Polyhedral Complex Derivation from Piecewise Trilinear Networks
Authors: Jin-Hwa Kim
Venue: NeurIPS 2024 poster (2024)
Keywords: Polyhedral Complex
Neural Radiance Fields
3D Mesh
Primary area: generative_models
TL;DR: Mesh extraction from neural surfaces with trilinear interpolating methods through the eikonal constraint of hypersurface planarity.

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
Jin-Hwa Kim NAVER AI Lab & SNU AIIS j inhwa. Batbba.kin@navvercorrr
Republic of Korea
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 encod- ing, we present theoretical insights and an analytical mesh extraction, showing the transformation of hypersurfaces to flat planes within the trilinear region un- der the eikonal constraint. Moreover, we introduce method for approximating intersecting points among three hypersurfaces contributing to broader applica- tions. 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: //gtth///tttttt com naver nwwer-ai http://oopccaalenniirrr pytorch.
Introduction
Recent advancements in visualizing deep neural networks [1-4] significantly contribute under- standing 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, egg., ReLU neural networks, each region of the input space represented as convex polyhedron. The assembly of these sets constructs polyhedral complex that delineates the decision boundaries of neural networks [5].
Building upon these breakthroughs, we explore the exciting prospect of analytically extracting mesh representation from neural implicit surface networks [6-8]. The extraction of mesh not only offers a precise visualization and characterization of the network's geometry but also does so efficienlly through the utilization of vertices and faces, in contrast to sampling based methods [9, 10]. The sampling-basee 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 signed distance function (SDF) by utilizing ReLU neural networks. However, recent approaches incorporate non-linear positional encoding techniques, such as trigono- metric functions []]] 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 videly-adopted trilinear interpolating methods, we present novel theoretical insights and practical methodology for precise mesh extraction. Following novel review
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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Harsin SAALE
(2A) LENEAR SUBDRVISION Ox THE CKID EDGE
(2s) BILINDAR SUBDIVISION THE GRID PLANE
[2e] TRILINARR
(1) VEneicns ANT FROM THE MARKS (SEc. 5)
SUBDWISION INSIDE THE CRID (TнM. 4.7]
(3) SKELETONIZE WITH LEVEL-SEL OF AN STIP (Sec. 4.3)
(4) EXTRACT FACES BY SURTING VERTICES (Sec. 4.3)
(2) CURVED EDGE SUIDIVISION oVER NEURONS (AtG. 2)
Figure 1: The mesh is analytically extracted from piecewise trilinear networks, comprising both HashGrid [12] and ReLU neural networks, that have been trained to learn a signed distance function with the eikonal loss. We start with initial vertices and edges defined using the grid marks (1), and its linear subdivision (2a); however, intersecting polygons (ref. Section 3.2) need bilinear subdivision (2b) and consequentially trilinear subdivision (2c). Note that the linear and bilinear subdivisions are specific instances of trilinear subdivisions with straightforward solutions.
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), 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 eurvature of these surfaces, achieving an exact solution is infcasible. In our approach, we utilize two hypersurfaces and a diagonal plane for a more feasible and precise approximation (Theorem 4.7). Additionally, we introduce 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 more detailed view, please refer to Figure 10 in Appendix.
Our contributions are summarized as follows:
1. We present novel theoretical insights and practical methodology for precise mesh extraction, from the piecewise trilinear networks using the HashGrid.
2. We provide theoretical analysis of the piecewise trilinear networks under the eikonal constraint revealing that, within trilinear region, a hypersurface transforms into plane.
3. 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 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] modern mesh packages, this exhaustive method often generates redundant complexities due to dis- cretization, 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 ap- proaches 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 mathemat- ical simplicity of hyperplanes defining linear regions. However, this restricts their applications in emerging domains of deep implicit surface learning that diverge from CPWA functions.

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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 feature space, respectively. However, these approaches cause the function represented by neural networks to deviate from CPWA functions, as trilinear interpolation is not affine transformation. Consequently, the complex regions are divided by intricate hypersurfaces.
Eikonal equation and SDE. An eikonal equation is 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:
f(x) = (d((z, 02) i E Ω
(d(xx00)
i E Ω2C
where aQ denotes the boundary of Ω. The metric d is defined (with slight notational abuse) as:
d(x,00 02) == inf d(x, y) yeas
(2)
where d represents the shortest distance from to 82 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 the boundary, ff(x)) = (==)
where the Jacobian of f represents the outward normal vector, or equivalently, [vf(x) = 1.
Preliminaries
We present overview of tropical geometry (Section 3.1) to provide 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. for less familiar readers. Later, we extend this idea to include considerations for trilinear interpolation in Section 4. You may choose skip these sections if you are already acquainted with these concepts.
3.1 Tropical geometry
Tropical geometry [25, 26) describes polynomials and their geometric characteristics. As a skele- tonized 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 non-differentiable, due tropical sum @ = max(-), ), is called tropical hypersurface:
Definition 3.1 (Tropical hypersurface). The tropical hypersurface of a tropical polynomial f(x) e/ c1 @
① c, Pr is defined as:
v(f) {x RRd q, = c,,0' f(x) for some as az}
a. where the tropical monomial ca with d-variate is defined as a 2 x1 d-variate a; = (0i,1,-... (i,a) and tropical product 3 =
asd with a x.
Definition 3.2 (ReLU neural networks). Assuming input and output dimensions are consistent, L-layer neural networks with ReLU activation are composition of functions defined as:
( (L)) oo (I-1) ( σ (1)
where (x)) max (x: 0) and p (0) (x) W 10 6()
"The eikonal is derived from the Greek, meaning image or icon.
The tropical named in honor of Hungarian-bon born Brazilian computer scientist Imre Simon [24].
3

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Although the neural networks v is generally non non-convex, it is known that the difference of two tropical signomials (allowing a real-valued in a tropical polynomial) can represent v [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 I-layer neural networks with ReLU activation v (z)) = O is a subset of or equals with the region where two tropical monomials, or two arguments of max, equal (Definition 3.1), as follows:
B S {~ : (x)
i (1, (I.....,,)) L), J (l...., H) if + L else (1))
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 single output for rather discussions an SDF (Definition 4.3). Remind that tropical operations satisfy the usual laws of arithmetic for recursive consideration.
(6) The piecewise hyperplanes corresponding to v2 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 v (ar) = O. 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 complex derivation from 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 xx where v(x*) = O. Then, the valid edges connecting those vertices would represent the decision boundary.
To elaborate with details, the order of subdividing 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 edge, v(zr)) and v(x1). which are proportional to the distances to the hyperplane by Thales's ' theorem [28]. The intersection point would simply be á = (1 w) + wab where = |dol/Ido di and
de- -
(1) (I)), upon the fact that do d1 < 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 tricky part that we need to find every pair of two vertices among ( J. 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 efficient way to find them.
Definition 3.4 (Sign-vectors). For the current hyperplane specified by the i" -th layer and its j*tth preactivation, we define the sign-vectors with a small positive constant e:
+1 if (ax) > tc if |f (mx)) $ +c
(x))o(()))
(4)
1 if vi (xx)
< -
for all i and j where (i.j)) S (("".')) and ((i.j) E (0) U is an ascending indexing function from lower layers, handling arbitrary hidden sizes of networks to vectorize a signed matrix.
In geometry. a polyhedron is a dimennimmaiional shape with polygonal faces, having straight edges and sharp vertices.

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Notice that a hyperplane divides 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 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 Hanin and Rolnick [20]. the number of linear regions is known for o(N'. /L!) where N is the total number of neurons, or of 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 И k²2xF where F is corner-feature dimension. Given weights w E [0, 1] the trilinear interpolation function vb is defined as:
2D
(w: H
Σ
wi w)
H. ERF
(5)
where the interpolating weight wii, w) E IR is defined using left-aligned and zero-trailing binariza- tion function B E 10, (ref. Appendix B.1) as follows:
D
wii w) = II (1 - -()) (1)((11w))--)) + B(t)
(6)
1=1
which is the volume of the opposite subsection the hypercube divided by the weight point w. For a general hypercube, scaling the weight for unit hypercube gets the same result. Going forward, assume D=3 for trilinear interpolation without explicitly stating it otherwise.
Lemma 4.2 (Nested trilinear interpolation). Let nested cube be inside of a unit cube, where its positions of the eight corners deviate -a, or +b,, such that aj by 2 O. from w; for each dimension 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 is identical.
Proof. Notice that trilinear interpolation is linear for each dimension to prove it. The detailed proof can be found in Lemma D.I.
Definition 4.3 (Piecewise trilinear networks). Let positional encoding module be IRD RF using the trilinear interpolation of spatially nearby learnable vectors on a three-dimensional grid, e.g., HashGrid [12] or TensoRF [13], where 7 is an instance of 1 in Definition 4.1. Usually, they transform the input coordinate æ to w as a relative position inside unit grid in a corresponding resolution. H is learnable parameter specified in the corresponding method. Then, define trilinear neural networks iz by prepending r to the neural networks from Definition 3.2. Here, the input and output dimensions of (LL) are F and (an SDF distance), respectively.
( = - Of
(7)
Note that is gridwise trilinear. Using Lemma 4.2, i is trilinear within the composite intersections of linear regions of v and trilinear regions r. (We discuss how access virtual cubic corner features where trilinear regions are not cubic in Section 5.) So, regard ū piecewise trilinear.
5

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4.2 Curved edge subdivision in trilinear space
In trilinear space, 7 projects line to curve except the lines on the grid. Notably, the diagonal line t= (1,6,4) 1)T where / E [0, 1] is projected to 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 (æo, Φτ) be the intersection of two hypersurfaces (x) (z(2) = 0, while In and 27 are on the two hypersurfaces. Then, the edge is defined as:
{x: : T (x)) = =1 - ((00))) + (T())) where x E Re and LE R].
Proof. The detailed proof using the piecewise linearity of v provided in Lemma D.4.
Notice that it does not decrease the number of variables to find a line solution since the trilinear interpolation t still forms hypersurfaces making complex cases. Yet, we theoretically demonstrate that the eikonal constraints D render them hyperplanes with linear solutions.
Theorem 4.5 (Hypersurface and eikonal constraint). A hypersurface T(x) = 0 intersects two points r(21)) T while r(x1..6) + for the remaining six points. These points form a cube, with an and 27 positioned on the diagonal of the cube. The hypersurface satisfies the eikonal constraint [ for all x E [0, 1/3. Then, the hypersurface of (x²) = is a plane.
Corollary 4.6 Araine-transformed hypersurface and eikonal constraint). Let vo(x) and 11 (x) be two affine transformations defined by WT b, E {0, 1). A hypersurface vo (r())) = 0 passing two points vo (xx))) Po ( (T(xT)
while vo (T( * satisfies the eikonal constraint (T=))) = for all æ E [o, 1]s. Then, the hypersurface of vo (T(x)) = 0 is a plane.
The proof's 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 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, 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 enror tolerated by the hyperparameter =le-4 as specified Definition 3.4 and Section 6. More discussions on this matter can be found in Appendix C.I.
illustrated in Figure 9, Appendix G. Thus, the new
Theorem 4.7 (Intersection of two hypersurfaces and diagonal plane). Let (x) = 0 and in (x) 0 be two hypersurfaces passing two points 2n and Φ7 such that (n(())) = i2 (xp) = and Do(a)) = ii (x7) = 0, P. == v0(xx) and Q; := Di (2;,, P. = [oo Pi; P4; P.). P3 = P2: P3; Po; P.]. and X = [[1 2);; x(1 - x): (1 - z)z; 2² Then, x E [0, 1] of the intersection point of the two
hypersurfaces and diagonal plane of z is the solution of the following quartic equation:
XTP
X (T - PPQ..) 0, while
(Pa Pp).
(8)
XT Pp)
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 be parallelized (ref. Lemma D.8). Given that the companion matrix remains small for a quartic equation in R the overall complexity remains O(VI). For the detailed complexity analysis on the proposed method, please refer Appendix E.
4.3 Skeletonization, faces, and normals
Skeletonization selects the vertices such that: > {z |0(x)| < <e, E V). and the edges Ex such that their two vertices are among Vd. Recall that because decision boundary is a subset the tropical hypersurface (Definition 3.1 and Proposition 3.3), this step ensures accurate mesh extraction.

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(a)
(b)
(c)
(d)
(c)
Figure 2: Trilinear regions in the xy-plane at 0.04, identified by sign-vectors (Definition 3.4), are represented with random colors. (a) Grids described in Section 5 and Algorithm 4. (b) The neurons the first layer representing folded hypersurfaces (blue arrow). (c) All neurons representing every nonlinear boundary. (d) Select all zero-set vertices and edges. (c) Skeletonized in Section 4.3.
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 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 each face has ambiguity with two opposite directions. Let T0,1.2 be the vertices forming a triangular face. The normal is defined using pross-prodduct with direction given by the right-hand rule. Please refer to Appendix B.2 for the details.
(x1 TD) X (x2 σo) where the normal is orthogonal to both vectors,
Implementation
We employ the HashGrid [12] for while discussion 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 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 input coordinate be [0, 1]3, and there is unit cube such that of its corners is at the origin. Taking into account multi-resolution grids, we obtain the marks m4. such that the grid planes are x m,, y = my, and m, 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 the associated grid vertices and edges, which serve as the initial sets for V and ε, respectively. Please refer Figure 2a where its multi-rees restiutesoo grids are visualized using randomized colors.
of Müller et al. [12]). Leveraging the obtained marks, we derive
Optimizing sign-vectors. The number of orthogonal planes may significantly surpass the number of neurons, making impractical to allocate elements for the sign-vector (Definition 3.4). To address this, leveraging the orthogonality of the grid planes, we store plane index along with an indicator of whether the input is on the plane (y 0) or not (? 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 outside of the region. For this, we replace
all ReLU activations using the mask m that: m
(xx) > e) (z) (wr) > e), where æo and 27 are the vertices of an edge of interest, for the calculated cubic corners Øo... This implies that
mff)
(z()(()))
exhibits linearity, except for instances where two vertices simultaneously deviate from linearity. For
P and Q, we select the last two pairs of and j such that (Zo) 700.)));7) = O, indicating two common hypersurfaces passing two points En and Ø7.
Experiment
Objective. For a given trilinear neural network with the eikonal constraint, our goal to get boundary mesh of the zero-set of the SDE, (Note that NeuS [6] showed how to convert the density
7

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Algorithm 1 Polyhedral Complex Derivation
Algorithm 2 Curved Edge Subdivision (CESub)
Input: Sets of vertices and edges & (from Section 5), # of layer L,, hidden size H, trilinear networks for i 0 to L do for j 0 to H1 - do V.E -
Input: Vertices V, Edges &, trilinear networks 87 layer index i, neuron index j
= i.vi, Vnew =
for in & do
if dea de
< 0 then
E,,
Section a Enww Intersect P, Q.e) D- Theorem 4.7
P Q Corners(e)
Algorithm 2
end for end for
V Vnew Vnew (Feww
e f Eo. Index (Inew)
V. E CESub(V, E,, L. L,
E t- E + {le Index(xnew))
V E Skeletonize &, v)
end if end for
Section 4.3
F = ExtractFaces(("".e ,E*)
V + V + Vnew
Section 4.3
t E+ FindPolygons(Vwel
Section 3.2
return V",E*,FF
return V.E
networks NeRFs to an SDF the frameworks of volume rendering.) To evaluate this, we utilize marching cubes with excessive sampling get sheudo-groung 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 e le-4 for the sign-vectors (ref. Definition 3.4). The weight for the eikonal loss is le-2. For HashGrid, the resolution levels of 4, feature size of 2, base resolution Nmin of 2, and max resolution Nma of 32 by default. Nuin and Nmax are doubled (x2) or quadrupled (x4) for Medium or Large settings in Figure 4. We use the official Python package of tinycudnn 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 Sn and S₁ be the two sets of points. Specifically, the bidirectional chamfer distance (CD) is defined as follows:
CD(Su. S2) [ (CD(S), S1) SD)s CD S1)
1
min yESi
| y1l2-
(9)
So
S
We randomly select 100K points the faces through ray-marching from the origin, directing toward a randomly chosen point on a unit sphere.
Table 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 ables parsimonious representations using the lower number of vertices having better CD), summarized by chamfer efficiency (CE) $ 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 CE of 47.8 versus 19.2 (MC).
CHAMFER DISTANCE
1E-2
7.4BE 08
QUBS MT NDC
5,00
3.000-0
900-03
211-00
1E-3
Q.61E-04
6328 04
4,87F-04
2 FAC-01
2.1E-01
1822 200E-44
1.51E:04
120K
Our method inherently holds an advantage over MC in captur- ing 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 a smaller interval, and it tends approach the plane. From the plotting of Small to
NUMBER
Figure 3: Chamfer distance for the bunny with the Large model com- paring with MC, MT, and NDC.
Phttps://githin http:///ggg.aaboooo//a////////// www.////ws/ttinneeeeeeeeeeeeeeee cuda-nn
$CE is defined as 100/(v /DD CD), reflecting the trade-off off between the number of vertices and the CD.

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Table 1: Chamfer distance (CD) and chamfer efficiency (CE) for the Stanford 3D Scanning repos- itory [31]. The CD (x (ele)) is evaluated using the marching cubes (MC) with 256 grid samples as the reference ground truth for all measurements. Please refer to Table and Table for the full results, and Table 4 for standard deviation and time spent in Appendix (Ours took 0.97 + 0.21 while
MC also took within sec for the bunny.)
Bunny
Dragon
Happy Buddha
Drill
Lucy
MC
M 4
CD4
CF 1
[V] 4
CD 4
CF. f
[
CD +
CF f
CF
CF t
CE f
32
1283
4106
19.0
1416
6330
11.2
1032
6951
13.9
11.2
7.5
24,9
5367
1371
13.6
6090
1936
8.5
4362
2465
9.3
19.2
128
21825
393
11.6
25236
542
7.3
18219
722
7.6
6.6
17.4
10.7
196
49569
14.3
57341
203
8.6
41351
276
8.8
7.5
15.3
11.7
Ours
4341
900
25.6
5104
1243
15.8
3710
1738
15.5
13.9
47.8
23.6
1E-02
0.08
CHAMPER DISTANCE (LOG)
MC-SMALL -CRLARGE ARGR
FLAINESS EIROR
0.001
0.06
0.01
OUBS-SMAL SMALL
0.1
X
0.04
OURS-LARGE
X
0.02
1E-04
20K
40K
60K
80K
100K
5
10
20
25
30
35
50
NUMBER OF THE VERTICES
LTERATION
Figure 4: Chamfer distances for the Stanford dragon varying the number vertices and the model sizes, showing consistent efficiency.
Figure 5: Effect of the weight of eikonal loss on the flatness error in Equation (11). For the plot, we conduct experiments using the unit sphere.
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 in Appendix F.1 and the right side of Figure 10 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 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 described in Section 4.3. The angular distance is defined as follows:
AD(A,, = E, 180 cos ((V)))) .N(()))
(10)
In Table in Appendix, the angular distances for the Stanford bunny shown. As expected, 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. 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 wo and æ7 two vertices consisting of diagonal edge, while 1...6 are its remaining cubic corners. We obtain ((x1...)) in piecewise trilinear region as described Section This evaluation conducted as follows:
Ana -1, [ (12. (| +
(11)
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 the eikonal loss results in elevated planarity errors, leading to the construction of an inaccurate mesh.

---

MC 256 (92K VERTICES)
OUBS
MC (5.6K VERTICES)
MT 32 (4.7K VERTICES)
NDC 64
(4.5K VERTICES)
(5.6K VERTICES)
Figure 6: This figure provides detailed visualization of Figure 10 in the Appendix and its competitors. MC 64, MT 32 (Marching Tetrahedra), and NDC (Neural Dual Contour [32]) suffer over-smoothing method
inaceuracy of the actual surface in the Small networks within the nose, whereas reflects these details (see the boundaries of a nose) with consistent normals (in colors). MT efficiently demands SDF values by utilizing six tetrahedra within grids get intermediate vertices. However, is inefficient in terms of the number of vertices extracted. NDC faces a generalization issue for a zero-shot setting for the given networks, producing unsmooth surfaces.
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 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. 1..61 seconds for 1.0.3+ vertices with optimized algorithm, compared 1.31 and 1 11.88 seconds for marching cubes with 256 and 512 samples per axis, respectively. We note that parallelizing over multiple GPUs would be reliable option to improve latency. For the discussions satisfying the eikonal equation in practice and the generalization to other positional encodings using trilinear interpolation, please refer Appendix C.1 and Appendix C.2, respectively.
7 Conclusions
In conclusion, our exploration of novel techniques 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 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.
10

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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. also express my sincere thanks to the anonymous reviewers for their help in improving the manuscript. The NAVER 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
14
A.1 Tropical geometry
14
A.2 Tropical hypersurface
15
A.3 Tropical algebra of neural networks
15
B Implementation details
16
B.I Zero-trailing binarization function
16
B.2 Sorting polygon vertices and HashGrid marks
16
Further discussions
Satisfying eikonal equation and approximation
Generalization to other positional encodings using trilinear interpolation
D Theoretical proofs
17
E
Complexity analysis
23
F Supplementary experiments
24
F.I
Complete results
24
F2 Mesh simplification using QEM
26
F3
Comparison with alternate approaches
26
F4
Error of the learned SDF
27
F5
Effect of model sizes
27
G Visualizations
28
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 the decision boundary for ReLU neural networks optimizing an SDF, resulting in the formation of a polyhedral complex. For this, we delve 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 for trilinear interpolation in Section 4. Given that Appendix A. 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.
serves
include considerations
A.1 Tropical geometry
Tropical geometry [25, 26] describes polynomials and their geometric characteristics. As skele- tonized 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 -co with the two operations tropical sum e and tropical product 3. Tropical sum and product are conveniently replaced with max(-) and +, respectively. Notice that -oo is the tropical sum identity, zero is the tropical product identity, and these satisfy associativity, commutativity, and distributivity. For example, classical polynomial x² 2zy + yd would represent max(3r, y + 2, 4y). The
14

---

classical polynomial is rewritten for a tropical polynomial by naively replacing + with @ and with C. respectively. After this, we rewrite by the definitions of tropical operations max(3z, z+y+2 4y). 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
Additionally, we present 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+2Y
2+X+Y
2+Y
2+X
1+2x
Figure 7: A classical polynomial 2²2 + 1/2 2xy + 2x + 2y + turns into a tropical polynomial as + 2 y, taking into account the implicit coefficient of 1. We redraw Figure of the tropical curve from Zhang et al. [27] as a region map. Each color indicates region where each tropical monomial has maximum.
max(1 + 2x, 1 + 2y, x y,
A.2 Tropical hypersurface
The set of points where a tropical polynomial is non-differentiabll, mainly due tropical sum, max(-). is called tropical hypersurface, denoted (ff). Note that V is an analogy to the vanishing set of polynomial.
Definition A.1 (Tropical hypersurface, restated). The tropical hypersurface of a tropical polynomial f(x) = c @ ... C cr is defined as:
v(f) = (a E IRd q: = c, = f(x) for some as + aa}
where the tropical monomial cæ with d-variate æ is defined as e, O x) 91.1 C ... @ x a.d with
d-variate a; = (0.,1...., (g,a) and tropical product == +. In other words, this is the set of points
where f (xx equals two or more monomials in
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 composition of functions defined as:
( (T) o o (('--)) .......... (1) (1)
(12)
where (ar) = max (xr, 0) and (00 (0 (())
Although the neural networks / is generally non-convex, it is known that the difference of two tropical signomials can represent v/ [27]. A tropical signomial takes the same form tropical polynomial (ref. Definition 3.1,, but it additionally allows the exponentials a to be real values rather than integers. With tropical signomials F, and
0 (z)
(00) (x)
(13)
n o 1 (xx) = (x) - (x)
(14)
15

---

where F G and H are as follows:
F (a) = max (x), (x))
(15)
G (a) = w G (x) : (x)
(16)
H0 (x) w ( p(t (x) w G () (x)
(x) +
(17)
while W -W w where w and w R.
Proposition A.3 (Decision boundary of neural networks, restated). The decision boundary for L- layer neural networks with ReLU activation vl5 (x) 0 is where two tropical monomials, two arguments of max, equal (Definition 3. 1), as follows:
_
(z) O, i (I..... (L)...)))
else (1))
where the subscript j denotes the j-th neuron, assuming the hidden size of neural networks is H, while the L-th layer has single output for rather discussions operations satisfy the usual laws of arithmetic for recursive consideration.
SDF. Remind that tropical
This implies that piecewise hyperplanes corresponding the preactivations {u,²) 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
input is continually changed, it moves another linear region. From 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
zero-trailing binarization function, consisting of D = bits. The zero-trailing means the number in bits is increasing left-aligned. For example, the number 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, to select the axis. Naturally, i and 7 indicate the opposite 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 the normal, the order of vertices follows counter-clockwisg arrangement. We determine the vertex order based on the normal derived from the Jacobian the mean of vertices. We identify the counter-clockwis arrangement of vertices using dot product to compute relative angles 0 E {- T and cross product for getting direction. Algorithm describes the algorithm in Section 5, calculating the HashGrid marks from the hyperparameters of a HashGrid.
Algorithm Sorting Polygon Vertices
Algorithm HashGrid Marks
Input: vertices F,, normal n. N [xx
Input: Resolution levels I,, base resolution Nmin max resolution Nmax
Σ; .N:NN
for i =
1
m {0, 1}
V: (w; - -11)
u|12
b = 1og,(Nmav) Chiaa//mmi///// /
c; = (v,, vn). d: = (0, x Vo. n)
for 1 L do
0, c2 1d,20 + 2 1d,
1/ (Nain exp2 b) -1)
end for
+- Max MAranggee - 1, s), 0)
return ArgSort(( (-0)))
end for
return Sort (Unique(m))
16

---

C Further discussions
C.1 Satisfying the eikonal equation and approximation
Theorem 4.7 handles where the cikonal equation does not make perfect hyperplanes. First, observed the characteristics of trilinear hypersurfaces as illustrated Figure in Appendix. Among 64 edge scenarios within 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 the x 2 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 ele-4 (Section 6) specified in Definition 3.4.
Figure also suggests empirical evidence. Empirically, the flatness error, derived from the proof of Theorem D.5, is effeectiveyy 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 higher learning rate of 0.1 (e.g., Yariv et al. [7]. Note that, in practice, the eikonal loss is only applied to near surrfce using clamps (if the target position is obviously far from a surface, ignore it; it is wise since we 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 curved surface in 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 Definition 4.1 (Trilinear interpolation), where the cubic corners are represented by H E DxF Since we do not rely on a specific mapping function HH,, H ... H7), we may apply 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 R1, R2, R3. in Eqn. 3 in Chen et al. [13], among the VM factorization, which can be cquivalently 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) low-rank factorization (TensoRFF. 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 near- surface 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 cikonal 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 nested cube be inside of a unit cube, where its positions of the eight corners deviate -a; or +bj, such that a,, b, 2 0, from w, 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.
17

---

Proof. Let a left-aligned and zero-trailing binarization function be B(-) {0, and = partial derivative of the trilinear interpolation with the nested cube H with respect to H7 and Ho would be:
= 3. The
@ w: H)
20
D
(1 - B(1)) (w) -a,) + (w,
Σ
II
(18)
0H1
aj 1 bj bo(wD ap) + (aD(WD) bp)
== 1=1
2
1-1 Σ
an + bn
D-1 B(i),) - a2)) + +B(i), + bb,)
(19)
a, + bj
j=1
20 30-1-1 11-1 (1 B()))by - a3) + B(i), (ung + +b²)
wp
Σ
(20)
aj + bj
1
j=1
D
II w = 8p(w; H)
(21)
0H,
j=1
and
0olw: H
2- D
(1 B(i),)by ww, + +a,) + +B(i),a,() w b,)
(22)
dHo
aj bj
i=0 5=1
20 1-1D-11
1- Bb w. + a,) B(i),a, w; - b,)
(1 - WD)
Σ
a, + bb
(23)
D = II w,)
do(w w: H)
(24)
HHo
respectively. In a similar way, we can describe the other cases H, to H3 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. I with the weights on the diagonal line t = (1,1, 1)T where t E [0, 1] is cubic Bézier curve.
Proof. We rewrite the trilinear interpolation from Definition 4.1 with the weight t as follows:
2
H1; =
Σ
w(i,t)
H.
(25)
1=0
2-11 3
Σ
II (1- BB))) H:
(26)
= j=1
(1 H) + (1 +)) + H2 + H4) + (1 (1 + H5 + Ho) +
H
(27)
(1 H + +3(1 t)2t( Hi H2 + H
H3 H5 + H
+ +3(1 t)t²
+ H
3
(28)
= (1 (1-t) + 3(1 t) t) +3(1 t)t P2 + P
(29)
Four points P. = Ho, P = Bézier curve.
H H H4 Ps
H Hs 3
and P3 = H, define the cubic
3
D
18

---

As t is changed from zero to one, the curve starts at Po going toward P1 and turns in the direction of P2 before arriving at Pz. Usually, the curve does not pass P1 nor P2; but these give the directional guide for the curve.
Lemma D.3 Hypeersuraaces in a trilinear space), In piecewise trilinear region where a lies, hypersurfaces of the piecewise trilinear networks are defined as follows,following the notation of Definition 4.1:
2
0 0=i(x)
Σ w(i, w) i(a;)) t=U
(30)
where un E 1) the relative position a grid cell and (ax) and {Hi) are the corresponding corner positions and representations, respectively, in a trilinear space.
Proof. In a piecewise trilinear region, we can use the linearity of v as follows:
;(x) = (v ( =
w(i w) H.)
(31)
1=0
2D
Σ
w(i, w) -v (HHH)
(32)
2
Σ
w(i, w) -(x.)
(33)
which concludes the proof.
Lemma D.4 (Curved edge of two hypersurfaces, restated). In piecewise trilinear region, let edge (Ig. be the intersection of two hypersurfaces Di(x)) = 8/ (x) = 0, while By and æ7 are on the two hypersurfaces. Then, the edge is defined as:
(x : (<)) (1-)))111))))))) tx77)
where x E R and te [0,
(34)
Proof. By Definition 4.3,
is (wo) = 7. (x7)
+> (T((O))) = (T((T)))
(35)
while i is linear in piecewise trilinear region. Since it similarly goes for Uj, the intersection line of two hyperplanes 24 and Dy is (1 (xx) tr(77) where E R. other words, the edge on the hypersurface 14 ((1 - (xx))))))))))))))))))))))))))) + (T+))) where E R. The edge is defined as follows:
{æ (xx) = (1 - (x + tT(r) where
E RR and t EER
(36)
which concludes the proof.
+ Notice that when (()) + T(x)2 + =())) +(()) = 0, at least one connecting edge can be identified. This occurs along the diagonal line t- where E [0, 1]. Along this line, r(x) represents a cubic Bézier curve, as shown in Proposition D.2 with the control points of Pa and P2 set to zeros, which is the line.
+
Theorem D.5 (Hypersurface and eikonal constraint, restated). A hypersurfuce T(x) intersects two points 7 (#o) = T(æ7)= 0 while T(E1....) & 0 for the remaining six points. These points form cube, with 2n and 27 positioned on the diagonal of the cube. The hypersurface satisfies the eikonal constraint (V(x())) forr all [0, 1]3. Then, the hypersurface of T r(x) 0 is plane.
19

---

Proof. By the definitions of 7 in Definition 4.3 and the eikonal constraint,
||Vr(=)) = ()π(Z))
ŌT(2))
(T(x))
2
+
=1
(37)
dr
Dy
dz
0 V
TT(x))
TT(x))
+ (x(x))
T
= 0
(38)
dr
dy
Oydr
Oz
dzdr
@ V
(T(xz)
+ dr(z)
0
(39)
Oy
Ox
Ordy
02
Ozdy
0| V|VVT=>)1B
2 (r(())
( (82((a))
(T(x))
(0er(x)
+
= 0
(40)
0z
0x
Ox0z
dy
Oydz
where
θr(x)
(1 -(x) ( (1) (T((2) - -(xo)) + = - -(xx1))
Oy
+ x (1 - 2) (T(xa) - T(x1)) + +T( - -((x)))
(41)
However, since T(0)) = ((07) = 0 while T T2x2) + 0 and
T (a:s) + 0, there is no solution 7 (
of (r(#) for all æ. Likewise, the same goes for dr(m) dz and (π()) dz
Additionally, the three partial
by
derivatives can be rewritten as follows:
8c + cB aC = a
(42)
where # 0, b 0, and * 0. By substituting B = -CC/c and A= -aCc, show that
-abC/c- abC/c = 0 + C 0, which makes == B = C = Therefore, we rewrite A, B, and
C as follows:
(r(x)
Oydz = (1 - x)( - T(x2) - r(24) + (xxs) + x + T(x,) - T(x3)) - r2xs))
(43)
Txx))
Ozdr =(1 - v)( - -((1)) r(x,) + (xx))) + y + T(x2) - r(x3) - T T(x6)
(44)
(T((x) =(1 (1- - 2)( - ) ( + T ((x)) + 2 + T - 7 (xs) - T r(z6))
(45)
Ordy
respectively. The following conditions would satisfy the eikonal constraint:
-T(2)) - T(x,) + T =
(46)
7 T(21) T(E3) r(xs)
(47)
T -T - 7(x1) + +r(s) =
(48)
+7(x2) T(x3) -(x6)
(49)
-T(1)) - 7(x2) + + =
(50)
+7 -(05) T
(51)
From Equation (48) and Equation (51),
T(x1)= = - (
(52)
From Equation (47) and Equation (50),
T(x22 = (x0ss
(53)
From Equation (47) and Equation (48),
T(x1)= = -T(xs)
(54)
From Equation (50), Equation (51), and Equation (54).
T + (1²2) + (xx4) =
(55)
7 (xxa) + (xx)) + ( = 0
(56)
20

---

Using Equation (52), Equation (53), Equation (54), Equation (55), and Equation (56),
T(2) = (z(1 y) - 2) - (1 (1- (1==))=)7(x))
-) + - z(1 -
-)))
+ ((1-x)(1-1)) -)= -yy( )
(57)
= (x a)) + (++aa)))+((((()((((((((aa+))))++ (2+
(58)
xT (x)) + (x2) 27 ( + (2x2) + T(x4))
(59)
.- + ( + (T =0
(60)
where a = 2ryz - zy yz z..
This is an equation of a plane where its normal vector is (T ((2)) r(z4)). Again, to satisfy the eikonal constraint,
7 7(x1) (202)2 + (
(61)
T (xx))² = + T(x6)²
(62)
which concludes the proof.
Corollary D.6
hypersurface and eikonal constraint, restated), Let 1o (x) and
(x)) be two affine transformations defined by W + b,, i E {0, 1}. A hypersurface uo (T = 0
passing two points vo (T(o))))) (T(T) O while vo vo(T(aI...)) & 0 sutisfies the eikonal [0, 1J3. Then, the hypersurface is plane.
constraint V (1 = for all
Proof. Similarly to Theorem D.5, we investigate the Jacobian of the eikonal constraint to be zero as follows:
δr Oy +007 ÔT Oz )² ( dv T((x)) + dv Ōr(x)) 1| (i) = δT dy δJ Oydr
(V (ann 0r(z))² 2 Oon or(z))² 2 Ou ŌT(x²) dr (0r dy 0-(z))
(63)
dun 8 TT(x))
= (64)
dx
δr Oz OJ Ozdx
Since 0v1/0 is a constant, using Lemma D.3 and replacing and T with vs vo- and vo respectively, give us the same constaints for vo (T(1....)) regardless of v1.
Theorem D.7 (Intersection of two hypersurfaces and a diagonal plane, restated). Let 80 (x)) = 0 and
5 (x) 0 be two hypersurfaces passing two points Do and wy such that Do (o(2))) 11 (20) 0 and
(o(x7)= = Pi (Φ7) = 0, P. = (0(x))) and Q; = Di (xx))
P Po; P1; P4; Po].
P3
P2i
Psi
Pai
Pr].
(65)
and
(1 x)² x(1 - x) (1 - x)x x²
X
(66)
Then, x E [0, 1] of the intersection point of the two hypersurfaces and a diagonal plane of = is the solution of the following quartic equation:
XPPPQq] X=
(67)
while
XTP
y XT(Pa - -P))
(Pa *
(68)
Proof. We rearrange (x) using x as follows:
v0(x)) = -1 - y) X Po: Pf: Pa; Po] +y XT [p:
P3; Po:
Pr.
(69)
= =(1- x) F + y 0
(70)
XTP,
+ = XT(Pa -p))
(Pa + Pa).
(71)
21

---

In similar way, we rewrite (x) as follows:
(P. - P3) = X x0 P .'Qua
(72)
= - X
(73)
= -
(74)
Notice that this is quartic equation of To find the coefficients of the general form of quadratic equation, we consider a mapping C from the interpolation coefficients т; to polynomial coefficients c;:
co
TO
0
TO
C
TI
2
1
1
0
71
(75)
02
-1
72
73
73
where To(1 - xr)² + TT0(1 x) + 72(1 - x)x +
T3x = co +
c1 + c222. Then, the coefficients of
the quartic equation of Equation (74) are as follows:
Coo
C02
C10
C11
C12
= C P3Q%)
(76)
C20
C21
C22
where the anti anti-diagonal sum of the result is the quartic coefficients, i.e.
Coo + (C10 f + (C2o + cn1 + Cuz mr)) (021 + (12)) c222
(77)
The solutions of the quartic equation such that [0, 1] are the valid intersection points, and choose one of them a 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 æ, and = T.
We reproduce the method 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
p(x) co
C1
Ca Ca-1x" n-1 + x
(78)
is defined as:
T
Cp
C1
C(p)
0 ...
C2
(79)
0
-n
The roots the characteristic polynomial p(x:) are the eigenvalues of (p)).
"The nonzero coefficient of the highest degree is one. If you are dealing with the equation, you can make it by dividing the coefficient of the highest degree for both sides.
22

---

E Complexity analysis
We provide detailed exposition of the complexity analysis in Section 3.2 ensure the comprehen- siveness 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 &, respectively. Therefore, our initial focus on the complexity associated with these inputs. Let [V/, [&), and | (E C &) be the number of vertices, edges, and splitting edges consisting of polygons (Section 3.2), at the iteration of and j.
1. During the initialization step (refer to Section 5), we determine [V] and |E| to construct grid in RDD In the case of having M HashGrid marks, the quantities are defined as MD and D(M 1) 1-M respectively. In case of the large M, we consider vertex and edge pruning for efficient computations inspired by pruning strategy proposed in Berzins [4]. If, for every future neuron, denoted as ((i,j) > <bi", , 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 [ to avoid an excessively dense grid, with a complexity of O(V)).
2. Evaluate i(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).
3. Identify splitting edges by comparing two signs per edge, with a complexity of O(EE).
4. Obtain P and Q for Theorem 4.7 with complexity of O(E)) in similar way.
5. Inference for Theorem 4.7 involves matrix-vector multiplications and the eigenvalue decom- position of companion matrix for each edge, with complexity of O(E)).. Considering the maximum size of for quartic equation, this operation is regarded as linear.
6. Find intersecting polygons by pairing two sign-vectors with two common zeros in shared region of the new vertices resulting from splitting, with complexity of O(Ê)).
In the study by Berzins [4], linear relationship between |V and |E| for 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 [ < /) using Theorem 5 in [20]. Consequently, all the steps outlined in Algorithm 2 demonstrate linear dependency the number of vertices, denoted as ((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.
23

---

Supplementary experiments
F.1
Complete results
Table 2 and Table are extended from the Stanford 3D Scanning dataset in Table 1. Table extended from the Stanford bunny of Table 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 We found that 256 samples were not enough for those. Table 5 shows the angular distance results. Tables to 5 use the Small model, while Table 8 uses the Large model (ref. Section 6). We conducted all experiments with NVIDIA V100 32GB.
Table 2: Chamfer distance (CD) and chamfer efficiency (CE) for the Part A of the Stanford 3D) Scanning repository [31] using the Small model the default setting (ref. Section 6). The chamfer distance (x 1e-6) is evaluated using the marching cubes (MC) with 256 grid samples as the reference ground truth for all measurements. The chamfer efficiency is defined as 100/(v x CD), providing concise representation of the trade-off between the number of vertices and the chamfer distance. The results are averaged over three random seeds.
Bunny
Dragon
Happy Buddha
MC #
[
+
CD 4
CE f
|
4
CD 4
CE
[V] +
CD |
CE
32
1283
4106
19.0
1416
6330
11.2
1032
6951
13.9
64
5367
1371
13.6
6090
1936
8.5
4362
2465
9.3
128
21825
393
11.6
25236
542
7.3
18219
722
7.6
192
49569
141
14.3
57341
203
8.6
41351
276
8.8
Ours
4341
900
25.6
5104
1243
15.8
3710
1738
15.5
Table 3: Chamfer distance (CD) and chamfer efficiency (CE) for the Part B of the Stanford 3D Scanning dataset [31].
Armadillo
Drill Bit
Lucy
MC #
| 4
CD
CE f
|
CD L
CE 1
| J
CD |
CE
32
1438
6212
11.2
144
91959
7.5
594
6753
24.9
64
6151
1960
8.3
873
5949
19.2
2701
2310
16.0
128
25164
605
6.6
3718
1542
17.4
11255
830
10.7
192
57306
231
7.5
8382
782
15.3
25589
335
Ours
4783
1499
13.9
694
3013
47.8
2406
1763
23.6
Table Chamfer distance (CD) and chamfer efficiency (CE) for the Stanford bunny are evaluated using the marching cubes with 2563 grid samples (+) as the reference ground truth. The standard deviations are provided after the notation, based on results obtained from three random seeds. which are used in training. Similar outcomes are observed for the remaining cases.
Method
Sample
Vertices |
CD (x1e6))) |
CE f
Time 4
32
1283 + 89
4106 + 202
19.0
0.00 + 0.00
48
2967 + 177
2178 + 94
15.55
0.01
0.06
56
4057
245
1539 =
16.0
0.01 0.00
Marching Cubes
64
5367 + 342
1371 + 93
13.6
0.02 0.00
128
21825 + 1336
393 = 19
11.6
0.10 +
192
49569 + 3043
141
14.3
0.44 0.10
224
67601
4146
114
12.9
0.60 0.03
Marching Cubes"
256
88492 + 5404
0.88 0.05
Ours
4341
25.6
0.97 0.21
24

---

Table 5: Angular distance (AD) for the Stanford bunny. Similarly to the chamfer distance, the angular distance (*) is also evaluated using the marching cubes with 256 grid samples (*) the reference ground truth for all measurements. The standard deviations are provided after the + notation, based on results obtained from three random seeds, which is used in training.
Method
Sample
Vertices
AD (*)
32
1283 + 89
6.97 = 0.45
48
2967 = 177
5.07 0.15
56
4057 = 245
4,23 = 0.25
Marching Cuhes
64
5367 + 342
4.10 0.17
128
21825 + 1336
2.47 -
192
49569 + 3043
1.63
67601 + 4146
1.53
Marching Cubes*
256
88492 1 253
Ours
4490 + 91 3.57 = 0.38
Table 6: Chamfer distance (CD) and chamfer efficiency (CE) for the Part A the Stanford 3D Scanning repository [31] using the Large model having 4x-resolution of HashGrid compared to the default setting (ref. Section 6). The chamfer distance (x le-6) is evaluated using the marching cubes (MC) with 5 5123 grid samples the reference ground truth for all measurements. The results are averaged over three random seeds.
Bunny
Dragon
Happy Buddha
MC #
|/ 4
CD 4
CE f
[ J
CD $
CE f
[V]
CD $
CE f
38395
4.12
25986
759
5.07
18621
1110
4.84
192
87245
299
3.83
58855
335
5.08
42793
533
4.38
224
119000
221
3.80
80470
268
4.64
58357
395
4.34
256
155484
182
3.53
105235
226
4.21
76183
322
4.07
Ours
135691
151
4.87
91147
157
6.99
67562
229
6.46
Table 7: Chamfer distance (CD) and chamfer efficiency (CE) for the Part B of the Stanford 3D Scanning dataset [31] using the Large model as the default setting (ref. Section 6).
Armadillo
Drill Bit
Lucy
MC
[V] 4
CD
CE f
[ 4
CD 4
CE 1
V 4
CD
CE
128
25309
887
4.45
3590
1342
20.8
12224
1304
6.28
192
57751
399
4.34
8092
625
19.8
27863
687
5.23
224
78703
315
4.04
11000
588
15.5
38337
524
4.97
256
103085
267
3.63
14339
496
14.1
49838
435
4.61
Ours
90567
186
5.94
10275
342
28.5
43631
358
6.40
Table 8: Chamfer distance (CD) and chamfer efficiency (CE) for the Stanford dragon using the Large model having 4x- rx-ouuoltt of HashGrid compared to the default setting (ref. Section 6) are evaluated using the marching cubes with 5123 grid samples (x) as the reference ground truth.
Method
Sample
Vertices 4
CD (x lc-6)
CE 1
Time 1
128
25986 1
121
759 1
64
5,07
0.29
0.01
Marching Cubes
192
58855 1
181
335
21
5,08
0.60
0.06
256
105235 1
4.21
1.16 0.04
Marching Cubes"
512
424311 + 1262
9.19 0.39
Ours
91147
203
6.99
14.9 1.66
25

---

1F-03
MC+QEM OURS+QEM
CHAMPER DISTANCE
0.86E-94aF-04
47E
1.84E:04
1.52E-04
1,21E-04 X
1F-04
0K
30K
60K
90K
120K
150K
180K
NUMBER OF VERTICES
Figure 8: Chamfer distance for the Bunny with the Large model applying QEM (a mesh simplifica- tion). Both benefit from QEM while method efficiently retains better chamfer distances.
F.2 Mesh simplification using QEM
In Figure 8, Quadric Error Metrics (QEM) [35] is a popular mesh simplification algorithm, could potentially improve both MC and our method for efficiency. Overall, as shown, the mesh simplification favors our method. Notice that QEM entails higher computational cost, least On log n) where is the number of vertices. Additionally, these methods can cause shape deformation or holes, sensitive hyperparameters. Therefore, QEM may be applied with caution for further mesh simplification, acknowledging the potential for unpredictable side effects. We used MeshLab's 's Quadric Edge Collapse Decimation with the default settings to halve the number of vertices for each step.
F.3 Comparison with alternate approaches
Marching Cubes (MC) remains the de facto sampling-basse based method. 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:
1. Analytical approaches for CPWA functions are limited by their inability to effectively manage spectral bias and their exponential computational cost, which struggles with expo- nentially growing linear regions (infeasible to run our benchmarks.) For example, the time complexity of Analytic Marching [2] is ((nn/)/((-)))))))
n L.?), which grows exponen- the networks, where [V/] represents the
tially with the number of layers L and the width number of vertices per face (Section 4.1 from Lei and Jia [2]). In contrast, our method has linear time complexity with respect to the number of vertices, as discussed Appendix E, allowing it to avoid visiting every linear region exponentially growing.
2. We also exclude optimization-basse methods 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 and qualitative Figure 6 analyses on Marching Tetra- hedra (MT) [9, 10] and Neural Dual Contour (NDC) [32], which serve as alternative representative approaches. MT is 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 better than MC in the same resolution, producing more vertices, though.) NDC faces generalization issue for 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.
26

---

F.4 Error of the learned SDF
In Equation (11) and Figure we confirmed our hypothesis that the eikonal loss induces piecewise linearity in trilinear networks. To further support this, we also present the emror 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-g soodod- truth. For this reason, one might wish to verify whether the predicted SDFs converge toward near-7ero.
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, Sine-graines trilinear regions from the dense grid marks (from the more dense mulli-resoouution grids) allow relatively small errors of 97 le-9 without the eikonal loss, although the eikonal loss of le-2 further decreases to 31 x 1e-9. Note that the stronger eikonal loss of le-1 proved ineffective due to over-regularization.
Table 9: Error the learned SDF Weight stands for the weight of the eikonal loss and SE stands for the squared error of predicted SDFs.
Method
Weight
SE (1e-9) |
MC64
le-2
2308
MC256
le-2
le-4
91
Ours
le-3
68
le-2
31
F.5
Effect of model sizes
In Table 10, explore the ablation study 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.
Table 10: Ablation study varying the number of layers and the width of networks for the Stanford bunny Small models. And, Note indicates the architecture of the decoding networks.
Layers
Width
Vertices
CD (1c-6)
Time (sec)
Note
4
4565
738
0.11
4863
0.13
16
4566
734
0.12
32
4577
754
0.22
64
4611
739
0.50
InstantNGP [12]
4
4547
666
0.12
8
4661
750
0.19
16
4628
724
0.65
Ours
32
4567
754
0.57
64
5588
759
2.86
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.
27

---

G
Visualizations
For Figure in the main paper (closer looks at bunny's nose), we want leave 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-groung truth), while MC64 fails 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 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 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, demon- strating variations the number 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 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 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.
28

---

0
No EDGE (1)
No EDGE (1)
No EDGE (6)
No EDGE (6)
XY, Y7-MLANR (2)
XY, YZ-PLANE (2)
YZ. XZ-PLANE (2)
Y7, X7-PLANT (2)
XY. XZ-PLANK (2)
XY, XZ-PLANK (2)
No FDOR (12)
No RDGR (6)
XY, YZ, XZ. (6)
XY, Y7, X7 (6)
XY, YZ. (Z)
No EDGE (2)
Figure 9: We depict the 64 scenarios the edge within a trilinear region, excluding isomorph cases specifying the count within parentheses. No edge indicates where diagonal zero corners not connected. The labels XY, YZ, and XZ refer to x = y. = 2, and 2 x diagonal planes, respectively, form the shared edge, connecting the two diagonal zero corners, with the hypersurface the schematic figure. nearly all cases, diagonal plane intersects with the hypersurface, except, most, four cases, depending the curvature determined by corner values.
29

---

Figure 10: The normal map is analytically generated from piecewise trilinear networks, comprising both HashGrid [12] and ReLU neural networks, that have been trained learn an SDF with the eikonal loss for the Stanford bunny [31]. Our approach dynamically assigns vertices based on the learned decision boundary. One notable observation that the hash grids influence the selection of vertices. Additionally, each grid can be subdivided into multiple polyhedra to accurately represent smooth curves. The Small (left) and Large (right) models (ref. Section 6) are used to learn the SDF.
MC 296 (90K Vinamness)
MC 125 [23 VIRTICES)
MC
MC 32 (LSK VERTICES)
MC 16 (308 Vismnces)
Ouns
(5.4K VERTICES)
(4.5 VERmncss)
MCC 256 [100K VEXTICES)
AIC 128 (25 VERTICES)
MC 66.1 Vieries)
MIC 32 (1.4 Veanoes)
MC 16 (209 Vasmicss)
Ocns (5.1K Visrnoss)
Figure 11: Visualizing normal maps for marching cubes varying the number of samplings, and ours, with the number of vertices denoted in parentheses. Low sampling has a risk omitting sharp object details, while dense sampling results in redundant vertices and faces. As our mesh is analytically extracted from the decision boundary apex points of networks, it represents an optimal choice in balancing detail preservation and efficiency.
30

---

(5:0 Vornun))
(6.K Vancace))
Figure 12: a face-to-face comparison, we examine marching cubes with 256 and 64 samplings, denoted as MC 256 and MC 64, respectively, and our method. Notably, MC 64 struggles accurately represent pointy areas of meshes due to its coarse sampling. In contrast, our approach has an advantage in this aspect, of networks. However, it is worth noting that our adaptive sampling introduces a disadvantage where some vertices in close proximity; specifically, proximity to others. We believe that optimizing the mesh can significantly improve the parsimony of the vertex count compared with marching cubes.
analytically extract the mesh from the decision boundary apex points
observed that 10.1% of vertices are within le-3
Figure 13: The extract mesh visualizations of the six from the Stanford 3D Scanning repository using the Large models. From left, Bunny, Dragon, Happy Buddha, Armadillo, Dril Bit, and Lucy.
31

---

NeurIPS Paper Checklist
1. Claims
Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
Answer: [Yes]
Justification: Our main claims made in the abstract and introduction are accurately reflected in theoretical and experimental expositions. Theorem 4.5 and Figure 5 provide the theoretical and experimental supports of the planarity constraint of the eikonal equation, Theorem 4.7 provides a plausible approximation for the curved edge subdivision, and Table shows the consistent chamfer efficiency across multiple meshes from the Stanford 3D Scanning repository, while Figure shows that the higher resolution of HashGrid is favorable to the planarity constraint, achieving lower chamfer distances.
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Reviews:
[{"review_id": "gSbI6AEDlR", "reviewer": "Reviewer_MSAr", "summary": "The 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.", "questions": "* 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?", "limitations": "The 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.", "rating": 5, "confidence": 2, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "* 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.", "weaknesses": "Here 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", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your detailed review and for pointing out the effective rebuttal points. We appreciate your time and effort in going through our work multiple times. \n\n**W1. Simplification of Language**\n\nWe are committed to making our paper more accessible. Based on your feedback, we will revisit the language and structure to ensure that the key ideas are presented as clearly as possible. Specifically, we will: 1) Simplify complex sentences and terminology where possible. 2) Add a brief introductory section that explains Polyhedral Complex Extraction in simpler terms before delving into the technical details. Please refer to Fig. 1, and Alg. 1 & 2 in the Appendix for visual and textual aids illustrating core concepts. We believe our new **Figure B & C (rebuttal PDF)** would also help readers.\n\n**W2. Pitfall to Over-Complex**\n\nThere is a *risk* that MC64 appears more similar to MC256. This is because we used small networks in this visualization to highlight the limits of MC with a limited number of vertices and edges in GT (simulating flat or high curvature/edged regions). Here, MC64 tends to oversample the vertices. When examining the nose of the rabbit in MC256, the same bright yellow color can be observed in areas where our method extracts flat facets. In contrast, MC64 oversamples around the edges, creating an over-smooth surface where it should not. For instance, when a cube is tilted at a 45-degree angle, its vertices may not be sampled as they lie within the sampling grids, leading to the creation of a multi-facet polytope. This is an innate weakness in sampling-based methods. Additionally, inaccuracies in estimating normal vectors from it will lead to incorrect light reflection outcomes. **Please refer to Figure A (rebuttal PDF)**, which shows detailed extracted vertices and edges and **Fig. F** for the normal errors in the Large models, for a visual explanation.\n\n**W3. Computational Complexity**\n\nAs mentioned in Section 4.2 and detailed in Appendix B, the time complexity of our method is linear with respect to the number of vertices, making it optimal for extraction. After submission, by utilizing PyTorch's `masked_scatter_` function to identify intersecting edges and others, we significantly reduced the processing time from 53.5 seconds to approximately 3 seconds for the Dragon (Large) model, which is in between MC and MT (Marching Tetrahedra), much closer to MC. We have provided both a theoretical analysis of the time complexity (Appendix B) and practical benchmarks using the Large models to validate our findings. *The code will be publicly released upon acceptance.*\n\nMoreover, we emphasize HashGrid allows us to use small decoding networks with hash tables of large parameters.\nNetworks using HashGrid (Muller et al., 2022) typically have **a small hidden size of 64 and only one hidden layer**, even in implicit neural surface learning, NeuS2 (Wang et al., 2023). This is because HashGrid contains a large number of learnable parameters in its hash tables, specifically (feature size 2) x (# of entries 2^19 = 524K) x (multi-resolution levels 4). In our experiments, we used a reasonable network capacity. For clarification, the input size of 3 is mapped to (levels 4 x features 2 = 8) by HashGrid, and then the decoding network maps 8 -> 16 -> 16 -> 1 with 3 mapping layers (as described in Sec. 6).\n\n**W4. Additional comparisons**\n\nUnfortunately, Analytic Marching is not feasible for our benchmarks due to its high time complexity. Specifically, its time complexity is $\\mathcal{O}((n/2)^{2(L-1)} |\\mathcal{V_P}| n^4 L^2)$, which grows exponentially with the number of layers L and the width n of the networks, where $|\\mathcal{V_P}|$ represents the number of vertices per face (Section 4.1 from Lei and Jia, 2020). In contrast, our method has a linear time complexity with respect to the number of vertices, as discussed in Section 3.2, allowing it to avoid visiting every linear region exponentially growing. Instead, we would like to direct you to **our global response G.2**, which includes supplementary experiments comparing our method with MT and NDC.\n\n**Q1. Why aren’t they (DeepSDF) limited to always produce flat plane reconstructions?**\n\nAs you might know, HashGrid (Muller et al., 2022) includes learnable parameters, the hash tables. The eikonal loss causes the hash entries to form a flat plane within trilinear regions. We denote the entries as $\\mathrm{H}$, which are learning parameters, in Definition 4.1 in Section 4.1. For this reason, the constant latent codes cannot be applied with Corollary 4.6. However, since a standard ReLU network (with a constant latent code) is piecewise linear, it may produce flat planes in linear regions. Nonetheless, we want to emphasize the spectral bias (Tancik et al., 2020) discussed in the Introduction, which arises without appropriate positional encodings, and it serves as our motivation. As a result, DeepSDF (Park et al., 2019) may be limited to simpler shapes and suffer from this spectral bias.\n\n**Q2. Quantitative metric to validate Corollary 4.6**\n\nWe appreciate the proposed metric; however, we must point out that our existing metrics already measure the suggested aspect. Firstly, using the Chamfer distance, we sample points on the surface of the extracted mesh and calculate the distance to the closest point (which is defined as the SDF value) on the ground truth mesh. This mesh is derived from the over-sampled MC, reflecting the learned surfaces of the networks. We believe this is essentially the same metric (under sufficient samplings and well-fitted SDF networks) as the one you suggested.\n\nFurthermore, we want to emphasize Equation 13 and Figure 3, which illustrate how flatness changes with respect to the weight of the eikonal loss using the flat constraints identified in the proof of Theorem D.5. We have confirmed that the eikonal loss is an effective method to ensure flatness and evaluate the extracted mesh using the Chamfer distance.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "9R94xoLmi7", "reviewer": "Reviewer_1zQb", "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.", "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", "rating": 5, "confidence": 1, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "- Thorough theoretical analysis\n- 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\n- No comparisons to any alternate approaches over marching cubes are provided -- e.g. marching cubes + QEM decimation etc.\n- No rigorous qualititative comparisons are provided, which could be important since chamfer does not always capture the shape quality", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your thorough review and constructive feedback. We explain our rationale for the evaluation and additional note for the angular distance we used in quantitative evaluations.\n\n**W1. While there are some improvements over marching cubes, the improvements are not quite substantial, and come at the cost of a higher runtimes**\n\n- Our method is a white-box approach that identifies apex vertices with a reasonable eikonal constraint, showing the improved Chamfer efficiency (Please refer to **our global response G.1** on this matter) across multiple benchmarks for six objects and both small and large models. After submission, by utilizing PyTorch's `masked_scatter_`function to identify intersecting edges and others, we significantly reduced the processing time from 53.5 seconds to approximately 3 seconds for the Dragon (Large) model, achieving similar time spent comparable to MC. This result highlights the potential for further improvements in efficient implementation. We have also provided a theoretical analysis of the time complexity (see Appendix B), which suggests that our approach is theoretically optimal. The code will be publicly released upon acceptance.\n\n\n**W2. No comparisons to any alternate approaches over marching cubes are provided -- e.g. marching cubes + QEM decimation etc.**\n\n- Thank you for your suggestions. Please refer to our **global responses G.2 and G.3** and the corresponding **new Figures A, D, E, and F in the rebuttal PDF**.  We provide supplementary experiments to compare with MT, NDC, and the QEM scenario.\n\n**W3. No rigorous qualitative comparisons are provided, which could be important since chamfer does not always capture the shape quality**\n\n- **Please refer to our Figures A & F in the rebuttal PDF, which show detailed extracted vertices and edges and normal errors for a visual explanation.** **Fig. A provides rigorous visualization with enlarged meshes and normal directions are visualized with plasma colors.**\nIn addition to the Chamfer distance, we present the angular distance in Table 5 in the Appendix. This measurement, derived from the extracted normal vector, a direction indicated by the normalized Jacobian of the SDF function, assesses how much the normal vectors deviate from the ground truth (detailed in Sec. 6). As anticipated, our method effectively estimates the faces using a relatively small number of vertices compared to the sampling-based approach. This trend is consistently observed across various objects. We believe this supplements the limited aspect of the chamfer distance.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "y9DHRHtCzH", "reviewer": "Reviewer_FmNY", "summary": "The 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.", "questions": "- 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?", "limitations": "The 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", "rating": 5, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 3, "strengths": "The 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).", "weaknesses": "Despite 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.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely appreciate your efforts and the detailed comments you provided. We are particularly pleased that the theoretical aspects of the work are considered a significant contribution.\n\n**At least briefly describing the former would make for a more rounded story.**\n\n- In Sec. 1, we introduce mesh extraction from Continuous Piecewise Affine (CPWA) functions, which are limited by spectral bias and typically use nonlinear positional encodings for this limit. Consequently, applying CPWA mesh extraction techniques becomes challenging. In Sec. 2, we discuss how many previous works face significant challenges due to the computational cost, with exponentially growing linear regions. Linear regions grow polynomially with the hidden size and exponentially with the number of layers (Zhang et al., 2018; Serra et al., 2018). These issues were our key motivations. \n- The elaboration on your \"handling volumetric complex\" would help provide specific feedback on that matter. \n- Also, could you refer to **our global response G.1** for \"Why does the chamfer efficiency matter?\"\n\n**This argument would be more convincing given a comparison to marching cubes with some basic mesh post-processing.**\n\n- We provided supplementary experiments to compare with MT, NDC, and using a mesh simplification, QEM (Quadric Error Metrics; Garland & Heckbert, 1997). Please refer to our global responses **G.2** and **G.3** and the corresponding new **Figures A, D, E, and F in the rebuttal PDF**. We present qualitative and quantitative results on them.\n\n**The work would also benefit from comparing the same procedure to piecewise linear networks...**\n\n- We are afraid you misunderstand. Our motivation stems from the issue of spectral bias (Tancik et al., 2020) in implicit neural surface learning methods, which necessitates the use of positional encoding, such as HashGrid (Muller et al., 2022). Therefore, exploring mesh extraction from these types of networks is a key focus of our work. Comparing our method to piecewise linear networks (first, it is hard to learn surface naively due to the spectral bias.) could be distracting and misguided, as our goal is not to argue that HashGrid improves quality in addressing spectral bias -- this has already been well-discussed and established in previous works.\n\n**The neural network size remains fixed (3x16)?**\n\n- Networks using HashGrid (Muller et al., 2022) typically have **a small hidden size of 64 and only one hidden layer**, even in implicit neural surface learning, NeuS2 (Wang et al., 2023). This is because HashGrid contains a large number of learnable parameters in its hash tables, specifically (feature size 2) x (# of entries 2^19 = 524K) x (multi-resolution levels 4). In our experiments, we used a reasonable network capacity. For clarification, the input size of 3 is mapped to (levels 4 x features 2 = 8) by HashGrid, and then the decoding network maps 8 -> 16 -> 16 -> 1 with 3 mapping layers (as described in Sec. 6).\n\n**Q1. In Theorem 4.5, is it true that \\tau is a trilinear function?**\n\n- Yes. We defined $\\tau$ as a trilinear function in Definition 4.3 in Sec. 4.1 (the previous section). In Definition 4.1, $\\psi$ represents trilinear interpolation. In Definition 4.3, we describe $\\tau$ as \"using the trilinear interpolation of spatially nearby learnable vectors on a three-dimensional grid.\" relating to $\\psi$ for instantiation. We will revise to have a formal definition of $\\tau$, enhancing clarity.\n\n**Q2. Could you clarify Theorem 4.5? My best understanding is that it states that the only trilinear function that satisfies the eikonal equation is a plane...**\n\n- Theorem 4.5 states that a trilinear function (within a piecewise trilinear region), which satisfies the eikonal equation, represents a plane. We do not argue that **only** trilinear function which satisfies the eikonal equation is a plane. We choose the trilinear networks using HashGrid since 1) it effectively handles the spectral bias, 2) it is one of the widely-used methods in NeRFs, and 3) it has a planarity opportunity using the eikonal constraint. Notice that we provided detailed proof in Theorem D.5. \n\n**Q3. In Line 221, do you perhaps mean $\\|\\tilde{v}(\\mathrm{x})\\| \\le \\epsilon$?**\n\n- Yes. Thank you for pointing out the typo.\n\n**Q4. 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\n- To eliminate Bayes error from underfitting. This work is solely focused on the extraction itself, not on surface learning. Notice that we used $512^3$ for large models, and confirm that it were sufficiently high resolution for meaningful evaluation.\n\n**Q5. Can you comment on the extraction of the volumetric mesh?**\n\n- The question seems unclear to us; but we would like to explain that before the skeletonization, the initial volumes (a cube of interest) are partitioned by grid marks (grid planes) and folded hypersurfaces. In tropical geometry, these are known as tropical hypersurfaces (Itenberg et al., 2009), which are sets of points where the function is not linear (or zero). Before the skeletonization step, we identify all non-linear boundaries and select the zero-set vertices and edges for our efficient extraction. **Please refer to Fig. C in the rebuttal PDF** for a visual aid.\n\n**Limitation**\n\n- Thank you for pointing that out. We want to draw your attention to Appendix B.1, which discusses \"Satisfying the eikonal equation and approximation.\" In this section, we examine the effectiveness of HashGrid in satisfying the constraint through the piecewise allocation of hash table entries and their locality. We will integrate this discussion into the Limitations section.\n\n\n**Errors and suggestions**\n\n- We sincerely appreciate your effort in providing a detailed list of errors and suggestions. We will incorporate your feedback to enhance the clarity of our work. Note that a discussion of HashGrid ( \\[12\\]) has been addressed above (model size).", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "skDXVahnho", "reviewer": "Reviewer_1oW4", "summary": "The 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.", "questions": "(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.", "limitations": "Yes, section 7 describes limitations.", "rating": 5, "confidence": 3, "soundness": 2, "presentation": 1, "contribution": 2, "strengths": "(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.", "weaknesses": "(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.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your thorough review and constructive feedback. We appreciate your recognition of the theoretical soundness of our framework. We would like to address your concerns regarding the presentation of our paper.\n\n**W1. I do not like the current presentation of the paper.**\n\n- We understand your concern about the paper's focus on lower abstraction details from the beginning. Our intent was to provide a solid theoretical foundation before delving into the empirical results, as we believe this balance is crucial for a comprehensive understanding. We are not simplpy \"moving the actual results to an Appendix.\" The main results are demonstrated in Table 1 and Figure 2, showcasing the improved Chamfer efficiency across multiple benchmarks for six objects and both small and large models. Tables 2 to 8 in the Appendix provide detailed numbers and are organized separately for readers who may wish to examine the specifics.\n\n**W2. This paper needs graphical intuitions to assist unexperienced readers.**\n\n- Our new **Figure B in the rebuttal PDF** could enhance clarity as you requested. It visualizes the regions of a tropical polynomial for inexperienced readers.\n\n**W3. Another example is the subdivision scheme.**\n\n- Although we have illustrated our curved edge subdivision scheme in Figure 1 (2a-c), **Figure C in the rebuttal PDF** shows a schematic of the subdivision progress, as you suggested. We hope this feedback may resolve your concerns.\n\n**W4. I also think a better overview image is needed... For example, $\\psi$ and $\\mathrm{H}$ are defined in Definition 4.1, and I could not find the motive for them in the main text.**\n\n- In Definition 4.1, $\\psi$ represents trilinear interpolation, and $\\mathrm{H}$ refers to the target features to be interpolated. In Definition 4.3, we describe $\\tau$ as \"using the trilinear interpolation of spatially nearby learnable vectors on a three-dimensional grid.\" HashGrid (Muller et al., 2022) assigns features from hash table entries to $\\mathrm{H}$ through hashing, with weights based on their relative positions in the grid. Thus, when $\\tau$ is instantiated with HashGrid, $\\psi$ in Equation 5 corresponds to $\\tau$, where $\\mathrm{w}$ represents a position in a normalized unit grid and $\\mathrm{H}$ consists of the hashed features. \n- Although HashGrid, which won the best paper award at SIGGRAPH'22, is well-known, we may have assumed readers would be familiar with its formal definition (our bad.) \n- Based on your suggestion, we will consider enhancing our manuscript with textual and graphical explanations. Meanwhile, we recommend you refer to Figure 3 in Muller et al. (2022) for a visual explanation of their use of trilinear interpolation.\n\n**W5. Discretization problems, probably because of the grid-based approach and interpolation.**\n- For mesh extraction, we would like to note that sampling-based methods, e.g., MC, severely suffer from discretization, especially in low-resolution regimes. For implicit surface learning, while sinusoidal positional embedding is also known for attenuating the spectral bias, it suffers slow training/rendering time, necessitating heavy density/SDF networks compared to HashGrid. For ours, one of the possible solutions is carefully adjusting multi-resolution levels and resolution growing factors to spread multi-resolution grids more evenly.\n\n**W6. 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\n- We are happy to supplement on this matter. We provided experiments to compare with MT, NDC, and using a mesh simplification, QEM (Quadric Error Metrics; Garland & Heckbert, 1997). For details, please refer to our global responses **G.2** and **G.3** and the corresponding new **Figures A, D, E, and F in the rebuttal PDF**. We present both qualitative and quantitative results to highlight the advantages of our method. \n- The \"several improvements over the years\" appear to be tailored to each method's specific niche, each with its own pros and cons (refer to G.2). In our benchmarks, MC, utilizing our modernized library PyMCubes, outperforms others in terms of efficiency, establishing it as the de facto method.\n\n**Q1. Typo in line 426**\n\n- Actually, this is not typo; but it embraces ambiguity. We want to say the *classical* polynomial $x^3 + 2xy + y^4$ would be rewritten for a *tropical* polynomial by naively replacing + with $\\oplus$ and $\\cdot$ with $\\odot$, respectively: $x^3 + 2xy + y^4$ -> $x \\odot x \\odot x \\oplus 2 \\odot x \\odot y \\oplus y \\odot y \\odot y \\odot y$. After this, we rewrite by the definitions of tropical operations as $\\max(3x, x + y + 2, 4y)$. This example (ours with max notation) comes from the Wikipedia (https://en.wikipedia.org/wiki/Tropical_geometry) and we believe this is a convention how they refer the *classical*/*tropical* polynomials. But, we would like to clarify that in revision.\n\n**Q2. Notation in Proposition 3.3**\n\n- Yes, the component j is from the resulting vector $v_j^{(i)}(x)$. We stated \"the subscript j denotes the j-th element (neuron index; the j-th element in the intermediate output), assuming the hidden size of neural networks is H.\" Since we define j in \\[1, H\\] we thought readers would follow. Your suggestion is valuable and will be incorporated into the update.\n\n**Q3. Also in Proposition 3.3 I would add an intuition commentary afterwards saying that...**\n\nGreat point! We appreciate that.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:
Please refer to the **rebuttal PDF (below PDF button)** attached for **Figures A-F** mentioned in the author's feedback. This PDF is high-resolution; please enlarge the PDF if you needed. 

* **Fig. A.** Detailed mesh visualizations for Ours, MC, MT, and NDC.
* **Fig. B.** A visual explanation for a tropical polynomial and how a space is divided by it.
* **Fig. C.** A schematic diagram for 2D subdivision of our method.
* **Fig. D.** A plot of chamfer distances for Ours, MC, MT, and NDC.
* **Fig. E.** A plot of chamfer distances for Ours and MC applying QEM (gradual mesh simplification).
* **Fig. F.** Visualization of face normal errors on the surfaces.
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In summary, our work received the following positive reviews:

* **Motivation**:
  - "A very reasonable extension" (MSAr)
  - "Well motivated and holds a lot of promise," "first attempt" (FmNY)
* **Findings and Insights**: 
  - For Thm 4.5 (Coro 4.6), "seem surprising," "great insight" (MSAr)
  - "Theoretical aspect of the work is in my view the most clear contribution" (FmNY)
  - "The theoretical framework is interesting and it is a good application of tropical algebra" (1oW4)
* **Results**: Proved mesh efficiency (MSAr, 1zQb)
* **Presentation**: 
  - "Self-contained" (MSAr)
  - "Thorough theoretical analysis" (1zQb)
  - "Mathematical notation is good overall, with good choices for the symbols and their use" (1oW4)
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We recognize that, as an interdisciplinary work bridging algebraic geometry and neural 3D, we respect the raised issues and have carefully considered them, understanding the need for thorough analyses and clear explanations. We summarized our global feedback on the core raised issues:

**G1. Why does the chamfer efficiency matter?** (1zQb, FmNY)

- Although we found our method achieved the *lowest* chamfer distance in the Large models (see Tab. 6-7 in Appendix and **Fig. D**), we argued that the chamfer efficiency is *our highlight* since: 
  1) sampling-based methods suffer exponential memory/compute costs to find optimal, 
  2) for light-weight deploy of meshes (e.g., WebGL), 
  3) and to get accurate normals for light modeling (see angular distance in Tab. 5 and **Fig. F**).

**G2. Additional Comparisons to Alternate Approaches (MT, NDC)** (MSAr, 1zQb, FmNY, 1oW4)

- 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 the evaluation**:

    1. 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).
    2. 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 (Remelli et al., 2020), Deep Marching Tetrahedra (Shen et al., 2021), and FlexiCubes (Shen et al., 2023). Additionally, these methods result in prolonged computational costs.

- **MT & NDC results:** To be faithful to the reviewers' requests, we provide both **rigorous qualitative** (**Fig. A & F**) and **quantitative** (**Fig. D**) analyses on **Marching Tetrahedra (MT)** and **Neural Dual Contour (NDC, Chen et al., 2022)** in the rebuttal PDF. 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 (Shen et al., 2021) for MT and the official code of NDC. And PyMCubes for our *modernized* MC library.

**G3. Applying QEM** (1zQb, FmNY, 1oW4)

-  QEM is a popular mesh simplification algorithm, it could potentially improve both MC and our method for efficiency, which is validated in **Fig. E**. Overall, mesh simplification favors our method, as shown. Notice that QEM entails a higher computational cost, at least $\mathcal{O}(n \log n)$ (with $n$: the number of vertices). Additionally, these methods can cause shape deformation or holes, sensitive to hyperparameters. We used MeshLab's Quadric Edge Collapse Decimation with the default settings to halve the number of vertices for each step.

**G4. Better Presentation** (MSAr, 1oW4)

- As an interdisciplinary work, from tropical geometry to deep learning to implicit neural surface learning, we respond that the content needs visual aids to illustrate core concepts and provide clear explanations for inexperienced readers (1oW4). We are committed to making our paper more accessible. Specifically, we will: 1) simplify complex sentences and terminology where possible. 2) add a brief introductory section that explains Polyhedral Complex Extraction in simpler terms before delving into the technical details. For those who might miss it, please refer to Fig. 1, and Alg. 1 & 2 in the Appendix for visual and textual aids illustrating core concepts and procedures. Supplementarily, our new **Figure B & C (rebuttal PDF)**, B) a visual explanation for a tropical polynomial and how a space is divided by it, and C) a schematic diagram for 2D subdivision of our method would also help readers.

Meta-review: The paper proposes a new method for mesh extraction from piecewise trilinear networks, grounded in a theoretical grounding in trilinear interpolation and polyhedral complex derivation.  The method is shown to be more accurate and efficient in comparison to MC and other alternative approaches.

The paper received four reviews with unanimity in their (post discussion) ratings. The reviewers carefully engaged with the paper and the author response and participated in ensuing discussions. They found the paper to be well motivated, and a reasonable, theoretically grounded, extension of previous work on complex polyhedral complex extraction from ReLU networks to architectures with positional encodings implemented using trilinear interpolation. The reviewers particularly appreciated the theoretical grounding of the proposed approach, with interesting insights, and holding a lot of promise. 

The initial reviews were mixed around the borderline. The primary concerns are around (a) the organization of the paper “focusing on lower abstraction details”, lacking supporting figures and visualizations compromising readability, and making the paper difficult to understand; (b) for a mathematical paper, riddled with typos, imprecise statements, poor phrasing, and undefined terms (definition before usage); (b) with unconvincing experimental evaluation – both qualitatively and quantitatively; inadequate comparison with alternate approaches, and, (d) an inadequate limitations section. 

The authors addressed many of these concerns during the discussion resulting in two reviewers revising their ratings upward, resulting in a unanimous rating of ‘borderline accept’ from all reviewers. I recommend that the paper be accepted subject to the authors incorporating the clarifications and results from their rebuttal and responses, as well as other promised changes improving the presentation of the paper.
Decision: Accept (poster)