Neural Vector Fields for Implicit Surface Representation and Inference

Implicit fields have been very effective to represent and learn 3D shapes accurately. Signed distance fields and occupancy fields are the preferred representations, both with well-studied properties, despite their restriction to closed surfaces. Several other variations and training principles have been proposed with the goal to represent all classes of shapes. In this paper, we develop a novel and yet fundamental representation by considering the unit vector field defined on 3D space: at each point in $\mathbb{R}^3$ the vector points to the closest point on the surface. We theoretically demonstrate that this vector field can be easily transformed to surface density by applying the vector field divergence. Unlike other standard representations, it directly encodes an important physical property of the surface, which is the surface normal. We further show the advantages of our vector field representation, specifically in learning general (open, closed, or multi-layered) surfaces as well as piecewise planar surfaces. We compare our method on several datasets including ShapeNet where the proposed new neural implicit field shows superior accuracy in representing any type of shape, outperforming other standard methods. The code will be released at https://github.com/edomel/ImplicitVF


Introduction
Representing 3D surfaces efficiently and conveniently has long been a challenge in computer graphics and 3D vision. A 3D surface representation should be accurate, while being suitable for any downstream task. Shape analysis such as 3D shape correspondences [22,30,64], 3D registration and deformations [18,40,61], or generation [7,23,37,44] rely on having the right representations. Recently some of these difficult questions have been answered with implicit neural representations (INR) [11,33,41,46], which use the classical signed distance field [38,39] or related binary occupancy field, learned with neural networks. Implicit fields are used in Neural Radiance Fields (NeRF) methods [34,59,62], surface representation from point clouds [2,4], image synthesis and processing [1,9,17,47], learning on videos [10,32] etc. On the left, the actual field representation proposed. On the right, the discrete divergence computation with div(f ) + 2, which yields 0 on the surface (blue), high (4) in the middle between surfaces (red) and around 2 everywhere else.
Two different types of implicit representations are commonly used. Distance based representations [13,41] associate to each point in space the distance from the closest surface of the object. This is either the signed distance field (SDF) [11,41,46] or the unsigned one (UDF) [13]. The former gives a negative sign to points inside the object and positive outside. The second type of representation is binary occupancy [33], where each point in space is classified to be either inside or outside the object. In both cases, a scalar field f (x) is learned to represent the surface at f (x) = 0. In general, SDF is the prevalent representation used in shape correspondences [64], Neural Radiance Fields [59], geometric regularization [2][3][4], etc. Recent work has proposed to supplement SDF with gradient information [50]. This adds regularization to surface reconstruction using the normals. Similar work also uses binary occupancy fields [28,35,36]. The neural UDF was recently shown to be suitable for learning open surfaces [13], with the downside being that it is difficult to discover the surface as a level set in the presence of prediction noise. Alternate strategies such as ray-based sampling methods [19] and pairwise point sampling [60] also solve open surface representation using a very different representation and sampling [19] or added training/architectural complexity [60].
We introduce a novel field for shape representation analogous to SDF. Inspired by image boundary representations in [42], we map each point in the R 3 3D space to a unit vector pointing towards the closest surface point. We call such representation simply Vector Field (VF). We mathe- matically demonstrate that the surface points can be recovered at the -2 level set of the vector divergence of the VF. Additionally, the proposed vector field allows us to represent both open and closed surfaces while being fast in inference. A further advantage is that it directly encodes the surface normal, a physical property of the surface, which can be used for surface regularization. In standard implicit functions, surface normals can be obtained only through a differentiation step [4,25]. We show the potential application of surface normals to the representation of piecewise planar surfaces through VF, thus improving the accuracy for certain object classes. Moreover, we perform experiments and show comparisons with VF on the standard ShapeNet [8] subsets with additional classes as well as on open cloth surfaces from the MGN dataset [5]. We obtain state-of-the-art results for surface representation.
In summary our contributions are threefold: • We propose a vector field representation, VF, for implicit representation of 3D shapes with mathematical proofs and empirical evidence for its representation capability.
• We show that the surface normal directly encoded in VF can be used to efficiently represent piecewise planar surfaces.
• We perform extensive tests for standard and general shapes, showing the strong performance of our method, achieving superior accuracy and generalization in multiple testing set-ups.

Related Work
We briefly review modern implicit functions for shape representation and analysis, followed by discussions on different modifications of the original approaches.
Standard representations. When comparing to traditional voxel-based representations for 3D shapes, the INR [12,33,41,46] have been able to solve, to a large extent, the high memory issues and limitations on downstream tasks [57,59,64]. Recent methods in INRs generally either use SDF [11,41,64] or binary occupancy [28,33,36], with very little known consensus on where and how they differ. Nonetheless, SDF and binary occupancy have the additional advantage that they represent closed surfaces and can be used to generate watertight meshes with the fast Marching Cubes (MC) algorithm. However, while the representations ensure watertight surfaces, the same property does not allow representing general shapes. Open surfaces, nonmanifold structures, multi-layered objects or objects with non-orientable surfaces, such as the Klein bottle, cannot be defined using the inside-outside separation used by SDF and binary occupancy. An equally fundamental representation, the distance field [39], was proposed as a solution for neural implicit representation of open surfaces [13]. On the other hand, [13] achieves it by forgoing quick and simple mesh inference by MC. A recent work [60] tackles open surface representation through classifying whether point-pairs are on the same side of the surface while enabling fast inference, at the cost of increased training complexity. In some cases, the standard representations are augmented with positional encoding [34,52] or sinusoidal activations [4,48] replacing ReLU layers in the neural architecture.
INR in NeRF and 3D Reconstructions. It is important to note that implicit fields are also used to construct Neural Radiance Fields (NeRF) [34,59], where the 3D surface is defined by a term called volume density σ. However, these methods are trained for accurate view synthesis rather than accurate scene representation. Several works [25,54,59] have proposed the use of a proper SDF in NeRF in order to improve the implicitly represented 3D scene. On the other hand, a ray-based implicit representation [19] was recently proposed, inspired from the so-called light field networks [49]. Another recent work [50] proposes to use SDF along with its gradient for the task of 3D reconstruction. It is true that the gradient information, similar to [3,4], can be used for surface regularization; however, the gradients themselves do not contribute to surface localization contrary to what we do.
Grid-based representations and modifications. An important class of 3D shape representations used the voxel grid, which is essentially the discretized binary occupancy. [26,27] directly extend the pixel-like image structure to volumetric data and have been used with straightforward extensions of image processing techniques, such as convolutions, to 3 dimensions. However, the clear drawback of traditional voxel-based methods is that they scale cubically with the resolution in terms of memory usage and computation. Therefore, the first methods proposed [14,53,56] could only work with 32 3 voxel grids. It has been later improved to 128 3 [55,63] with drawbacks in terms of network sizes and training speed. Recently voxel-grids are used in implicit representation, particularly in NeRF, by using octree-based sparse voxels [31], sparse spherical harmonic functions on the grid [62], voxel occupancy and hashing [35] or simply interpolating voxel occupancy [51]. Such grid-based interpolation and proxy functions may help improve the standard representations discussed above.

Method
We first formally define VF and list the properties which make it suitable for the representation task. We outline how it can be used in practice and compare its properties to earlier representations. Finally, we show how using VF allows us to impose a planar prior on the predicted shapes, which is useful for several object/scene classes.

Vector Field Representation
Suppose we have a 3D object or scene in a subset of 3D space, Ω ⊆ R 3 , embedding the set of 2D surfaces, Π ⊂ Ω, with surface point x S ∈ Π. In order to define the VF representation, we map each point x ∈ Ω to the normalized direction v ∈ Γ ⊂ R 3 pointing towards the closest surface point x S . Given the definition, the points {x S } are discontinuities in the field as positions where the direction towards the surface flips. At the surface points, the field v is mapped to either of the two opposite directional normals of the surface, by arbitrarily considering the limit going from higher to lower coordinate values. Generalizing to non-differentiable surface points where no surface normal is defined, this is equivalent to treating the point x S as a point x ′ = x S + ϵ s.t. x ′ / ∈ Π and ∥ϵ∥ 2 → 0 and computing the field values for point x ′ .
The VF representation is thus the mapping from positions in space to directions f : Ω → Γ. Figure 2 visualizes our representation for an example object. We define the VF representation formally as follows: Here ∥.∥ is the ℓ 2 norm operator. Eq. (1) defines the representation that maps each point in the subset Ω of space to a normalized direction vector v ∈ Γ. Here ϵ ∈ R 3 is an infinitesimal displacement. We note that when multiple pointŝ x S satisfy the infimum condition, one of them can be chosen arbitrarily. Similarly, the point x ′ is chosen as a point with higher co-ordinate value thanx S , with the only constraint that x ′ / ∈ Π. This particular choice is also an arbitrary one. We now need to ensure that VF can properly represent surfaces embedded in R 3 . Specifically, we establish a oneto-one map between surface points and the −2 level set of divergence computed on VF. This can be done by establishing the following three properties about the VF; for their proof, please refer to the supplementary material.
A point x ∈ R 3 then is a surface point, x ∈ Π, if and only if it belongs to the zero level set of g(x) + 2, i.e., The result just obtained holds for piecewise smooth surfaces [38] and can be proved by measuring flux in and out. However, in practice, the field divergence can slightly change on the surface. For example, the divergence can go lower than −2 at non continuously differentiable surface points. Nonetheless, the thresholding operation required for surface inference remains unaffected. Thanks to these generalized properties, VF can be modeled directly as an INR to represent any type of surface.

Field Creation and Training
VF can be used on top of any INR architecture without significant changes as we show in the experiments (Section 4). We follow standard practice [41,64] and measure the performance of VF for shape inference using the auto-decoder architecture. The field prediction at each coordinate is conditioned on a high dimensional embedding vector. At training time, the embedding vector is optimized together with the network parameters to a shape class; at test time, on the other hand, only the embedding vector is optimized on a small set of observed points. This allows us to test the reconstruction accuracy, while also considering the generalization capabilities. Our architecture only differs in that the output dimension is now 3D instead of 1D. Mathematically, we parameterize the neural VF prediction as follows: Eq. (4) describes the neural VF prediction with the network parameters θ and the embedding vector c ∈ R M . During training both θ and c are optimized together for a category. Strictly speaking, div f (x)+2 is the scalar-valued implicit function that defines the surface. However, a discrete divergence operation can easily be performed at the postprocessing step, and therefore we use the term INR more broadly to discuss the original function f itself. For training and optimization, the ℓ 1 loss 1 is computed between the predicted vectorsv with components x, y, and z and the ground truth vectors v gt : Here v gt is the ground truth VF andv is the prediction. More specifically, for each object, it is computed on a set of coordinates randomly sampled in space with a higher density around the surface [41]. For inference, surfaces can be defined as points in space with divergence lower than a fixed threshold α. Given the theoretical value of −2 for surface points, we set α = −1.5 to account for small prediction inconsistencies. Divergence computation allows us to adapt the marching cubes (MC) algorithm to run on a VF voxel grid. With this, similarly to a concurrent work [24] which uses the gradient of UDF, it is then possible to obtain accurate meshes. The α threshold on divergence identifies the voxels containing a surface and the VF values are used to assign the sides to which the vertices belong. For more details about our implementation, refer to the Supplementary Material.

Piecewise Planar Prior on VF
Applying surface priors on an INR requires discovery and regularization of the surface normals, which are con-1 ℓ 2 or cosine loss may also be used with a similar outcome. ventionally obtained in SDF and UDF through differentiation. In contrast, VF representation allows one to enforce priors on surface normals directly on the prediction. We show this advantage of VF by using it to represent piecewise planar objects. Piecewise planar surface representation or reconstruction is an application well explored in the literature [16,20,43,58] as large low-textured planar parts are common in man-made environment. The Manhattan-world assumption [15] was recently used in INR [25] to obtain multiview 3D reconstruction of Manhattan worlds. We show that, using VF, it is straightforward to extend the Manhattan assumption to arbitrary directions and numbers of planes which can be specific to the object in a category.
Due to the explicit normals in the VF implicit representation, it is possible to jointly predict a set of bases and select between them at each point in space to obtain the final VF. The direction bases represent then a very strong prior on the dominant directions that are present in the scene. The new architecture is thus described by the following: More specifically, we use a similar auto-decoder f with the latent embedding c, and an additional small network branch h 2 , whose input is a different latent vector c b , and the output is the set of bases b ∈ R 3×k , where k is the number of bases. We fix k = 10 for our implementation in this work. The main branch f takes then as input the latent vector c and the predicted bases b, together with the query position, and selects which of the bases to use at the queried position x using the binary one-hot vector output s b . In this way, it is possible to learn shape-specific bases, which are then selected to form the VF prediction.
The main branch of the network is trained as a classification network -with the number of classes equal to k -to predict the best matching basis b i using the Cross Entropy loss. Basis prediction, instead, is trained with an ℓ 1 loss on the basis that best matches the ground truth VF. The final loss is given by: Eq. (7) converges to the correct basis and VF such that they match the closest basis vector. We fix the scalar hyperparameter w b = 0.1.

Discussion on Implicit Representations
Unlike SDF and UDF [39], the VF divergence as implicit function is not discussed in the literature. Here we provide more insights into the main differences with existing representations.
VF, SDF and binary occupancy. The properties in Section 3.1 show that VF can represent any piecewise smooth surface, including open and non-orientable surfaces, proving its expressiveness.
Besides the stronger representation power, we show in Section 4 that VF is able to outperform SDF [41] and binary occupancy [33] also on watertight meshes, as subtly hinted by Figure 4. The difference can still be partially attributed to the more general representation power. A very thin object, even though represented as watertight, can be better modeled as a thin-shelled surface. When modeled as watertight objects, small prediction errors may lead to holes in the surface. We show this effect on the inference of a lamp from ShapeNet [8] in Figure 3. Only VF can fully preserve the lampshade and the wire structure.
Additionally, the different performance can be attributed to the effect of extracting the surfaces by thresholding the divergence of VF rather than the predicted level in itself as in SDF and binary occupancy. We conduct a small experiment with a very simple shape shown in Figure 4. We use a small 4-layered network to represent a cube. As it is visible from the edges, VF can best preserve the sharpness of the shape. Furthermore, in Figure 4e we show the effect of artificially adding Gaussian noise on the field predictions. We plot the reconstruction error as rescaled Chamfer distance while increasing the noise standard deviation. As it can be seen, VF and binary occupancy are less sensitive to noise, which makes them less affected by small errors in the prediction. The worse performance of VF compared to binary occupancy at high noise levels is expected, and is partly due to the addition of noise on three components, which make the likelihood of flipping at least one direction far higher; at lower realistic levels, no discernible difference can be noted. However, it is not clear how the lack of surface context in binary occupancy affects the overall learning process in comparison to others. At each point, away from the surface, the field value contains information regarding the surface distance (in SDF) and direction (in VF), no such information are encoded in binary occupancy.
VF and open surface representations. While considering distance based representations, the UDF, used by [13], can represent any type of surface, similar to VF. At a first glance, one may expect VF and UDF to perform similarly, after all they are related by Property 3.1. However, as shown by [60], predicting UDF [13] suffers from slow inference times and is prone to producing results with holes in the reconstructed shapes due to prediction noise, a major hindrance in UDF.
However, there are some major differences when comparing the two methods in practice. The presence of noise means that inferring a surface with UDF requires computing its gradient [13], which makes the process more timeconsuming and memory intensive than SDF/VF. Furthermore, evaluating the error as ℓ 1 distance on the cube reconstruction in Figure 4, it shows that the directions obtained by differentiating UDF are more noisy than explicitly predicting VF. VF has an average ℓ 1 error of 0.125, while the directions obtained differentiating UDF give an ℓ 1 error of 0.149. This results in lower accuracy at the edges in Fig  Additionally, as UDF is not symmetric around 0, it suffers from a bias towards the middle of the prediction range between 0 and the maximum distance threshold. This is a factor for the high level of noise on UDF prediction as shown in [42] for 2D boundary detection. The same can be seen in the cube toy experiment with an error in UDF prediction at the surface that goes up to the size of 3 voxel sides in a 64 3 grid. We provide further analysis on this aspect in the Supplementary Material.
Point-pair training for surface detection, proposed by [60], can represent any type of surface as VF. However, it requires a more complex network as it predicts UDF together with the surface detection. Furthermore, for each candidate cube at inference, it requires computing the boundary presence for each 2 vertices couples. The prediction with point couples does not also allow to trivially apply prior knowledge on the surface properties, something that is straightforward with VF. Finally, as shown in the Section 4, it achieves quantitatively lower performance compared to competitor methods. A hypothesis for this difference, might be the higher complexity of learning a function that relates point couples, when compared to single point predictions as SDF, binary occupancy, UDF, and VF.

Experiments
First, we describe the experimental details for network structure and the training set-up, followed by the tasks and the metrics used. We then provide the results on each task comparing the performance of VF to other representations. Finally, we show more qualitative results and provide a discussion on the performance.

Network Structure and Methods
In order to fairly evaluate our method against others, we follow standard architecture and training methods. For shape reconstruction, we use the popular architecture proposed in [41] in all methods, together with its training and sampling protocol, unless a change is required for the compared method. It is a fully connected auto-decoder network that takes as input a latent vector together with the queried position to predict the field. Both the network and the latent code are optimized at training time, while only the latent vector is optimized at test time. The network is trained for 2000 epochs with samples from 64 scenes in each batch and 16384 points per scene. Each method uses a latent vector of size 256 which is optimized for 800 iterations on 800 points each, at test time. For further details concerning the network structure and the training, please refer to the original work [41].
The auto-decoder network structure is used together with the SOTA representations currently available, namely the binary occupancy representation (OccNet [33]), the thresholded signed and unsigned distance field (DeepSDF [41] and NDF [13]), the point-pair method of surface detection (GIFS [60]), and the proposed VF representation. Each method is trained with its proposed loss; specifically, Occ-Net [33] uses the binary cross-entropy loss, DeepSDF [41], NDF [13], and GIFS [60] the ℓ1 loss, which we use for VF (a) GT (b) NDF [13] (c) GIFS [60] (d) DeepSDF [41] (e) OccNet [33] (f) VF Figure 5. Watertight shape representation: VF can represent a wide range of shapes with accuracy and aesthetically pleasing results. In thin surfaces and straight angles, VF achieves the best performance preserving the complete surface and without oversmoothing. See, for example, the separation between cushions of the sofa. We highlight the major differences in blue rectangles.
as well. The threshold value for the signed and unsigned distance field is also set based on previous work [13,41]. At inference, we first sample the fields on a 256 3 voxel grid and then apply marching cubes (MC) algorithm on it. OccNet [33] and DeepSDF [41] can be directly used with the standard version of MC [29] to provide the resulting mesh. GIFS [60] uses an adapted MC algorithm available in their public code. NDF [13] is evaluated with their proposed evaluation protocol which projects points in space to the surface based on the predicted NDF (for more details refer to [13]); additionally, for fair comparison with the other methods, we also evaluate NDF [13] with the developed adaptation of MC and call this method MC NDF. The same MC variant is applied to our method, VF, to obtain a resulting mesh.

Tasks, Metrics, and Datasets
We apply the INR methods on three different tasks: watertight shape reconstruction, open shape reconstruction, and piecewise planar objects reconstruction. In each task, the network is first trained on a shape class, and then used to reconstruct a set of unseen objects belonging to the same class.
All reconstructions are evaluated using the symmetric Chamfer distance (CD) on 30000 points with the results written as CD × 1000. Specifically, 30000 points are uniformly sampled on the ground truth and the predicted mesh, and the average distances from each point in the set to the closest in the other are summed. To reduce the random effect of points sampling, each result is obtained by averaging the results over three different samplings. Additionally, each method is evaluated also in terms of F1-score [21].
We use multiple classes of the popular ShapeNet [8] dataset for evaluation. We measure watertight mesh reconstruction performance on the chairs, lamps, planes, sofas, and cars classes separately; chairs, lamps, planes, sofas are made watertight following the protocol in [41] and, for cars, we use the popular [14] split. Open shapes are evaluated on the unprocessed mesh from the ShapeNet [8] classes, lamps, cars, buses and the clothes of the MGN dataset [6]. For every task, each class is divided into a training, validation and testing set, with respectively 70%, 10%, and 20% of the data.

Shape Reconstruction
First, in Table 1, we evaluate the different representations on watertight shape reconstruction from different ShapeNet [8] categories. Second, in Table 2, inference is run on nonwatertight meshes from ShapeNet [8] and MGN [6] dataset. Table 1 and 2 show that VF is overall the best performing method, outperforming any other representation in most cases. The differences can be attributed to the higher capability of VF to preserve sharp details and its robustness as discussed in Section 3.3. Furthermore, the better performance of VF compared to NDF [13], together with the worse results achieved when applying MC to it (MC NDF), suggests that directly predicting directions is more effective than deriving them from a distance function.

Qualitative Results
Overall, we can see that every method is able to accurately predict the target shapes with smooth results as shown in Figure 5. Furthermore, VF, together with NDF [13] and GIFS [60] are able to generalize to open and more complex (a) GT (b) NDF [13] (c) GIFS [60] (d) VF shapes, as in Figure 6. Looking at Fig. 5 more closely, it can be seen that DeepSDF [41] and OccNet [33] struggle in predicting thin surfaces as in the plane reconstruction example. They also struggle to predict sharp angles in the chair and sofa examples; highlighted in blue. In comparison to NDF [13], VF can predict smoother surfaces and does not suffer from small holes or artifacts, with a possible reason being the higher level of noise on the UDF gradient with respect to VF as described in Section 3.3. Despite smoother than NDF [13], GIFS [60] still suffers from inconsistencies in the surface details and struggles to accurately reconstruct tubular parts as in chair and lamp examples. We provide additional visualizations in the supplementary material. Here, we evaluate qualitatively and quantitatively the performance of applying the planar prior on VF -called Planar VF -on piecewise planar shapes and compare it to VF. This is done on the raw meshes from the bookshelves, cabinets and laptops classes of Shapenet dataset [8]. In Figure 7, we qualitatively compare the meshes generated with VF and Planar VF. Even though both methods are able to represent shapes accurately, Planar VF preserves even small planar regions better and can represent square angles sharply, showing the potential surface regularization capability of VF normals. This can be seen for example in the legs of the cabinet or in the details of the edges. The higher qualitative accuracy is also supported by the quantitative evaluation in Table 3. These results show an additional advantage of being able to explicitly model directions, which allows one to adapt to the desired target shapes effectively.

Piecewise Planar Shape Representation
Limitations. Despite the significant results, VF does not ensure watertight meshes even when required and at the same time necessitates a non-standard MC algorithm for mesh inference. Furthermore, compared to the decades old binary occupancy and half a century old SDF, many of its properties and potential benefits/drawbacks are yet to be understood, which may be important in many downstream tasks.

Conclusion
In this paper we revisit the INRs for representing 3D shapes and analyze the drawbacks of currently available representations. Popular representations, such as the SDF or binary occupancy, lack the expressive power to represent open or non-watertight surfaces. On the other hand, previous INR methods for open surfaces have problems with fast inference, accuracy, or training complexity. We proposed to solve open surface representation by considering the novel VF and proved its theoretical suitability. We further showed the possibility to easily apply a shape prior on VF, allowing us to effectively fine-tune the method for piecewise planar objects. Our INR method using VF on standard architecture outperforms existing representations by a significant margin and opens up several new avenues in surface regularization and reconstruction.

A. Proofs of VF Properties
In this Section, we provide the proofs of Properties stated in the main text in Section 3.1. For clarity, we restate the representation definition and the properties followed by the actual proofs.
We define VF (see Section 3.1) as follows.
x + ϵ; x ′ / ∈ Π and ϵ > 0. (1) The arbitrary constraint ϵ > 0 implies that each component is greater than 0. Based on the definition of VF, we prove the following properties, listed in Section 3.1. Proof Consider u(x) = ∥x −x S ∥. Except at the discontinuities on the surface, ∇u(x) is defined as, The gradient of the distance field is given by, ∇u(x) = x−x S ∥x−x S ∥ , thus ∇u(x) = −v for points outside the surface. Please refer to [38], page 9 regarding a discussion on the gradient of the distance field. The gradient points to the direction of increasing distance. The result is valid for all smooth surfaces. For piece-wise smooth surfaces, additional discontinuities may arise, specifically where the vector field v diverges. Note that at the discontinuities, we can still consider the property to be true provided that we use the same one hand-side limit, thanks to the definitions 4.1 and 5.1 in Rudin [45]. Proof For any implicit function F (x) = 0, the gradient is equal to the surface normal as x approaches the surface (please refer to [38], pages 9 -12). Note that at the surface, there are two opposite directions depending on the limit chosen, and both are equally valid for our purpose. (1), and the following transform: A point x ∈ R 3 then is a surface point, x ∈ Π, if and only if it belongs to the zero level set of g(x) + 2, i.e., Π = L 0 (g + 2). (3) Proof Let us start proving that, if a point belongs to the surface, then it is in the zero level set of Eq. (3). We measure the divergence as the negative inward flux of the field. In the infinitesimal sphere around any surface point, using Property 3.2, the only component of VF present is normal to the surface, with tangential components approaching to 0. Around such an infinitesimal neighborhood, the only change observed will be the direction flip of the normals, the tangential change approaching to 0. Thus the normal fields on the two sides of the surface, pointing in opposite direction, are summed with concordant signs. These two set of normals pointing at the opposite direction create a divergence flow of −2, as they are pointing to each other and with norm equal to 1 by definition. Therefore, we have demonstrated that it belongs to the zero level set L 0 (g + 2).
To prove the implication that only surface points are in the level set, we can identify two cases of pointsx not belonging to the surface. We consider case I when the given pointx is equidistant to multiple surface points, multiple outward vectors stem from it, consequently producing a positive divergence value. We consider case II when a point x has a single closest surface point. Given that VF is oriented towards the closest surface point from Eq. (1), following the field direction from any point, it remains constant unless it encounters a surface. Hence, the divergence amounts to a value approximately 0. A more rigorous measure can be obtained by trying to mimimize the divergence measure for the given case II. By elimination, it can be seen that the minimum divergence in case II occurs when multiple fields in the infinitesimal region aroundx are converging to a single point as shown in Figure 9. To obtain a divergence measure we can consider the infinitesimal surface around the pointx drawn with green color. Note that the surface is made of spherical domes and planar sections in 3D and forms a closed volume as required by the divergence theorem. For such a closed surface the field is either parallel (spherical domes) or perpendicular (conic surface) to the field at any point. The flux through the conic surface is parallel and thus 0. The divergence for the infinitesimal closed surface and thus the pointx can be minimized by maximizing the difference of surface area between the spherical domes. To establish the minimum divergence, we first establish the areas of the spherical domes. The surface Figure 8. General surface visualizations. The proposed VF representation can encode any type of surface and can be converted to a mesh representation. Figure 9. Illustration of case II for the proof. We illustrate the case in order to measure the divergence around the pointx which has a single closest point on the surface Π. S1 and S2 are two different spherical domes with radii r1 and r2 respectively, not to be confused with curves. area of a spherical dome defined by the solid angle θ and radius r is given by, This gives the following areas for the two spherical domes: We now consider the proof by contradiction and assume the that the divergence measured by the infinitesimal surfaces S 1 , S 2 is -2, as follows: Eq. (11) emerges from the fact that the divergence is the difference of flux between the two domes and that the field vectors are perpendicular unit vectors on those surfaces. Thus, the divergence is the measure of the difference of the surface areas. Finally combining Eq. (10) and Eq. (11), we obtain the following condition.

B. Qualitative Results Analysis
Here we provide additional visualizations of the results obtained with VF and compare to the other representations. Figure 8 shows the results achieved by VF on some challenging open and multi-layered examples to further demonstrate the ability of VF to generate accurate predictions in every shape class. In particular it is worth noting the accuracy in reconstructing the very thin parts in the lamp structure and the inside of the car and the mini-bus.
In Figure 10, we provide additional qualitative comparison to the other existing methods. Similarly to the results shown previously, this highlights the different properties of the methods. As volumetric representations, like DeepSDF [41] and OccNet [33], predict smooth watertight meshes, they achieve high-quality results in smooth classes such as chairs. On the other hand, this property shows to be a limitation when representing thin object parts, such as in the sofas or planes example. Overall, they also struggle in predicting sharp details as highlighted throughout every class.
More example comparison between VF, NDF [13], and GIFS [60] can be seen in Figure 11. Here again, we can notice the similar expressive power between methods, with VF that can outperform the other representations at the detail level and with sharper results. In particular, NDF [13] is significantly more affected by noise compared to VF, which leads to small holes and artifacts. As shown in the main paper and in Section D, a probable cause being the higher level (a) GT (b) NDF [13] (c) GIFS [60] (d) DeepSDF [41] (e) OccNet [33] (f) VF Figure 10. Qualitative reconstruction. We compare each representation method in reconstructing shapes from each used ShapeNet [8] class. VF performs significantly better than the other representations in any shape and on par with DeepSDF on the chair example.
Regarding the qualitatively poor performance of DeepSDF [41] and OccNet [33] in the sofas class (fourth row), we point out that the shown example is a challenging shape as it is empty inside and thin (see GT shape), which does not allow it to be easily represented as watertight. Regarding the performance on cars, it needs to be noted that the watertight shape includes the engine space visible from the front grid. For this reason, DeepSDF [41] and OccNet [33], struggle to reconstruct the thin bonnet accurately.
of noise observed in the UDF 3 and its gradient. GIFS [60], on the other hand, is less affected by noise but is still less effective in representing details. This might be caused by the higher complexity of having 2 separate predictions and of learning a function that relates point pairs. Furthermore, it does not allow to easily model surface properties, such as planarity, restricting its possible future applications.

C. Implementation Details and Ablation
In this section, we show the specifics of the network structure used for VF and Planar VF and its training. We further ablate the network structure showing that VF can benefit from larger architectures, such as the one proposed by [13] on the cars ShapeNet [8] category.
As described in the main text, the architecture structure used for the experiments with VF and every other method is an auto-decoder network. The specifics of its number 3 The representation used by NDF [13] of layers and size can be seen in Figure 12a. It is trained for 2000 epochs with samples from 64 scenes in each batch and 16384 points per scene. The optimizer used is Adam, with a starting learning rate of 0.001 which is decreased by a factor of 2 every 500 epochs. The same optimizer and scheduler is used also for the latent vector with a starting learning rate of 0.0005. Additionally to the ℓ 1 loss used to train VF, a ℓ 2 regularization loss is applied to the latent vector to keep its norm small. Weight normalization and 0.2 dropout are applied to each network layer, as well as the ReLU activation except for the output layer.
During inference, only the latent vector is optimized for each shape. It is initialized as a zero-mean Gaussian random vector with 0.01 standard deviation, and then optimized for 800 iterations with initial learning rate of 0.005, decreased by a factor of 10 after 400 iterations. In each iteration, 800 randomly sampled points are selected for the optimization. In order to obtain the reconstructed mesh, the auto-decoder is queried using each position on a voxel grid, together with (a) GT (b) NDF [13] (c) GIFS [60] (d) VF Figure 11. General shape representation: VF can represent both open and multi-layered shapes. VF can outperform the other representations across every shape class. We note VF reconstructs with accuracy even the inside of the shown car.  Figure 12b. It is trained and optimized in an equivalent manner as explained for VF with the same hyperparameters highlighted for the network and latent vector optimization. As an important training detail, we note that the bases vectors b are first shuffled in their order before being fed to the main network branch. This allows the network to learn to select the appropriate basis and not overfit on the most common set of basis. Table 4 shows the results obtained using the larger network proposed in [13]. It is significantly larger compared to the standard auto-decoder architecture commonly used for the task, and requires significantly longer training -2× increase in both memory and training time. From Table 4, it can be seen that VF benefits from the larger network, and keeps an edge over the compared methods in most metrics.

D. Shape Representation Properties
In this section, we show experiments on simple shapes as in Section 3.3 to support the validity of VF and compare against UDF. To highlight the differences, we try to reconstruct with a small network two tetrahedron-based shapes and analyze the accuracy and noise levels on them. The first toy experiment uses a tetrahedron overlapped with its dual shape; the second, which can be considered a stress test of the representations capabilities, consists of trying to reconstruct a very complex shape, obtained by overlapping multiple rotated tetrahedrons (as explained in Fig. 14).
The network used, takes as input the 3D coordinates, has 5 fully connected (FC) layers with a hidden size of 512, and outputs the VF or UDF prediction. Figure 13 shows the performance on the 2 overlapped tetrahedrons. We observe that on this slightly more complex shape, UDF struggles significantly more than VF in preserving all the flat surfaces and the edges accurately. This is reflected on the accuracy in predicting the correct directions towards the surface. VF achieves an average ℓ 1 error of 0.1831, while the UDF gradients performs almost 3 times worse with an error of 0.4862. Figure 13d shows the effect of the bias towards the mid range of the predicted values that affects UDF, as noted in the main text. It is evident that the model predicts values in the middle of the range much more often than they actually appear. Instead, the number of predictions of values at the extremes of the range is significantly smaller than in the ground truth. This reduced capability to predict accurately, is a further reason for the difference in performance between VF and UDF. Figure 14 shows the stress test on a very complex toy Effect of UDF bias on the prediction. It shows the number of UDF predictions in a small range divided by the gt number of UDF in the same range. Values above 1 at a certain distance implies that the number of predictions around that value is higher than the actual number of points at that distance; the opposite is true for values below 1. For a perfect UDF surface extraction, one desires the value at distance 0 to be 1 (necessary condition). Instead the value is much lower indicating an unknown bias.  Figure 14. Stress test on a tetrahedron-based complex shape. It is obtained by taking a tetrahedron and its dual (oriented with top and bottom on the z axis) and replicating the shape by rotating of 60 • around the x axis multiple times until returning to the original position. VF strongly outperforms UDF being able to effectively grasp the complexity of the shape.
shape. Qualitatively we can see a significant difference in the representation ability of VF when compared to UDF. This is reflected in the noise value, with the average ℓ 1 error of VF at 0.3085, much lower than the one of the UDF gradient at 0.7952. Regarding Figure 14, we note that the high complexity is not just given by the exterior of the shape, but also by all the intersection of the planes under the surface; both these make the function representing the object extremely hard to learn.

E. Adapted Marching Cubes
A challenging part of implicit 3D representation is an easy transformation to standard mesh representation. Mesh allows rendering and manipulation of 3D shapes using standard graphics tools. For the purpose of going from implicit to mesh representation, a traditionally successful algorithm has been Marching Cubes (MC). Current MC algorithms are developed to produce smooth surfaces without holes and with continuously changing normals. This produces visually appealing results but constitutes a challenge when adapting the algorithm for different types of representation.
The standard MC algorithm is applied to a scalar field to produce triangles at the positions where the scalar field crosses a predefined value. Considering SDF, the value used is 0 and the resulting polygonal surface approximates the zero level set of SDF. This is done taking by first voxelizing the space and evaluating the field values in these voxels. Furthermore, in order to choose between the multiple possible mesh configurations that arise from the vertex assignments, neighboring voxels are used to ensure surface continuity. To secure smoothly changing normals, the MC algorithms compute them based on a neighborhood around each surface position and locate the mesh faces in each voxel with a trilinear interpolation of the field. Figure 15. MC comparison on VF with and without the use of its norm. VF norm constitutes an effective measure of the distance from the surface and helps smooth the reconstructed meshes To adapt the MC algorithm to VF, we now assess the three aspects just highlighted: • Level set definition: as the vector field does not have a level set that can be used to apply MC, we need to define a way to indicate the voxels which have a surface. For this, we use Property 3 and identify the surface voxels as the ones that have a divergence smaller or equal to −2. The discrete divergence can be computed in a highly parallel manner, either with convolutions on the voxel grid or inside the MC computation.
• Vertices clustering: for each surface voxel, the directions of the vertices are then clustered into two groups, based on their cosine similarity. This replaces the inside-outside direction computation in the case of SDF. In this case, the two clusters can be identified by taking the two vectors among the 8 inside the voxel with the lowest cosine similarity between each other. The other vectors are then associated to the cluster with which they share the highest cosine similarity. This is an effective clustering technique with very low computational cost.
• Smooth normal predictions: the final step to adapt is the use of values for the vertices to exploit the trilinear interpolation. In the case of VF, there is no exact representation of the distance to the surface as it does not explicitly encode it. However, we can use the continuity property of the function represented through an INR. As the surface is defined by points where field directions flip, we observe that the norm of the field is reduced around the surface points. We note that, even though this measure cannot represent the actual value of the distance, it monotonically changes close to the surface and hence can be used for our purpose. The same effect is also used when applying the traditional MC algorithm on the binary occupancy field [33]. Figure 15 shows that this can be effectively used for the purpose, as the resulting mesh is much smoother.
For brevity, our modified MC method is described in Algorithm 1.