Title: Polyhedral Complex Derivation from Piecewise Trilinear Networks
Venue: NeurIPS 2024
Accepted: yes
Keywords: Polyhedral Complex

Neural Radiance Fields

3D Mesh
Abstract: Recent advancements in visualizing deep neural networks provide insights into their structures and mesh extraction from Continuous Piecewise Affine (CPWA) functions. Meanwhile, developments in neural surface representation learning incorporate non-linear positional encoding, addressing issues like spectral bias; however, this poses challenges in applying mesh extraction techniques based on CPWA functions. Focusing on trilinear interpolating methods as positional encoding, we present theoretical insights and an analytical mesh extraction, showing the transformation of hypersurfaces to flat planes within the trilinear region under the eikonal constraint. Moreover, we introduce a method for approximating intersecting points among three hypersurfaces contributing to broader applications. We empirically validate correctness and parsimony through chamfer distance and efficiency, and angular distance, while examining the correlation between the eikonal loss and the planarity of the hypersurfaces.

Polyhedral Complex Derivation from Piecewise Trilinear Networks

Abstract

Recent advancements in visualizing deep neural networks provide insights into their structures and mesh extraction from Continuous Piecewise Affine (CPWA) functions. Meanwhile, developments in neural surface representation learning incorporate non-linear positional encoding, addressing issues like spectral bias; however, this poses challenges in applying mesh extraction techniques based on CPWA functions. Focusing on trilinear interpolating methods as positional encoding, we present theoretical insights and an analytical mesh extraction, showing the transformation of hypersurfaces to flat planes within the trilinear region under the eikonal constraint. Moreover, we introduce a method for approximating intersecting points among three hypersurfaces contributing to broader applications. We empirically validate correctness and parsimony through chamfer distance and efficiency, and angular distance, while examining the correlation between the eikonal loss and the planarity of the hypersurfaces. The code is available at https://github.com/naver-ai/tropical-nerf.pytorch .

1 Introduction

Recent advancements in visualizing deep neural networks [1-4] significantly contribute to understanding their intricate structures. This progress provides valuable insights into the expressivity, robustness, training methodologies, and distinctive geometry of neural networks. By leveraging the inherent piecewise linearity in the Continuous Piecewise Affine (CPWA) functions, e.g. , ReLU neural networks, each region of the input space is represented as a convex polyhedron. The assembly of these sets constructs a polyhedral complex that delineates the decision boundaries of neural networks [5].

Building upon these breakthroughs, we explore the exciting prospect of analytically extracting a mesh representation from neural implicit surface networks [6-8]. The extraction of a mesh not only offers a precise visualization and characterization of the network's geometry but also does so efficiently through the utilization of vertices and faces, in contrast to sampling-based methods [9, 10]. The sampling-based methods are limited by their black-box nature, which can obscure the underlying structure and lead to inefficiencies in capturing fine geometric details.

These networks typically learn a signed distance function (SDF) by utilizing ReLU neural networks. However, recent approaches incorporate non-linear positional encoding techniques, such as trigonometric functions [11] or trilinear interpolations using hashing [12] or tensor factorization [13], to mitigate issues like spectral bias [14], ensuring fast convergence, and maintaining high-fidelity. Consequently, applying mesh extraction techniques based on CPWA functions becomes challenging.

Focusing on the successful and widely-adopted trilinear interpolating methods, we present novel theoretical insights and a practical methodology for precise mesh extraction. Following a novel review

(1) Vertices and

edges from the grid

marks (Sec. 5)

(3) Skeletonize with

the zero level-set of

an sdf (Sec. 4.3)

(4) Extract faces by

sorting vertices

(Sec. 4.3)

(2b) Bilinear

subdivision on

the grid plane

(2a) Linear

subdivision on

the grid edge

(2c) Trilinear

subdivision inside

the grid (Thm. 4.7)

Trilinear

cubic region

represented

by a neuron

(2) Curved edge subdivision over neurons (Alg. 2)

of the edge subdivision method [4], an efficient mesh extraction technique for CPWA functions, within the framework of tropical geometry (Section 3), we demonstrate that ReLU neural networks incorporating the trilinear module are piecewise trilinear networks (Definition 4.1). Then, we establish that the hypersurface within the trilinear region becomes a plane under the eikonal [15] constraint (Theorem 4.5), a common practice during the training of SDF.

Building upon this observation, we propose a method for approximating the determination of intersecting points among three hypersurfaces. Given the inherent curvature of these surfaces, achieving an exact solution is infeasible. In our approach, we utilize two hypersurfaces and a diagonal plane for a more feasible and precise approximation (Theorem 4.7). Additionally, we introduce a methodology for arranging vertices composed of faces to implicitly represent normal vectors. We experimentally confirm accuracy and simplicity using chamfer distance, efficiency, and angular distance, simultaneously exploring the relationship between the eikonal loss and the planarity of the hypersurface. In Figure 1, we present the analytically generated skeleton and normal map produced by our method using piecewise trilinear networks composed of HashGrid and ReLU neural networks. For a more detailed view, please refer to Figure 10 in Appendix.

Our contributions are summarized as follows:

We present novel theoretical insights and a practical methodology for precise mesh extraction, from the piecewise trilinear networks using the HashGrid.

We provide a theoretical analysis of the piecewise trilinear networks under the eikonal constraint revealing that, within a trilinear region, a hypersurface transforms into a plane.

We validate its correctness and parsimony through chamfer distance and efficiency, and angular distance, showing the correlation between the eikonal loss and hypersurface planarity.

2 Related work

Mesh extraction in deep neural networks. A conventional black-box approach is to assess a given function at sampled points to derive the complex of the function, commonly through marching cubes [9] or marching tetrahedra [10, 16]. Despite the inclusion of mesh optimization techniques [17] in modern mesh packages, this exhaustive method often generates redundant complexities due to discretization, increasing the computational load by using unnecessary mesh elements ( e.g. , fragmented planars). Aiming for exact extraction, initial efforts focused on identifying linear regions [18] and employing Analytical Marching [2] to exactly reconstruct the zero-level isosurface. Contemporary approaches involve region subdivision, wherein the regions of the complex are progressively subdivided from neuron to neuron and layer to layer, allowing for the efficient calculation of the exponentially increasing linear regions [4, 19-21]. Nevertheless, many previous studies have explored CPWA functions consisting of fully connected layers and ReLU activation, primarily owing to the mathematical simplicity of hyperplanes defining linear regions. However, this restricts their applications in emerging domains of deep implicit surface learning that diverge from CPWA functions.

1

Positional encoding. The spectral bias of a multi-layer perceptron (MLP) hinders effectively learning high frequencies, both theoretically and empirically [14]. To address this limitation, Fourier feature mapping, employing trigonometric projections similar to those used in the Transformer architecture [22], proves successful in representing complex 3D objects and scenes. Furthermore, to address its slow convergence and enhance rendering quality, novel positional encoding methods using trilinear interpolations, e.g. , HashGrid [12] or TensoRF [13], are introduced. Both techniques generate positional features through trilinear interpolation among the eight nearest corner features derived from pre-defined 3D grids. These features are obtained by hashing from multi-resolution hash tables or factorized representations of a feature space, respectively. However, these approaches cause the function represented by neural networks to deviate from CPWA functions, as trilinear interpolation is not an affine transformation. Consequently, the complex regions are divided by intricate hypersurfaces.

Eikonal equation and SDF. An eikonal 1 equation is a non-linear first-order partial differential equation governing the propagation of wavefronts. Notably, this finds application in the regularization of a signed distance function (SDF) in the context of neural surface modeling [6, 8, 23]. If Ω represents a subset of the space X equipped with the metric d , the SDF f ( x ) is defined as follows:

where ∂ Ω denotes the boundary of Ω . The metric d is defined (with a slight notational abuse) as:

where d represents the shortest distance from x to ∂ Ω in the Euclidean space of X . In the Euclidean space with a piecewise smooth boundary, the SDF exhibits differentiability almost everywhere, and its gradient adheres to the eikonal equation . Specifically, for points x on the boundary, ∇ f ( x ) = N ( x ) where the Jacobian of f represents the outward normal vector, or equivalently, |∇ f ( x ) | = 1 .

3 Preliminaries

Wepresent an overview of tropical geometry (Section 3.1) to provide a formal definition of the tropical hypersurface and the tropical algebra of neural networks. Then, we discuss the edge subdivision algorithm [4] for extracting the tropical hypersurface (Section 3.2). We supplement Appendix A.1 for less familiar readers. Later, we extend this idea to include considerations for trilinear interpolation in Section 4. You may choose to skip these sections if you are already acquainted with these concepts.

3.1 Tropical geometry

Tropical 2 geometry [25, 26] describes polynomials and their geometric characteristics. As a skeletonized version of algebraic geometry, tropical geometry represents piecewise linear meshes using tropical semiring and tropical polynomial ( ref. Appendix A.1). The set of points where a tropical polynomial f is non-differentiable, due to tropical sum ⊕ := max( · ) , is called tropical hypersurface :

Definition 3.1 (Tropical hypersurface) . The tropical hypersurface of a tropical polynomial f ( x ) = c 1 x α 1 ⊕··· ⊕ c r x α r is defined as:

̸

where the tropical monomial c i x α i with d -variate x is defined as c i ⊙ x a i, 1 1 ⊙ · · · ⊙ x a i,d d with d -variate α i = ( a i, 1 , . . . , a i,d ) and tropical product ⊙ := + .

Definition 3.2 (ReLU neural networks) . Assuming input and output dimensions are consistent, L -layer neural networks with ReLU activation are a composition of functions defined as:

where σ ( x ) = max( x , 0) and ρ ( i ) ( x ) = W ( i ) x + b ( i ) .

Although the neural networks ν is generally non-convex, it is known that the difference of two tropical signomials (allowing a real-valued α in a tropical polynomial) can represent ν [27] ( ref. Appendix A.3). Given this observation, we proceed to examine the following proposition:

Proposition 3.3 (Decision boundary of neural networks) . The decision boundary for L -layer neural networks with ReLU activation ν ( L ) ( x ) = 0 is a subset of or equals with the region where two tropical monomials, or two arguments of max , equal (Definition 3.1), as follows:

̸

where the subscript j denotes the j -th neuron, assuming the hidden size of neural networks is H , while the final L -th layer has a single output for rather discussions as an SDF (Definition 4.3). Remind that tropical operations satisfy the usual laws of arithmetic for recursive consideration.

The piecewise hyperplanes corresponding to ν ( i ) j are the candidates shaping the decision boundary. It enables for-loop iterations over the neurons across layers, as in Algorithm 2, not visiting every exponentially growing linear region [18, 27]. The equality in Proposition 3.3 generally does not hold, but we can filter them by checking if ν ( L ) 1 ( x ) = 0 . For more details, please refer to Berzins [4]. We note that our formal explanation through tropical geometry is unprecedented in the previous work [4].

3.2 Edge subdivision for tropical geometry

Polyhedral 3 complex derivation from a multitude of linear regions is inefficient and may be infeasible in some cases. Rather, Berzins [4] argue that tracking vertices and edges of the decision boundary sequentially considering the polynomial number of hyperplanes is particularly efficient, even enabling parallel tensor computations for edge subdivisions.

Initially, we start with a unit cube of eight vertices and a dozen edges, denoted by V and E , respectively. For each piecewise linear, or folded in their term, hyperplane, we find the intersection points of each one of the edges and the hyperplane (if any) to add to V . The divided edges by the intersection are also added to E . Note that we also find new edges of polygons, the intersections of convex polyhedra representing corresponding linear regions [5], and the folded hyperplane. Repeating these for all folded hyperplanes, before selecting all vertices x ⋆ where ν ( x ⋆ ) = 0 . Then, the valid edges connecting those vertices would represent the decision boundary.

To elaborate with details, the order of subdividing is invariant as stated in Proposition 3.3; however, we must perform the subdivision with all hyperplanes in the previous layers in advance if we want to keep the current set of edges not crossing linear regions. One critical advantage of this principle is that we can find the intersection point by evaluating the output ratio of the two vertices of an edge, ν ( x 0 ) and ν ( x 1 ) , which are proportional to the distances to the hyperplane by Thales's theorem [28]. The intersection point would simply be ˆ x 0 , 1 = (1 -w ) x 0 + w x 1 where w = | d 0 | / | d 0 -d 1 | and d k = ν ( l ) j ( x k ) , upon the fact that d 0 · d 1 < 0 if the hyperplane divides the edge.

Updating edges needs two steps: dividing the edge into two new edges and adding new edges of intersectional polygons by a hyperplane for convex polyhedra representing linear regions. The latter is a tricky part that we need to find every pair of two vertices among { ˆ x · , · } , both within the same linear region and on the two common hyperplanes, obviously including the current hyperplane. They argue that the sign-vectors for the preactivations provide an efficient way to find them.

Definition 3.4 (Sign-vectors) . For the current hyperplane specified by the i ∗ -th layer and its j ∗ -th preactivation, we define the sign-vectors with a small positive constant ϵ :

for all i and j where ϕ ( i, j ) ≤ ϕ ( i ∗ , j ∗ ) , and ϕ ( i, j ) ∈ { 0 } ∪ N is an ascending indexing function from lower layers, handling arbitrary hidden sizes of networks to vectorize a signed matrix.

Notice that a hyperplane divides a space into two half-spaces, two linear regions. Collectively, if the sign-vectors for the i -th layer and its j -th preactivation of two vertices have the same values except for zeros, which are wild cards for matching, we can say that the two vertices are within the same linear region in the current step ( i ∗ , j ∗ ) . For the second, the sign-vector has at least three zeros specifying a point by at least three hyperplanes. If two zeros are at the same indices in the two sign-vectors, the two vertices are on the same two hyperplanes, forming an edge. Notice that our principle asserts edges should be within linear regions.

Berzins [4] argue that the edge subdivision provides the optimal time and memory complexities of O ( |V| ) , linear in the number of vertices, while it can be efficiently computed in parallel. According to Hanin and Rolnick [20], the number of linear regions is known for o ( N L /L !) where N is the total number of neurons, or ϕ ( L, 1) in our notation.

4 Method

4.1 Piecewise trilinear networks

Definition 4.1 (Trilinear interpolation) . Let a unit cube be in the D -dimensional space, where its corners are represented by H ∈ R 2 D × F where F is a corner-feature dimension. Given weights w ∈ [0 , 1] D , the trilinear interpolation function ψ is defined as:

where the interpolating weight ω ( i, w ) ∈ R is defined using a left-aligned and zero-trailing binarization function B ∈ { 0 , 1 } D ( ref. Appendix B.1) as follows:

which is the volume of the opposite subsection of the hypercube divided by the weight point w . For a general hypercube, scaling the weight for a unit hypercube gets the same result. Going forward, we assume D =3 for trilinear interpolation without explicitly stating it otherwise.

Lemma 4.2 (Nested trilinear interpolation) . Let a nested cube be inside of a unit cube, where its positions of the eight corners deviate -a j or + b j , such that a j , b j ≥ 0 , from w j for each dimension j , using the notations of Definition 4.1. The eight corners of the nested cube are the trilinear interpolations of the eight corners of the unit cube. Then, the trilinear interpolation with the unit cube and the nested cube for w is identical.

Proof. Notice that trilinear interpolation is linear for each dimension to prove it. The detailed proof can be found in Lemma D.1.

Definition 4.3 (Piecewise trilinear networks) . Let a positional encoding module be τ : R D → R F using the trilinear interpolation of spatially nearby learnable vectors on a three-dimensional grid, e.g. , HashGrid [12] or TensoRF [13], where τ is an instance of ψ in Definition 4.1. Usually, they transform the input coordinate x to w as a relative position inside a unit grid in a corresponding resolution. H is a learnable parameter specified in the corresponding method. Then, we define trilinear neural networks ˜ ν , by prepending τ to the neural networks ν from Definition 3.2. Here, the input and output dimensions of ν ( L ) are F and 1 (an SDF distance), respectively.

Note that τ is gridwise trilinear. Using Lemma 4.2, ˜ ν is trilinear within the composite intersections of linear regions of ν and trilinear regions of τ . (We discuss how to access virtual cubic corner features where trilinear regions are not cubic in Section 5.) So, we regard ˜ ν as piecewise trilinear.

4.2 Curved edge subdivision in a trilinear space

In a trilinear space, τ projects a line to a curve except the lines on the grid. Notably, the diagonal line t = ( t, t, t ) ⊺ where t ∈ [0 , 1] is projected to a cubic Bézier curve ( ref. Proposition D.2). We aim to generalize for the curved edge subdivision for hypersurfaces .

Lemma 4.4 (Curved edge of two hypersurfaces) . In a piecewise trilinear region, let an edge ( x 0 , x 7 ) be the intersection of two hypersurfaces ˜ ν i ( x ) = ˜ ν j ( x ) = 0 , while x 0 and x 7 are on the two hypersurfaces. Then, the edge is defined as:

Proof. The detailed proof using the piecewise linearity of ν is provided in Lemma D.4.

Notice that it does not decrease the number of variables to find a line solution since the trilinear interpolation τ still forms hypersurfaces making complex cases. Yet, we theoretically demonstrate that the eikonal constraints on ˜ ν render them hyperplanes with linear solutions.

̸

Theorem 4.5 (Hypersurface and eikonal constraint) . A hypersurface τ ( x ) = 0 intersects two points τ ( x 0 ) = τ ( x 7 ) = 0 while τ ( x 1 ... 6 ) = 0 for the remaining six points. These points form a cube, with x 0 and x 7 positioned on the diagonal of the cube. The hypersurface satisfies the eikonal constraint ∥∇ τ ( x ) ∥ 2 2 = 1 for all x ∈ [0 , 1] 3 . Then, the hypersurface of τ ( x ) = 0 is a plane.

̸

Corollary 4.6 (Affine-transformed hypersurface and eikonal constraint) . Let ν 0 ( x ) and ν 1 ( x ) be two affine transformations defined by W ⊺ i x + b i , i ∈ { 0 , 1 } . A hypersurface ν 0 ( τ ( x ) ) = 0 passing two points ν 0 ( τ ( x 0 ) ) = ν 0 ( τ ( x 7 ) ) = 0 while ν 0 ( τ ( x 1 ... 6 ) ) = 0 satisfies the eikonal constraint ∥∇ ν 1 ( τ ( x ) ) ∥ 2 2 = 1 for all x ∈ [0 , 1] 3 . Then, the hypersurface of ν 0 ( τ ( x ) ) = 0 is a plane.

The proofs using its partial derivatives can be found in Theorem D.5 and Corollary D.6.

Since we aim for a practical polyhedral complex derivation that piecewise trilinear networks represent, we replace one of the hypersurfaces forming the curved edge with a diagonal plane in a piecewise trilinear region. We choose this option not only for its mathematical simplicity but also due to the characteristics of trilinear hypersurfaces, as illustrated in Figure 9, Appendix G. Thus, the new vertices lie on at least two hypersurfaces, and the new edges exist on the same hypersurface, while the eikonal constraint minimizes any associated error from this approximation. This error is tolerated by the hyperparameter ϵ = 1e-4 as specified in Definition 3.4 and Section 6. More discussions on this matter can be found in Appendix C.1.

Theorem 4.7 (Intersection of two hypersurfaces and a diagonal plane) . Let ˜ ν 0 ( x ) = 0 and ˜ ν 1 ( x ) = 0 be two hypersurfaces passing two points x 0 and x 7 such that ˜ ν 0 ( x 0 ) = ˜ ν 1 ( x 0 ) = 0 and ˜ ν 0 ( x 7 ) = ˜ ν 1 ( x 7 ) = 0 , P i := ˜ ν 0 ( x i ) and Q i := ˜ ν 1 ( x i ) , P α = [ P 0 ; P 1 ; P 4 ; P 5 ] , P β = [ P 2 ; P 3 ; P 6 ; P 7 ] , and X = [(1 -x ) 2 ; x (1 -x ); (1 -x ) x ; x 2 ] . Then, x ∈ [0 , 1] of the intersection point of the two hypersurfaces and a diagonal plane of x = z is the solution of the following quartic equation:

̸

Proof. Please refer to Theorem D.7 for the detailed proof, which uses linear algebra to rearrange two trilinear equations with two variables to get a solution.

Note that finding roots of a polynomial is the eigenvalue decomposition of a companion matrix [29], which can be parallelized ( ref. Lemma D.8). Given that the companion matrix remains small for a quartic equation in R 4 × 4 , the overall complexity remains O ( |V| ) . For the detailed complexity analysis on the proposed method, please refer to Appendix E.

4.3 Skeletonization, faces, and normals

Skeletonization selects the vertices such that: V ⋆ = { x | | ˜ ν ( x ) | ≤ ϵ, x ∈ V} , and the edges E ⋆ such that their two vertices are among V ⋆ . Recall that because a decision boundary is a subset of the tropical hypersurface (Definition 3.1 and Proposition 3.3), this step ensures accurate mesh extraction.

(a)

(b)

(c)

(d)

(e)

Faces are formed by edges lying on the same plane and sharing a common region. This is identified by evaluating the sign-vectors ( ref. Definition 3.4 and Section 5). Note that faces are not inherently triangular; however, if needed, they can be triangulated via triangularization for further analysis.

Normals are conventionally described by the order of the vertices as each face has ambiguity with two opposite directions. Let x 0 , 1 , 2 be the vertices forming a triangular face. The normal is defined using cross-product as n = ( x 1 -x 0 ) × ( x 2 -x 0 ) where the normal is orthogonal to both vectors, with a direction given by the right-hand rule. Please refer to Appendix B.2 for the details.

5 Implementation

We employ the HashGrid [12] for τ , while our discussion on its generalizability is provided in Appendix C.2. We illustrate our method in Algorithm 1, 2, and Figure 2, inspired by Hanin and Rolnick [30] for its visualization of linear regions. We describe the implemental details of the algorithms in the following sections and the corresponding caption of Figure 6.

Initialization. While the original edge subdivision algorithm [4] starts with a unit cube of eight vertices and a dozen edges, we know that the unit cube is subdivided by orthogonal planes consisting of the grid. Let an input coordinate be x ∈ [0 , 1] 3 , and there is a unit cube such that one of its corners is at the origin. Taking into account multi-resolution grids, we obtain the marks m i , such that the grid planes are x = m i , y = m i , and z = m i , following Algorithm 4 in Appendix. Notice that we carefully consider the offset s/ 2 , which prevents the zero derivatives from aligning across all resolution levels ( ref. Appendix A of Müller et al. [12]). Leveraging the obtained marks, we derive the associated grid vertices and edges, which serve as the initial sets for V and E , respectively. Please refer to Figure 2a where its multi-resolution grids are visualized using randomized colors.

Optimizing sign-vectors. The number of orthogonal planes may significantly surpass the number of neurons, making it impractical to allocate elements for the sign-vectors (Definition 3.4). To address this, leveraging the orthogonality of the grid planes, we store a plane index along with an indicator of whether the input is on the plane ( γ = 0 ) or not ( γ = 1 ).

Piecewise trilinear region. In Theorem 4.7, we need the outputs of corners P and Q in a common piecewise trilinear region; however, some corners may be outside of the region. For this, we replace all ReLU activations using the mask m that: m ( i ) j = (˜ ν ( i ) j ( x 0 ) > ϵ ) | (˜ ν ( i ) j ( x 7 ) > ϵ ) , where x 0 and x 7 are the vertices of an edge of interest, for the calculated cubic corners x 0 ... 7 . This implies that ν exhibits linearity, except for instances where two vertices simultaneously deviate from linearity. For P and Q , we select the last two pairs of i and j such that γ ϕ ( i,j ) ( x 0 ) = γ ϕ ( i,j ) ( x 7 ) = 0 , indicating two common hypersurfaces passing two points x 0 and x 7 .

6 Experiment

Objective. For a given trilinear neural network with the eikonal constraint, our goal is to get a boundary mesh of the zero-set of the SDF. (Note that NeuS [6] showed how to convert the density

Algorithm 1 Polyhedral Complex Derivation

Algorithm 2 Curved Edge Subdivision (CESub)

networks of NeRFs to an SDF in the frameworks of volume rendering.) To evaluate this, we utilize marching cubes with excessive sampling to get a pseudo-ground truth mesh to remove Bayes error caused by underfitting. We explicitly specify the samplings in the results.

Hyperparameters. We used the number of layers L of 3 and hidden size H of 16 for the networks, and ϵ of 1e-4 for the sign-vectors ( ref. Definition 3.4). The weight for the eikonal loss is 1e-2. For HashGrid, the resolution levels of 4, feature size of 2, base resolution N min of 2, and max resolution N max of 32 by default . N min and N max are doubled (x2) or quadrupled (x4) for Medium or Large settings in Figure 4. We use the official Python package of tinycudnn 4 for the HashGrid module.

Chamfer distance and efficiency. The chamfer distance is a metric used to evaluate the similarity between two sets of sampled points from two meshes. Let S 0 and S 1 be the two sets of points. Specifically, the bidirectional chamfer distance (CD) is defined as follows:

We randomly select 100K points on the faces through ray-marching from the origin, directing toward a randomly chosen point on a unit sphere.

Table 1 and Figure 4 show the results for the Standford 3D Scanning repository [31]. Varying the number of samples in marching cubes [9], we plot the baseline with respect to the number of generated vertices. Given that our method extracts vertices from the intersection points of hypersurfaces, it enables parsimonious representations using the lower number of vertices having better CD, summarized by chamfer efficiency (CE) 5 in Table 1, and ours are located in the lower-left region compared with the baselines in Figure 4. In particular, we observe a strong CE in a simple object, e.g. , Drill Bit, which achieved a CE of 47.8 versus 19.2 (MC).

Our method inherently holds an advantage over MC in capturing optimal vertices and faces, especially in cases where the target surfaces exhibit planar characteristics. When dealing with curved surfaces, the smaller grid size of the Large model in Figure 4 views a curved surface in a smaller interval, and it tends to approach the plane. From the plotting of Small to

Number of the vertices

0K

60K

120K

180K

240K

300K

360K

2.76E-03

3.99E-03

5.20E-03

7.48E-03

4.67E-04

8.61E-04

1.21E-03

1.90E-03

1.86E-04

1.82E-04

2.21E-04

2.99E-04

6.32E-04

1.51E-04

Ours

MC

MT

NDC

Chamfer distance

1e-2

1e-3

1e-4

Chamfer distance (log)

1E-04

1E-03

1E-02

Number of the vertices

20K

40K

60K

80K

100K

MC-small

MC-medium

MC-large

Ours-small

Ours-medium

Ours-large

Chamfer distance (log)

1E-04

1E-03

1E-02

Number of Vertices

20K

40K

60K

80K

MC-small

MC-medium

MC-large

Ours-small

Ours-medium

Ours-large

100K

Flatness error

0.02

0.04

0.06

0.08

Iteration

5

10

15

20

25

30

35

40

45

50

0.000

0.001

0.01

0.1

Flatness error

0.02

0.04

0.06

0.08

Iteration

5

10

15

20

25

30

35

40

45

50

0.000

0.001

0.01

0.1

Large , our CDs (colored crosses) are decreasing to zero, consistently with better CEs, confirming this speculation. The complete results of the Large models and its visualization can be found in Tables 6 to 8 in Appendix F.1 and the right side of Figure 10 in Appendix, respectively. We also compare ours with Marching Tetrahedra (MT) [10, 16] and Neural Dual Contour (NDC) [32] in Figure 3, where our method achieved the lowest chamfer distance and the most efficiency method with respect to the number of vertices. For the rationale behind our selection, please refer to Appendix F.3.

Angular distance. The angular distance measures how much the normal vectors deviate from the ground truths. Analogous to the chamfer distance, we sample 100K points on the surface and calculate the normal vectors as described in Section 4.3. The angular distance is defined as follows:

In Table 5 in Appendix, the angular distances for the Stanford bunny are shown. As we expected, our approach efficiently estimates the faces using a parsimonious number of vertices compared to the sampling-based method. We confirm this trend is consistent across other objects.

Eikonal constraint for planarity. To assess the efficacy of the eikonal constraint, which enforces the planarity of hypersurfaces as outlined in Theorem 4.5, we quantify the flatness error associated with the planarity constraints presented in the proof of Theorem D.5. Let x 0 and x 7 be two vertices consisting of a diagonal edge, while x 1 ... 6 are its remaining cubic corners. We obtain ˜ ν ( x 1 ... 6 ) in a piecewise trilinear region as described in Section 5. This evaluation is conducted as follows:

In Figure 5, we empirically validate that the eikonal loss enforces planarity in the hypersurfaces within piecewise trilinear regions, as described in Theorem 4.5. The absence of the eikonal loss results in elevated planarity errors, leading to the construction of an inaccurate mesh.

MC 256

(92K Vertices)

MC 64

(5.6K Vertices)

Ours

(4.5K Vertices)

MT 32

(4.7K Vertices)

NDC 64

(5.6K Vertices)

Visualization. We provide the qualitative analyses in Figure 6 comparing with MT [10, 16] and NDC[32], along with MC. While the competitors exhibit over-smoothing or inaccuracies, particularly in the nose region, our method preserves surface details with consistent normals and outperforms in both accuracy and generalization. In addition, we present Figures 10 to 13 in Appendix G encompassing the comparison between the Small and Large models (Figure 10), visualizing the marching cubes varying the number of samples (Figure 11), detailed comparisons (Figure 12), and the gallery of the Stanford 3D Scanning meshes from the Large models (Figure 13).

Limitations. Although we implement the algorithm using pre-defined parallel tensor operations in PyTorch [33], achieving time and memory complexity of O ( |V| ) as detailed in Appendix E, our method inherently requires forward passing the intermediate vertices during edge subdivisions iterating over neurons, which depends on previous steps (Algorithm 1). For the Stanford bunny using the Small model, the intermediate counts for vertices and edges exceed 11K and 19K, respectively, although they are eventually reduced to 4.3K vertices for face extraction. This overhead is negligible for the Small models, taking 0.97 seconds; however, for the Large models, the process takes 4.15 ± 0.61 seconds for 140.3 ± 5.6K vertices with our optimized algorithm, compared to 1.31 and 11.88 seconds for marching cubes with 256 and 512 samples per axis, respectively. We note that parallelizing over multiple GPUs would be a reliable option to improve latency. For the discussions on satisfying the eikonal equation in practice and the generalization to other positional encodings using trilinear interpolation, please refer to Appendix C.1 and Appendix C.2, respectively.

7 Conclusions

In conclusion, our exploration of novel techniques in visualizing trilinear neural networks unveils its intricate structures and extends mesh extraction applications beyond CPWA functions. Focused on trilinear interpolating methods, our study yields novel theoretical insights and a practical methodology for precise mesh extraction. Notably, Theorem 4.5 shows the transformation of hypersurfaces to planes within the trilinear region under the eikonal constraint. Empirical validation using chamfer distance and efficiency, and angular distance affirms the correctness and parsimony of our methodology. The correlation analysis between eikonal loss and hypersurface planarity confirms the theoretical findings. Our proposed method for approximating intersecting points among three hypersurfaces, coupled with a vertex arrangement methodology, broadens the applications of our techniques. This work establishes a foundation for future research and practical applications in the evolving landscape of deep neural network analysis and toward real-time rendering mesh extraction.

Acknowledgments

I would like to express my sincere appreciation to my brilliant colleagues, Sangdoo Yun, Dongyoon Han, and Injae Kim, for their contributions to this work. Their constructive feedback and guidance have been instrumental in shaping the work. I also express my sincere thanks to the anonymous reviewers for their help in improving the manuscript. The NA VER Smart Machine Learning (NSML) platform [34] had been used for experiments.

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Appendix

Table of contents

A Tropical geometry and tropical algebra of neural networks

In this section, we present an overview of tropical geometry (Appendix A.1) to provide a formal definition of the tropical hypersurface (Appendix A.2). This hypersurface is significant as it serves as the decision boundary for ReLU neural networks optimizing an SDF, resulting in the formation of a polyhedral complex. For this, we delve into the examination of the tropical algebra of neural networks (Appendix A.3). Furthermore, we discuss the edge subdivision algorithm [4] for extracting the tropical hypersurface (Section 3.2), and we extend this discussion to include considerations for trilinear interpolation in Section 4. Given that Appendix A.1 to A.3 offer insights into tropical geometry, illustrating how each neuron contributes to forming the polyhedral complex, you may choose to skip these subsections if you are already acquainted with these concepts.

A.1 Tropical geometry

Tropical geometry [25, 26] describes polynomials and their geometric characteristics. As a skeletonized version of algebraic geometry, tropical geometry represents piecewise linear meshes using tropical semiring and tropical polynomial .

The tropical semiring is an algebraic structure that extends real numbers and -∞ with the two operations of tropical sum ⊕ and tropical product ⊙ . Tropical sum and product are conveniently replaced with max( · ) and + , respectively. Notice that -∞ is the tropical sum identity, zero is the tropical product identity, and these satisfy associativity, commutativity, and distributivity. For example, a classical polynomial x 3 + 2 xy + y 4 would represent max(3 x, x + y + 2 , 4 y ) . The

classical polynomial is rewritten for a tropical polynomial by naively replacing + with ⊕ and · with ⊙ , respectively. After this, we rewrite by the definitions of tropical operations as max(3 x, x + y +2 , 4 y ) . Roughly speaking, this is the tropicalization of the polynomial, denoted tropical polynomial or simply f . Note that this example (ours with max notation) comes from the Wikipedia 6 .

Additionally, we present a graphical diagram in Figure 7 illustrating how the tropical polynomial divides the 2D space for clearer understanding. Please see the caption for further details.

1+2x

1+2y

2+x+y

2+x

2+y

2

A.2 Tropical hypersurface

Figure B.

A

classical polynomial

x 2  + y 2  + 2xy + 2x + 2y + 2 turns into a tropical polynomial as max(1+2x, 1+2y, 2+x+y, 2+x, 2+y,  2), taking  into  account  the The set of points where a tropical polynomial f is non-differentiable, mainly due to tropical sum, max( · ) , is called tropical hypersurface , denoted V ( f ) . Note that V is an analogy to the vanishing set of a polynomial.

implicit  coe ffi cient of  1 ,  from Zhang et al.  (2018).  Each  color Definition A.1 (Tropical hypersurface, restated) . The tropical hypersurface of a tropical polynomial f ( x ) = c 1 x α 1 ⊕··· ⊕ c r x α r is defined as:

̸

where the tropical monomial c i x α i with d -variate x is defined as c i ⊙ x a i, 1 1 ⊙ · · · ⊙ x a i,d d with d -variate α i = ( a i, 1 , . . . , a i,d ) and tropical product ⊙ := + . In other words, this is the set of points where f ( x ) equals two or more monomials in f .

A.3 Tropical algebra of neural networks

Definition A.2 (ReLU neural networks, restated) . Assuming input and output dimensions are consistent, L -layer neural networks with ReLU activation are a composition of functions defined as:

where σ ( x ) = max( x , 0) and ρ ( i ) ( x ) = W ( i ) x + b ( i ) .

Although the neural networks ν is generally non-convex, it is known that the difference of two tropical signomials can represent ν [27]. A tropical signomial takes the same form as a tropical polynomial ( ref. Definition 3.1), but it additionally allows the exponentials α to be real values rather than integers. With tropical signomials F , G , and H ,

where F ( l ) , G ( l ) , and H ( l ) are as follows:

Proposition A.3 (Decision boundary of neural networks, restated) . The decision boundary for L -layer neural networks with ReLU activation ν ( L ) ( x ) = 0 is where two tropical monomials, or two arguments of max , equal (Definition 3.1), as follows:

̸

where the subscript j denotes the j -th neuron, assuming the hidden size of neural networks is H , while the L -th layer has a single output for rather discussions as an SDF. Remind that tropical operations satisfy the usual laws of arithmetic for recursive consideration.

This implies that piecewise hyperplanes corresponding to the preactivations { ν ( i ) j } are the candidates shaping the decision boundary. Each linear region has the same pattern of zero masking by ReLU ( · ) , effectively making it linear as an equation of a plane Ax + b = 0 for some A and b . In turn, if the masking pattern changes as an input is continually changed, it moves to another linear region. From a geometric point of view, this is why the number of linear regions grows polynomially with the hidden size H and exponentially with the number of layers L [18, 27].

B Implementation details

B.1 Zero-trailing binarization function

B is zero-trailing binarization function, consisting of D = 3 bits. The zero-trailing means the number in bits is increasing left-aligned. For example, the number one is ' 100 ', while the numbers two and four are ' 010 ' and ' 001 ', respectively. The number three is ' 110 '. From the left, each bit indicates x, y, and z axis in our notation. This convention seems to be widely used in the CUDA implementation in C language. Notice that we use the j subscript of B j to select the axis. Naturally, i = 0 and i = 7 indicate the opposite two corner points.

B.2 Sorting polygon vertices and HashGrid marks

Algorithm 3 describes the algorithm in Section 4.3, which are sorting polygon vertices implicitly representing the direction of faces. If the viewing direction aligns opposite to the normal, the order of vertices follows a counter-clockwise arrangement. We determine the vertex order based on the normal derived from the Jacobian of the mean of vertices. We identify the counter-clockwise arrangement of vertices using dot product to compute relative angles θ ∈ [ -π, π ] and cross product for getting direction. Algorithm 4 describes the algorithm in Section 5, calculating the HashGrid marks from the hyperparameters of a HashGrid.

Algorithm 3 Sorting Polygon Vertices

Algorithm 4 HashGrid Marks

C Further discussions

C.1 Satisfying the eikonal equation and approximation

Theorem 4.7 handles where the eikonal equation does not make perfect hyperplanes. First, we observed the characteristics of trilinear hypersurfaces as illustrated in Figure 9 in Appendix. Among 64 edge scenarios within a trilinear region, a diagonal plane intersects with the hypersurface in nearly all cases. Moreover, we can take advantage of it since we can eliminate one variable by assuming the curved edge lies on the x = z plane. In this light, Theorem 4.7 used linear algebra to rearrange two trilinear equations to get Equations (71) and (74). As we mentioned in the manuscript, the new vertices lie on at least two hypersurfaces, and the new edges exist on the same hypersurface, while the eikonal constraint minimizes any associated error. This error is tolerated by the hyperparameter ϵ =1e-4 (Section 6) as specified in Definition 3.4.

Figure 5 also suggests empirical evidence. Empirically, the flatness error, derived from the proof of Theorem D.5, is effectively controlled by the learning rate of the eikonal loss. In the experiments, we used 0.01 to give our results, while many other neural implicit surface learning methods were used to exploit a higher learning rate of 0.1 ( e.g. , Yariv et al. [7]). Note that, in practice, the eikonal loss is only applied to near-surface using clamps (if the target position is obviously far from a surface, ignore it; it is wise since we are concerned with the surfaces, not wanting exact SDFs for all space) for effective learning, which is sufficient to our extraction method.

Moreover, the smaller grid size of the Large model ( ref. Section 6) in Figure 4 views a curved surface in a smaller interval, and it tends to regard it as a plane. From the plotting of Small to Large , our CD (colored crosses) is decreasing to zero, consistently with better CE, confirming this speculation.

C.2 Generalization to other positional encodings using trilinear interpolation

We can say that any encoding using trilinear interpolation (that) we defined can be applied in our method. Here, the definition is implicitly referred to Definition 4.1 (Trilinear interpolation), where the cubic corners are represented by H ∈ R D × F . Since we do not rely on a specific mapping function f : x → ( H 0 , H 1 . . . H 7 ) , we may apply our extraction method to any encoding using trilinear interpolation that we defined. Specifically, InstantNGP [12] used a hashing function to map a corner point to the arbitrary entry of hash tables, allowing gracious collisions with online adaptivity (refer to their Sec. 3), while TensoRF [13] can be interpreted that it used a mapping function to the summation of the multiplications of two vectors from the vector-matrix factorization. Therefore, if we have the full ranks for R 1 , R 2 , R 3 , in Eqn. 3 in Chen et al. [13], among the VM factorization, which can be equivalently represented by the HashGrid with a sufficient number of hash table sizes.

Then, the discussion should be directed toward their encoding characteristics, which impact our extraction method due to their online adaptivity (HashGrid) or low-rank factorization (TensoRF). When we train an SDF, the optimization to global minima is generally difficult to achieve. Instead, previous works on neural implicit surface learning methods resort to optimizing the SDF on nearsurface since we are interested in modeling the zero-set surfaces, allowing some errors for the points that are far from the surfaces. Practically, this is done by defining a loss function using clamps and ignoring far-surface points. For this reason, the online adaptivity of HashGrid or low-rank factorization of TensoRF can exploit this nature: focused learning of the surfaces adaptively allocating their learnable parameters to the near-surface, which means there might be enough room for the regularization by the eikonal equation. The investigation of the efficiency of representing the implicit neural surfaces using various positional encodings is seemingly out of the scope of this work since our main interest lies in the theoretical exposition of the mesh extraction from it and setting up a feasible bridge between theory and practice.

D Theoretical proofs

Lemma D.1. (Nested trilinear interpolation, restated) Let a nested cube be inside of a unit cube, where its positions of the eight corners deviate -a j or + b j , such that a j , b j ≥ 0 , from w j for each dimension j , using the notations of Definition 4.1. The eight corners of the nested cube are the trilinear interpolations of the eight corners of the unit cube. Then, the trilinear interpolation with the unit cube and the nested cube for w is identical.

Proof. Let a left-aligned and zero-trailing binarization function be B ( · ) ∈ { 0 , 1 } D and D = 3 . The partial derivative of the trilinear interpolation with the nested cube ˆ H with respect to H 7 and H 0 would be:

and

respectively. In a similar way, we can describe the other cases H 1 to H 6 of the nested trilinear weights since there are two cases for each dimension inside the product.

Proposition D.2 (Cubic Bézier curve) . Trilinear interpolation from Definition 4.1 with the weights on the diagonal line t = ( t, t, t ) ⊺ where t ∈ [0 , 1] is a cubic Bézier curve.

Proof. We rewrite the trilinear interpolation from Definition 4.1 with the weight t as follows:

Four points P 0 := H 0 , P 1 := H 1 + H 2 + H 4 3 , P 2 := H 3 + H 5 + H 6 3 , and P 3 := H 7 define the cubic Bézier curve.

As t is changed from zero to one, the curve starts at P 0 going toward P 1 and turns in the direction of P 2 before arriving at P 3 . Usually, the curve does not pass P 1 nor P 2 ; but these give the directional guide for the curve.

Lemma D.3 (Hypersurfaces in a trilinear space) . In a piecewise trilinear region where x lies, hypersurfaces of the piecewise trilinear networks are defined as follows, following the notation of Definition 4.1:

where w ∈ [0 , 1] D is the relative position in a grid cell and { x i } and { H i } are the corresponding corner positions and representations, respectively, in a trilinear space.

Proof. In a piecewise trilinear region, we can use the linearity of ν as follows:

which concludes the proof.

Lemma D.4 (Curved edge of two hypersurfaces, restated) . In a piecewise trilinear region, let an edge ( x 0 , x 7 ) be the intersection of two hypersurfaces ˜ ν i ( x ) = ˜ ν j ( x ) = 0 , while x 0 and x 7 are on the two hypersurfaces. Then, the edge is defined as:

Proof. By Definition 4.3,

while ν i is linear in a piecewise trilinear region. Since it similarly goes for ν j , the intersection line of two hyperplanes ν i and ν j is (1 -t ) τ ( x 0 ) + tτ ( x 7 ) where t ∈ R . In other words, the edge is on the hypersurface ν i ( (1 -t ) τ ( x 0 ) + tτ ( x 7 ) ) = 0 where t ∈ R . The edge is defined as follows:

which concludes the proof.

Notice that when τ ( x ) 1 + τ ( x ) 2 + τ ( x ) 4 = τ ( x ) 3 + τ ( x ) 5 + τ ( x ) 6 = 0 , at least one connecting edge can be identified. This occurs along the diagonal line x = t · 1 , where t ∈ [0 , 1] . Along this line, τ ( x ) represents a cubic Bézier curve, as shown in Proposition D.2 with the control points of P 1 and P 2 set to zeros, which is the line.

̸

Theorem D.5 (Hypersurface and eikonal constraint, restated) . A hypersurface τ ( x ) = 0 intersects two points τ ( x 0 ) = τ ( x 7 ) = 0 while τ ( x 1 ... 6 ) = 0 for the remaining six points. These points form a cube, with x 0 and x 7 positioned on the diagonal of the cube. The hypersurface satisfies the eikonal constraint ∥∇ τ ( x ) ∥ 2 2 = 1 for all x ∈ [0 , 1] 3 . Then, the hypersurface of τ ( x ) = 0 is a plane.

Proof. By the definitions of τ in Definition 4.3 and the eikonal constraint,

where

̸

However, since τ ( x 0 ) = τ ( x 7 ) = 0 while τ ( x 2 ) = 0 and τ ( x 5 ) = 0 , there is no solution τ ( x 1 ... 6 ) of ∂τ ( x ) ∂y = 0 for all x . Likewise, the same goes for ∂τ ( x ) ∂x and ∂τ ( x ) ∂z . Additionally, the three partial derivatives can be rewritten as follows:

̸

where a = 0 , b = 0 , and c = 0 . By substituting B = -bC/c and A = -aC/c , we can show that -abC/c -abC/c = 0 ⇔ C = 0 , which makes A = B = C = 0 . Therefore, we rewrite A, B, and C as follows:

̸

̸

̸

respectively. The following conditions would satisfy the eikonal constraint:

From Equation (48) and Equation (51),

From Equation (47) and Equation (50),

From Equation (47) and Equation (48),

From Equation (50), Equation (51), and Equation (54),

Using Equation (52), Equation (53), Equation (54), Equation (55), and Equation (56),

where α = 2 xyz -xy -yz -zx .

This is an equation of a plane where its normal vector is ( τ ( x 1 ) , τ ( x 2 ) , τ ( x 4 ) ) . Again, to satisfy the eikonal constraint,

̸

which concludes the proof.

Corollary D.6 (Affine-transformed hypersurface and eikonal constraint, restated) . Let ν 0 ( x ) and ν 1 ( x ) be two affine transformations defined by W ⊺ i x + b i , i ∈ { 0 , 1 } . A hypersurface ν 0 ( τ ( x ) ) = 0 passing two points ν 0 ( τ ( x 0 ) ) = ν 0 ( τ ( x 7 ) ) = 0 while ν 0 ( τ ( x 1 ... 6 ) ) = 0 satisfies the eikonal constraint ∥∇ ν 1 ( τ ( x ) ) ∥ 2 2 = 1 for all x ∈ [0 , 1] 3 . Then, the hypersurface is a plane.

Proof. Similarly to Theorem D.5, we investigate the Jacobian of the eikonal constraint to be zero as follows:

Since ∂ν 1 /∂τ is a constant, using Lemma D.3 and replacing ν 1 and τ with ν 1 ◦ ν -1 0 and ν 0 ◦ τ , respectively, give us the same constaints for ν 0 ( τ ( x 1 ... 6 ) ) regardless of ν 1 .

Theorem D.7 (Intersection of two hypersurfaces and a diagonal plane, restated) . Let ˜ ν 0 ( x ) = 0 and ˜ ν 1 ( x ) = 0 be two hypersurfaces passing two points x 0 and x 7 such that ˜ ν 0 ( x 0 ) = ˜ ν 1 ( x 0 ) = 0 and ˜ ν 0 ( x 7 ) = ˜ ν 1 ( x 7 ) = 0 , P i := ˜ ν 0 ( x i ) and Q i := ˜ ν 1 ( x i ) ,

and

Then, x ∈ [0 , 1] of the intersection point of the two hypersurfaces and a diagonal plane of x = z is the solution of the following quartic equation:

̸

Proof. We rearrange ˜ ν 0 ( x ) using x = z as follows:

̸

while

In a similar way, we rewrite ˜ ν 1 ( x ) as follows:

Notice that this is a quartic equation of x . To find the coefficients of the general form of quadratic equation, we consider a mapping C from the interpolation coefficients τ i to polynomial coefficients c i :

where τ 0 (1 -x ) 2 + τ 1 x (1 -x ) + τ 2 (1 -x ) x + τ 3 x 2 = c 0 + c 1 x + c 2 x 2 . Then, the coefficients of the quartic equation of Equation (74) are as follows:

where the anti-diagonal sum of the result is the quartic coefficients, i.e. ,

The solutions of the quartic equation such that x ∈ [0 , 1] are the valid intersection points, and we choose one of them x if there are more than one solution. Note that we can compute a batch of quartic equations using the parallel computing of the eigenvalue decomposition with Lemma D.8.

We determine y by Equation (71) using x , and z = x .

We reproduce the method to find polynomial roots from companion matrix eigenvalues [29].

Lemma D.8 (Polynomial roots from companion matrix eigenvalues, reproduced) . The companion matrix of the monic polynomial 7

The roots of the characteristic polynomial p ( x ) are the eigenvalues of C ( p ) .

is defined as:

E Complexity analysis

We provide a detailed exposition of the complexity analysis in Section 3.2 to ensure the comprehensiveness of our work. Notice that our complexity analysis closely aligns with the approach outlined in Berzins [4], as in their Appendix A.

Algorithm 2 operates on the vertices and edges, V and E , respectively. Therefore, our initial focus is on the complexity associated with these inputs. Let |V| , |E| , and | ˆ E| ( ˆ E ⊂ E ) be the number of vertices, edges, and splitting edges consisting of polygons (Section 3.2), at the iteration of i and j .

During the initialization step (refer to Section 5), we determine |V| and |E| to construct a grid in R D . In the case of having M HashGrid marks, the quantities are defined as M D and D ( M -1) M D -1 , respectively. In case of the large M , we consider vertex and edge pruning for efficient computations inspired by a pruning strategy proposed in Berzins [4]. If, for every future neuron, denoted as ϕ ( i, j ) > ϕ ( i ∗ , j ∗ ) , the two vertices of an edge exhibit identical signs, the edge can be safely pruned as it will not undergo splitting. We make the assumption of that M D < |V| to avoid an excessively dense grid, with a complexity of O ( |V| ) .

Evaluate ˜ ν ( V ; i, j ) with a complexity of O ( |V| ) , assuming negligible inference costs as constants. The intermediate results from the inference can be used for the sign-vectors (Definition 3.4).

Identify splitting edges ˆ E by comparing two signs per edge, with a complexity of O ( |E| ) .

Obtain P and Q for Theorem 4.7 with a complexity of O ( | ˆ E| ) in a similar way.

Inference for Theorem 4.7 involves matrix-vector multiplications and the eigenvalue decomposition of a companion matrix for each edge, with a complexity of O ( | ˆ E| ) . Considering the maximum size of 4 for a quartic equation, this operation is regarded as linear.

Find intersecting polygons by pairing two sign-vectors with two common zeros in a shared region of the new vertices resulting from splitting, with a complexity of O ( | ˆ E| ) .

In the study by Berzins [4], a linear relationship between |V| and |E| for a mesh is empirically observed (see Figure 8a in Berzins [4]). It is worth to note that the number of splitting edges can be effectively upper-bounded as | ˆ E| < |E| D/ϕ ( i, j ) using Theorem 5 in [20]. Consequently, all the steps outlined in Algorithm 2 demonstrate a linear dependency on the number of vertices, denoted as O ( |V| ) .

To obtain faces, we sort the vertices within each region based on the Jacobian with respect to the mean of the vertices (see Algorithm 3). Considering triangularization, where each face has three vertices, the complexity remains O ( |V| ) , assuming the negligible cost of calculating the Jacobian, especially when performed in parallel.

F Supplementary experiments

F.1 Complete results

Table 2 and Table 3 are extended from the Stanford 3D Scanning dataset in Table 1. Table 4 extended from the Stanford bunny of Table 1, including standard deviations, and pseudo-ground truth results, obtained through marching cubes with 256 samples. Notice that we used the more excessive 512 samples for the Medium and Large models in Figure 4. We found that 256 samples were not enough for those. Table 5 shows the angular distance results. Tables 2 to 5 use the Small model, while Table 8 uses the Large model ( ref. Section 6). We conducted all experiments with an NVIDIA V100 32GB.

Chamfer distance

1E-04

1E-03

Number of the vertices

0K

30K

60K

90K

120K

150K

180K

3.66E-04

2.08E-04

1.47E-04

1.21E-04

1.13E-04

3.58E-04

2.37E-04

1.87E-04

1.64E-04

1.52E-04

MC+QEM

Ours+QEM

F.2 Mesh simplification using QEM

In Figure 8, Quadric Error Metrics (QEM) [35] is a popular mesh simplification algorithm, it could potentially improve both MC and our method for efficiency. Overall, as shown, the mesh simplification favors our method. Notice that QEM entails a higher computational cost, at least O ( n log n ) where n is the number of vertices. Additionally, these methods can cause shape deformation or holes, sensitive to hyperparameters. Therefore, QEM may be applied with caution for further mesh simplification, acknowledging the potential for unpredictable side effects. We used MeshLab's Quadric Edge Collapse Decimation with the default settings to halve the number of vertices for each step.

Figure D.

for the

Chamfer distance (CD) Figure  E. CD  for  the  Bunny F.3 Comparison with alternate approaches

Large with the

Large model applying

model. Our method achieved the lowest chamfer  distance  and  the most  e ffi ciency method  with QEM  (mesh  simplification). Both  benefit  from  QEM  while our  method  e ffi ciently  retains Marching Cubes (MC) remains the de facto sampling-based method. In contrast, our method is a white-box method grounded in the theoretical understanding of trilinear interpolation and polyhedral complex derivation. Here is our rationale for this evaluation:

respect to the number of vertices.

better chamfer distances.

Analytical approaches for CPWA functions are limited by their inability to effectively manage spectral bias and their exponential computational cost, which struggles with exponentially growing linear regions (infeasible to run our benchmarks.) For example, the time complexity of Analytic Marching [2] is O (( n/ 2) 2( L -1) |V P | n 4 L 2 ) , which grows exponentially with the number of layers L and the width n of the networks, where |V P | represents the number of vertices per face (Section 4.1 from Lei and Jia [2]). In contrast, our method has a linear time complexity with respect to the number of vertices, as discussed in Appendix E, allowing it to avoid visiting every linear region exponentially growing.

We also exclude optimization-based methods as they typically rely on MC or its variants initialization in their pipelines (our method can also be integrated into them), such as MeshSDF [36], Deep Marching Tetrahedra [10], and FlexiCubes [37]. Additionally, these methods result in prolonged computational costs.

We provided both rigorous quantitative Figure 3 and qualitative Figure 6 analyses on Marching Tetrahedra (MT) [9, 10] and Neural Dual Contour (NDC) [32], which serve as alternative representative approaches. MT is a popular variant of MC that utilizes six tetrahedra within each grid to identify intermediate vertices. NDC is a data-driven approach (a pretrained model) that uses Dual Contour. In short, MT efficiently requires SDF values but is less efficient regarding the number of vertices extracted. (MT is better than MC in the same resolution, producing more vertices, though.) NDC faces a generalization issue for a zero-shot setting, producing unsmooth surfaces. We used the code from Deep Marching Tetrahedra [10] for MT and the official code of NDC. And PyMCubes for our modernized MC library.

F.4 Error of the learned SDF

In Equation (11) and Figure 5, we confirmed our hypothesis that the eikonal loss induces piecewise linearity in trilinear networks. To further support this, we also present the error analysis of the learned SDF. Although the Chamfer distance virtually evaluates the SDF values based on sampling, concerns may arise regarding noise in the pseudo-ground truth. For this reason, one might wish to verify whether the predicted SDFs converge toward near-zero.

We randomly sampled 100K points on the surface of pretrained networks (the Stanford bunny Large model) and calculated the average of squares for predicted SDF (model's outputs) on the surface. As we expected, the predicted SDFs on the reconstructed surface converge to zero as the weight of the eikonal loss increases, and the number is close to zero ≈ 3e-8 as shown in Table 9. Note that, for a Large model, fine-grained trilinear regions from the dense grid marks (from the more dense multi-resolution grids) allow relatively small errors of 97 × 1e-9 without the eikonal loss, although the eikonal loss of 1e-2 further decreases to 31 × 1e-9. Note that the stronger eikonal loss of 1e-1 proved ineffective due to over-regularization.

F.5 Effect of model sizes

In Table 10, we explore the ablation study to determine whether model size impacts model fitting, which in turn affects the GT mesh (we used MC256 for the pseudo-ground truth of the Small models). With a smaller model size, the learned mesh becomes simpler due to underfitting. As our evaluation focuses on how accurately the extraction methods capture the underlying zero level-set decision boundary of the networks, the change in CD remains small.

Notice that the model with the number of layers of 3 and the width of 64 has the longest runtime of 2.80, primarily because the number of intermediate vertices exceeds 80K. The time complexity is more sensitive to the number of vertices than to the number of layers or the width, which are only indirect factors. Note that the model capacity of the decoding networks is comparable to InstantNGP [12], where the representational efficiency is driven by the HashGrid.

G Visualizations

For Figure 6 in the main paper (closer looks at bunny's nose), we want to leave a note about the small facets along the left-side of the nose line in the highlighted region. We assume these narrow facets are embedded in the facets of MC256 (pseudo-ground truth), while MC64 fails to capture the left and right edge of the nose and represents this with more facets cutting these two sides of edges - not aligned with nose lines and rather assign more facets in here.

In Figure 9, we explain the rationale behind adopting the diagonal plane assumption in Theorem 4.7 and Section 4.2. We illustrated 64 edge scenarios within a trilinear region, omitting isomorphic cases.

In Figure 10, we showcase the analytical generation of a normal map from piecewise trilinear networks, utilizing HashGrid [12] and ReLU neural networks. Our approach dynamically assigns vertices based on the learned decision boundary. Notably, the configuration of hash grids influences vertex selection, and each grid can be subdivided into multiple polyhedra for accurate representation of smooth curves.

In Figure 11, we present comparative visualizations of normal maps for marching cubes, demonstrating variations in the number of samplings alongside our method, with the respective vertex counts indicated in parentheses. This analysis highlights the trade-off between detail preservation and computational efficiency, showcasing the optimality of our analytically extracted mesh from decision boundary apex points.

In Figure 12, we compare marching cubes (MC) with 256 and 64 samplings to our method. MC 64 shows limitations in representing pointy areas, while our approach excels with analytical extraction from decision boundary apex points. Adaptive sampling leads to some vertices being in close proximity (10.1% within 1e-3), prompting potential mesh optimization for improved efficiency.

In Figure 13, the gallery of the six Stanford 3D Scanning meshes from the Large models is shown.

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1

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper proposes a theoretical link between the tropical algebraic interpretation of ReLU-based neural networks and surfaces represented by signed distance functions trained on those networks. Based on that theoretical interpretation, a surface extraction algorithm is proposed.\n\nSTRENGTHS:\n(1) The mathematical notation is good overall, with good choices for the symbols and their use.\n\n(2) The theoretical framework is interesting and it is a good application of tropical algebra.\n\nWEAKNESSES:\n(1) I do not like the current presentation of the paper. I understand that it is theoretical, but its focus on lower abstraction details from the very beginning makes it a niche paper in NeurIPS. For example, the Experiments section focus on the details of the empirical settings, moving the actual results to an Appendix. Another example is that there is little effort in providing an overview of the method. Overall this is the core design of the paper: focus on the definitions and details before properly motivating them. In its current form, this paper is more aligned with an applied mathematics journal. This is the core reason of my current rating and fixing that problem is not possible without rewriting the paper. However, I think the theoretical framework is sound so I am not recommending rejection. All other following weaknesses derive from this presentation problem.\n\n(2) This paper needs graphical intuitions to assist unexperienced readers. For example, showing the plot of a simple tropical polynomial, with the hypersurface points highlighted, would give a good intuition of the rationale behind Definition 3.1 (i.e. the linking points between each piece-wise linear region must be where two monomials are equal). If space for the images is a problem, I would advise to include them in the supplementary material, with references in the main text. This more didactical approach would increase the public that could appreciate the paper.\n\n(3) Another example is the subdivision scheme. Showing an schematic image of the subdivision would help understanding.\n\n(4) I also think a better overview image is needed. Figure 1 does not make a good overview because its abstraction level is too high. A better image should refer to core symbols in the definitions. Not having a proper overview makes understanding the motivations behind some definitions more difficult. For example, $\\psi$ and $H$ are defined in Definition 4.1, and I could not find the motive for them in the main text. They are only used at Proposition D.2, at the Appendix. This kind of graphical intuition is even more important in a geometrical problem, which is the case of this paper.\n\n(5) The extracted surfaces have noise and discretization problems, probably because of the grid-based approach and interpolation. Grid-based methods are known to introduce discretization problems (Neural Geometric Level of Detail - NGLOD, Instant Neural Geometric Primitives - Instant-NGP).\n\n(6) The evaluation is lacking. The paper only compares against classic marching cubes. Being a classic algorithm, several improvements were introduced by more recent papers over the years.\n\nQUESTIONS:\n(1) Typo in line 426: $x^3 + 2xy + y^4$ should be $x^3 \\oplus 2xy \\oplus y^4$. This is important because an unexperienced reader may get stuck in there.\n\n(2) I believe the notation $v_j^{(i)}(x)$ in Proposition 3.3 is a little bit ambiguous. It is not immediately obvious that the $j$-th component is from the result of the layer computation $v^{(i)}(x)$ or if it is the $j$-th weight in layer $(i)$. In the second case $v_j^{(i)}(x)$ would have a weird interpretation, but I can see readers being stuck in there. Changing the notation probably would have consequences, but at least saying that the component $j$ is from the resulting vector $v^{(i)}(x)$ in the commentary that follows the definition would avoid that ambiguity.\n\n(3) Also in Proposition 3.3 I would add an intuition commentary afterwards saying that $v_j^{(i)}(x)$ is a tropical polynomial, thus $v_j^{(i)}(x) = 0$ is an hyperplane composed of the points where two monomials are equal, intuitively connecting Proposition 3.3 with Definition 3.1.\n\nLIMITATIONS:\nYes, section 7 describes limitations.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 2, "DATE": "2024-07-09"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper extends the idea of extracting the polyhedral complex from ReLU networks to extracting a more general cell complex from ReLU networks with a positional encoding, specifically, trilinear encodings, such as HashGrid and TensoRF, commonly used in neural field representations of geometries or scenes.\nIt first theoretically extends the concept of piecewise linear to piecewise trilinear networks. It then proposes a modified *curved* edge subdivision algorithm to extract a polyhedral mesh approximating the neural implicit surface. The key assumption governing the approximation error is the eikonal loss, which ensures that the isosurfaces of the trilinear functions are close to planar.\nThe algorithm is validated on several shapes qualitatively and quantitatively, comparing to marching cubes in terms of the mesh size, accuracy, and time. It demonstrates sparser meshes at an increased computational cost.\n\nSTRENGTHS:\nThe attempt to generalize the insights from piecewise linear networks to ReLU networks with positional encodings is well motivated and holds a lot of promise. To the best of my knowledge this is the first attempt at such a characterization as well as a practical implementation. The theoretical aspect of the work is in my view the most clear contribution (however, I could not fully verify the details).\n\nWEAKNESSES:\nDespite the method being able to handle volumetric complex of the input space, the the work only considers the \"skeletonized\" complex, i.e., the boundary mesh. At least briefly describing the former would make for a more rounded story. Instead, the skeletonized method is framed as a potential alternative to marching cubes, which unfortunately is not convincing from the results. The only demonstrated advantage is in terms of the mesh size. This argument would be more convincing given a comparison to marching cubes with some basic mesh post-processing. The work would also benefit from comparing the same procedure to piecewise linear networks and the underlying edge subdivision method in terms of training time, extraction time, mesh size, and accuracy.\n\nSimilarly, while the method is evaluated for different positional encodings, the neural network size remains fixed (3x16) making it hard to judge how the method scales in practice.\n\nThe topic and the proposed method are nuanced and technical, so it does not help that there are many imprecise statements, including in theorems, and poor phrasing, which render the manuscript hard to follow (see questions and suggestions below). The authors also overlook the chance to help the reader with more visual aids.\n\nOverall, I think the work has great potential which is not fully realized and publication-ready due to the unrefined presentation and somewhat uninspiring application.\n\nQUESTIONS:\n- In Theorem 4.5, is it true that $\\tau$ is a trilinear function? While $\\tau$ is defined elsewhere in 4.3, this assumption is missing in the theorem itself which is confusing. You should also clarify in the manuscript how trilinearity is used in the proof, e.g. in (41) - (43).\n- WIth that, could you clarify Theorem 4.5? My best understanding is that it states that the only trilinear function which satisfies the eikonal equation is a plane, but a counterexample to this is easy to conceive.\n- In Line 221, do you perhaps mean $ |\\tilde{\\nu}(\\mathbf{x})| \\leq \\epsilon$? Otherwise, you select the edges of the sub-level set, instead of the level set.\n- Why do you use the $256^3$ marching cubes mesh as a ground-truth instead of using the input mesh on whose SDF you train?\n- Can you comment on the extraction of the volumetric mesh? Is my understanding correct that this is what you have before the skeletonization step? Does this and your method more broadly offer a way to explain the success of trilinear positional encodings in terms of the underlying complex?\n\nLIMITATIONS:\nThe limitations paragraph focuses solely on the runtime of the method. The section would benefit from reiterating and discussing the reliance on the eikonal constraint, which is known to be hard to satisfy globally. I would also urge the authors to consider discussing the limitations of the experiments as described in the weaknesses.\n\n\n**Errors and suggestions**\n\n- The motivation of discussing the normals in 4.3 is unclear. \n- All the left-hand terms in 40-43 should be squared. The rhs in (40) is the square of the gradient norm, not the gradient norm.\n- The work would benefit from a discuss on the size of the NN and the encoding for practical applications. To my surprise, [12] uses similarly sized networks to your 3x16 with decent performance.\n\n- Line 19: \"polyhedral\" -> 'polyhedron\"\n- Line 50: \"This provides a theoretical exposition of the eikonal constraint\" is unclear\n- Line 59: you should give some examples of where marching cubes are used in a learning pipeline and explain what you mean by redundancy (probably flat regions which are described by many triangles?)\n- Equation 2: I would not recommend overloading the notation for $d$\n- Definition 3.2: what do you mean by \"without loss of generality\" when defining the NN?\n- Line 109: \"subscription\" -> \"subscript\"\n- Line 110: \"for rather discussions as an SDF\" is entirely unclear\n- Lines 112-114 are very unclear\n- Line 127: \"subdivising\" -> \"subdividing\"\n- Definition 4.1: what is $\\psi$? You should also define/explain $F$ before using it. Why is the definition given for general $D$ if this is a trilinear interpolation?\n- Line 203: \"also the\" -> \"also due to the\" ?\n- Line 248: \", similarly\" is not needed\n- Line 259: Can you be more precise with the objective? The goal is to get a boundary mesh of the zero-set of the SDF. \n- Line 262: reformulate to use a verb\n- Line 276: CE is not defined in the main text.\n- Line 294: \"plenary\" -> \"planarity\"?\n- Line 560: do not write \"by the way\"\n- Overall, please check the text with a spell and/or style checker", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 2, "DATE": "2024-07-12"}

comments: SUMMARY:
The paper presents a theoretical and practical framework to extract meshes from piecewise trilinerar networks using hash grid representation. They further show that hypersurfaces within the trilinear regions become planes under the eikonal constraint. A method for approximating the intersection between 3 hypersurfaces is proposed (approximated as 2 hypersurfaces and a diagonal). Evaluation is performed against marching cubes resolutions using chamfer distance, chamfer efficiency and angular distance.

STRENGTHS:
- Thorough theoretical analysis
- Proof of concept practical evaluations show improvement over naive marching cubes at similar complexity of meshes

WEAKNESSES:
- While there are some improvements over marching cubes, the improvements are not quite substantial, and come at the cost of a higher runtimes
- No comparisons to any alternate approaches over marching cubes are provided -- e.g. marching cubes + QEM decimation etc.
- No rigorous qualititative comparisons are provided, which could be important since chamfer does not always capture the shape quality

QUESTIONS:
Would be nice if the authors can add some qualitative comparisons and also against more than just marching cubes. Further current evaluation is only performed on a handful of meshes, would be nice if an evaluation on a bigger dataset can be performed

LIMITATIONS:
Yes
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 3
DATE: 2024-07-12

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe work addresses the problem of exact iso-surface extraction given an implicit surface represented by a neural network. The idea behind this line of work is to closely analyze the architecture of the neural network to efficiently extract the true (or at least an approximate of) learned iso-surface. Previous work titled 'Polyhedral Complex Extraction from ReLU Networks using Edge Subdivision' develops the analysis for ReLU networks. However, many newer works use local latent codes to achieve better reconstruction quality. These latent codes are usually obtained via trilinear interpolation from a grid of learnable latent codes. The previous method does not trivially apply to iso-surface extraction in this case. The current work continues the analysis for ReLU networks that now have trilinear positional encoding as input. The paper first shows that under the eikonal constraint, trilinear regions decode into flat planes. Then, it derives a polynomial defining the intersection of two hypersurfaces. With this, the paper suggests an algorithm for zero-level set extraction given the trained neural network and the positional encoding.\n\nSTRENGTHS:\n* The paper extends previous work from “Polyhedral Complex Extraction from ReLU Networks using Edge Subdivision” to architectures with positional encoding (PE) implemented via trilinear interpolation. This is a very reasonable extension, as many implicit representation methods in practice use grids of local embeddings.\n* Theorem 4.5 (and Corollary 4.6), which prove that the trilinear regions of PE decode into flat planes under the eikonal constraint, seem surprising. If true, this is a great insight into the performance of these models.\n* The suggested algorithm can extract a good approximation of the learned surface and matches the accuracy of Marching Cubes at much smaller resolutions.\n* The paper is self-contained and presents interesting insights into what neural implicit representation methods are capable of reconstructing.\n\nWEAKNESSES:\nHere are the main concerns I found in the paper in order of importance (with the first being the most important):\n\n* I found the paper very hard to understand. I am quite familiar with neural implicit representations (though not with Polyhedral Complex Extraction), but it still took me at least three full passes before I started understanding the general idea of the paper. The paper is self-contained and formal, which is great, but I would consider simplifying the language for the sake of understandability.\n\n* The work is positioned as a better way of iso-surface extraction, yet there is no visualization of extracted meshes in the main paper. It is very hard to say if the mesh extracted with the proposed method is actually better than the ones extracted with Marching Cubes. In my view, MC64 seems to be actually closer to MC256 than the mesh extracted with 'ours' in Figure 7. A much more detailed visual comparison of extracted meshes seems necessary in the main paper given the claims.\n\n* The method is shown to work only with relatively small neural networks: the experiments are performed on a model with three layers and a hidden size of 16, while DeepSDF, for instance, operates on eight layers, each with a width of 512. It seems that it can be used only as a source of understanding the behavior of these architectures, not as a practical method of iso-surface extraction.\n\n* Comparison with only Marching Cubes seems very limited; it would be great to see a comparison with 'Analytic Marching' as well.\n\nI will consider adjusting the final rating if weakness #2 and question #1 are addressed in the rebuttal\n\nQUESTIONS:\n* I am quite confused about the following (probably due to my limited understanding of the math in the paper): A standard ReLU network with a constant latent code as input (i.e., DeepSDF) can be seen as a trivial trilinear positional encoding (i.e., the entire domain is trilinear). Given that such networks are usually trained to satisfy the eikonal constraint, why aren’t they limited to always produce flat plane reconstructions, given Corollary 4.6?\n\n* It seems that a natural measure of how accurately the algorithm extracts the learned iso-surface would be:\n\n        1. Sample N points on the surface of the extracted mesh\n        2. Predict SDF on each of them\n        3. Calculate sum of squared SDF values\nit should be 0 if we extracted the learned surface precisely. Would be interesting to see this metric. Can we confirm Corollary 4.6 with it?\n\nLIMITATIONS:\nThe limitations are well discussed. I would also add a discussion on the depth and width of the neural networks for which the method is practical to use.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 2, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-13"}