G EOMETRY -C ONSISTENT N EURAL S HAPE R EPRESENTATION WITH I MPLICIT D ISPLACEMENT F IELDS
Anonymous authors
Paper under double-blind review
A BSTRACT
We present implicit displacement ﬁelds , a novel representation for detailed 3D geometry.
Inspired by a classic surface deformation technique, displacement mapping, our method represents a complex surface as a smooth base surface plus a displacement along the base’s normal directions, resulting in a frequency-based shape decomposition, where the high-frequency signal is constrained geometrically by the low-frequency signal.
Importantly, this disentanglement is unsupervised thanks to a tailored architectural design that has an innate frequency hierarchy by construction.
We explore implicit displacement ﬁeld surface reconstruction and detail transfer and demonstrate superior representational power, training stability, and generalizability.
I NTRODUCTION
Neural implicit functions have emerged as a powerful tool for representing a variety of signals.
Compared to conventional discrete representations, neural implicits are continuous and thus not tied to a speciﬁc resolution.
Recently, neural implicits have gained signiﬁcant attraction in a variety of applications ranging from 3D reconstruction Sitzmann et al.
; Niemeyer et al.
; Yariv et al.
; Peng et al.
(2020), neural rendering Mildenhall et al.
Pumarola et al.
image translation Skorokhodov et al.
; Chen et al.
(2020a) to deformation approximation Deng et al.
In this paper, we focus on neural implicit representations for 3D geometry.
While neural implicits can theoretically model geometry with inﬁnite resolution, in practice the output resolution is dependent on the representational power of neural nets.
So far, the research community approaches the problem from two main directions.
The ﬁrst is to partition the implicit function using spatial structures Chabra et al.
; Jiang et al.
Liu et al.
; Takikawa et al.
(2021), thus making the memory and computation demands dependent on the geometric complexity.
The other direction focuses on improving networks’ ability to represent high-frequency signals, either in a preprocessing step (referred to as positional encoding)
Mildenhall et al.
(2020) or by using sinusoidal representation networks ( SIREN ) Sitzmann et al.
However, training these networks is very challenging, as they are prone to overﬁtting and optimization local minima.
Inspired by the classic computer graphics technique, displacement mapping Cook (1984); Cook et al.
(1987), we propose a novel parameterization of neural implicit functions, implicit displacement ﬁeld , abbreviated as IDF, to circumvent the above issues.
Our method automatically disentangles a given detailed shape into a coarse base shape represented as a continuous, low-frequency signed distance function and a continuous highfrequency implicit displacement ﬁeld, which offsets the base iso-surface along the normal direction.
The key novelty of our approach lies in extending the classic displacement mapping, which is discrete and lies only on the base surface, to a continuous function in the R 3 domain and incorporating it into contemporary neural implicit representations, ergo achieving a disentanglement of geometric details in an unsupervised manner.
Our main technical contribution includes
a principled and theoretically grounded extension of explicit discrete displacement mapping to the implicit formulation, 2.
a neural architecture that creates a geometrically interpretable frequency hierarchy in the neural implicit shape representation by exploiting the inductive bias of SIREN s, and 3.
introducing transferable implicit displacement ﬁelds by replacing the common coordinates input with carefully constructed transferrable features, thus opening up new opportunities for implicit geometry manipulation and shape modeling.
Systematic evaluations show that our approach is signiﬁcantly more powerful in representing geometric details, while being lightweight and highly stable in training.
R ELATED WORK
Hierachical neural implicit shape representation.
Neural implicit shape representation was initially proposed by several works concurrently (Park et al., 2019; Chen & Zhang, 2019; Mescheder et al., 2019), and since then many works have sought to introduce hierarchical structures into the neural representation for better expressiveness and generalizability.
The majority of these methods focus on spatial structures.
Chabra et al.
(2020) and Saito et al.
(2019; 2020) use sparse regular voxels and dense 2D grid, respectively, to improve detail reconstruction.
In the spirit of classic approaches, e.g . Frisken et al.
; Ohtake et al.
Liu et al.
; Takikawa et al.
; Martel et al.
(2021) store learned latent codes in shape-adaptive octrees, leading to signiﬁcantly higher reconstruction quality and increased rendering speed.
A common disadvantage of these methods is that the memory use and model complexity are directly tied to the desired geometric resolution.
In parallel, other proposed methods learn the spatial partition.
Some of these methods decompose the given shape using parameterized templates, such as anisotropic Gaussians (Genova et al., 2019), convex shape CVXNet (Deng et al., 2020; Chen et al., 2020b) or simple primitives (Hao et al., 2020), while others represent local shapes with small neural networks and combine them together either using
Gaussians (Genova et al., 2020) or surface patches (Tretschk et al., 2020).
Due to limitations of template functions and delicate spatial blending issues, these methods can only handle very coarse geometries.
Concurrently, Li & Zhang (2021) propose a two-level neural signed distance function for singleview reconstruction.
Exploiting the fact that most man-made shapes have ﬂat surfaces, it represents a given shape as a coarse SDF plus a frontal and rear implicit displacement map for better detail construction.
Besides having entirely different applications – we focus on representing signiﬁcantly higher geometry resolutions – our implicit displacement is grounded in geometry principles and applies to general shapes.
High-frequency representation in neural networks As formally explained by Rahaman et al.
; Basri et al.
(2020), neural networks have a tendency to learn low-frequency functions.
To combat this issue, Mildenhall et al.
(2020) incorporate “positional encoding” for neural rendering and demonstrate remarkable progress in terms of detail reconstruction, which is a sinusoidal mapping for the input signal, a practice later theoretically justiﬁed by Tancik et al.
Alternatively,
SIREN also shows impressive advances in detail representation by replacing ReLU activation with sin functions.
With these new networks gaining popularity, a few works delve deeper and apply a coarse-to-ﬁne frequency hierarchy in the training process for deformable shape representation (Park et al., 2020b) and meshing (Hertz et al., 2021).
In our method, we also create a frequency hierarchy by leveraging this new form of networks – not only in the training scheme but also explicitly in the construction of the networks to reﬂect our geometry-motivated design principles.
Detail transfer Detail transfer refers to transplanting the disentangled geometric details from a source shape onto a target object with high ﬁdelity and plausibility.
Classic detail transfer methods represent surface details as normal displacements (Botsch et al., 2010; Zhou et al., 2007; Sorkine & Botsch, 2009).
The majority of them are parametric (Ying et al., 2001;
Biermann et al., 2002;
Sorkine et al., 2004;
Zhou et al., 2006; Takayama et al., 2011), relying on a consistent surface parameterization between the source and the target shape.
Non-parametric approaches (Chen et al.,
[Figure]
2012; Berkiten et al., 2017), on the other hand, ﬁnd best-matching surface patches between the source and target, and copy the details iteratively from coarse to ﬁne.
These classic approaches produce high quality results, but often require a pre-deﬁned base surface or abundant user inputs.
In the “deep” realm, DeepCage (Yifan et al., 2020) proposed a neural deformation method that maps solely the coarse geometry, hence allowing detail transfer without tackling detail disentanglement.
Hertz et al.
(2020) learn the coarse-to-detail correspondence iteratively from multi-scale training data, while Chen et al.
(2021) synthesizes details by upsampling a coarse voxel shape according to a style code of another shape using GANs.
All of these approaches use explicit representations, hence they are subject to self-intersection and resolution limitations.
D 2 IM-Net
Li & Zhang, 2021) uses two planar displacement maps to transfer surface details by mapping the coordinates of the source and target shapes using part segementation, thus limiting the application to man-made rigid shapes.
In comparison, our method does not require any correspondence mapping.
M ETHOD
[Figure]
We represent a shape with ﬁne geometric details using two SIREN networks of different frequencies in the activation functions.
The SIREN with lower frequency describes a smooth base surface; the SIREN with higher frequency adds microstructure to the base iso-surface by producing an implicit displacement ﬁeld along the base’s normal direction (see Figure 2).
In this section, we ﬁrst formally deﬁne implicit displacement ﬁeld by generalizing the classic explicit and discrete displacement mapping in Sec 3.1, then in Sec 3.2 we introduce the network architectures and training strategies that are tailored to this deﬁnition, ﬁnaly in Sec 3.3 we extend the implicit displacement to address transferability.
3. I MPLICIT DISPLACEMENT FIELDS
In classic displacement mapping as shown in Figure 1, high-frequency geometric details are obtained on a smooth base surface by taking samples from the base surface and offsetting them along their normal directions by a distance obtained (with interpolation) from a discrete height map.
Two elements in this setting impede a direct adaptation for implicit shape representation: 1.
the displacement mapping is deﬁned discretely and only on the base surface, whereas implicit surface functions are typically deﬁned continuously on the R 3 domain; 2.
the base surface is known and ﬁxed, whereas our goal is to learn the base surface and the displacement jointly on-the-ﬂy.
Addressing the above challenges, we ﬁrst deﬁne implicit displacement ﬁelds (IDF), which are continuous analog to height maps that extend displacement mapping to the R 3 domain.
Deﬁnition 1.
Given two signed distance functions f
and ˆ f and their respective iso-surfaces at a given value τ ∈ R , S τ = (cid:8) x ∈ R 3 | f ( x ) =
τ (cid:9) and ˆ S τ = (cid:110) x ∈ R 3 | ˆ f ( x ) = τ (cid:111) ,
an implicit displacement ﬁeld d: R 3 → R deﬁnes the deformation from S τ to ˆ S τ such that
[Equation]
This deﬁnition is schematically illustrated in Figure 3, where the iso-surface S 0 and S τ are mapped to ˆ S 0 and ˆ S τ with the same implicit displacement ﬁeld d .
Notably, the height map in classic displacement mapping is a discrete sampling of IDF for the limited case τ = 0 .
In the context of surface decomposition, our goal is to estimate the base surface [-S 
-]f and the displacement d given an explicitly represented detailed surface ˆ S 0 .
Following equation 1, we can do so by minimizing the difference between the base and the ground truth signed distance at query points x ∈ R 3 and their displaced position ˆ x = x + d ( x ) n , i.e ., min | f ( x ) − ˆ f GT (ˆ x ) | .
However, this solution requires evaluating ˆ f GT (ˆ x ) dynamically at variable positions ˆ x , which is a costly operation as the detailed shapes are typically given in explicit form, e.g ., as point clouds or meshes.
Hence, we consider the inverse implicit displacement ﬁeld ˆ d , which deﬁnes a mapping from ˆ S τ to S τ , f (cid:16) ˆ x + ˆ d (ˆ x ) n (cid:17) = ˆ f (ˆ x ) , as depicted in Figure 3.
Assuming the displacement distance is small, we can approximate n , the normal after inverse displacement, with that before the inverse displacement, i.e .
[Equation]
This is justiﬁed by the following theorem and corollary, which we prove in the Appendix A.
Theorem 1.
If function f : R n → R is differentiable, Lipschitz-continuous with constant L and
Lipschitz-smooth with constant M , then (cid:107)∇ f ( x + δ ∇ f ( x ))
− ∇ f ( x ) (cid:107) ≤ | δ | LM .
Corollary 1.
If a signed distance function f satisfying the eikonal equation up to error (cid:15) > 0 , |(cid:107)∇ f (cid:107) − 1 | < (cid:15) , is Lipschitz-smooth with constant M , then (cid:107)∇ f ( x + δ ∇ f ( x ))
− ∇ f ( x ) (cid:107) <
(1 + (cid:15) ) | δ | M .
Given n = ∇ f ( x ) (cid:107)∇ f ( x ) (cid:107) , ˆ n = ∇ f
(ˆ x ) (cid:107)∇ f (ˆ x ) (cid:107) , and ˆ x = x
+ d ( x ) n , let δ
= d ( x ) (cid:107)∇ f ( x ) (cid:107) , we can show (cid:107) ˆ
n − n (cid:107) ≤ 1 + (cid:15) 1 − (cid:15) | δ | M ( c.f . Appendix A).
In other words, the difference of ˆ n and n is bounded by a small constant.
Thus we obtain the approximation in equation 2, which allows us to presample training samples { ˆ x } and use precomputed ˆ f GT (ˆ x ) or its derivatives (see Sec 3.2) for supervision.
3. N ETWORK DESIGN AND TRAINING
The formulation of (inverse) implicit ﬁeld in the previous section is based on three assumptions: (i) f is smooth, (ii) d is small, (iii) f satisﬁes the eikonal constraint up to an error bound.
In this section, we describe our network architecture and training technique, with emphasis on meeting these requirements.
Network architecture.
We propose to model f and ˆ d with two SIREN s denoted as N ω B and
N ω D , where ω B and ω D refer to the frequency hyperparameter in the sine activation functions x (cid:55)→ sin ( ω x ) .
Evidently, as shown in Figure 4, ω dictates an upper bound on the frequencies the network is capable of representing, thereby it also determines the network’s inductive bias for smoothness.
Correspondingly, we enforce the smoothness of f and detail-representing capacity of ˆ d using a smaller ω B and a larger ω D .
Empirically, we ﬁnd that ω B = 15 and ω D = 60 ﬁts our needs in most cases.
Moreover, we add a scaled tanh activation to the last linear layer of N ω D , i.e .
α tanh ( · ) , which ensures that the displacement distance is smaller than the constant α .
Networks containing high-frequency signals, e.g . N ω D , require large amounts of accurate ground truth data for supervision to avoid running into optimization local minima (Park et al., 2020b).
Consequently, when dense and accurate ground truth SDF values are not available, high-frequency signals often create artifacts. This is often the case in void regions when learning from point clouds, as only implicit regularization and fuzzy supervision is applied (see the ﬁrst and last terms of equa-
ν
= 0 .ν = 0 .ν
= 0 .
N ω B ( x ) χ tion 4).
Hence, we apply an attenuation function χ ( N ω B
[Equation]
=
[Figure]
from the base surface, where ν determines the speed of attenuation as depicted in Figure 5.
Combining the aforementioned components, we can compute the signed distance of the detailed shape at query point x in two steps:
[Equation]
Training.
We adopt the loss from SIREN , which is constructed to learn SDFs directly from oriented point clouds by solving the eikonal equation with boundary constraint at the on-surface points.
Denoting the input domain as Ω (by default set to [ − 1 , 1] 3 ) and the ground truth point cloud as P = { ( p i , n i ) } , the loss computed as in equation 4, where λ { 0 , 1 , 2 , 3 } denote loss weights:
[Equation]
As the displacement and the attenuation functions depend on the base network, it is beneﬁcial to have a well-behaving base network when training the displacement (see Sec 4.2).
Therefore, we adopt a progressive learning scheme, which ﬁrst trains N ω B , and then gradually increase the impact of N ω D .
Notably, similar frequency-based coarse-to-ﬁne training techniques are shown to improve the optimization result in recent works (Park et al., 2020b; Hertz et al., 2021).
We implement the progressive training via symmetrically diminishing/increasing learning rates and loss weights for the base/displacement networks.
For brevity, we describe the procedure for loss weights only, and we apply the same to the learning rates in our implementation.
First, we train N ω B by substituting ˆ f in the loss equation 4 with f , resulting a base-only loss denoted L f .
Then, starting from a training percentile T m ∈ [0 , 1] , we combine L f and L ˆ f via κ L f + (1 − κ )
L ˆ f with κ =  (cid:16) 1 + cos (cid:16) π ( t − T m ) (1 − T m ) (cid:17)(cid:17) , where t ∈
[ T m , 1] denotes the current training progress.
3. T RANSFERABLE IMPLICIT DISPLACEMENT FIELD .
In classic displacement mapping, the displacement is queried by the UV-coordinates from surface parameterization, which makes the displacement independent of deformations of the base surface.
We can achieve similar effect without parameterization by learning query features , which emulate the UV-coordinates to describe the location of the 3D query points w.r.t .
the base surface.
We construct the query features using two pieces of information: (i) a global context descriptor, φ ( x ) , describing the location of the query point in relation to the base surface in a semantically meaningful way, (ii) the base signed distance value f ( x ) , which gives more precise relative location with respect to the base surface.
Since both are differentiable w.r.t .
the euclidean coordinates of the query point, we can still train N
ω
D using derivatives as in equation 4.
Our global context descriptor is inspired by Convolutional Occupancy Networks (Peng et al., 2020).
Speciﬁcally, we project the sparse on-surface point features obtained using a conventional point [Figure] cloud encoder onto a regular 3D (or 2D, c.f .
Sec 4.3) grid, then use a convolutional module to propagate sparse on-surface point features to the off-surface area, ﬁnally obtain the query feature using bilinear interpolation.
We use normals instead of point positions as the input to the point cloud encoder, making the features scale-invariant and translation-invariant.
Note that ideally the features should also be rotation-invariant.
Nevertheless, as we empirically show later, normal features can in fact generalize under small local rotational deformations, which is sufﬁcient for transferring displacements between two roughly aligned shapes.
We leave further explorations in this direction for future work.
N ω D is tasked to predict the displacement conditioning on φ ( x ) and f ( x ) .
However, empirical studies (Chan et al., 2020) suggest that SIREN does not handle high-dimensional inputs well.
Hence, we adopt the FiLM conditioning (Perez et al., 2018; Dumoulin et al., 2018) as suggested by Chan et al.
(2020), which feeds the conditioning latent vector as an afﬁne transformation to the features of each layer.
Speciﬁcally, a mapping network M converts φ ( x ) to a set of Cdimensional frequency modulators and phase shifters { γ i , β i } , which transform the
i -th linear layer to (cid:0) 1 + 12 γ i (cid:1) ◦ ( W i x
+ b i )
+ β i , where W i and b i are the parameters in the linear layer and ◦ denotes element-wise multiplication.
Finally, since SIREN assumes inputs in range ( − 1 , 1) , we scale f using ¯ f ( x ) = tanh (cid:0)
ν f ( x ) (cid:1) to capture the variation close to the surface area, where ν is the attenuation parameter described in section 3.2.
Figure 6 summarizes the difference between transferable and non-transferable displacement ﬁelds.
Formally, the signed distance function of the detailed shape in equation 3 can be rewritten as
[Equation]
R ESULTS
We now present the results of our method.
In Sec 4.1, we evaluate our networks in terms of geometric detail representation by comparing with state-of-the-art methods on the single shape ﬁtting task.
We then evaluate various design components in an ablation study in Sec 4.2.
Finally, we validate the transferability of the displacement ﬁelds in a detail transfer task in Sec 4.3.
Extended qualitative and quantitative evaluations are included in section B.
Implementation details.
By default, both the base and the displacement nets have 4 hidden layers with 256 channels each.
The maximal displacement α , attenuation factors ν , and the switching training percentile is set to T m are set to 0 .
02 and 0 .
2 respectively ; The loss weights λ { 0 , 1 , 2 , 3 } in equation 4 are set to 5, 400, 40 and 50.
We train our models for 120 epochs using ADAM optimizer with initial learning rate of 0.0001 and decay to 0.00001 using cosine annealing (Loshchilov & Hutter, 2016) after ﬁnishing 80% of the training epochs.
We presample 4 million surface points per mesh for supervision.
Each training iteration uses 4096 subsampled surface points and 4096 offsurface points uniformly sampled from the [ − 1 , 1] 3 bounding box.
To improve the convergence rate, we initialize SIREN models by pre-training the base model N ω B (and baseline SIREN , c.f .
Sec 4.1) [Table] [Figure] to a sphere with radius 0 . 5 .
This initialization is optional for our training but is critical for baseline SIREN s.
Data.
We test our method using 16 high-resolution shapes, including 14 from Sketchfab (ske, 2021) and 2 from Stanford 3DScanRepo (sta, 2021).
Our transferable displacement model is tested using shapes provided by Berkiten et al.
(2017), Yang et al.[-(2020), and Thingi
-]
and Zhou & Jacobson (2016).
D ETAIL REPRESENTATION .
We compare our approaches with 5 baseline methods. 1) FFN (Tancik et al., 2020) with SOFTPLUS activation and 8 frequency bands progressively trained from coarse-to-ﬁne, where a total of 8 hidden layers each of size 256 are used to match our model size; additionally we apply a skip-connection in the middle layer as proposed in DeepSDF Park et al.
2) NGLOD (Takikawa et al., 2021) trained using 4 and 6 levels of detail (LODs) corresponding to 64 3 and 256 3 spatial resolution respectively, with LOD4 comparable with our model in terms of the number of parameters.
3) baseline SIREN , for which we trained three different variations in hope of overcoming training divergence issues;.
4) direct residual, where we compose the signed distance value simply as the sum of base and displacement nets, i.e .
ˆ f ( x ) = N ω B
( x )+ N ω D ( x ) .
5) D-SDF (inspired by Pumarola et al.
Park et al.
(2020a)), which represents the displacement as a R 3 vector, i.e .
ˆ f = f ( x + ∆) , where ∆ ∈ R 3 is predicted in the second network.
We follow network specs of D-Nerf Pumarola et al.
(2020), which contains two 8-layer MLP networks with R E LU activation and positional encodings.
Among these, NGLOD, direct residual and D-SDF requires ground truth SDF for supervision, the rest are trained using our training loss.
Two-way point-to-point distance and normal cosine distance are computed as the evaluation metrics on 5 million points randomly sampled from meshes extracted using marching cubes with 512 3 resolution.
As shown in Table 1 and Figure 7, our method outperforms the baseline methods with much higher reconstruction ﬁdelity. NGLOD with 6 LODs is the only method onpar with ours in terms of detail representation, however it requires storing more than 300 times as many as parameters as our model.
SIREN networks with larger ω have severe convergence issues even with sphere initialization ( c.f . Implementation Details) and gradient clipping.
Direct residual doesn’t enforce displacement
[Table]
directions and produces large structural artifacts.
D-SDF yields qualitatively poor results, as the displacement net is unable to learn meaningful information (more analysis is shown in section B.1.2).
A BLATION STUDY
[Table]
We study the contributions of different design components, namely the displacement scaling α tanh , the attenuation function χ and the progressive training.
As Table 2 shows
[Table]
all the test modes converge within comparable range, even for the model with the least constraints.
This shows that our model is robust against violations of theoretical assumptions speciﬁed in Sec 3.1.
At the same time, the performance rises with increasingly constrained architecture and progressive training, suggesting that the proposed mechanisms further boost training stability.
Table 3 shows that in a reasonable range of α and ν there is very little variance in reconstruction quality, indicating the robustness w.r.t . parameter selection.
If α is too small, the displacement may no longer be sufﬁcient to correct the difference between the base and detailed surface, causing the slight increase of chamfer distances in the table for α = { 0 . 01 , 0 . 02 } .
When ν is too large (0.2), i.e . the high frequency signal is not suppressed in the void region, which leads to higher chamfer distances due to off-surface high-frequency noise.
4. T RANSFERABILITY
We apply our method to detail transfer in order to validate the transferability of IDF.
Speciﬁcally, we want to transfer the displacements learned for a source shape to a different aligned target shape.
In the ﬁrst test scenario, the base shape is provided and lies closely to the ground truth detailed surface.
In the second scenario, we are only provided with the detailed shapes and thus need to estimate the base and the displacements jointly.
The pipeline consists of the following steps: 1) train N ω B by ﬁtting the source shape (or the source base shape if provided), 2) train T ω D , M and the query feature extractor φ jointly by ﬁtting the source shape using equation 5 while keeping N ω B ﬁxed, 3) train N ω B new by ﬁtting the target shape (or the target base shape if provided), 4) evaluate equationby replacing N ω B with N ω B new .
To prevent the base network from learning high-frequency details when the base is unknown, we use ω B = 5 and three 96-channel hidden layers for N ω B .
Example outputs for both scenarios are shown in Figure 10, where the base shapes are provided for the shorts model.
We use a 32 3 and a 128 2 grid (for the frontal view), for the shorts and face model respectively in φ to extract the query features.
Our displacement ﬁelds, learned solely from the source shape, generate plausible details on the unseen target shape.
The transferred details contain high-frequency signals ( e.g . the eyebrows on the face), which is challenging for explicit representations.
However, for the second scenario the performance degenerates slightly since the displacement ﬁeld has to compensate errors stemming speciﬁcally from the base SDF.
In additional, we evaluate the design of the transferable IDF model by removing the mapping net and the convolutional context descriptor φ .
For the former case, we drop the FiLM conditioning and simply use concatenation of φ ( x ) and ¯ f ( x ) as the inputs to N ω D ; for the latter we directly use the normal at the sampled position as the context descriptor, i.e . φ ( x ) =
∇ f ( x ) .
As Figure 9 shows, the removal of mapping net and φ lead to differ- [Figure] [Figure] ent degrees of feature distortions.
We also compare with the DGTS (Hertz et al., 2020), which fails completely at this example since it only consumes local intrinsic features.
Furthermore, the effect of scaling ¯ f is shown in Figure 8, where using unscaled f as input to T ω D leads to artifacts at the boundary.
C ONCLUSION AND LIMITATIONS
In this paper, we proposed a new parameterization of neural implicit functions for detailed geometry representation.
Extending displacement mapping, a classic shape modeling technique, our formulation represents a given shape by a smooth base surface and a high-frequency displacement ﬁeld that offsets the base surface along its normal directions.
This resulting frequency partition enables the network to concentrate on regions with rich geometric details, signiﬁcantly boosting its representational power.
Thanks to the theoretically grounded network design, the high-frequency signal is well constrained, and as a result our model shows convergence qualities compared to other models leveraging high-frequency signals, such as SIREN and positional encoding.
Furthermore, emulating the deformation-invariant quality of classic displacement mapping, we extend our method to enable transferability of the implicit displacements, thus making it possible to use implicit representations for new geometric modeling tasks.
A limitation of our detail transfer application is the necessity to pre-align the two shapes.
In future work, we consider exploring sparse correspondences as part of the input, which is a common practice in computer graphics applications, to facilitate subsequent automatic shape alignment.
P ROOFS
Theorem 1.
If function f : R n → R is differentiable, Lipschitz-continuous with constant L and
Lipschitz-smooth with constant M , then (cid:107)∇ f ( x + δ ∇ f ( x ))
− ∇ f ( x ) (cid:107) ≤ | δ | LM .
Proof.
If a differentiable function f is Lipschitz-continuous with constant L and Lipschitz-smooth with constant M , then
[Equation]
Corollary 1.
If a signed distance function f satisfying the eikonal equation up to error (cid:15) > 0 , (cid:12)(cid:12)(cid:12) (cid:107)∇ f (cid:107) − 1 (cid:12)(cid:12)(cid:12) < (cid:15) , is Lipschitz-smooth with constant M , then (cid:107)∇ f ( x + δ ∇ f ( x ))
− ∇ f ( x ) (cid:107) <
(1 + (cid:15) ) | δ | M .
Proof.
(cid:12)(cid:12)(cid:12) (cid:107)∇ f (cid:107) − 1 (cid:12)(cid:12)(cid:12) <
(cid:15) ⇒ (cid:107)∇ f (cid:107) < (cid:15)
+ 1 .
This means f is Lipschitz-continuous with constant (cid:15) + 1 .
Then by Theorem 1, (cid:107)∇ f ( x + δ ∇ f ( x )) − ∇ f ( x ) (cid:107) < | δ | (1 + (cid:15) ) M .
Finally we show the upper bound for the normalized gradient, i.e .,
[Equation]
(8) where n = ∇ f ( x ) (cid:107)∇ f ( x ) (cid:107) , ˆ n = ∇ f
(ˆ x ) (cid:107)∇ f ( ˆ x ) (cid:107) and ˆ x = x + d ( x ) n with d ( x ) denoting the small displacement.
Proof.
[Equation]
For brevity
[Equation]
we denote ∇ f (ˆ x ) and ∇ f ( x ) as u and v .
Without loss of generality, we assume (cid:107) u (cid:107) ≤ (cid:107) v (cid:107) .
Then
[Equation]
by Eikonal constraint
[Equation]
Since | d ( x ) | is a small and (cid:107)∇ f ( x ) (cid:107) is close to 1, we can set δ = d ( x )
[Equation]
f ( x ) (cid:107) .
Thereby using
Corollary 1, we conclude
[Equation]
thus
[Equation]
Eq.
equation 11 can be proved as follows
[Equation]
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:13) (cid:13) (cid:13) , (17) which depicts the distance of the unit sphere projections of u and v .
Obviously, as shown in
Figure 11, (cid:107) d (cid:107) ≤ (cid:107) d (cid:48) (cid:107) if (cid:94) ( d , e ) ≥ 90 ◦ .
Since e = ( 1 (cid:107) u (cid:107) − 1 (cid:107) v (cid:107) ) v and ( 1 (cid:107) u (cid:107) − 1 (cid:107) v (cid:107) ) ≥ 0) , to show that (cid:94) (cid:104) d , e (cid:105) ≥ 90 ◦ is the same as to show that (cid:94) (cid:104) d , v (cid:105) ≥ 90 ◦ .
Indeed:
[Figure]
[Equation]
by Cauchy-Schwarz
[Equation]
B A DDITIONAL EXPERIMENTS AND RESULTS . B. A DDITIONAL INFORMATION TO THE COMPARISON IN S EC 4. B.1. A DDITIONAL EVALUATION RESULTS
Below we show the per-model Chamfer distances in addition to the average shown in Table 1.
More qualitative results are shown in Figure 15.
[Table]
[Table]
SIREN
[Table]
= [Table]
SIREN
[Table]
=
[Table]
SIREN
[Table]
=
[Table] [Table]
1.25 / 0.34 compressor 5.55/1.
1.39 / 0.23 dragon 5.10/4.
dragon warrior 5.94/7.
1.47 /4.56 dragon wing 5.68/5.
1.31 / 1.49 dragon china 6.20/2.
1.40/ 0.51 dragon cup 4.39/3.
1.10 / 0.51 helmet 4.79/1.
[Table]
B.1. A NALYSIS
We provide further diagnosis for the underwhelming results from the direct residual and the D-SDF models.
Direct residual composes the ﬁnal SDF as a simple sum of the base SDF and a residual value.
We train this model also with the attenuation and scaled tanh activation, and supervise both the base SDF and the composed SDF to stablize the base prediction.
However, as shown in
Figure 12a, the composed SDF often contain large structural errors.
Further inspections show that during the training, such structural errors change quickly and the reconstruction oscillates in scale.
These indicate that the without the directional constraints on the displacement, the two networks are not sufﬁciently coupled and interfere with each other during training.
[Table]
In D-SDF the displacement is not enforced to be along the normal direction.
D-SDF yields competitive quantitative result, but as Figure 12b shows, the predicted displacement vectors are homogeneous, indicating that due to the lack of constraints the displacement network is not incentivised to return meaningful outputs
. B. S TRESS TESTS
While we used dense and clean sampled point clouds as inputs in the paper, as our focus is on detail representation, we examine the behavior of our method under noisy and sparse inputs.
Speciﬁcally, we train our network with 400 thousand and 40 thousand sampled points (10% and 1% of the amount in our main experiment, respectively), and added σ = 0 .
002 and σ = 0 .
005 Gaussian noise on both the point normals and the point positions.
From the qualitative and quantitative results shown in
Figure 13 and Table 5, we can see that our method recovers geometric details better than Poisson reconstruction given sufﬁcient training data (c.f. the left half of the ﬁgure).
When the training sample is sparse, our method tends to generate more high-frequency noise as a result of overﬁtting.
B. D ETAIL TRANSFER .
The proposed transferable IDF makes it possible to deploy φ and T ω D a new base surface without any ﬁne-tuning or adaptation.
In other words, to transfer details to multiple targets, we only need to ﬁt a new base network N ω B new .
The composed SDF ˆ f ( x ) can be computed simply by replacing N ω B with N ω B new .
We show the result of a multi-target detail transfer in Figure 14.
It is worth mentioning
[Figure]
l e a r n e d t a r g e t b a s e t a r g e t s w i t h d e t a il f r o m t
h e s o u r ce t a r g e t s that even though the point extractor is trained on a single source shape, it is able to generalize across different identities thanks to the built-in scale and translation invariance.
B. I NFERENCE AND TRAINING TIME
Training the models as described in the paper takes 2412 seconds ( 40 minutes), which amounts to 120 epochs, i.e., around 20 seconds per epoch, where each epoch comprises 4 million surface samples and 4 million off-surface samples.
In comparison, the original implementation of NGLOD takes 110 minutes to train 250 epochs, where each epoch comprises 200000 surface samples and 300000 off-surface samples.
As for inference, using the same evaluation setup, NGLOD6 takes 193.9s for 512 3 points, while our inference takes 250.06 seconds for 512 3 query points.
All benchmarking is performed on a single Nvidia 2080 RTX GPU.
These timings could be improved by optimizing the model for performance, which we did not.
For example, instead of using autodiff for computing the base surface normals, one could exploit the fact that the differentiation of SIREN is also a SIREN and explicitly construct a computation graph for computing the base surface normals.
[Figure] [Figure]