A Lennard-Jones Layer for Distribution Normalization
Authors: Mulun Na
Jonathan Klein
Biao Zhang
Wojtek Palubicki
Soren Pirk
Dominik Michels
Venue: Accepted by TMLR ()

Abstract: We introduce the Lennard-Jones layer (LJL) for the equalization of the density of 2D and 3D point clouds through systematically rearranging points without destroying their overall structure (distribution normalization). LJL simulates a dissipative process of repulsive and weakly attractive interactions between individual points by considering the nearest neighbor of each point at a given moment in time. This pushes the particles into a potential valley, reaching a well-defined stable configuration that approximates an equidistant sampling after the stabilization process. We apply LJLs to redistribute randomly generated point clouds into a randomized uniform distribution. Moreover, LJLs are embedded in the generation process of point cloud networks by adding them at later stages of the inference process. The improvements in 3D point cloud generation utilizing LJLs are evaluated qualitatively and quantitatively. Finally, we apply LJLs to improve the point distribution of a score-based 3D point cloud denoising network. In general, we demonstrate that LJLs are effective for distribution normalization which can be applied at negligible cost without retraining the given neural network.

Published in Transactions on Machine Learning Research (09/2024)
A Lennard-Jones Layer for Distribution Normalization
Mulun Na
mubun.nai@kautt..tuu.u
Computational Sciences Group KAUST
Jonathan Klein
Amaathnn.leenn@aaeee
Computational Sciences Group
KAUST
Biao Zhang
biao.zhang@@oout..com edu.
Visual Computing Center
KAUST
Wojciech Palubicki
wwjjeiehh.patchcckeemm...kk
Natural Phenomena Modeling Group
AMU
Sören Pirk
sp@informatikk.nii uni-kiel.
Visual Computing and Artificial Intelligence Group CAU
Dominik L. Michels
fommnnik.michesss@t.... exuu.
Computational Sciences Group
KAUST
Reviewed OpenReview: https: openrevieee net/ por/ffrmm? id= emGl 7
Abstract
We introduce the Lennard-Jones layer (LJL) for the equalization of the density of 2D and 3D point clouds through systematically rearranging points without destroying their overall structure (distribution normalization). TJJ. simulates dissipative process of repulsive and weakly attractive interactions between individual points by considering the nearest neighbor of each point at given moment in lime. This pushes the particles into polential valley, reaching a well-defined stable configuration that approximates the stabilization process. We apply LJLs to redistribute randomly generated point clouds into a randomized uniform distribution. Moreover, LJLs are embedded in the generation process of point cloud networks by adding them at later stages of the inference process. The improvements in 3D point cloud generation utilizing LJLs are evaluated qualitatively and quantitatively. Finally, apply LJISs to improve the point distribution of score-based 3D point cloud denoising network. In general, demonstrate that LJLs are elfective for distribution normalization which can be applied at negligible cost without retraining the given neural network.
equidistant sampling after
Upon request, we are happy to share the source code to generate the results presented in this paper. Please contact the first or the last. author of this manuscript.
Source Code:
Keywords: Distribution Normalization, Generative Modeling. Lennard-Jones Potential, Particle Simulation, Point Clouds.

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Published in Transactions on Machine Learning Research (09/2024)
X
X
X
X,
X., X_ /eopp (an)
K,
X
X
X
X
Figure 1: An overview of the integration of LJLs into the inference process of well-trained generative models. A well-trained model generates a meaningful point cloud from random noise in a sequential way (bottom row). LJLs are inserted after certain internediate-generatioo steps with damping step size Åt, and LJ potential parameters c and (top row). Embedding LJI.s in generative models can improve the generation results by normalizing the point distribution.
1 Introduction
Next triangle meshes, point clouds are one of the most commonly used representations of 3D shapes. They consist of an unordered set of 3D points without connections and are the native output format of 3D scanners. They also map well to the structure of neural networks, either in generative or processing (e.g. denoising) tasks. A major problem with point clouds is, that their entropy heavily depends on the distribution of the points. In areas with holes, not enough information about the shape is expressed, while clustered areas waste storage without adding additional details. We introduce a so-called Lennard-Joner layer (I.J..) for distribution normalization of point clouds. We use the term distribution normalization to describe the process of systematically rearranging the points' positions in order to equalize their density across the surface without changing the overall shape. More formally, distribution normalization describes the property of maintaining global structure while maximizing minimum distances between points. The Lennard- Jones (IJ) potential is widely considered an archetype model describing repulsive and weakly attractive interactions of aloms or molecules. It was originally introduced by John Lennard-Jones (Jones, 1924a;b; Lennard-Jones, 1931) who is nowadays recognized as one of the founding fathers of computational chemistry. Applications of LJ-based numerical simulations can be found in different scientific communities ranging from molecular modelling (Jorgensen et al., 1996) and soft-matter physics (Al-Racei & El-Daher, 2019) to computational biology (Hart & Istrail, 1997). For a given point cloud, we aim for transformation from the initial distribution into a blue noise-type arrangement by simulating the temporal evolution of the associated dynamical system. Each point of the point cloud is interpreted as particle in dynamic scene to which the L.J potential is applied as moving force. In each iteration of the temporal integration process, the point cloud is decomposed into a new set of independent subsystems, each containing a single pair of particles. Each of these subsystems is then simulated individually, which greatly increases stability. As we also include dissipation into our system, the point configuration of the particle system will be stabilized and form randomized uniform point distribution after a sufficient number of iterations. Learning based point clouds related rescarch has been explored for years, especially in the fields of generation and denoising. Existing models are solely trained to generate and denoise the point cloud without considering the overall point distribution and therefore exhibit the common problems of holes and clusters. By embedding L.JLs into well-trained architectures, we are able to significantly improve their results in terms of point distribution while at the same time introducing minimal distortion of the shape as shown in Fig. 1. We also circumvent resource- and time-intensive retraining of these models, as I.J.. are solely embedded in the inference process. This way, LJLs are ready-to-use plug-in solution for point cloud distribution optimization.
2 Related Work
2.1 Molecular Dynamics
Simulating molecular dynamics an essential tool for applications such as understanding protein fold- ing (Creighton, 1990) designing modern drugs (Durrant & McCammon, 2011) well as state-of-the-art
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materials (Van Der Giessen et al., 2020). While the motion of atoms and molecules can in principle be obtained by solving the time-dependent Schrödinger equation, such a quantum mechanics approach remains too computationally prohibitive to investigate large molecular systems in practice. Instead, classical molec- ular dynamics uses Newtonian mechanics treating the muclei as point particles in a force field that accounts for both, their mutual as well as electronic interactions. This force field is derived from a potential en- ergy function (e.g- the IJ potential) that is formulated either from expectation values of the quantum system or using empirical laws (Rapaport, 2004). Given the typically large number of atoms and molecules involved in molecular dynamics, analytical solution of the resulting Newtonian mechanical system is usually out of reach. Consequently, numerical methods that evaluate the position of each nucleus at (fixed or adaptive) time intervals are used to find computational approximations to the solutions for given initial conditions (Hochbruck & Lubich, 1999). Classical molecular dynamics simulators are LAMMPS Molecular Dynamics Simulator (2014), RATTLE (Andersen, 1983) and SHAKE (Ryckaert et al., 1977); in which the strong covalent bonds between the muclei are handled as rigid constraints. Modern approaches allow for the more efficient numerical simulation of large molecular structures as elficiency has further increased, either through algorithmic improvements (Michels & Desbrun, 2015) or by leveraging specialized hardware for parallel computing (Walters et al., 2008). Among others, Alharbi et al. (2017) provided Eurographics tu- torial on real-time rendering of molecular dynamics simulations. More recent work also addresses immersive molecular dynamics inmulations in virtual (Bhatia et al., 2020; Bimiieson-Biniiie et al., 2020) and augmented reality (Eriksen et al., 2020).
2.2 Blue Noise Sampling
Colors of noise have been assigned to characterize different kinds of noise with respect to certain properties of their power spectra. In visual computing, blue noise is usually used more loosely to characterize noise without concentrated energy spikes and minimal low-frequency components. Due to its relevance in practical applications such as rendering, simulation, geometry processing, and machine learning, the visual computing community has devised a variety of approaches for the generation and optimization of blue noise. Several methods can maintain the Poisson-disk property, by maximizing the minimun distance between any pair of points (Cook, 1986; Lloyd, 1982; Dunbar & Humphreys, 2006; Bridson, 2007; Balzer el al., 2009; Xu et al., 2011; Yuksel, 2015; Ahmed et al., 2017). Schlömer et al. (2011) introduced farthest-point optimization(FPO) that will maximize the minimum distance of the point set by iteratively moving every point to the farthest distance from the rest of the point set. de Goes el al. (2012) successfully generated blue noise through optimal tBAOS) Their method has widely been accepted as the benchmark for best-quality blue noise, which was recently surpassed by Gaussian blue noise(GIBN) (Ahmed al., 2022). GBN is classified as kernel- based methods (Öztireli et al., 2010; Fattal, 2011; Ahmed & Wonka, 2021) that augment each point with a kernel to model its influence. There are several previous works that we consider methodologically closely related to ours as the authors applied the physical-base based particle simulation to synthesize blue noise. The method proposed by Jiang et al. (2015), is based on which takes different effects caused by pressure, cohesion, and surface tension into account. This results in a less puristic process compared to our work. Both Schmaltz et al. (2010) and Wong & Wong (2017) employ electrical field models, in which electronically charged particles are forced to move towards an equilibrium state after numerical integration because of the electrostatic forces. Adaptive sampling can be achieved by assigning different amounts of electrical charge to particles.
hydrodynamics(SPP) fluid simulation
2.3 Generative Models for 3D Point Cloud
Research related to 3D point clouds has recently gained a considerable amount of attention. Some meth- ods aim to generate point clouds to reconstruct 3D shapes from images (Fan et al., 2016) from shape distributions (Yang et al., 2019). Achlioptas et al. (2017) propose an autoencoder for point cloud reconstruc- tion, while others aim to solve point cloud completion lasks based on residual networks (Xie el al., 2020), with multi-scale (Huang et al., 2020) or cascaded refinement networks (Wang et al., 2020) or with a focus on upscaling (Li et al., 2019). Some methods solve shape completion tasks by operating on point clouds without structural assumptions, such as symmetries or annotations (Yuan et al., 2018), or by predicting shapes from partial point scans (Gu et al., 2020). 3D shape generation methods either use point-to-voxel
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representations (Zhou et al., 2021) rely shape latent variables to directly train on point clouds (Luo & Hu, 2021a; Zeng et al., 2022). by either employing normalixing flows (Rezende & Mohamed, 2015; Dinh et al., 2017) or point-voxel convolutional neural networks (Liu et al., 2019). ShapeGF (Cai et al., 2020) is an application of score-based generative models (Song & Ermon, 2019) for 3D point cloud generation, which uses estimated gradient. fields for shape generation where a point cloud is viewed as samples of a point distribution on the underlying surface. Therefore, with the help of stochastic gradient ascent, sampled points are moving toward the regions near the surface. Luo & Hu (2021a) first attempt apply denoising diffusion probabilistic modeling (DDPM) (Sohl-Dickstein et al., 2015; Ho et al., 2020) in the field of point cloud generation. Points in the point cloud are treated as particles in the thermodynamic system. During the training process, the original point cloud is corrupted by gradually adding small Gaussian noise at each forward diffusion step, while the network is trained to denoise and reverse this process.
2.4 Denoising Techniques for 3D Point Cloud
Approaches for denoising point clouds aim to reduce perturbations and noise in point clouds and facilitate downstream tasks like rendering and remeshing. Similar to images, denoising for point clouds can be solved with linear (Lee, 2000) and bilateral (Fleishman et al., 2003; Digne & de Franchis, 2017) operators, or based on sparse coding (Avron et al., 2010). Although these methods can generate less noisy point clouds, they also tend to remove important details. Duan et al. (2018) leverages the 3D structure of point clouds by using tangent planes at each 3D point to compute denoised point cloud from the weighted average of the points. Many existing approaches employ neural network architectures for point cloud denoising purposes, by building graphs or feature hierarchies (Pistilli et al., 2020; Hu et al., 2020), by employing differentiable rendering for point clouds (Yifan et al., 2019), or by classifying and rejecting outliers (Rakotosaona et al., 2020). Hermosilla et al. (2019) propose an unsupervised method for denoising point clouds, which combines a spatial locality and a bilateral appearance prior to mapping pairs of noisy objects to themselves. The majority of barning-based denoising methods directly predict the displacement from noisy point positions to the nearest positions on the underlying surface and then perform reverse displacement in one denoising step. Luo & Hu (2021b) propose an iterative method where the model is designed to estimate a score of the point distribution that can then be used to denoise point clouds after sufficient iterations of gradient ascent. More specifically, clean point clouds are considered as samples generated from the 3D distribution p on the surface, while noise is introduced by convolving p with noise mode n. The score is computed as the gradient of the log-probability function Vlog(p n). They claim that the mode of p n is the ground truth surface, which means that denoising can be realized by converging samples to this mode.
3 Lennard-Jones Layer
The LJ polential V is defined as
V(r) = 12 - ((9))-())
(1)
which is a function of the distance between a pair of particles simulating the repulsive and weakly attractive interactions among them. The distance r is measured by the Euclidean metric. The first part (.)12 attributes the repulsive interaction while the second part (-)6 attributes the soft attraction. With the help of the parameters e and o, the LJ potential can be adjusted to different use cases. Fig. 2 illustrates the role of these parameters: The potential depth given by the parameter 6 >> which determines how strong the attraction effect is, and a is the distance at which the LJ potential is zero.
The corresponding magnitude of the LJ force can be obtained by taking the negative gradient of Eq. (1) with respect to which is Fr)(r) - (r). The direction of LJ force each particle is always along the line connecting the two particles, which point towards each other when two particles are attracted and vice versa. Equilibrium distance re is where Fr.s(re) = 0 and the LJ potential reaches the minimum value. For the distance r < TE, the repulsive part dominates, while for r > TE the particles start to attract to each other. When r gets bigger, the attractive force will docrcase to zero, which means that when two particles are far enough apart, they do not interact. If we assume a uniform mass distribution among all particles and normalize with the mass, the corresponding equations of motion acting on pair of particles whose positions
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Viee
0.9
LJ
2116
(e, o]
A
Figure 2: Illustration of the LJ polential and the LJL. The strength and position of the repulsive and weakly attractive zones in the LJ potential are controlled by the hyperparameters e and Ø, while At controls the damping step size of the LJL. Applying the LJLL a point cloud leads to a more uniform distribution of the points.
are described by 21, 22 E IR2 or R& can be obtained by
D2 31 FL)( 31 22
$2 FzJ
x2 L1
Given a 2D or 3D input point cloud in Euclidean space, we aim for a transformation from a white noise- into a blue noise-type arrangement by simulating the temporal evolution of the associated dynamical system. In this regard, we use a formulation of the T.J potential where pairs of particles are grouped as a pair potential to simulate the dynamic LJL procodure. The computations within LJL are summarized in Alg. 1. Within system containing a number of particles, the potential energy is given by the sum of LJ potential pairs. This sum shows several local minima, each associated with a low energy state corresponding to the system's stable physical configuration. It is essential aspect of our method, that we do not compute the complete numerical solution of the resulting N-body problem (in which N denotes the number of involved particles). We only take into account the closest. neighbor (A: = neighborhood) of each particle within every ileration of the temporal integration process. Note that being the closest neighbor of a particle is not necessarily a mutual property of two particles within the whole particle system. However, the whole system of particles can be decomposed into several independent subsystems by iteratively assigning the closest neighbor to each particle and, throughout the process, not considering particles which have already been assigned as closest neighbor. Following this approach, every subsystem just contains single pair of particles. In each iteration, different subsystems (i.e., new pairs) can be formed. As the subsystems are not interacting within each iteration, the sum of the LJ potentials has just a single (global) minimum if no changes to the nearest neighbors are present, see Appendix A.
LJL creates pairwise independent subsystems using nearest neighbor search (NNS) by assigning the nearest points to the current point set X,. We simulate this process by repetitively integrating forward in time using decaying time step At which dampens exponentially according to
At(-) exp(-B(-)),
(2)
in which a > 0 refers to the initial step size, and > 0 defines the damping intensity. For each pair, the position update is computed independently by integrating the equation of motion numerically using a basic Störmer Verlet scheme. Within this numerical integration process, we intentionally do not make use of the particles' velocities from the previous step as this corresponds a velocity reset (setting all velocities identically to zero) causing desired dissipation effect for the final convergence. Moreover, V(r) increases rapidly when < a. we clamp the LJ potential from = 0.9o to r = 100g and employ the hyperbolic tangent as an activation function restricting the gradient (r)(() to the interval [-1, 1] to avoid arithmetic overflows. Following this method, we can redistribute any randomly generated point cloud into a uniform distribution by itcratively applying I.JLs. This process stops as soon as a stable configuration is reached, which means that the difference between the previous and current generated point clouds is below the tolerated threshold. Consequently, this spatial rearrangement of particles prevents the formation of clusters and holes.
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The Lennord-Jonel layer (LJE). NNS denotes the nearest neighhor search. Input: Point cloud X, and point cloud NNS)..).
ALGORITHM 1:
Output: Point cloud X++1-
4 - At(i)
7 [|a NNS(X,))l, III, 0., 100c)
Ar + tanh( (r)) /2
X(+1 t X1 + (X; | NNS(X.)
(b)
(c)
(d)
Figure Influence of the parameter a. The rectangle denotes the unit square in which 1024 points are initialized. (a) 0.2r' leads clusters; (b) a' distributes points well and keeps them close to the
boundary; (c) 10o' spreads points out of the boundary extremely; (d) 10 with fixed boundary
conditions. of denotes the estimated optimal value of o.
4 Applications
4.1 Numerical Examples on 2D Euclidean Plane
The LJ potential itself has two paramcters: The repulsion distance a and the attraction strength c. Similar to the time step Al, scales the gradient, however, is applied before computing the tanh-function (see Alg. 1). In practice, we find that setting = 2 results in well-behaving LJ gradient. The repulsion distance corresponds to the optimal distance that all particles strive to have from their neighbors. To derive the proper value for we take inspiration from the point configuration of the hexagonal lattice. The largest minimum distance r between any two points is given by r = 22/(v/N) for N points over a unit square (Lagae & Dutré, 2005). Fig. shows the result of different values of a under identical initial conditions where points are initialized in unit square. An appropriate a results in a uniform point distribution that does not exceed the initial square too much. a' 22/(v3.)) denotes the estimated optimal value of Small a's lack rodistribution ability o, particles tend to interact with all the points nearby and are spread out of the region excessively. After exploring the effect of a on the point distribution over the unbounded region, we introduce fixed boundary this constraint scenario, LJL is robust with respect to a as long as it sufficiently large. When it comes to the damping step size At in Eq. 2, we want. LJL updates with decreasing intensity controlled by the damping parameters a and 3 in order to converge samples in later iterations. Empirically, we a 0.5 and 8 0.01 ensure a proper starting step size and slow damping rate. We analyze the behavior of LJLs in different situations such considering different numbers of neighbors & and with or without altraction term in LJ potential, see Appendix B.1.
the point set they lead to limited effect of LJL particles. Given large
conditions where points will stop at the boundaries when they tend to exceed.
4.1.1 Blue Noise Analysis
For many applications visual computing, blue noise patterns are desirable. Because LJJLs have the ability normalize the point distribution, apply IJJLs to redistribute samples to randomized uniform distribution

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184
frequency
frequency
frequency
frequency
(a) GBN
(b) BNOT
(c) FPO
(d) LJL
Figure 4: Spectral analysis of different blue noise generation methods. The rows from top to bottom display the input point set, power spectrum, radially averaged power spectrum, and anisotropy.
solcly from white noise distribution and analyze the corrosponding blue noisc properties. Here we synthesize the blue nose distribution in a 2D unit square with periodic boundaries. Inside the region, initial random points are generated and iteratively rearranged by LJLs to generate blue noise. The spectral analysis for blue noise includes the power spectrum which averages the periodogram of distribution in the frequency domain and two corresponding 1D statisties: radial power and anisotropy. Radial power averages the power spectrum over different radii of rings and the corresponding variance over the frequency rings is anisotropy indicating regularity and directional bias. We use the point set analysis tool PSA (Schlömer & Deussen, 2011) for spectral evaluation of different 2D blue noise generation methods (1024 points) in Fig. 4. The blue noise generated by the L.JL closely resembles the results produced by the FPO method.
4.2 Redistribution of Point Cloud over Mesh Surfaces
As motivating example for real applications, use LJTs to evenly distribute points over mesh surfaces. Initially, the point cloud is randomly generated inside a 3D cube [_1, To ensure the generality of LJL parameters in 3D cases, mesh sizes are normalized to fit into the cube. The LJL parameters a = 0.5, 3 = 0.01, and e = are the same as the previous example. We set a = 50' due to the addition of the third dimension, with the factor of determined through hyperparameter tuning. the following four cases illustrated Fig. 5, we show the importance of applying surface projection in-between LJL iterations. Without the help of LJLs, random points are directly projected the sphere shown in Fig. 5(a). When LJL is only used once pre-processing step before the projection, points located the surface but still non-uniformly distributed (sce Fig. 5(b)). When using LJL as post-processing after the projection, additional noise is introduced and points are shifted away from the surface shown in Fig. 5(c). Due to the fact that LJJLs iteratively rearrange points, in the last case, we project points to the underlying surface each intermediate LJJ. step. The resulting point configuration is not only evenly distributed,, but also lies on the sphere shown in Fig. 5(d). The computations involved in LJL-guided point cloud redistribution are summarized in Appendix B.2.

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(a) LJL
(b) Pre-procesing
(c) Post-processing
(d) Iterative LJLL
Figure 5: Rodistribution of random 3D point cloud over a unit sphere. (a) Direct projection of random points.
(b) Apply LJL before the projection as pre-processing (c) Apply LJL after the projection as post-processing,
(d) Apply projection in between LJL iterations.
Dist Score: 0.02012
Dist Score:0.01966
Dist Score: 0.01806
Dist Score: 0.03960
Dist Score: 0.03302
Dist score: 0.03671
Figure 6: Projecting random 3D point clouds mesh surfaces with (right) and without (left) IJJLs (3000 sample points).
For geometric analysis, we use the mean value of Euclidean distances from each point X; to its nearest neighbor NNS; denoted as DistanceScore to evaluate the point distribution of N points shown in Eq. 3. The nearest neighbors are those who satisfy the intersection angles of their normals less than to ensure they are on the same side of the surface. A higher distance score indicates the uniform point distribution and fewer holes and clusters in the results. The score is computed by
N
DistanceScore 1
X, NNS (X+)l.
The results of TJI-guuded redistribution demonstrate significant improvements in point distribution, shown
in Fig. 6.
4.3 LJL-embedded Deep Neural Networks
Clusters and holes are consuming the limited number of points without contributing additional information to the result. In the following two applications, LJLs are plugged into the 3D point cloud generation and denoising networks that work in an iterative way (e.g. Langevin dynamics). With the help of LJ we can enforce points to converge to a normalized distribution by embodding LJIss into refinement steps (generation and denoising). The evaluation metric NoiseScore is defined as the mean value of the minimum Euclidean distances from the point X; to the underlying surface M, i.e.,
NoiseScore
Σ
min I
x/|
(4)
N
aEM
where x any point on the surface. We use the mean value of Euclidean distances from each point X, its nearest neighbor NNS (Schlömer et al., 2011) denoted Score to evaluate the point distribution
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Eg
Latest Space
Do + Wl
Ar
(,,
Noir Searo
Dastance Score
Figure 7: A pipeline for searching LJL parameters through well-trained autooncodar-decodeer
of N points in Eq. 5. We want LJL-embedded results to have as small as possible noise score increments while having large DistanceScore increments.
DistanceScore
X NNS(X;))1
(5)
1=0
4.3.1 LJL-embedded Generative Model for 3D Point Cloud
In this section, we show that LJLs can be applied to improve the point distribution of point cloud generative models ShapeGF (Cai et al., 2020) and DDPM (Luo & Hu, 2021a) without the need to retrain them. It has been shown that the aforementioned two types of probabilistic generative models are deeply correlated (Song et al., 2021). The inference process of both models can be summarized as an iterative refinement process shown in the bottom row of Fig. 1. Initial point cloud Xo is sampled from Gaussian noise N(0, 7). The re- cursive generation process produces each intermediate point cloud X,, according to the previously generated point cloud x Even though there is randomness introduced in cach iteration to avoid local minima, the generation process depends heavily the initial point positions. Every point moves independently without considering the neighboring points, thus generated point clouds tend to form clusters and holes.
The top row of Fig. 1 provides an overview of the intogration LJIss into the inference process well-trained generative models. Since the sampling procedure of these generative models performed recursively, one can insert LJI.s into certain intermediate steps and guide the generated point cloud have normalized distribution. The generator and LJLs have different goals: the former aims converge all points a predicted surface ignoring their neighboring distribution, which is also can be seen denoising process. The latter aims to prevent the formation of clusters and holes to improve the overall distribution. It is therefore important to obscrve whether LJLs are inhibiting the quality of the generation process. The trade-off between distribution improvement and surface distortion needs to be measured.
To find LJL parameters that achieve a favorable trade-off, we apply LJLS in the well-trained autoencoder- decoder models. In Fig. 7, given a point cloud Y, the autocncoder Eg encodes it into latent code Z, then the decoder Do which is the generator will utilize iterative sampling algorithms to reconstruct the point cloud conditioned Z. We fix the encoder E, and insert LJLS solely in decoder Dg. To determine proper parameters for LJLs in generation tasks, the reconstructed results need to have a good point distribution and preserve the original shape structures according to distance and noise scores. The computation related to this task is summarized in Alg. 2.
During the generation process, LJLs are activated from starting steps (SS) till ending steps (T"). This implies that in each valid generation step, the LJL undergoes multiple iterations until convergence. A systematic parameter searching for SS is shown in Appendix C.2.1. It is not necessarily advantageous to perform L.I.s at every iteration, since LJLs slow down the convergence of the generation in the beginning steps. LJL- embedding starts in the second half of the generation steps. We also circumvent embedding LJLLs in the last few steps to ensure that no extra noise is introduced. Point clouds are normalized into the cube [-1, 1J3, and we set c = and a = 50' due to the additional third dimension. The decaying time step At in i-th generation step set as
At(1) = i
max X; X,, xxxpl/exp(-331
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IJL. parometer searching vão astoencoder Eg and decoder Dg. Point cloud Y, starting 88 and ending T < steps. Output: Reconstructed point cloud Xr.
ALGORITHM 2:
Input:
Z' EY No I)
FOR LE I...T
IF SSSIST'
X - D0(X)-1,2)
X; - (JLa(X,, NNS(X.)
ELSE:
X1 t- (0(X1-1,2)
RETURN XT
Figure 8: Comparison of generation-only (red) and WJ.-embedded generation (blue).
where At(i) is scaled by the maximum difference between the current X, and the previous X,-1 point cloud, such that each position modification AX caused by LJLss maintains similar scale. Furthermore, At(i) is divided by the iteration number lo diminish the redistribution effect as generation steps increase. In order to determine a and 3, we perform a systematic parameter searching based on distance and noise scores, see Appendix C.2.2. We found that a = 2.5 and 3 0.01 meet the requirements of less noise and better distribution.
Incorporating the optimal parameters, tested L.J.-embedded generative models ShapeNet (Chang et al., 2015) shown in Fig. 8. We evaluate the chair and airplane datasets using DDPM model, while the car dataset is evaluated with ShapeGF model. There is a 99.2% increase in the DistanceScore and a 7.8% increase in the noise score for the car dataset. The point distribution is improved by 42.8% while the noise score is only increased by 2.8% for airplancs. Finally, we get a 207.9% increase DistanceScore, and 19.1% increase in the noise score for the chair dataset, shown in Table 1. More details about LJL parameter settings and generation results can be found in Appendix C.2.
4.3.2 LJL-embedded Denoising Model for 3D Point Cloud
Given noisy point cloud x (a:N} which is normalized into the cube [-1, the score-based model (Luo & Hu, 2021b) utilizes stochastic gradient ascent to denoise it step by step. In each intermediate step, the denoising network (x) predicts the gradient of the log-probability function (score) which is vector field depicting vectors pointing from each point position towards the estimated underlying surface. The gradient ascent denoising process will iteratively update cach point position to converge to the predicted surface,
Table 1: Evaluation of L.JL-embedded shape gen- eration by measuring the introduced error (Nois- eScore) and improvement in distribution (Dis- tanceScore). Smaller noise and larger distance scores are better.
Table 2: Evaluation of L.JL-embedded denoising for two different. levels of initial noise after dif- ferent amounts of iterations of denoising steps. Smaller noise and larger distance scores are bet- ter.
Noise Score |
Distance Score 1
Noise Score
1
Distance Score f
Airplane
+ 2,8%
+ 42.8%
Noise Level:
Low
High
Low
High
Chair
19.1%
+207.9%
30 Iterations:
0.00%
42.63%
8.79%
100771%
Car
+ 7.8%
+ 99.2%
60 Iterations:
-0.22%
+1.37%
+ +13.04%
+16.85%
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(a) Noisy input.
(b) Denoise-only.
(c) LJL--embedeed denoising.
Figure Comparison between denoising without and with LJLs.
Noise Scale
few out 12:30
Meee Gove
23011
Meke Soee a010
lave Oel 3740
More
core
foo poeel
New fooe teeil
kow
30
Iterations
Figure 10: Comparison of denoise-only (red) and LJL-embedded denoising (blue) for 30/60 denoising itera- tions and different input noise scales.
where we can oblain the denoised result X The denoising step can be described follows:
(+ (t-1) So (t-)), t I,...,T
where Y is denoising step size for t-th itcration. LJLs are inserted after each intermediate denoising step except the last few iterations to avoid extra perturbation to the denoised results. LJJ. parameter settings and evaluation metrics used in this task are inherited from previous generation tasks, where a = 55 and = 2. We keep 3 = 0.01 and choose a 0.3 which will minimize the ratio of the noise and distance score increment rates, see Appendix C.1.1. The only difference is that the position update scale (max | - X;-)ll) for the denoising step different from that in the generation task. The denoising results then do not only converge to the predicted surface with less distortion but are also distributed uniformly shown in Fig. 9.
We continue to determine appropriate denoising iterations and the performance of denoising models (with and without LJls) on different noise scales (more and less) shown in Fig. 10. We apply isotropic Gaussian noise with a higher noise scale of 3% and a lower noise scale of 1% of the shape's bounding sphere radius as the standard deviation. This is because denoising tends to progressively smooth the surface and wash out the details. This phenomenon is shown in Fig. 11.
It clear that no matter the noise scales, for the denoise-only model, the noise score increases when the number of iterations goes up. While for the TJJ--embedded denoising model, the noise remains steady
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2.033
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2.016
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1/915
0.014
6.014
5.022
0.003
8.913
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0.003
(a)
Figure 11: (a) Denoising results of lower noise scale point clouds with respect to increasing denoising itera-
tions. (b) Denoising results of higher noise scale point clouds with respect increasing denoising iterations.
for the first 60 denoising iterations and the distance scores of both models decrease when itcration times increase regardless of noise scales. We further evaluate the LJL-embedded denoising model on the test set from Luo & Hu (2021b) shown in Table 2. By combining LJLs with the denoising network, we find that it improves DistanceScore with negligible increments of NoiseScore. Morcover, we boost the performance of existing denoising models without extra training, More LJL-embedded denoising results and evaluations are shown in Appendix C.I.
5 Conclusion
In this contribution, we have extended the concept of the LJ potential describing pairwise repulsive and weakly attractive interactions of atoms and molecules to variety of tasks that require equalizing the density of a set of points without destroying the overall structure. It has been shown that LJLs are conceptually and practically capable of generating high-quality blue noise distributions. To demonstrate the benefit of applying LJTs in the context of the 3D point cloud generation task, we incorporated L.J.s into the generative models ShapeGF (Cai et al., 2020) and DDPM (Luo & Hu, 2021a) aiming for the generation of point clouds with an improved point distribution. By embodding LJLss in certain intermediate generation steps, the clusters and holes are decreased significantly because LJLs increase the chances of points converging cover the entire shape. Finally, we combine LJLs with a score-based point cloud denoising network (Luo & Hu, 2021b) such that the noisy point clouds are not only moved towards the predicted surfaces but also maintain the uniform distribution. Consequently, this rearrangement prevents the formation of clusters and holes. Since adding LJLs in these cases does not require retraining the network, the cost of applying LJLs is negligible.
The presented work offers several avenues for future work. From an algorithmic perspective, the k-nearest neighbor classification performance could be improved through supervised metric learning methods such as neighborhood components analysis and large margin nearest neighbor. The distribution normalization on surfaces could be addressed by means of considering tangential bundles. By doing this, we force particles not to leave ial manifolds during I.JL, interactions and in a perfect case to move solely along the geodesics of the implied surfaces. This would require that we do not distort the underlying surfaces while performing distribution normalization Geodesical distances between points on surfaces also refer to a uniform sampling, which seems more adequate than Euclidean distances of embedding spaces. The reconstruction of geodesics from point clouds has been explored by Crane et al. (2013). From an analytical point of view, the presented LJL for distribution normalization has a discrete, piecewise character as it takes only the closest neighbors into account which change from iteration to iteration. Most common approaches in theoretical mechanics aim for continuous description, e.g., based Hamiltonians and their analytical solution numerical integration. It is theorctically not yet clear if it is possible to obtain a blue noisc-type arrangement of particles utilizing pure Hamiltonian mechanics. For future work, we would like to further investigate this question. On a slightly different trajectory, we would like to explore the possibilities of relativistically moving particles. If such particles move within a medium with a suitable refraction index, then the power density of radiation grows linearly with the frequency. This effect is well studied in the literature and known as Chhilo-Chheeekov radiation (Cherenkov, 1934; Jackson, 1999).
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Acknowledgements
This work has been supported by KAUST through the baseline funding of the Computational Sciences Group. The authors are grateful to Abdalla G. M. Ahmed for sharing Gaussian blue noise data and to Dmitry A. Lyakhov for helpful discussions about properties of the Lennard-Jones potential. The reviewers valuable comments are gratefully acknowledged.
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A Convergence Properties of the Lennard-Jones Layer
The LJL's output is computed by repetitively integrating forward in time while damping the motion of each particle over time. As a consequence of the damping effect, we observe that almost changes to the nearest neighbors are present at. the later stage of the process. Assumiig that there are changes to the nearest neighbors, we can theoretically prove the convergence of the LJL to the global potential minimum as outlined in the following theorem.
Theorem: The LJL''s output is uniquely defined matching the single (global) minimum of the corresponding potential under the assumption that there are no changes the nearest neighbors throughout the process.
Without loss of generality, let us consider set which is composed of even number of N particles. It is processed by the LJL, ie., the dynamical process of each pair of particles is governed by the LJ potential defined in Eq. (1). The total potential V controlling the process carried out by the LJL is given as the sum of the LJ potentials of the N/2 pairs, i.e.,
Proof:
N/2 Σn Vi with V v=== 46 (())) 17)) :=1
12
Vrn - TN/2)
(6)
in which rı, i i= 1,...,,/13. denotes the Euclidean distance of the two particles composing the i-th pair. The derivatives V of Eq. (6) with respect to rj, j = 1, I....N/22 are given by
d ((r .........NNNN/)) TN/2)) 24e
6
7 2 o 12
Vi(rs)
(7)
dr;
From (r,), 0, obtain the candidate rf o/ for local extremum saddle particle. The second partial derivative test results in Vj" (rz) > for which reason, the potential V has a local minimum at (ri....,rN (//22) 82 (o, .) with V(ri,..., (xy2)) (e)....:)) This minimum is further the only local extremuu and the global minimum of V which we can easily see by considering lim,y Vz(r,) = oo and lom Vg(r,) = 0, j = N/.
In other words, the potential energy of each pair has only a single energy valley and cach pair is independent by construction, the LJL converges to the unique minimum of the potential V.
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B Parameter Configuration of the Lennard-Jones Layer
We investigate the behavior of LJL in detail by evaluating the point configuration of point sets 2D Euclidean plane and 3D mesh surfaces and propose strategies for LJL parameter searching for practical applications.
B.1 Numerical Examples on 2D Euclidean Plane
Before applying LJLs to practical applications, now analyze the behavior of LJLs in different situations (i.c., different numbers of neighbors k, with or without attraction term in LJ potential). We set LJL parameters a 0.5, 3 = 0.01, = 2, and a = 5o'. All different cases are initialized with uncorrelated randomly distributed points inside a unit square region.
B.1.1 K-NN
The interaction acting individual particles according LJ potential can be summed with varying number of nearest neighbors. Our theoretical finding guarantees convergence only for the consideration of single nearest neighbor. The experiment shown in Fig. 12 indicates point configurations of LJLs considering a different number of nearest neighbors Cases with more than one neighbor increase the difficulty of reaching uniform point distribution.
L
&
(b)
1: 10
k 499
Figure 12: LJJL evaluated with a varying amount of nearest neighbors (500 points in total).
B.1.2 The Impact of the Attraction Term
We continue to discover the impact altraction term in LJ polential. Two experiments, shown in Fig. 13 run in identical settings except for the appearance of the attraction term. Intuitively, the repulsive term alone should achieve an effect of redistribution. Without attraction term, however, particles are casily pulled out of the initial boundaries and spread further. LJLs with the attraction term can enable the overall control of the point set and prevent particles from spreading out, which leads to compact and equal distribution.
B.2 Redistribution of Point Cloud over Mesh Surfaces
Au
In Fig. 14, two particles and B form a cluster when they are directly projected on the surface (A' and B'). With the help of LJL rearrangement, particles are moved to new positions (Ar, and BiJ) and further projected back the surface with proper distance (Ay,, and B, In this case, the redistributed points are not only located on the surface but also have blue noise properties. The computations involved in LJL-ggudedd point cloud redistribution are summarized in Algorithm 3, where we start with a random 3D point cloud X, initialized inside the cube [-1, 1|3 and normalized mesh surface M. In order to select nearest. neighbors the same side of
Bai
A
B'a
Figure 14: An illustration of how LJILs perform redistribution of clus- tered particles.
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Figure 13: LJL with activated (left) and deactivated (right) attraction term.
the surface, LJLs only act on pairs that satisfy the condition that the intersection angle between their normals less than - In each
LJL iteration, the redistributed points are projected back onto the surface. After sufficient amount of LJL iterations, random point clouds can be rearranged to a normalized distribution and efficiently avoid the formation of clusters and holes.
ALGORITHM 3: Distribution normalization of 3D point cloud over the mesh surface M.
Input: Point cloud X, and tolerated threshold tol. Output: Point cloud X,,
0
repeat
N, < Normal xx,)
Nº < Normal(NNS(X..) X,.))
if Angle(Nn, N²,) I
Xn+1 LJL(X,,, NNS().))
else:
Xat1 t- X,
Xn+1* t Project-to-Meshh(XX
10
AX + |x+11 X,.ll
11
n1 4- n +
12
until AX < tol
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C LJL-embedded Deep Neural Networks
In this section, we show more details of LJL-embedded networks' parameter searching and results. We test models on NVIDIA GeForce RTX 3090 GPU if not specified otherwise. Two well-trained point cloud gencrative models used in the papers are from ShapeGF (Cai et al., 2020) and DDPM (Luo & Hu, 2021a). The well-trained point cloud denoising model is from Luo & Hu (2021b).
C.1 LJL-embedded Denoising Model for 3D Point Cloud
C.1.1 Systematic Parameter Searching for
We keep = 0.01 inherited from the generation task and search for by finding the minimun ratio of noise and distance score increment rate, shown in Fig, 15. We choose 0.3 for the denoising task.
Noise/Distance incremeen
0.04
0.02
0.00
0.25
0.50
0.75
1.00
1.25
1.50
Figure The ratio of noise and distance score increment rale. Parameter searching for optimal a by finding the global minima a 0.3.
C.1.2 More LJL-embedded Denoising Results
LJL-eebedded denoising results with different denoising iterations and noise scales are shown in Fig. 16.
Notse Scale
boo Abick
Bolddwe boitl
Lirld
More
LEKA
LSB1
Bondow Hien
60
Herations
Figure 16: Comparison of denoisc-only (red) and LJL-embedded denoising (blue) for 30/60 denoising itera- tions and different input noise scales.
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C.2 LJL-embedded Generative Model for 3D Point Cloud
C.2.1 Systematic Parameter Searching for Starting Steps
Assuning that it takes T 100 steps to generate point clouds, continue to search for good starting steps SS to insert LJLLs shown in Fig. 17. SS = O means LJJLs are applied from the very beginning, whereas SS = 101 means generation without L.JLs. In order to balance the generation speed and LJJL normalization effects, we choose SS = 60, meaning embedding should start in the second half of the generation steps.
A
A
S 30
90
101
Figure 17:
The optimal starting step (SS) to embed LJLs in the generation process (i.c. T=100). We select SS = 60 in the act generation lask.
C.2.2 Systematic Parameter Searching for and 8
In order to determine and 8, we perform systematic parameter searching based on distance and noise scores, shown in Fig. 18. We found that 2.5 and B 0.01 meet the requirements of less noise and better distribution.
C.2.3 Point Cloud Generation Process with Different Numbers of Points
We especially focus on the task of using as few points possible in generation tasks. For large number of points, incfficient point distributions are less of a concern since even for low-density regions, there are sufficient points to describe the underlying surface. Fig. 19 shows how LJILs behave in the generation process with different numbers of points N. Note that our choice of o = 50' implicitly accounts for the change in N. In the case generation tasks with fewer points, the LJL--ebeddded generator can redistribute the limited number of points even as possible while maintaining the global structure. Combining LJLss with generative models enables points to have the ability to converge the different. parts of the shape.
with
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Published in Transactions on Machine Learning Research (09/2024)
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Figure 18: Parameter searching for decaying factors a and B, where 0j 0.1, 02
2.5, 03
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31 0.001, B2 = 0.01, A3 0.1. The optimal parameters are 2.5 and B 0.01.
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Figure 19: Examples of generation results with regard the increasing number of sample points N. Red: Raw generation results. Blue: T.J.-embedded generation results.
C.2.4 More LJL-embedded Generation Results
More I.J--embedded generation results are shown in Fig. 20, Fig. 21, and Fig. 22.
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Published in Transactions on Machine Learning Research (09/2024)
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Figure 20: Comparison of raw generation (red) and LJI--emeddded generation (blue).
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Published in Transactions on Machine Learning Research (09/2024)
Figure 21: Comparison of raw generation (red) and L.JL-emeddded generation (blue).
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Published in Transactions on Machine Learning Research (09/2024)
Figure 22: Comparison of raw generation (red) and LJI--mmeddeed generation (blue).
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Reviews:
[{"review_id": "gTMBVXGAVG", "reviewer": "Reviewer_iFqM", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n1. The method is conceptually very simple and easy to understand, and has an attractive physics-based justification that may allow it to be more widely used. The physical interpretation of the $\\sigma$ and $\\epsilon$ parameters also makes it easier to select reasonable values for them based on the problem setting without requiring extensive hyperparameter tuning.\n\n2. While I am not familiar with the application of distribution normalization for point clouds, the results presented show the method works fairly well for distribution normalization, particularly on 3d structures. It is also effective at increasing the minimum distance between points when applied as a part of an iterative generative or denoising model, while increasing the noise score of the point cloud with respect to the mesh by a relatively small amount.\n\n3. As someone who is not so familiar with the literature on point cloud distribution normalization, I appreciated the detail of the literature review in section 2, though there were some mistakes/omissions in the molecular dynamics section which I am more familiar with.\n\nWeaknesses:\n1. In some points of this manuscript, there is a distinct lack of details in the description of the method itself or the analysis. One particularly blatant example is section 4.1.1., where the authors simply describe the how the blue noise analysis is used, but do not talk about the results of this analysis at all. This is particularly notable since the accompanying Figure 4 also does not contain any details in the caption or figure itself, even though it contains 16 subfigures, and according to the results shown in this figure, which once again are not discussed at all almost every other distribution normalization method seems to produce similar or better results to the LJL.\nThis is not the only case where this lack of details is present. The authors do not explicitly define the noise score they use as one of the main measure of distribution normalization quality, and in the cases where the LJL was incorporated in the iterative generative and denoising models, it is not clear to me whether only a single LJL update is applied, or whether the LJL is applied multiple times until convergence.\nAnother such example is the case where the authors add a factor of 5 to the $\\sigma$ parameter of the 3D case compared to the value used for the 2D case. While the choice of $\\sigma$ for the 2D model has a mathematical justification, the authors do not elaborate on the choice of the factor 5 for the 3D case or provide any ablation showing how this was selected optimized.\n\n2. The experiments and results section in this manuscript should also be presented better and be somewhat expanded. Besides the already mentioned example of Figure 4, in the beginning of section 4.3.1., the authors mention that they are testing whether their model can improve the ShapeGF model and the DDPM models for point cloud generation, however, when presenting the results in Figure 8 and Table 1, the authors show only one set of values for the pure generative models. While the authors mention previously that these two models are highly correlated, they should still specify which of the two models was used for the results, or provide the scores for both models for the highest amount of transparency.\nAnother very important omission is the lack of information about the selection or sizes of the training, validation and test sets that were used to tune the \\alpha and \\beta for the timestep decay of the LJL. This is important information that should be included in the main text, but it is not even mentioned in the appendix.\nAdditionally, considering that the LJL involved finding the nearest neighbor for each point, which can be quadratic, and then updating the position of each point at every step, the authors should include an analysis showing the computational overhead that including the LJL incurs on the baseline models, so we can have a better understanding of it's impact on computation time.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The authors introduce an approach that aims at producing a uniform blue noise distribution of point clouds on 2d and 3d structures. The approach is inspired from the Lennard-Jones potential (LJP) commonly used in computational chemistry, which is used to model how to particles repel or attract each other as a function of their distance.\nTo apply this notion for distribution normalization of point clouds, the authors implement a so-called Lennard-Jones layer (LJL) that selects the closest neighbor to each point, calculates the attractive or repulsive force between the each point and its nearest neighbor as the negative gradient of the LJP between these two points, and adjusts the position of the node accordingly using numerical integration of the Newton equations. This procedure is repeated in an iterative manner, with the time interval used for the numerical integration being decayed, until point cloud positions converge based on a given threshold. The procedure can also be incorporated as a part of the iterative process of generative and denoising models, by applying the LJL after certain steps of the iterative process of the generative/denoising model.", "claims_and_evidence_yn": "No", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "- Move figure 1 to a place further in the manuscript. It is first mentioned in section 4.3.1. on page 9, but it currently appears on page 2.\n- In section 2.1. the text 'LAMMPS Molecular Dynamics Simulator (2014), which utilizes “velocity Verlet” integration' seems to imply that LAMMPS only uses velocity Verlet, however it also can perform integration with the rESPA integrator (Tuckerman et al. 1992), so the authors should either add this or remove the reference to velocity Verlet to avoid any incorrect or misleading statements.\n- In section 3, the authors claim that by applying the LJL only based on the nearest neighbor of each node, the resulting sum of the LJ potentials has a global minimum, which allows their numerical integration procedure to converge to this minimum. The authors should provide an explicit proof for this in the appendix.\n- A discussion is of the results presented in section 4.1.1. and Figure 4 is needed, where other methods seem to outperform LJL when it comes to generating blue noise. Also, the caption of figure 4 needs to be much more detailed and explain the contents of the subfigures in much more detail, or the subfigures themselves need to be captioned and labeled more clearly. Additionally, the font of the labels on the bottom row of figure 4 is currently too small, this needs to be increased so that is it more easily readable.\n- A mathematical definition for the noise score needs to be provided in the main text.\n- The authors should explain or justify (either theoretically or by hyperparameter tuning) the choice of factor 5 for the \\sigma parameter in the 3D case as compared to the 2D case.\n- A clear labeling in needed in sections 4.3.1. and 4.3.2. regarding which model was used both for the baseline and in conjunction with LJL. Additionally, in the case of section 4.3.1., it would be ideal for both results for ShapeGF and DDPM to be provided, unless the authors can credibly show that the results are equivalent.\n- It should be clearly specified what amount of noise is present in the different noise scales used for the results in section 4.3.2. The current use of 'more' and 'less' noise is in no way rigorous and completely useless without any notion of relative and absolute scale.", "broader_impact_concerns": "No concerns.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "aC1pCDut2I", "reviewer": "Reviewer_Xzde", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n\n- The paper introduces the problem well, highlighting the benefits of well-distributed point clouds for a range of downstream applications and acknowledging the challenge of uniformly sampling point clouds.\n\n- The approach to minimize the Lennard-Jones (LJ) potential is both simple and effective. The computation is made efficient by considering only the nearest neighbor (NN) for connectivity.\n\n- Demonstrations across various applications showcase the versatility of integrating LJ potential minimization within a network during inference. Additionally, spectral analysis on the LJ minimization verified its blue noise-like properties.\n\nWeaknesses:\n\n- The paper treats Lennard-Jones minimization as a \"layer\" without verifying its differentiability for training. Experiments primarily use LJ minimization as post-processing, leaving its integration into training as unverified. It would be beneficial for the authors to demonstrate training diffusion models with the LJ layer, potentially by training a denoising network for the last few iterations.\n\n- When integrating LJ minimization into score-based models, the distribution formulation corresponding to the LJ potential remains unknown. Given that the gradient of the data distribution is implicit and the gradient of the LJ potential is explicit, incorporating this probability into the generation process would further verify the proposed method formally.\n\n- The choice of NN for connectivity lacks justification. Alternatives like connecting all points or using Delaunay triangulation for connectivity could offer benefits. Both of the strategies will generate a positive-definite linear system at each iteration, potentionally give better results than the NN solution. In addition, the Delaunay triangulation also creates a sparse system for efficient solving.\n\n- The paper does not address how close the final potential is to a global minimum that indicates an optimal solution. The authors claim a single (global) minimum due to non-interacting subsystems within each iteration, yet it's unclear if this minimum leads to a better solution. Comparing the minimized potential energy from the proposed solution with that of an \"oracle\" solution, especially in a fully connected system with initialization from the proposed method, would provide valuable insights.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper focuses on integrating Lennard-Jones potential minimization as a layer into neural networks. The minimization of the Lennard-Jones potential normalizes the distribution of given point clouds and results in a more uniformly distributed point clouds on potential surfaces. To demonstrate the effectiveness of the proposed algorithms, the authors conduct experiments across three applications: redistributing point clouds on mesh surfaces, point cloud generation, and point cloud denoising. They achieved both quantitative and qualitative improvements over the baselines that do not utilize Lennard-Jones potential minimization layer.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Please refer to the weakness for request changes.", "broader_impact_concerns": "There is no ethnic concern of the proposed method, since point cloud is a very abstractive geometry data format.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "ioGgDtC5Ut", "reviewer": "Reviewer_4rJz", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "## Strengths\n- The paper is particularly well-written, providing a smooth and enjoyable reading experience.\n- It demonstrates scientific rigor by qualitatively and quantitatively validating the newly introduced approach.\n- Leveraging a fundamental concept from computational chemistry, the paper effectively refines 3D point cloud structures. Despite originating from a disparate field, the Lennard-Jones potential proves to be a fitting tool for this task.\n- The paper exemplifies how principles from physics and chemistry can inspire robust algorithms across different domains, such as computer vision.\n- Detailed and comprehensive, the introduction and related work sections ensure accessibility for readers unfamiliar with point-cloud generation tasks.\n\n## Weaknesses\n- Upon thorough examination, I could not identify any **significant** weaknesses to highlight. Using the Lennard-Jones attractive-repulsive tradeoff for refining point-cloud structures is both elegant and effective, as demonstrated through the experiments. See the following paragraph for suggestions for improvement.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper introduces a novel layer aimed at improving the redistribution of point clouds, drawing inspiration from the Lennard-Jones potential, a well-established law in physics and chemistry that elucidates the interplay between attractive and repulsive forces among pairwise particles in atomistic systems. This innovative layer is termed the Lennard-Jones Layer (LJL) for brevity. When integrated with conventional denoising and generative algorithms, this newly introduced layer facilitates distribution renormalization. In other words, it enables the redistribution of elements within a point cloud system in a manner that maximizes the distance between each pair of points (akin to particles in the Lennard-Jones framework), while minimizing perturbations to the existing point cloud configuration during the convergence phase. This yields two significant advantages: firstly, it mitigates the formation of voids and clusters within the point-cloud structure, thereby converging towards improved point-cloud representations. The optimization objective is quantified by the \"Distance Score\" (refer to equations (3-4)), wherein a higher value signifies a more uniform distribution with reduced instances of voids and clusters. Another pivotal metric is the \"Noise Score,\" which should ideally be minimized to preserve the overall structure. The authors employ a heuristic approach to set the hyperparameters for LJLs via a grid search and optimize them using a well-trained autoencoder-decoder structure. The LJL algorithm and hyperparameter optimization procedures are delineated in Algorithms 1 and 2, respectively. To circumvent excessive repulsion caused by a large parameter σ in the Lennard-Jones potential, a periodic boundary condition constraint is enforced.\n\nIn summary, LJLs represent a promising approach inspired by the Lennard-Jones potential for refining generated 2D and 3D point-cloud structures, ensuring a more even distribution with fewer voids and clusters while preserving the overall structure of the point-cloud mesh.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "None of the following requested changes are critical for securing my recommendation for acceptance; rather, they represent some minor adjustments that I believe would improve the paper.\n\n- I find it challenging to interpret Table 1 and 2. Specifically, in Table 1, it may seem that higher Noise and Distance values are beneficial, suggesting that higher values are better. However, based on my understanding, this is not the case. Including a few more words in the caption would be beneficial, as well as carefully explaining the content and implications of the results shown in the table. On the other hand, Table 2 is not self-contained. The results in this table cannot be understood without carefully reading the main manuscript. This lack of clarity is not ideal for a paper where figures and tables should always be explanatory and self-contained. Therefore, I strongly recommend the authors include a few more words in the caption to describe both tables and the data therein more accurately.\n\n- Is there any metric to quantitatively assess the quality of a sample generated with and without LJLs? For instance, is there a way to numerically quantify *how much better* the blue point clouds are compared to the red ones shown in Figure 8? As far as I can tell, the experiments shown in the main manuscript currently focus only on the modified distance and noise scores after the application of the LJLs. However, it is not clear to me whether this actually represents a good metric for *absolute* success. I am not too familiar with the task of point cloud generation in computer vision, and I wonder whether there’s a more general metric to assess how well a generated point-cloud structure reproduces a target mesh. Performing an ablation study in this sense, comparing the proposed approach and existing ones to different benchmarks, would be a valuable addition to strengthening the results of the paper.\n\n- I am curious about the computational overhead introduced by using LJLs in combination with the generative models ShapeGF and DDPM analyzed in the manuscript. Adding a paragraph and some ablation study on the runtime performance would be interesting to see and, again, strengthen the conclusion of the manuscript. Specifically, in the conclusion, the authors state that the cost of applying LJLs is negligible since they do not require network retraining. However, in Figure 7 and Algorithm 2, a framework using an encoder-decoder structure to optimize the hyperparameters of the LJLs is introduced. I wonder how the pipeline necessary for setting optimal parameters for the LJLs impacts the overall computational cost.\n\n- I would suggest adding labels on the y-axis in Figure 11. My opinion is that plots should always explicitly report axis labels, even though those might be clear from the legend.\n\n- Typo: On page 9 (middle page), “[…] conditioned on Z. we fix encoder […]” should be corrected to “[…] conditioned on Z. **We** fix **the** encoder […].”", "broader_impact_concerns": "I could not see any concerns regarding the ethical implications of the submitted work.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: After submission, there was a brief issue with regards to visibility of the authors' responses, leading to two of the reviewers basing their recommendations on the response to only one reviewer. Based on this initial information, reviewers had split recommendations. A primary reason for the rejection was: (a) a lack of a proof of the global optimality of the method (raised by two reviewers), and (b) a lack of a revision. The paper then had additional discussion in a somewhat odd state where recommendations were in, but reviewers were still discussing the paper with the AE and were not fully convinced. 

While the discussion hadn't concluded, the AE thinks it's in the best interest to a conclusion. Having examined the paper and discussion, the AE recommends accepting with minor revisions and asks the authors to prepare an updated version of the manuscript handling the following two main things that the reviewers were concerned about:

**Request 1: The proof of global optimality.**

There is a sketch in the responses that partially addresses the reviewer concerns about global optimality. However, the AE understands why one of the reviewers wants to see a revision: the sketch would be stronger if it directly addressed Algorithm 1 and the variables involved and was a bit more detailed. For example, a more formal version could clear up this question: it is clear that the pairs all have a unique minimum and each pair is independent. However, since the points move, the nearest neighbors may change over time from timestep to timestep. Is the statement of global optimality purely about the global minimum at each timestep i? Is it about the full process? Is it about the full process assuming that there are no changes to the nearest neighbor? Regardless of which it is, having a more formal statement would be immensely helpful. **The AE asks the authors to please put the more formal proof in the appendix as discussed in the response**

**Request 2: The promised revisions.**

One reviewer was interested in a revision, which is fairly common in journals for ensuring that the reviewers know that the promised changes will be made in a way that addresses the concerns. **The AE asks the authors to please make the promised changes from their response to the reviews**

Once the revision is in, the AE will then double check the responses against the revision to ensure that the reviewer concerns are addressed correctly. 

The AE thanks all for their patience after the visibility mishap, and especially thanks the reviewers for continuing engagement even after their recommendations were due.
Decision: Accept (poster)