On the Effectiveness of Weight-Encoded Neural Implicit 3D Shapes   
Authors: Thomas Ryan Davies
Derek Nowrouzezahrai
Alec Jacobson
Venue:  (2021)

Abstract: A neural implicit outputs a number indicating whether the given query point in space is inside, outside, or on a surface. Many prior works have focused on _latent-encoded_ neural implicits, where a latent vector encoding of a specific shape is also fed as input. While affording latent-space interpolation, this comes at the cost of reconstruction accuracy for any _single_ shape.  Training a specific network for each 3D shape, a _weight-encoded_ neural implicit may forgo the latent vector and focus reconstruction accuracy on the details of a single shape. While previously considered as an intermediary representation for 3D scanning tasks or as a toy-problem leading up to latent-encoding tasks, weight-encoded neural implicits have not yet been taken seriously as a 3D shape representation. In this paper, we establish that weight-encoded neural implicits meet the criteria of a first-class 3D shape representation. We introduce a suite of technical contributions to improve reconstruction accuracy, convergence, and robustness when learning the signed distance field induced by a polygonal mesh --- the _de facto_ standard representation. Viewed as a lossy compression, our conversion outperforms standard techniques from geometry processing. Compared to previous latent- and weight-encoded neural implicits we demonstrate superior robustness, scalability, and performance.

Under review as a conference paper at ICLR 2021
ON THE EFFECTIVENESS OF WEIGHT-ENCODED NEURAL IMPLICIT 3D SHAPES
Anonymous authors
Paper under double-blind review
ABSTRACT
A neural implicit outputs a number indicating whether the given query point space is inside, outside, encoded neural implicits, where a latent vector encoding of a specific shape is also fed input. While affording latent-space interpolation, this comes at the cost of reconstruction accuracy for any single shape. Training a specific network for each 3D shape, weight-encoded neural implicit may forgo the latent vector and focus reconstruction accuracy on the details of a single shape. While previously consid- ered as intermediary representation for 3D scanning tasks or as toy-problem leading up to aatent-encoding tasks, weight-encoded neural implicits have not yet been taken seriously as 3D shape representation. In this paper, we establish that weight-encoded neural implicits meet the criteria of a first-class 3D shape repre- sentation. We introduce a suite of technical contributions to improve reconstruc- tion accuracy, convergence, and robustness when learning the signed distance field induced by a polygonal mesh - the de facto standard representation. Viewed as a lossy compression, conversion outperforms standard techniques from geome- try processing. Compared to previous latent- and weight-encoded neural implicits demonstrate superior robustness, scalability, and performance.
surface. Many prior works have focused on latent-
INTRODUCTION
While 3D surface representation has been a foundational topic of study in the computer graphics community for over four decades, recent developments in machine learning have highlighted the potential that neural networks can play as effective parameterizations of solid shapes.
The success of neural approaches to shape representations has been evidenced both through their ability of representing complex geometries as well as their utility in end-to-end 3D shape learning, reconstruction, and understanding and tasks. These approaches also make use of the growing avail- ability of user generated 3D content and high-fidelity 3D capture devices, c.g., point cloud scanners.
For these 3D tasks, one powerful configuration is to represent 3D surface S as the set containing any point 7 E 3 for which an implicit function (i.e., a neural network) evaluates to zero:
S = {I E R
fo
2) 0}.
(1)
where 0 E 1 are the network weights and ī E 1 is an in- put latent vector encoding a particular shape. In contrast to the de facto standard polygonal mesh representation which explicitly dis- cretizes a surface's ' geometry, the function f implicitly defines the shape S encoded in Σ. We refer to the representation in Eq. (1) as a datent-encoder neural implicit.
Explicit (mesh)
Implicit
Park al. (2019) propose to optimize the weights 0 so each shape S; E D in dataset shape distribution D is encoded into a corresponding latent vector ž, successfully trained, the weights 0 of their DEEPSDF implicit function As always with supervision, reducing the training set from DD will affect f's ability to generalize and can lead to overfitting. Doing may seem, first, be an ill-fated and uninteresting idea.
be said to generalize across the "shape space" of D.
Our work considers an extreme case when the training set is reduced to a single shape S,. We can draw simple but powerful conclusion: in this setting, can completely forgo the latent vector

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(i.e., k = 0). From the perspective of learning the shape space of D, we can "purposefully overfit" a network to a single shape S, S == E
where Og now parameterizes a weight-encoded neural implicit for the single shape S..
In the pursuit of learning the "space of shapes." representing a single shape as a weight-encoded neural implicit has been discarded basic validation check or stepping stone toward the ultimate goal of generalizing over many shapes (see, c.g.. (Chen & Zhang, 2019; Park et al., 2019; Atzmon & Lipman, 2020a;b)). Weight-encoded neural implicits, while not novel, have been overlooked as valuable shape representation beyond learning and computer vision tasks. For example, the original DEEPSDF work briefly considered and nearly immediately discards the idea of independently encoding each shape of a large collection:
"Training specific neural network for each shape is neither feasibe nor very useful."
Park et al. (2019)
We propose training a specific neural network for each shape and will show that this approach is both feasible and very useful.
We establish that weight-encoded neural implicit meets the criteria of a first-class representation for 3D shapes ready for direct use in graphics and geometry process- ing pipelines (see inset table) While common solid shape representations have some important features and miss others, neural implicits provide a new and rich constella- tion of features. Unstructured point clouds are often raw output from 3D scanners, but do not admit straightfor- ward smooth surface visualization (I). While meshes are the de facto standard representation, conducting signed distance queries and CSG operations remain non-trivial (II). Signed distances or occupancies stored on a regular grid admit fast spatial queries and are vectorizeable just like 2D images, but they wastefully sample space uniformly rather than compactly adapt their storage budget to a particular shape (III). Adaptive or sparse grids are more cconomical, but, just as meshes will have different number of vertices and faces, adaptive grids will different storage profiles and access paths precluding consistent data vectorization (IV).
III
IV
Point cloud
X/o
Mesh
X
X
Regular grid
X
Adaptive grid
Neural implicit
While previous methods have explored weight-encoded neural implicits sentation for scene reconstruction (e.g., (Mildenhall et al., 2020)) and noisy point-cloud surfacing tasks (e.g., (Atzmon & Lipman, 2020a;b)), we consider neural implicits as the primary geomet- ric representation. Beyond this observational contribution, our technical contributions include a proposed architecture and training regime for converting the (current) most widely-adopted 3D ge- ometry format polygonal meshes into weight-encoded neural implicit representation.
intermediary repre-
We report on experiments' with different architectures, sampling techniques, and activation func- fions-including positional encoding (Mildenhall et al., 2020) and sinusoidal activation approaches (Sitzmann et al., 2020b) that have proven powerful the context of neural implicits. Compared to existing training regimes, we benefit from memory improvements (directly impacting visualization performance), stability to perturbed input data, and scalability to large datasets.
Weight-encoded neural implicits can be treated as an efficient, lossy compression for 3D shapes. Increasing the size of the network increases the 3D surface accuracy (see Figure 1) and, compared to standard graphics solutions for reducing complexity (mesh decimation and storing signed distances on a regular grid), we achieve higher accuracy for the same memory footprint as well as maintaining SIMD representation: shapes can be represented as n weight-vectors for a fixed architecture.
The benefits of converting an existing mesh to a neural implicit extends beyond compression: in approximating the signed distance field (SDF) of the model, neural implicits are both directly usable for many tasks in graphics and geometry processing, and preferable in many contexts compared to traditional representations. Many downstream uses of 3D shapes already mandate the conversion of meshes to less accurate grid-based SDFs, due to the ease and efficiency of computation for SDFs: here, neural implicits serve as drop-in replacement.
Source code, data, and demo (anonymized) repo: https//ggithh..cc///////CCCCO
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Many works explore latent-encoding methods (e.g., (Park et al., 2019; Atzmon & Lipman, 2020a;b)), taking advantage of interpolation in latent space (learedd) proxy for exploration in the "space of shapes". We show that this flexibility comes direct cost of other de- sirable proprieties. In particular, we show that latent- encoded neural implicits scale poorly as a representation for individual shapes both at training and inference time. Existing latent-encoded neural implicits are sensitive to the distribution of train- ing data: while they may perform well for large datasets of limited subclass of shapes (e.g.. "jet airplanes"), we show that training fails with more general 3D shape datasets. Even within class, existing methods rely on canonical orientation alignment (see Figure 2) in order to alleviate some of this difficulty such orientation are notably (and notoriously) not present in 3D shapes captured or authored in the wild and, as a result, latent-encoded neural implicits will fail to provide meaning- ful results for many real-world and practical shape datasets. Fitting latent-encoded neural implicits to each shape independently complicates shape space interpolation, rendering it difficult though not impossible (Sitzmann et al., 2020a). In contrast, weight-encoded neural implicits leverage the power of the neural network function space without the constraints imposed by the requirement of generalizing across shapes through latent sampling.
Encoding:
Latent
Weight
Interpolation:
trivial
non-trivial
Scalability:
poor
excellent
Stability:
poor
excellent
METHOD
Neural implicits soared in popularity over the last year. While significant attention has been given to perfecting network architectures and loss functions in the context of latent-encoding and point- cloud reconstruction, there is relatively little consideration of the conversion process from 3D sur- face meshes to weight-encoded neural implicits (c.g., both Park et al. (2019) and Sitzmann et al. (2020b) consider this task briefly). We focus identifying setup to optimize weight-encoded neural implicits for arbitrary shapes robustly with small number of parameters while achieving a high surface accuracy. Once successfully converted, consider how the weight-encoded neural implicit representation compares to standard 3D model reduction techniques and how choosing this representation impacts downstream graphics and geometric modeling operations.
2.1 SIGNED DISTANCE FIELD REGRESSION
In general, the value of an implicit function f away from its zero-isosurface can be arbitrary. In shape
learning, many previous methods have considered occupancy where (z) outputs the likelihood of being inside of a solid shape (and extract the surface as the 50%sis)suraace) Mildenhall et al. (2020);
Mescheder al. (2019); Littwin Wolf (2019); Chen Zhang (2019); Maturana Scherer (2015); Wang et al. (201 8). We instead advocate that should approximate the signed distance field (SDF) induced by given solid shape. Learning properties aside (see, e.g., (Park et al., 2019)), SDFs are more immediately useful in graphics and geometry processing applications.
Given a surface S aV of volumetric solid C R.3, the signed distance field Js R IR induced by S is a continuous function of space that outputs the distance of query point ø E R modulated by +1 depending on whether 7 is inside or outside of the solid:
{
E
gs = signg( ( min | -ll.
where s
PES
otherwise.
Our goal is to regress feed-forward network fa to approximate the SDF of given surface S:
fo ( gs (z).
If successfully trained, the weights 0 E IR" encode neural implicit representing S.
2.1.1 ARCHITECTURE
Our proposed architecture is a feed-forward fully connected network with layers, of hidden size H. Each hidden layer has ReLU non-linearities, while the output layer is activated by tanh.
Increasing the depth and width this network will generally improve accuracy but at the cost of increasing the memory footprint and, for example, the time required to render the surface. The
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Maan Error
Parameters
Render Time (s)
16 32 04 120 256 512
512
10
22
14
126 255 312
Figure I: We visualize the role that varying the number of network layers and hidden layer sizes: (left to right) average reconstruction error, memory footprint and first-frame render time (DeepSDF, other setups, and our defaults in red, gray, and blue, respectively).
weight-encoded neural implicit's s rendered in Figures 2, 4, and 8 all share a common architecture of just 8 fully connected layers with a hidden size of 32 (resulting in just 7553 weights, or 59 kB in memory). Through experimentation on a subset of 1000 meshes from Thingi10k (Zhou & Jacob- son, 2016), we find that this configuration yields good balance between reconstruction accuracy, rendering speed, and memory impact (Figure 1). While maintaining acceptable surface quality, our default architecture has a 99% reduction in number of parameters and 93% speed up in "time to render first frame" compared to the default weight-encoding architecture of (Park et al., 2019).
Excited by the recent work exploring methods to overcome MLP's bias to learn low frequency signals faster, we performed experiments using both positional encodings (Tancik et al., 2020) and SIREN activations (Sitzmann et al., 2020b). Both perform well when the network architecture is sufficiently wide (e.g., H > 64), but introduce surface noise with our more compact architecture. See Appendix A.3 for detailed experimental setup and findings.
By increasing N and H. our network could in theory (Hornik et al., 1989) learn to emulate any arbitrary topology shape with infinite precision. In reality, like any representation, there are trade- offs. The network complexity can be increased over our base configuration for smaller surface reconstruction error, or decreased for faster rendering speeds depending on the application. sample of geometries produced at number of configurations can be seen in Figure 1.
2.1.2 INTEGRATED Loss >> IMPORTANCE SAMPLING
Particularly choices of pointwise loss functions have been well explored by previous papers (Park al., 2019; Atzmon & Lipman, 2020a;b; Gropp et al., 2020; Sitzmann et al., 2020b), in our ex- periments we find that a simple absolute difference fā) loss after the fact via ad hoc sampling (near- (eear))sufface sample process (Park al., 2019; Atzmon & Lipman, 2020a;b) leaves an unclear notion whether the total loss can be expressed as an integral and hides possibly unwanted bias. We focus instead on how to integrate this pointwise loss over space.
(z)) works well. Defining the total
Sampling based on mesh vertices (Littwin & Wolf, 2019; Sitzmann et al., 2020b) reduces accuracy in the middle of triangle edges and faces and introduces bias near regions of the mesh (inset: Vertex) with denser vertex distributions regardless of the geometric complexity or saliency of the region.
Similarly, sampling from Gaussians centered on the surface Park et al. (2019); Chen & Zhang (2019); Atzmon & Lipman (2020a;b) will place over emphasis in regions of high curvature, in thin solid/void regions (inset: Surface).
Vigs
Vertex
Surface
Ours
In contrast to ad hoc samplings, we define the total loss directly as integral over space,
Awway
(( = w(x) (fo(z) gT( dz.
(5)
p3
where R + Rgo is non-negative weighting function with finite integral over R33
Methods which randomly sample within a bounding box around a given shape (Mescheder et al., 2019: Tancik al., 2020) can be understood as choosing to be the characteristic function of the box. As Park et al. (2019) already observe, this is wasteful if we care most that f is accurate near the shape's 's surface (i.e., where gs 0).
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We achieve this directly without yet invoking sampling by choosing exponentially dis- tance to grows, specifically:
w(T) = e -
(6)
where B 2 0 can be adjusted from uniform sampling (B 0)
for surface-only sampling.
Attempting to sample space and measure the integrand of Eq. (5) directly leads to many samples having little to numerical effect during training. For example, if 30 and we consider point unit distance away from the surface, the weighting term itself closes in on machine double precision 9e 14. By resisting the urge prematurely sample until after we have written our total loss function an integral, we can instead apply importance sampling (Kahn & Harris, 1951) to construct proportional approximation:
L( R
Σ 1f((#) - ZED
where D. is a distribution over R with probabilities proportional to w.
We sample from D., in practice via a simple subset rejection strategy. Starting with large (e.g., 10M) pool of uniform samples within a loose bounding sphere around the shape, we re-sample (with replacement) smaller (e.g., 1M) subset with probability according to w. Further improvements may be possible by incorporating advanced sampling patterns à la Xu et al. (2020).
Compared to uniform sampling, weighting by our choice of w leads to faster convergence and re- duced surface reconstruction when validating against subset of 1000 geometries from Thingi (96 epochs with surface error of 0.00231). Compared to the sampling of Park al. (2019), we match convergence speed (86 epochs each) and demonstrate 5% improvement in surface error.
Perhaps the most valuable property of our im- portance sampling scheme to be its flexibility.
Ground truth
Standard sampling
Region-based sampling
Our method has effectively removed all unin-
tended bias present in previous approaches, and enables complete user control on intended bias to the sampling process. The importance met- ric, (i)) can be modified to explicitly bias im- portance toward regions of high curvature, minimum feature size (emulating the hidden bias of Park et al. (2019)). or near user annotations (see inset where w(x) is additionally scaled according to user selection). This flexibility allows for greater use of the network's capacity on areas important the user, without increasing overall network complexity radically changing the sampling protocol.
biis points
2.2 ROBUST LOSS FUNCTION FOR MESHES IN THE WILD
The input should be the boundary a solid region R3; that is, a closed, consistently oriented, son-self-intersecting surface. Ignoring "two-sided" meshes that not intended to represent the boundary of solid shape (c.g., clothing), many not most meshes found online which intend represent solid shape would not qualify these strict pre-conditions. Zhou & Jacobson (2016)
observe that nearly 50% Thingil0k's solid models for 3D printing fail one criteria an- other. The failure point in terms of our equa- tions so far is the definition of the signing func- tion sign(T) in Eq. (3) which relies on deter- mining whether point T lies inside V.
Original geometry
Visual hull reconstruction
Winding number reconstruction
with slice plane
To determine insideness, previous approaches either require watertight inputs, use error-prone voxel flood-flling (Mescheder et al., 2019) or use inaccurate visual hulls as a proxy (Park et al., 2019) (see inset where visual hill sign- ing can be shown to "close off" internal struc- ture. Virtox (left) under CC BY. ). Altema-- tively, Atzmon & Lipman (2020a;b) advocate for loss function based unsigned distances.
Messy input mosh
Unsigned distance field
Winding number field
Robust signed distance field.
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This introduces unnecessary initialization and convergence issues, that can be avoided if we assume that the input mesh intentionally oriented to enclose a solid region (as is the case for nearly all of ThingilOk), but may suffer from open boundaries, ielf-intersecttions, non-manifold elements, etc. Under these assumptions, the generalized winding number (Jacobson et al., 2013) computes correct insideness for solid meshes and gracefully degrades inset). Using the tree-based fast winding numbers of Barill et al. (2018) and a bounding volume hi- erarchy for (unsigned) distances, we can construct our IM-point sample efficiently and optimize weights f for even the most problematic meshes (see inset) in an average of 90 seconds per shape.
fractional value for messy input shapes (see
2.3 EFFICIENT VISUALIZATION
Our weight-encoded neural implicit representation can be treated as its classical counterpart (SDF) and rendered efficiently using taherr-tracing (Hart, 1996). Sphere tracing is a common technique for rendering implicit fields where rays are initialized in the image plane and iteratively "marched" along by a step size equal to the signed distance function value at the current location. The ray declared to have hit the surface when sufficiently close (< e). For more details, see Morgan McGuire's 's comprehensive notes at casual-effects.com
We trivially adapt traditional sphere-tracing by initializing the starting position of each ray to be its first (if any) intersection with the similarity transformed unit sphere, since all weight-encoded neural implicits are normalized lay within. As rays of the image will converge different times, we employ a dynamic batching method that composes batches of points for inference based on a mask buffer which tracks rays that have converged to the surface or reached the maximum number of steps. Local shading requires the surface normal at the hit point. For SDFs, the unit normal vector is immediately revealed as the spatial gradient (i.c., f This can be computed by finite differences or back propagation through the network.
IMPLEMENTATION AND RESULTS
We implement weight-encoded neural implicit networks in Tensorflow (Abadi et al. (2015)) with point sampling and mesh processing implemented in libigl (Jacobson al. (2016)). We train model for up to 102 epochs and allow early stopping for quickly converging geometries. We use the ADAM optimizer (Kingma & Ba (2014)) with fixed learning rate of 10-4. These settings generalized well across a wide range of geometries (see Figures 4 and 5).
3.1 SURFACE VISUALIZATION AND CSG
visualization and shading kernels in CUDA, using CUTLASS (Kerr et al. (2018)) linear algebra libraries for efficient matrix multiplication at inference-time.
We implement
We achieve average display frame rate of 34 Hz for the large subset of the Thingi 10k dataset we visualize- when rendering a single neural implicit at 512 x 512 resolution on an Nvidia P100 GPU. This significant performance improvement over previous learnt implicit inference and dis- play pipelines, attributed in large part to our compact representation. Liu et al. (2020) present a specialized renderer capable of a Hz display rate, however at the price of many conservative opti- mizations: these include overstepping along all rays by a factor of 50%, increasing the convergence criteria (early stopping), and implementing coarse-to-fine display strategy. While these additional optimizations could further improve our rendering speed (at the cost of reduced visual quality), we opt rely on a simpler (and very efficient) standard
SDF renderer.
Indeed, as our representation is a learnt representation of the SDF, we also inherit other important benefits of traditional implicit function repre- sentations. Weight-encoded neural implicits admit robust shape manipulation and modifi- cation using constructive solid geometry oper- ations (CSG) - by directly modifying the in- ferred distance values (see inset and accompa- nying video). Weight-encoder neural implicits admit SIMD evaluation and, given their compact- ness, many neural implicits can be rendered in parallel interactive rates modern GPUs.

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3.2 STABILITY AND SCALE
Training deep neural networks on large geometric datasets can be cumbersome and time consuming. For our weight-encoded neural implicit representation to be effective, we must be able to convert any 3D shape into its weight-encoded form in a reasonable amount of time, Due to our relatively simple base network architecture (8 layers of 32 neu- rons each) we find that we can overfit our model to any 3D shape in 90 seconds, As this requires only 59 kB of memory, we can train many models/shapes concurrently modern GPUs without approaching any prac- tical memory limitations this case of train- ing is uncommon to other learning- based shape representations. Converting the entirety of the 10,000 models in the Thingil0k dataset Zhou & Jacobson (2016) on an Nvidia Titan RTX only took 16 hours a single GPU, or four hours on four Nvidia Titan RTX cards.
average.
DeepSDP
8-00
==:5*
8 - 10"
Ours
Figure 2: Unlike our representation, DeepSDF re- construction quality degrades quickly for geome- tries not aligned to default, per-class orientations. See accompanying video for animation.
Converting the Thingi Ok dataset from mesh format to weight- encoded neural implicit format reduces the overall storage from 38.85 GB to 590 MB a 1:66 compression rate. While DeepSDF network Park et al. (2019) trained on the same dataset could compress this dataset to impressive MB footprint, the latent-decoded geometries it produces are of comparably lower quality. This comparison is representative, as Thingi 10k real-world mesh dataset of objects obtained "in the wild". The dataset neither contains geometries aligned frame of reference nor comprises objects nearing no semblance of inter-class categorization. These two properties make it diffi- cult for any latent-encoded neural implicit network to converge to reasonable result during training.
common
Figure 3: Latent-encoded SDFs (red) struggle to reconstruct "unique" features (grey, plane's tail) despite training on a single class of objects (planes). Our representation (blue) does not. gpvillamil under CC BY.
We further support these claims using two experiments. First, we attempt to train DeepSDF on the Thingi dataset, and sec- ond we experiment with DeepSDF's ability reconstruct shapes with slight perturbations from the shapenet (Chang et al., 2015) common shape orientation. Here, DeepSDF does not converge on the 10,000 model Thingi 10k dataset, producing incoherent re- constructions when exploring the latent space of shapes it has learned. Moreover, if we further limit DeepSDF to training with single class of objects, it is not able to reconstruct features on the tails of the inter-class distribution (inset, right). Secondly, we evaluate DeepSDF's ability to reconstruct geometries not aligned to the common orientation. Here, we retain single-class DeepSDF training and reconstruct the same input shape at orientations differing from the default (Figure 2). This test validates latent-encoding's 's reliance on having consistently aligned datasets, immediately precluding their use with large, real-world datasets.
3.3 REPRESENTATION COMPACTNESS
All of the shapes in Figure 4 were rendered with weight-encoded neural implicits generated using base network architecture, resulting in total of 7553 weights for each shape's implicit function. At just 59 kB of memory we find that our lightweight representation can capture complex geometric topologies at high resolution compared uniform signed distance grids adaptively decimated meshes with similar memory footprints.
The comparisons in Figure 8 use geometry converted weight-encoded neural implicit in our base configuration, visualized next to the rendered result of uniformly sampled SDF grid with 20 samples as well as with the original mesh adaptively decimated (Garland & Heckbert, 1997) down to 7600 floats (i.e., vertex and face data). Compared to decimated meshes (our base- line non- non-uniform format), we observe that weight-encoded neural implicits have similar surface

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quality but with smoother reconstructions due to the continuous (versus piecewise linear) na- ture of the implicit. Compared to SDFs stored on a grid (our baseline uniform format) we ob- serve far better quality equal memory. Fur- thermore, we notice that our approach better captures high frequency surface detail com- pared to both these representations, often pro- ducing results that more closely match the cur- vature of the original shape.
We measure our method's robustness by con- verting the Thingi ((Zhou & Jacobson, 2016)) dataset and measuring the average sur- face error (1/NE:) (p()) and training loss. We report mean training loss for errors between the true and predicted SDF values points sampled using our importance metric (Section 2.1.)). This surface error is the sum of errors at points along the shape's O-isocontour. These metrics measure both the error the surface and within the shape's 's bounding volume. Errors within the bounding volume decrease rendering performance and/or lead to hole artifacts in the shape during visualization. Surface errors are more evident after meshing the implicit SDF using, i.e., marching cubes. We sample 10" surface points when measuring surface error, and compute loss against a training set of 1M points. We visualize results on the entire Thingi 10k dataset in Figure 5. We find that, at our base configuration, 93% of the 103 Thingi 10k shapes reach a surface error below 0.003, and no model exceeds 0.01 (worst case of 0.0097; see Fig. 6).
Figure 4: Thingi 10k models compressed to 59kB, reducing the dataset from 38.85 GB to 590 MB.
4 LIMITATIONS AND FUTURE WORK
Our default architecture will fail to satisfyingly approxi- mate very topologically or ge- ometrically complex shapes. While increasing the size of the network will generally al- leviate this (see Figure 6), it would be interesting to con- sider cascading or adaptively sized networks. Our L' loss function encourages the net- work to match the values of a shape's signed distance field, but not necessarily its derivatives (cf. Gropp et al. (2020); Sitzmann et al. (2020b)). True SDFs satisfy an Eikonal equation (j0g/dī = 1) and this property is sometimes important for downstream tasks. For future work, we would like to investigate whether Eikonal satisfaction can be ensured exactly by construction. With respect to single-shape accuracy, latent-encodings work well in spe- cialized scenarios (e.g., large-networks trained on canonically aligned specialized classes). With respect to shape-space learning, latent-encodings lie in simpler continuous space than weights, which suffer from transposition and reordering non-injectivities (i.e., multiple weight vectors repre- sent the same implicit). Nevertheless, weight-encodings allow us faithfully prepare large diverse datasets of 'real-world' shapes into vectorizeable representation. We have shown this is simply not possible with existing latent- encodings. We include the full Thingi10k dataset converted weight- encoded neural implicits vectors as a data release?. This vectorized data is ripe for meta-learning future work. Indeed, concurrent work already exploring this direction Sitzmann et al. (2020a). We hope our consideration of weight-encoded neural implicits as a first-class shape representation encourages their use in computer graphics, geometry processing, machine learning, and beyond.
Training Loss
Surface Error
2000
2000
1500
1500
1000
1000
500
500
0,025
0,001
0,0035
010073
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0,016
Moce
0,0005
0,00055
CL001
0.09.25
0,0015
0.00733
0002
0,0025
02025
009255
More
Figure 5: Loss and surface error distributions over the entirety of
the Thingi dataset.
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preprint
A
APPENDIX
A.1 THE WEIGHT-ENCODED NEURAL IMPLICIT FILE FORMAT
Our compact weight-encoded neural implicit is designed to be effortlessly consumed and integrated into existing graphics and geometry processing pipelines. For each trained model, the chosen archi- tecture and similarity transformaton matrix (since all geometries are normalized to the unit sphere) are written as the first bytes before encoding the learned weights 0 into an HDF5 format file.
For fixed architecture, the instructions to evaluate the estimated SDF is the same for any point and any shape. This SIMD property allows multiple geometries to be evaluated in parallel. The fixed storage profiles and memory layout of our learned implicit functions provide consistent query and rendering speeds. We store our model weights using the HDF5 format. This allows easy in- tegration into Tensorflow (below) which can load our model natively. We additionally support the loading of arbitrary weight-encoded neural implicit through the "High Five" HDF5 C++ library for rendering and meshing.
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import tensorflow tf import numpy as np
# load model "key" dictating architecture. SIMD. sdfModel tf. m model_from_mm_pppmmmmmm (open( key json )))
# load specific weight for Standford bunny geometry sdfModel load weightss ('bunny. h5')
# generate 128x/28x/28 grid for SDF queries
K np. linspcce (-100,1 128))
grid [[x.y.2] for in K for in K for z in K]
# infer SDF at each point
S = sdfModel. predict
A.2 ERROR DRIVEN CONVERSION
We fix the architecture during the Thingil0k dataset conversion, resulting in a constant and compact memory footprint. If, however, maintaining target surface reconstruction quality is of more impor- tance to a fixed memory cost, we can instead shift to error driven surface fitting approach (much like classical approaches (Ohtake et al., 2005)), scaling network architecture complexity based on the input geometry. As each generated weight-encoded neural implicit encodes its own architec- ture, such an approach results in smaller architectures for simpler geometries and larger ones for topologically-come complex geometries. We visualize the effect of eπrоr driven optimization in Figure 6, where we perform simple grid search until reaching a user-desired surface error threshold.
Based on conversion of the Thingi 10k dataset, find that majority of models are well rep- resented using our base configuration (Fig. 5) - if desired, geometries that fall within the tails of the complexity distribution can be retrained with larger architectures, again until reach desired surface fidelity. This decision can be further informed by whether SIMD and fixed memory access patterns beneficial the underlying application.
Original
Decimated
Base
Increased
Figure 6: With only 7553 parameters, our base weight-encoded neural implicit format can lack the representative power converge on highly complex geometries (similar decimated mesh with same memory footprint). Increasing the network capacity to equal the memory impact of the original mesh results in near perfect reconstructions. tbuser (left) under BY.
A.3 SIREN AND FOURIER FEATURES
In an effort to improve the reconstruction quality of our weight-encoded neural implicits we ex- plored recent work focused on improving MLPs ability to represent high frequency signals. We ex- perimented with three methods: namely we investigated using the SIREN (Sitzmann et al., 2020b) activations, positional encoding (Mildenhall et al., 2020), and Fourier features (Tancik et al., 2020). Each of these approaches have lead to impressive resuts for high-fidelity reconstructions of 3D sur- faces mitagating the known problem that MLPs learn low frequency signals faster (Rahaman et al., 2019).
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Mildenhall al. (2020) define their positional encodings as,
(p) (sin(2"mpp) cos(22pp))p_oo... st (2 (200(22-110)) Tp)
(8)
where 7 is mapping from II into the higher dimensional space 2R
While Tancik al. (2020) expands on this approach with random gaussian features yielding the mapping function,
x(p) = (2 Bp),
where each entry in B E Rmxd is sampled from (100 o²), and a is left as a hyperparameter specific each problem.
We evaluate both of these approaches by mapping each axis (x.y.z) of our sampled points to the higher dimensional space. We find that when the network architecture is of sufficient width these mappings work exceptionally well. We evaluated using y with various L configurations ranging from 10. Unfortunatly, we find that our light weight (and intentionally underparamerterizee)) architecture struggles to learn from the augmented input signal. We visualize the affect of positional encodings when L 10 Figure 7. Similarly, we see drastic degredation of quality when em- ploying À for mapping
default embedding size of 256 (not shown as we were unable to march). These approaches are clearly practical methods for reconstructing focus on minimizing the number of parameters the cost of mapping to higher dimension input too high.
fidelidiyyy surfaces, but with
Our experimental setup for evaluating Sitzmann et al. (2020b) periodic activation consisted of mod- ifying an existing tensorflow (Abadi et al., 2015) implementation to accept our spatial queries as input and signed distances target. We train the SIREN model 200 epochs with learning rate of 5 and the same loss as our own configuration. Interestingly, we find that the SIREN model produces smoother approximations of the armadillo's 's surface (see Figure 7) but lacks fine detail. Once again, when increasing our model complexity to just 8 layers of 64 hidden units, we start to see the benefits of the periodic activation yielding much better approximations our relu activation. For our base configuration of just 7553 parameters we choose to continue us- ing RELU activation, but where high-fidelity weight- eeggh--encoding neural implicits are required, SIREN should be employed.
the surface then
Ground truth
Sitzmann et al. Mildenhall et al. (2020)
Ours
(2020)
Figure 7: Results of using Mildenhall et al. (2020) and Sitzmann et al. (2020b)
A.4 REPRESENTATION COMPACTNESS
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Figure 8: Our learnt weight-encoded neural implicit format (right) can be shown to better approx- imate the original surface (grey, inset) compared to adaptive decimation of the original triangle mesh Garland & Heckbert (1997) (left) and uniform signed distance grid (middle) with equal mem- ory impact. gpvillamil (skull), Makerbot (whale), morenaP (frog), artec3d (dragon), JuliaTruch- sess(octopus) under CC BY.
13


Reviews:
[{"review_id": "zN-Lcbc6IUl", "reviewer": "AnonReviewer1", "summary": "", "questions": "", "limitations": "", "rating": 7, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "\nThis paper presents a simple (and convincing) idea that neural networks can be employed to \"memorize\" the surface geometry of 3D objects. Such memorized networks -- dubbed neural implicits -- exhibit significantly lower memory requirements, while allowing for better surface approximation compared to alternative prior approaches.\n\n## Strengths\n\n**S1** This paper pursues a very timely research direction: several interesting approaches over the past year [A, B, C] (to list a few) have been proposed that leverage the overparameterized nature of neural nets to \"memorize\" the surface geometry of objects. While prior approaches require the neural network to learn a \"distribution\" over multiple shapes of a category and \"generalize\" to unseen shapes at test time, this paper poses a different question: can neural nets learn to reproduce one specific shape (by overfitting). This essentially turns a bug (i.e., overfitting) into a feature (i.e., neural implicits), which is a neat idea!\n\n**S2** The paper is extremely well-written, and a thoroughly enjoyable read. Most design choices are well-motivated, and where applicable, empirically/qualitatively justified.\n\n**S3** I find the discussion about various sampling strategies insightful. This section (2.1.2) makes apparent several intricacies that usually fly under the radar. Although I do have a few reservations (see discussion under \"Weaknesses\"), I think the paper does a good job of explaining how sampling strategies such as [A] result in unintended bias. It is worth noting that there isn't consensus in the implicit-representation-learning community about what sampling strategies work better than the others (eg. [A] and [B] present conflicting accounts on this matter).\n\n**S4** In general, I find the experiment setup convincing. The code and/or data release might see a lot of community interest in following up on this work.\n\n## Weaknesses\n\nI do think this work can benefit from a number of additional discussions and/or explorations, which I list below.\n\n**W1** It appears that the focus of this work is in reconstructing the geometry of the surface. Often 3D meshes have other important attributes that are essential for downstream tasks (face normals, color and/or texture information, material properties). This feels like an important limitation that could benefit from discussion. (Do we expect learning textures, normals, etc. to be \"trivial\"? In which case it'd greatly strengthen the paper. If not, explicitly highlighting this could result in interesting follow-up work).\n\n**W2** While the paper paints the rosy side of neural implicits, it might benefit from quantifying the \"ineffectiveness of weight-encoded neural implicit 3D shapes\". How well do surface continuity properties hold (C0, C1, and higher) hold? What can possibly be done to mitigate such ill-effects (if any)? While Sec. 4 talks about these questions being interesting future directions, it might add value to the current submission to quantify the gap (eg. Eikonal constraint violation, watertight-ness or manifold-ness violations) to be comprehensive.\n\n**W3** Traditionally, multiple representations have flourished in the computer graphics community, primarily because each of them possesses a complementary set of strengths and weaknesses. I'd assume that to hold true for neural implicits too. For instance, rendering neural implicits, computing higher order information (curvature? derivatives?) and above all, interactive editing applications might be more inefficient with neural implicits?\n\n**W4** I feel the discussion about sampling strategies could greatly benefit from an accompanying empirical analysis. While most arguments in this section (2.1.2) appear intuitive, they're often hand-wavy.\n\n**W5** Since one of the primary goals of neural implicits is to reduce memory requirements for storing and processing polygon meshes, I hoped to see more discussion on \"classical\" data compression techniques used in the computer graphics community. A seminal reference is [D].\n\n**W6** Discussion of a few other closely related papers, such as [E, F] can be added.\n\n## Minor remarks\n\n**M1** The following remarks have had zero-impact on my overall score, and as such, I do not expect the authors to respond on these issues.\n\n**M2** The abstract contains a few (minor) grammatical errors that cound benefit by a proofread.\n\n**M3** A concurrent recent work [G] seems to follow a fairly similar idea, and it would be nice to discuss and contrast the current submission wrt this (particularly for the camera-ready version).\n\n\nReferences\n==========\n\n[A] Mescheder, Lars, et al. \"Occupancy networks: Learning 3d reconstruction in function space.\" Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. 2019.\n\n[B] Park, Jeong Joon, et al. \"Deepsdf: Learning continuous signed distance functions for shape representation.\" Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. 2019.\n\n[C] Sitzmann, Vincent, et al. \"Implicit neural representations with periodic activation functions.\" arXiv preprint arXiv:2006.09661 (2020).\n\n[D] Pierre Alliez and Craig Gotsman. Recent Advances in Compression of 3D Meshes. Advances in multiresolution for geometric modelling, 2005.\n\n[E] Hanocka, Rana, et al. \"Point2Mesh: A Self-Prior for Deformable Meshes.\" arXiv preprint arXiv:2005.11084 (2020).\n\n[F] Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains. Matthew Tancik*, Pratul Srinivasan*, Ben Mildenhall*, Sara Fridovich-Keil, Nithin Raghavan, Utkarsh Singhal, Ravi Ramamoorthi, Jonathan T. Barron, Ren Ng.\n\n[G] Overfit Neural Networks as a Compact Shape Representation. arXiv. 2020.", "extra_scores": {}, "extra_text": {}}, {"review_id": "OS-6mLf7SGO", "reviewer": "AnonReviewer2", "summary": "", "questions": "", "limitations": "", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "This paper discusses implicit SDF representations encoded by neural networks. \n\nThe paper is written partly as an opinion paper, partly as if it was an enormous contribution and completely new idea to overfit a network to a simple shape (thus not requiring a latent vector) compared to using a latent code to obtain a common latent space for a set of shapes. It is thus constantly written in opposition to Park et al. 2019, which I found useless and tiring to read, takes far too much space to my taste and I think is detrimental to the overall quality and interest. The idea of encoding an implicit function in the weights of a network without latent code is not new at all, and in particular common practice in all the NERF literature. The fact that it will lead to better reconstructions than learning a common latent space is obvious.\n\nTo me, the most interesting parts of the paper is the tools it proposes to deal with implicit SDFs, that are, independently of this paper, becoming more important in 3D vision: the use of generalized winding numbers to encode general meshes and efficient visualization tools. Admittedly, these are known in the graphics community, which may make the paper hard to publish in a top tier conference, but I believe modern tools compatible with recent DL libraries would be valuable and a paper introducing them could have a good impact. Similarly, a pipeline allowing CSG operations could have been presented and analyzed (further than a few qualitative examples)  The current version of the paper however provides too little details and evaluation on these and on the contrary spends a lot of time on discussion which I felt were not any real contribution.\n\nIn more details, I from my point of view\n- sections discussing relation to deepSDF (1 and 2.1.1) could be drastically compressed (and section 1 could be rewritten completely)\n- the paper is missing a related work \n- the interest of 2.1.2 is very limited. Yes, sampling is equivalent to weighting, this is more or less importance sampling. Some papers prefer to define the sampling, this one prefers to define the weight. The results to support this strategy seem anecdotical and insufficient (one would have to carefully cross-validate the parameters in both to draw any conclusion) Moreover, it is far more costly since need to start by sampling a much larger set of points.\n- I find 2.2 and 2.3 of potential interest, but they are not much developed nor evaluated\n- 3.1 is missing the crucial evaluation of the rendering quality\n- 3.2 focusses on comparison with deepSDF which I think does not bring much since it is a weak baseline/not designed for the same purpose + should be quantitative, not qualitative\n- I did not find 3.3 to be very informative\n-  the paper is missing clear evaluations and quantitative analysis (which is related to the fact its contribution is unclear)\n- the conclusion (4) finally discusses the issue of normals, which I believe should have been addressed earlier\n\n\n\nSome more comments: \n- I think the \"neural implicit\" term introduced near the first inset is not well chosen (implicit is an adjective) but in any case it should not be used so much in the intro before its introduction\n- the quality of the reconstructions is actually not great, this is not very visible in a printed version because all images are small, but zooming on the bunny for example actually shows it is quite coarse. This is an important practical problem, and one could imagine how to jointly work efficiently with representation of different quality (even if that sounds more like a graphics paper than an ICLR paper) Also, from the video it seems training is not super stable (which is surprising, maybe decreasing the LR after a number of epochs would lead to better results)\n- the MLP encodes a linear by part function, the zero-level-set   (which does not change after the tanh) is thus a set of planes, resulting in a planar by part approximation of the shape, which is very visible in the qualitative results and should be discussed. (this also implies that the same quality for a similar budget can be obtained with a mesh)\n- 90 seconds to obtain a representation is still very expensive. Indeed, if one were, as suggested in the paper, to use the representation to perform CSG, one would need to either save the full construction process or convert often into the implicit representation. It's also unclear how errors would accumulate when re-encoding several time a given shape. ", "extra_scores": {}, "extra_text": {}}, {"review_id": "FUsSgVgj4uw", "reviewer": "AnonReviewer3", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "The paper proposes a weight-encoded neural implicit representation for 3D shapes. The idea is to encode every shape in the network weights of its own designated small MLP network, instead of trying to learn a latent space of shapes. This leads to a really compact shape representation based on signed distance fields that could be interesting for many applications. The approach uses importance sampling to speed up training and robust losses.\n\nThe paper evaluates the representation in terms of visualization efficiently, stability, and the compactness of the representation on a large dataset of shapes. The approach outperforms latent encoded shape representations such as DeepSDF in terms of compression and accuracy.\n\nI believe the idea of encoding shapes using signed distance fields and MLPs for compression is really exciting. Such implicit representations of shapes are becoming really popular at the moment and have huge potential in the future. In terms of novelty, I have seen many shape representation approaches based on MLPs lately, but none so far focussed on the compression aspect.\n\nIn summary, I believe that the compression aspect of this work is really interesting and will inspire follow up works along those lines. The method achieves impressive compression ratios and can decompress/render the shapes still at high speeds.", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: The paper proposes how weight-encoded neural implicit can be strong 3D shape representations. A neural network is trained such that it overfits over a single shape, and the weights of such network is a great representation for the 3D shape. Results are shown on signed distance field (SDF) generation from meshes.

Strengths:
- an interesting idea for generating compact representations of 3D shapes
- Will further foster several conversations within the deep learning community

Weaknesses:
- Very limited evaluation to support the authors  claims, particularly against other traditional learnable 3D representations
Decision: Reject