometry
regularization methods employed.
Figure 12: The diagram presents results from the
ORCa model and its inputs. It converts objects
with unknown geometry into radiance-field cameras
by modeling multi-view reflections as projections of
a 5D radiance field. Additionally, it transforms the
object surface into a virtual sensor to capture this
radiance field, enabling depth and radiance estima-
tion of the surrounding environment. The model
then queries this radiance field to perform novel view
synthesis beyond the field-of-view. ORCa consists
of three steps: modeling the object’s geometry as a
neural implicit surface, converting the object’s sur-
face into a virtual sensor, and projecting the environ-
ment’s radiance field along these virtual cones. The
learned radiance field of the environment allows visu-
alization of occluded areas in novel viewpoints. Fig-
ure adapted from [56].4.3 Multiresolution INRs
Multiresolution is a well-studied concept in classical
geometry processing [28]. Given this, and inspired by
traditional multi-resolution theory for images [60, 25],
Lindell et al. [20] introduce BACON, an architec-
ture aimed at addressing the confinement to single-
scale signal representation for coordinate-based net-
works. BACON proposes the creation of an analyti-
cal Fourier spectrum, enabling controlled behavior at
unsupervised points and allowing for design driven by
the signal of spectral characteristics. This method
also supports multiscale signal representation with-
out needing supervision at every scale. It has been
applied to the neural representation of images, radi-
ance fields, and 3D scenes using SDFs, demonstrating
its effectiveness in representing signals across various
scales. Figure 13 shows an overview of the method.
Figure 13: The BACON architecture generates in-
termediate results with a specific spectral bandwidth
chosen during initialization. When trained with high-
resolution data, the network learns to decompose out-
puts across multiple resolutions, useful for tasks like
fitting 3D shapes using signed distance functions.
The characteristics of the network are defined by
its Fourier spectrum, ensuring constrained behavior
even in unsupervised scenarios. Figure adapted from
[20].
In addressing the limitations of BACON, it is noted
that its cut-off of the Fourier spectrum introduces
artifacts, particularly the ringing effect observed in
images and noise on surfaces. This limitation stems
from the inherent constraints of band-limiting, which,
11
Method Input Geometric regularization
PANDORA [7] Multi-view and polarized RGB Eikonal Constraint
PhySG [70] Multi-View RGB Eikonal Constraint
ORCa [56] Multi-View RGB Eikonal Constraint, Mean Curvature Constraint
NeuS [58] Multi-View RGB Eikonal Constraint,
NeuS2 [59] Multi-View RGB Eikonal Constraint
Neuralangelo [19] Multi-View RGB Eikonal Constraint, Mean Curvature
Neural Warp [6] Multi-View RGB Eikonal Constraint
Geo-NeuS [10] Multi-View RGB Eikonal Constraint
VolSDF [63] Multi-View RGB Eikonal Constraint
MonoSDF [68] Multi-View RGB, Depth/Normals Eikonal Constraint, Monocular Supervision
Table 2: Breakdown of SDF Reconstruction from multi-view images. We classify them based on the type
of input and geometric regularization approaches.
while facilitating certain advantages in signal repre-
sentation, can also result in undesirable visual arti-
facts that affect the quality of the output in appli-
cations involving high-frequency detail. Other tech-
niques such as BANF [48], build on BACON, but
does so with filtering during optimization, which al-
lows it to have a band-limited frequency decomposi-
tion. This allows our application of SDF, to have an
SDF in multiple levels of detail.
Other techniques allow numerous level-of-detail for
rendering different images, for example Takikawa
et al. [53], allow for rendering in multiple levels-of-
detail and MINER [43] allows the training on multi-
ple levels of scale.
4.4 Dynamic INRs
INRs serve as efficient and versatile geometry repre-
sentations, encoding both functional and differential
data of the underlying object compactly. However,
they lack intuitive control over shape editing and an-
imations. Here we explore methods to manipulate
implicit surfaces by deformations, whether rigid or
not, shape interpolation, and animations. We also
list interesting examples of works that do not strictly
fit the definition of INR, but exploit differential prop-
erties of smooth networks in their objective function.
Niemeyer et al. [33] presents the first approach to
leverage deep learning for 3D surface animation.Oc-cupancy flowextends Occupancy Networks [30] by
learning a vector field in addition to the occupancy
for each point inR3continuously along time, thus
animating objects in the scene. Their work achieves
good results without discretizations and shape tem-
plates, both usual techniques incorporated in most
contemporaneous methods. More recently Yang et al.
[61] proposed to leverage INRs for geometry pro-
cessing tasks, specifically smoothing and sharpening,
in addition to more complicated rigid deformations,
such as twists and bends. The authors do not pro-
pose their method for animation specifically, but ge-
ometrical deformations, thus we classify them under
the umbrella of dynamic INRs. The authors propose
approximating a local surface of a level set by utiliz-
ing the derivatives of the underlying field. By solely
relying on the field derivatives, it is possible to use
intrinsic geometric properties of the level set, such
as curvatures. This enables the construction of loss
functions that capture surface priors like elasticity or
rigidity. This is made possible by exploiting the in-
herent infinite differentiability of specific neural fields
which facilitates the optimization of loss functions
involving higher-order derivatives through gradient
descent methods. Consequently, unlike mesh-based
geometry processing algorithms that rely on surface
discretizations to approximate these objectives, this
strategy can directly optimize the derivatives of the
field. Their method can apply transformation like
12
rotation or translation on complex 3D objects as we
can see in Figure 14.
Figure 14: Deformation results from Yang et al. [61]:
(A) Input shape, (B) Baseline and (C) their method.
Their method applies multiple transformation on the
Armadillo. Figure adapted from [61].
Similarly to Yang et al. [61], Mehta et al. [29] pro-
poses the use of sinusoidal MLPs to apply smooth
deformations, smoothing, and sharpening to surfaces
parameterized as INRs. However, as in Yang et al.
[61] they still need supervision during the intermedi-
ate time-steps to learn a deformation of the base sur-
face. This supervision is done by converting the im-
plicit object to an explicit representation via march-
ing cubes. While their work achieves good results, it
does not achieve smoothness along time, since each
time-step irreversibly modifies the INR, making it im-
possible to walk through the deformation after train-
ing.
Novello et al. [35] proposes to incorporate the dif-
ferential equation directly into the loss function, thus
removing the need for intermediate-time discretiza-
tions. As in [33], Novello et al. [35] expands the do-
main to include the time parameter (f:R3×R→R
instead off:R3→R), resulting in a smooth rep-
resentation in both space and time. The authors
proposed to leverage the Mean Curvature Flow for
smoothing and sharpening of the base INR, similarly
to [61] and [29] as we can see in Figure 15.
Additionally, they propose to exploit the Level Set
equation for shape interpolation, as in [29], achiev-
ing fully continuous deformations parameterized by
a single neural network. See Fig. 16 for interpolation
and deformation examples.
An interesting application of dynamic INRs is on
the face-morphing task, where the goal is to create
a transition between two or more existing faces, or
Figure 15: Smoothing and Sharpening the Armadillo
with the method from Novello et al. [35].
different expressions of the same face. In this con-
text, Zheng et al. [73] proposes to leverage INRs
by creating expression and identity deformations be-
tween previously captured faces. They expand on
DeepSDF [37], and create two separate latent spaces:
one for expressions , and one for identities. A third
network encodes a template shape, which is the aver-
age face of the dataset. The authors further expanded
their method by incorporating a refined displacement
grid to capture finer facial details [74]. Both works
incorporate the Eikonal term as a regularizer for the
loss function in addition to a normal alignment term.
Using a separate network for the flow, as done in an-
other domain in Schardong et al. [44], holds promise
13
Figure 16: Deformation and interpolation of INRs.
Top row shows the evolution of the zero-level sets of
an INR according to a vector field with a source and
a sink. The SDF of the Spot is the initial condi-
tion att= 0 (middle). The sink/source is inside the
head/body of the Spot. Bottom row shows an exam-
ple of shape interpolation between Bob, positioned at
t= 0 and Spot, att= 1. Images adapted from [35].
for this field.
Wang et al. [59] expands NeuS [58] to dynamic
scene reconstruction from videos, by improving the
training performance of NeuS with HashGrids in-
troduced by InstantNGP [32]. They exploit a key
property that the main object remains reasonably
static on successive frames and propose an incremen-
tal training scheme from a base INR built from the
first video frame. Additionally, they regularize their
loss term using an Eikonal term, similarly to Novello
et al. [34].
5 Conclusion
The paper reviewed INR approaches for solving the
problem of implicit surface reconstruction by apply-
ing geometric regularization to the INR’s level sets.
To present these approaches, we introduced ageo-
metric INRframework, outlining a commonly used
pipeline in this field. We defined the key components
of this pipeline, includinginput data, which can con-
sist of either oriented point clouds or posed images,
and ageometric loss function, which leverages the dif-
ferential properties of the function for regularization
and sampling during training.Additionally, we explored several applications that
leverage the differentiability of neural networks and
the discrete geometry of oriented point clouds to fit
the INR’s zero-level set to the data. Our review
covers prominent geometric INRs, such as SIREN
and IGR, along with state-of-the-art methods from
image-based reconstruction literature, like NeuS and
Neuralangelo. These methods demonstrate that in-
corporating differential geometry concepts into the
INR loss function offers significant potential for re-
constructing surfaces from posed images.
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