NeuroGauss4D-PCI: 4D Neural Fields and Gaussian Deformation Fields for Point Cloud Interpolation

Authors: chaokang  jiang
dalong  du
jiuming  liu
siting  zhu
zhenqiang  liu
zhuang  ma
zhujin  liang
jie  zhou
Venue: NeurIPS_cc_2024_Conference
Published: 2024-05-13 18:29:32

Abstract: Point Cloud Interpolation confronts challenges from point sparsity, complex spatiotemporal dynamics, and the difficulty of deriving complete 3D point clouds from sparse temporal information. This paper presents NeuroGauss4D-PCI, which excels at modeling complex non-rigid deformations across varied dynamic scenes. The method begins with an iterative Gaussian cloud soft clustering module, offering structured temporal point cloud representations. The proposed temporal radial basis function Gaussian residual utilizes Gaussian parameter interpolation over time, enabling smooth parameter transitions and capturing temporal residuals of Gaussian distributions. Additionally, a 4D Gaussian deformation field tracks the evolution of these parameters, creating continuous spatiotemporal deformation fields. A 4D neural field transforms low-dimensional spatiotemporal coordinates ($x,y,z,t$) into a high-dimensional latent space. Finally, we adaptively and efficiently fuse the latent features from neural fields and the geometric features from Gaussian deformation fields.
NeuroGauss4D-PCI outperforms existing methods in point cloud frame interpolation, delivering leading performance on both object-level (DHB) and large-scale autonomous driving datasets (NL-Drive), with scalability to auto-labeling and point cloud densification tasks.

Main paper: 
 introduction: 
 Point cloud frame interpolation (PCI) [1; 2] aims to estimate intermediate frames given two or more point cloud frames. This task enables the generation of temporally smooth and continuous point cloud sequences at arbitrary timestamps, which is crucial for applications such as autonomous driving [3; 4] and virtual reality [5; 6; 7] . PCI can be expressed with the following formula:F Θ ({P ∈ R Ni×3 } t=0,4,8••• , T t=0,4,8••• Training Data ),F Θ (P t=i , T t=i , T t=j Inference InputEQUATION)PCI faces several challenges due to the unique characteristics of point cloud data and the complexity of modeling spatiotemporal dynamics: 1) Point clouds are inherently sparse and unordered, lacking the regular structure of images. For instance, NeuralPCI [3] simply concatenates spatial and temporal coordinates as inputs to an MLP, struggling to adequately represent the motion and correlation of multiple unordered point clouds over time. 2) PCI involves modeling the spatiotemporal dynamics of point clouds, requiring the interpolation model F Θ to capture the spatial structure and temporal evolution of the scene from 4D training data ({P ∈ R Ni×3 } t=0,4,8••• , T t=0,4,8••• ), and model the non-rigid deformations and non-linear trajectories of discrete 3D points. This leads to the linear motion assumption of PointINet [8] , which uses bidirectional scene flow to warp input frames for estimating intermediate frames, failing to capture complex non-linear motion. Moreover, 3D scene flow estimation methods, such as 3DSFLabelling [9] , PV-RAFT [10] , and NSFP [11] , can only express the motion field between two frames but cannot robustly represent higher-order dynamic scenes over longer time spans.3) The inference process in PCI faces the challenge of generalizing from sparse temporal samples. The model F Θ generates an accurate point cloud P P red t=j ∈ R N ×3 for time t = j based on P t=i and time frame at T t=i and the time frame at T t=j . This demands strong interpretability and 4D modeling capabilities from the model to accurately predict the interpolated point cloud from minimal information. IDEA-Net [2] addresses the correlation between two input frames by learning a one-to-one alignment matrix and refining the linear interpolation results using a trajectory compensation module. However, for large-scale, occluded autonomous driving point cloud scenes, IDEA-Net's one-to-one correspondence assumption struggles to accurately predict the point cloud at interpolated timestamps from sparse temporal samples.For the challenges of point cloud frame interpolation, we propose a novel 4D spatio-temporal modeling method based on Gaussian representations, NeuroGauss4D-PCI. It first captures the geometric structure of point clouds through iterative Gaussian soft clustering, providing structured representations for unordered data. Then, the temporal radial basis function Gaussian residual module interpolates over discrete time steps, learning the continuous dynamics of Gaussian parameters to capture the spatio-temporal evolution of point clouds. The 4D Gaussian Deformation Field further models the spatio-temporal variation trends of Gaussian parameters, effectively overcoming the limitations of linear interpolation. Finally, the attention fusion module adaptive fusion of latent neural representations and explicit geometric representations, enhancing the modeling capability of spatio-temporal correlations. As shown in Fig. 1 , this compact and efficient method demonstrates superior point cloud sequence generation performance on multiple datasets.Contributions: • We propose NeuroGauss4D-PCI, a novel 4D spatio-temporal modeling method for point cloud frame interpolation. It captures geometric structures through iterative Gaussian soft clustering, providing structured representations for unordered point clouds, while adaptively fusing the latent neural field features. • A Temporal Radial Basis Function Gaussian Residual module is designed to interpolate over discrete time steps, learning continuous Gaussian parameter dynamics to effectively model the spatio-temporal evolution of point clouds. • An innovative 4D Gaussian deformation field is introduced to model the spatio-temporal variations of Gaussian parameters, achieving smooth and realistic point cloud deformations through 4D Gaussian graph convolutions and Gaussian representation pooling. • Our method outperforms others on multiple datasets, generating accurate point cloud sequences with minimal errors. It can be easily extended to tasks like lidarcamera temporal synchronization and point cloud densification. 
related work: 
 Neural Field In computational geometry, latent neural fields [12; 11; 13] encode complex 3D scenes with the principle f Θ (ϕ(χ)) = o. χ denotes the input parameters, such as coordinates in a 3D space. The function ϕ is a feature transformation applied to χ, which enhances the neural network's ability to model complex patterns by providing a high-dimensional representation of the input space. The neural network f Θ , parameterized by Θ, then maps these transformed features to the output o.This output can represent various aspects of the 3D scene, such as color [14] , density [15] , or surface normals [16] , depending on the task. GPNF [17] presents a method using neural fields for efficient and topology-flexible geometry processing, outperforming traditional mesh methods in shape editing tasks. NSFP [11] introduces a novel implicit regularizer based on neural scene flow priors, utilizing coordinate-based MLPs for robust scene flow optimization. Implicit neural representations [18] ensure differentiable interpolation for 3D point clouds, facilitating end-to-end learning [19] , yet their performance is limited by a dependency on high-quality training data [9] and computational demands [20] , with inherent trade-offs in interpretability.Recent studies have demonstrated that 3D Gaussian Splatting (3D GS) [21; 22; 23] leverages an explicit radiance field for efficient:EQUATIONwhere I(p) represents the rendered intensity or color at pixel location p, N is the total number of 3D Gaussian models in the scene, w i is the weight of the i-th Gaussian model affecting its contribution to the final image, G(p; µ i , Σ i ) is the 3D Gaussian function defined as exp-1 2 (p -µ i ) T Σ -1 i (p -µ i) , µ i is the center of the Gaussian representing the position in 3D space, and Σ i is the covariance matrix that determines the shape and scale of the Gaussian distribution. 3DGS [21] innovatively employs 3D Gaussians to represent scenes, optimizing volumetric radiance fields with minimal computational waste, enabling high-quality real-time novel-view synthesis. Recent works [24; 25; 26] focused on creating 4D scene representations using Gaussian splatting from video to model geometry and appearance changes across frames. Few explored modeling temporal 3D motion changes from raw, sparse 3D point clouds, requiring tracking spatial coordinates over time and dynamically adjusting Gaussian parameters to accurately reflect temporal deformations.Point Cloud Interpolation (PCI) Existing PCI models are categorized into three types: based on 3D scene flow [10; 20; 27; 8; 28] , neural fields [11; 3] , and feature alignment [29] or trajectory regression [2] . 3D scene flow methods estimate correspondences between frames and use linear interpolation for intermediate frames. PointINet [8] calculates bidirectional 3D scene flow to warp point clouds and insert new frames. IDEA-Net [2] applies aligned feature embeddings to manage trajectory nonlinearities, enhancing interpolation. NeuralPCI [3] implicitly learns the nonlinear temporal motion characteristics of multi-frame point clouds through a neural field. However, these methods struggle with modeling 4D geometric deformations in point cloud sequences, impacting robustness. Our method combines 4D neural fields and Gaussian deformation fields to effectively model and predict dynamic point cloud changes, capturing nonlinear motion and deformation accurately.3 The AlgorithmQuestion: Does the paper discuss the limitations of the work performed by the authors? 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overview of 4d neural field and gaussian deformation field: 
 As depicted in Figure 2 , temporal point cloud coordinates are encoded using Fourier basis functions to encapsulate periodic information as follows:EQUATIONTo enhance the modeling capability for periodic features, the original coordinate features are augmented by incorporating their sine and cosine values. Features from Posenc(P|(x, y, z, t)) are further encoded using a 4D neural field. The iterative Gaussian cloud soft clustering module employs Gaussian statistics to extract characteristics such as mean, covariance and geometric features from the point cloud, offering a statistical description of internal dynamic changes. Furthermore, the proposed temporal radial basis function Gaussian residual (RBF-GR) module employs radial basis functions to interpolate these Gaussian parameters across different time steps. The designed 4D Gaussian deformation field module learns the temporal evolution of point cloud Gaussian parameters, generating a continuous spatiotemporal deformation field. Finally, an fast latent-geometric fusion module adaptively fusion features, enabling the generation of point clouds through a prediction head. (T pred ) (E T ) ∑ 𝒄𝒐𝒗 𝜇 𝜱 𝒇𝒆𝒂𝒕 Sec. 3.3 Q K V t=6 t=7 t=8F G = Dgcnn(P[A == i]) ▷ Eq. 4. 15: F G att = SelfAttention(F G ) ▷ Eq. 5. 16: Φ G f eat .append(F G att ) 17: end forThe iterative Gaussian cloud soft-clustering representation, given input point coordinates P, number of Gaussians M , consists of three steps: 1) Soft-clustering of point clouds to Gaussian distributions is achieved by introducing soft assignment weights S ij , where S ij represents the probability of the i-th point belonging to the j-th Gaussian, calculated based on the Euclidean distances D ij between points and Gaussian centers C, overcoming the limitations of traditional hardclustering methods. 2) The Gaussian means µ and covariance matrices Σ are iteratively optimized over κ iterations, with µ updated by minimizing the distances D ij and Σ calculated based on the distribution of points P m relative to the Gaussian means, adaptively fitting the spatial distribution of point clouds.3) Multi-scale Gaussian features Φ G f eat are extracted by applying DGCNN [30] and selfattention mechanisms to points assigned to each Gaussian, learning rich local and global features. Finally, the point cloud is assigned to the corresponding Gaussians through the index matrix A.The raw point cloud coordinates lack topological structure and contain noise and redundancy. We propose an iterative Gaussian soft-clustering point cloud representation module (Algorithm 1) that converts point clouds into structured Gaussian representations. The module introduces DGCNN [30] (Eq. 4) to extract local geometric features, where x i is the i-th point, x j is its k-nearest neighbor, Γ graph is the graph convolution operation. θg is the convolution parameter. ⊎ represents tensor concatenation; Self-Attention (Eq. 5) to capture global contextual information, where W Q , W K , W V are the weight parameters of the query, key, and value matrices, respectively, and d k is the dimension of the key matrix. t1→t2 , and features ∆Φ f eat t1→t2 between time t 1 and t 2 . The covariance matrix Σ t2 is then updated using the learned rotation residual. W rbf = σ sof tmax (M LP (Φ f eat t1 )), where W rbf denotes the attention weights for RBF activations, adaptively adjusted based on Gaussian features. The Temporal-RBF-GR employs radial basis functions (RBFs) with learnable centers c i and scales s i to capture the temporal evolution of Gaussian parameters.EQUATIONEQUATIONExcept for Gaussian cloud representation, we introduce 4D neural fields, parameterizing a continuous, latent spatio-temporal field function using multi-layer perceptron (MLP) networks, as shown in Fig. 2 . This maps low-dimensional spatio-temporal coordinates (x, y, z, t) to a high-dimensional latent feature space, representing the motion and changes of point clouds in the spatio-temporal domain.Compared to Gaussian point cloud representations, deep networks can fit more complex non-linear spatio-temporal correlations.Point cloud data from LiDAR includes noise, which, while not affecting subsequent perception tasks, impedes the learning of temporal consistency. We apply statistical outlier removal [31] during preprocessing to eliminate noise by calculating each point's average distance to its neighbors and identifying outliers using global mean and standard deviation. Experiments show (Table 2 ) this method markedly decreases interference during network training.This module accomplishes the interpolation and updating of Gaussian distribution parameters in the continuous time domain, which can be primarily divided into three parts (Fig. 3 ): RBF Activator:The RBF Activator converts discrete time steps into continuous time representation by mapping scalar time t to radial basis function activation values ζrbf i (t). RBF networks encode time locally, with each kernel function centered at a specific time c i and exponentially decaying activations. The RBF activation values ζrbf i (t) reflect the similarity between time t and center times c i , enabling smooth interpolation of Gaussian parameters. Gaussian Interpolator: The Gaussian Interpolator interpolates the mean µ t and rotation matrix R t of the Gaussian distribution using RBF activation values. Gaussian parameters estimated at discrete time steps are interpolated to obtain parameter residuals ∆µ (θ) t1→t2 and ∆R (θ) t1→t2 from time t 1 to t 2 . The interpolation process exploits the linearity and closure property of Gaussian distributions. Feature Interpolator: The Feature Interpolator interpolates Gaussian features using RBF activation values to obtain a continuous feature representation in time. The module enables smooth updating of Gaussian distribution parameters in continuous time, allowing effective fitting and generation of time-varying 3D point cloud sequences.The temporal Gaussian graph convolutional (TG-GCN) deformation field plays a key role in capturing the spatiotemporal features from Gaussian point cloud representations (Fig. 4 ). It takes as input Based on the features extracted from the deformation field, the motion field head and deformation field feature head respectively predict the motion flow and feature field (Fig. 4 ), with the core principle being the learning of a continuous deformation field, utilizing the time information in the Gaussian point cloud representation and the spatial dependencies captured by graph convolution, to achieve smooth and realistic deformation of the point cloud, enabling accurate modeling of 4D point cloud sequences and interpolation between time steps.The Gaussian representation pooling module first projects the original point cloud coordinates and latent features onto Gaussian ellipsoids centered at each point, leveraging the previously established index mapping between the points and the Gaussian ellipsoids (Sec. 3.2). It then applies maxpooling to extract the most prominent features within each Gaussian ellipsoid, enhancing the model's perception of deformation flow and features. However, this Gaussian ellipsoid projection and maxpooling operation inevitably leads to a certain degree of information loss. To compensate for the potential information loss in the feature extraction process of the Gaussian representation pooling module, we further introduce a neural field module to learn and refine the point cloud deformation flow and feature expressions.The proposed module fuses the complementary advantages of the latent features F L from point cloud latent representations and the geometric features F G from 4D Gaussian deformation fields via an attention-based mechanism. F L serves as the query, while F G serves as the key and value. Through linear projections (W q , W k , W v ), the features are mapped into a shared latent space. An attention score is computed between the projected query and key, which is then normalized via softmax to obtain attention weights. These weights adaptively assign importance to different feature components, enabling effective fusion. The weighted sum of the projected value features is then computed, integrating multi-modal information. Finally, a residual connection with the original query F L yields the fused representation FL&G , preserving temporal information:EQUATIONThe module's efficiency stems from its simple linear operations and avoidance of iterative or recurrent mechanisms, enabling fast computation. Furthermore, the global attention mechanism provides a comprehensive receptive field, enhancing representational capacity compared to naive concatenation. By adaptively fusing geometric and temporal features, the module improves the model's expressiveness for capturing complex spatio-temporal patterns in large-scale dynamic LiDAR scenes.The 3D Point Cloud Prediction head accurately predicts the point cloud at the target time step by fusing multi-source spatio-temporal information, including features FL&G from the 4D neural field and Gaussian deformation field, time encoding Posenc(T ) representing the target time step, and the initial prediction f low + P. This fusion provides rich spatio-temporal priors, enabling the model to reason about the target point cloud's structure and motion patterns. Through a lightweight MLP, the fused features are directly mapped to the residual flow ∆f * , avoiding prediction from scratch and leveraging the initial interpolated flow field based on the Gaussian deformation field:EQUATIONAdding ∆f * to the current frame accurately predicts the point cloud at the target time step.The proposed NeuroGauss4D-PCI optimizes the weights of 4D neural fields and Gaussian deformation fields using three self-supervised losses, eliminating the need for labeled data. We employ a cumulative temporal loss to refine network parameters, facilitating efficient spatiotemporal modeling of 4D point clouds while ensuring smoothness and distribution consistency.Chamfer Distance Loss: Measures the bidirectional distance between the predicted point cloud P * 1 and the ground truth P 2 , promoting proximity between the predicted and ground truth points.EQUATION)Smoothness Constraint Loss: Enhances the smoothness of the predicted inter-frame flow field ∆f * , where N (∆f * i ) denotes the neighbors of ∆f * i . It encourages similarity in flow vectors among neighboring points, ensuring smoother transformations in large-scale scenes. This loss is not required when modeling Dynamic Human Bodies.EQUATION)Earth Mover's Distance: Calculates the minimum cost to move points from P * 1 to match the distribution of P 2 , where T P1→P2 is the optimal transport map. It promotes a distribution in the predicted point cloud that resembles the ground truth.EQUATION)The overall loss integrates these functions across all time steps t i , t j and temporal frames b i , b j , weighted by λ 1 , λ Datasets: Dynamic Human Bodies (DHB) dataset [2] contains 14 sequences of non-rigid human motions. The NL-Drive [3] dataset captures large-scale motion scenes from KITTI [32] , Argoverse 2 [33] , and Nuscenes [34] for autonomous driving. KITTIs [35; 32] and KITTIo [36; 32] are two versions of the KITTI Scene Flow dataset with and without occlusion masks, respectively.Metrics: Like existing methods [3; 2; 8], we use Chamfer Distance (CD) and Earth Mover's Distance (EMD) to measure the consistency between predicted and ground truth point clouds. For the 3D scene flow task, we employ two error metrics EP E 3D and Outliers and two accuracy metrics, ACC S and ACC R , to quantify performance.Experimental Setup: We sample the input points to 1024 for object-level scenes and 8192 for autonomous driving scenes. NeuroGauss4D-PCI consists of 5 components, as shown in Table 4 . Refer to the supplementary materials for the parameter settings of each component and more experimental details. 
result comparison on point cloud interpolation: 
 Table 1 demonstrates the superior performance of the proposed method, consistently outperforming others on CD and EMD metrics, achieving near-zero errors for sequences like Squat and Swing. The dynamic soft Gaussian representation effectively models non-rigid deformations in human motions by transforming raw point clouds into multiple Gaussian distributions, with radial basis functions learning their temporal residuals. Fusing 4D neural field features preserves per-point detail.The extremely low errors on the DHB dataset validate the model's excellence in non-rigid motion modeling and point cloud interpolation. Notably, the proposed method achieves the best performance with significantly fewer parameters (0.09M) compared to methods like PointINet [8] (1.30M) and NeuralPCI [3] (1.85M). This is attributed to the efficient Gaussian representation and deformation field design, enabling superior performance through a more compact model.On large-scale autonomous driving LiDAR datasets [3] , NeuroGauss4D-PCI demonstrates exceptional temporal point cloud prediction/interpolation performance, significantly outperforming uni-directional 3D flow [11; 10] and neural field method [3] , as shown in Table 2 . It effectively handles challenges in temporal LiDAR scenes, such as large-scale non-linear motions, occlusions, and non-uniform data distributions (Fig. 1 , and Fig. 5 ). NeuroGauss4D-PCI models the 4D temporal point cloud as a Gaussian deformation field with continuous and differentiable Gaussian representations, ensuring smooth interpolation and robust predictions. Temporal Gaussian graph convolutions capture local and global spatio-temporal correlations, enabling fine-grained motion prediction. Notably, Gaussian representations offer parameter efficiency, expressing complex geometric shapes and non-linear motions smoothly with fewer parameters compared to neural field methods.To demonstrate the effectiveness of the proposed point cloud interpolation model in capturing complex deformations and transient motion patterns between two frames, we evaluated its performance on 3D scene flow estimation using the Scene Flow KITTI dataset (both non-occluded [35; 32] and occluded [36; 32] ), using error and accuracy metrics. As shown in Table 3 , our method outperformed approaches based on feature pyramids (PointPWC [37] , Bi-PointFlow [38] ), complex 3D point cloud transformers (PT-FlowNet [39] , GMSF [40] ), and self-supervised learning (SPFlowNet [41] , SCOOP + [42] ) across multiple metrics. Unlike other methods that require learning inter-frame correspondences, NeuroGauss4D-PCI only needs a single point cloud and target timestamp to predict the corresponding point cloud during inference. In the learning phase, NeuroGauss4D-PCI accurately models complex non-rigid deformations and details in temporal point clouds by integrating latent neural fields, iterative Gaussian representations, and 4D deformation fields. Its compact iterative Gaussian representations significantly reduce parameters while effectively capturing continuous point cloud changes by learning temporal residuals of Gaussian distribution parameters through radial basis functions, exhibiting superior capability in inter-frame point cloud matching. Table 4 shows the effect of each component on model performance. Using just the neural field, the Chamfer Distance (CD) measures 0.58×10 -3 on the DHB dataset and 1.06 on the NL-Drive dataset, showcasing its capacity to capture spatio-temporal features. Adding the Gaussian point cloud representation slightly lowers the CD by 0.01 and 0.03 on DHB and NL-Drive, respectively, indicating limited benefits from this integration. Removing the neural field and relying solely on the Gaussian point cloud and T-RBF-GR module significantly worsens performance, emphasizing the neural field's critical role in modeling 4D spatio-temporal point clouds. However, integrating the 4D Gaussian Deformation Field with the Gaussian representation markedly enhances performance, with a 14% and 19% decrease in CD and a 17% and 6.5% reduction in Earth Mover's Distance (EMD) on the two datasets, respectively. This highlights the deformation field's effectiveness in accurately representing dynamic variations. Adding the T-RBF-GR module further improves performance, showcasing its utility in addressing dynamics and temporal correlations. By fusing the latent features and the explicit 4D deformation field features, we effectively reduce errors in point cloud prediction. The Fast-LG-Fusion module, in particular, achieves the best performance, attaining the lowest CD and EMD across datasets, except for a slight 0.02 increase in CD on the DHB dataset compared to direct concatenation. 
conclusion: 
 NeuroGauss4D-PCI is proposed for accurate 4D temporal point cloud modeling and interpolation. 4D neural fields encode latent spatio-temporal features, and an iterative Gaussian cloud representation structures point clouds. A temporal RBF Gaussian residual module smoothly updates Gaussian distribution parameters. 4D Gaussian deformation fields leverage temporal graph convolutions for robust dynamic point cloud modeling. Efficient fusion of latent spatio-temporal and robust Gaussian deformation features enables 3D point cloud prediction. Evaluated on multiple datasets, the method demonstrates significant superiority over existing approaches, particularly in complex dynamics scenarios. Ablation studies validate the contribution of each module to overall performance.Broad Implications: Our work is the first to successfully combine radial basis functions for predicting 3D Gaussian distribution residuals and introduce 4D Gaussian deformation fields into the temporal point cloud prediction task. As the supplementary materials indicate, this is a widely applied yet underexplored issue. Finally, we hope that the proposed NeuroGauss4D-PCI can provide valuable assistance to the community in exploring temporal point cloud understanding and modeling.Limitations: 1) Interpretability: The integration of various features and the opacity of deep neural networks pose challenges in understanding the decision-making process and the fundamental principles behind model predictions. 2) Efficiency: Similar to NeRF [43] , nearly 90% of the time is consumed by runtime scene optimization, with inference accounting for only about 10% of the time. 
safeguards: 
 Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?Answer: [NA]Justification: [NA] Guidelines:• The answer NA means that the paper poses no such risks.• Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters. • Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images. • We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort.12. Licenses for existing assets Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?Answer: [Yes] Justification: We use some of the code from NeuralPCI [3] , and all the code in this repository uses Apache License 2.0. And we have correctly cited this work in the main paper.Guidelines:• The answer NA means that the paper does not use existing assets.• The authors should cite the original paper that produced the code package or dataset.• The authors should state which version of the asset is used and, if possible, include a URL. • The name of the license (e.g., CC-BY 4.0) should be included for each asset.• For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided. • If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset. • For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided. • If this information is not available online, the authors are encouraged to reach out to the asset's creators. 13. New Assets Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets? Answer: [Yes] Justification: We provide detailed documentation on the model architecture, training procedures, and evaluation protocols in the main paper and supplementary material. In addition, we also release the code and pre-trained models in the supplementary material, enabling reproducibility and promoting future research in this area. We carefully anonymize assets during the submission process and will provide a publicly accessible zip file upon publication.Guidelines:• The answer NA means that the paper does not release new assets.• Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates. This includes details about training, license, limitations, etc. • The paper should discuss whether and how consent was obtained from people whose asset is used. • We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution. • For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review. 
 
 Appendix: Dynamic Human Bodies (DHB) [2] dataset consists of point cloud sequences of 14 non-rigid deformable human motions. Six sequences-Longdress, Loot, Redandblack, Soldier, Squat 2, Swing are designated for testing, and the remaining eight for training. The NL-Drive dataset [3] , integrating data from KITTI odometry [32] , Argoverse 2 sensors [33] , and Nuscenes [34] , focuses on capturing large-scale motion scenes and is divided into training, validation, and test sets in a 14:3:3 ratio. It specifically targets hard samples with significant inter-frame motion and large self-motion, employing detailed 6-DOF pose transformations and a filtering mechanism to support the training and testing of advanced autonomous driving algorithms. KITTIs [35; 32] and KITTIo [36; 32] , two versions of the KITTI Scene Flow dataset processed for 3D point cloud scene flow evaluation, exclude occluded points in the former, containing 142 training scenes, while the latter includes all points with occlusion masks, featuring 150 training scenes. Heterogeneous sensors like LiDAR and cameras often suffer from asynchronous data acquisition [44; 45] , posing challenges in fusing their complementary data streams. As illustrated in Fig. 6 , NeuroGauss4D-PCI offers a powerful solution by enabling the interpolation of arbitrary point cloud frames from continuous point clouds, facilitating precise multi-sensor time synchronization. This capability unlocks seamless integration of data from multiple sensors, enhancing the robustness and accuracy of perception systems in dynamic environments.Moreover, NeuroGauss4D-PCI can be effectively applied to 4D automatic annotation tasks, assigning point-wise labels to unlabeled intermediate frames based on sparse 3D labels. This automated labeling process significantly reduces the time and effort required for manual annotation, a critical bottleneck in many computer vision applications. Consequently, it accelerates the development and deployment of systems relying on labeled point cloud data.Furthermore, NeuroGauss4D-PCI demonstrates remarkable accuracy in LiDAR point cloud densification, generating high-resolution, dense point clouds from sparse inputs. Accurate dense point clouds serve as invaluable data sources for a wide range of autonomous driving perception tasks [46; 47; 48; 49] , including depth estimation [50; 51] , occupancy prediction [52] , and 4D scene reconstruction [53; 54; 55] . The ability to obtain high-fidelity representations of the environment from limited sensor data highlights the immense potential of NeuroGauss4D-PCI in enabling robust and reliable perception systems.In the qualitative comparison (Fig. 7 ) on the DHB dataset [2] , NeuroGauss4D-PCI demonstrates superior performance compared to state-of-the-art open-source models, including PointINet [8] , IDEA-Net [2] , and NeuralPCI [3] , in the task of temporal point cloud interpolation. The proposed method excels in capturing and modeling non-rigid human motions with high fidelity, as evident from the visual results. NeuroGauss4D-PCI represents the input point cloud as a set of Gaussian ellipsoids The overlap between green and pink points demonstrates our algorithm's accuracy in these complex scenarios, showcasing its robustness to data sparsity, occlusions, ambiguous temporal features from repeating structures, and large environmental displacements.architecture, combining Gaussian ellipsoid representations, temporal RBF residuals, 4D neural fields, and attention-based feature interpolation, endows it with a unique ability to accurately model and interpolate non-rigid point cloud sequences. The qualitative results on the DHB dataset demonstrate the method's superiority in handling complex human motions.The qualitative visualization (Fig. 8 ) on the NL-Drive dataset showcases NeuroGauss4D-PCI's exceptional performance in point cloud interpolation and prediction for autonomous driving scenarios. Compared to NeuralPCI, our method exhibits superior temporal consistency and local coherence, benefiting from the robust mathematical constraints imposed by the Gaussian parameterization. For unstructured and sparse point clouds, the 4D modeling approach enables more accurate predictions in sparse and unstructured regions. Additionally, NeuroGauss4D-PCI demonstrates robustness in handling occluded areas, as evident in the second row of the visualization.1. Claims Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? Answer: [Yes] Justification: The abstract and introduction accurately reflect the main contributions and scope of the paper. The paper proposes a novel point cloud interpolation method called NeuroGauss4D-PCI, which effectively represents point cloud data using iterative Gaussian clouds. The experimental section demonstrates and discusses the effectiveness of the proposed method. Guidelines:• The answer NA means that the abstract and introduction do not include the claims made in the paper. • The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers. • The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings. • It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper. 


Decision: This paper proposes a novel method for point cloud interpolation using Gaussian soft clustering and a 4D neural field. The experimental results on multiple datasets, including both indoor and outdoor environments, demonstrate the effectiveness of the proposed method.

Four reviewers (2x BA, 1x WA, 1x SA) acknowledged the strengths of this paper, and the authors' rebuttal successfully addressed most of the concerns raised by these reviewers.

Reviewer Z54i, however, provided a negative rating of Borderline Reject, concerning issues with "the poor color scheme of the figures," "the significance of the research direction," and "robustness to more challenging scenes." The authors responded to these concerns, but no further feedback was received from the reviewer.

After reviewing the paper, the reviews, and the rebuttal discussion, the AC has decided to recommend acceptance. For the camera-ready version, please improve the color scheme in the figures and revise the writing to enhance visibility and readability. Additionally, include the new experiments and discussions to further strengthen the manuscript.

Reviews: review: paper_summary: paper_summary: This paper proposes a method, NeuroGauss4D, to tackle the problem of 4d point cloud interpolation. NeuroGauss4D consists of the following 5 components:
1. "Iterative Gaussian Soft Clustering": a module to encode the scene to Gaussian representation with DGCNN features (map: X, Y, Z, T -> Gaussians(Mu, Cov, Feat)).
2. "4D Neural Field": a neural field with Fourier features and MLP to capture spatio-temporal features in latent space (map: X, Y, Z, T -> latent).
3. "Temporal RGB-GR": a module with temporal radial basis functions to interpolate Gaussian parameters in continuous time (map: Mu, Cov, Feat, T -> ΔMu, ΔCov, ΔFeat, ΔT).
4. "4D Gaussian Deformation Field": a module that take Gaussian parameters (Mu, Cov, Feat) and the latent features as input to predict the motion and feature deformation field (map: Mu, Cov, Feat, T -> deformation field).
5. "Fast-LG-Fusion": an attention mechanism that fuses the latent features (FL) and the geometric features (FG, from the deformation field) to predict the point cloud at the target time step (map: FL, FG, T -> point cloud).
strength_weakness: strength_weakness: 1. This method shows the application of dynamic Gaussian representation to the problem of 4D point cloud interpolation, which is a good idea. However, the execution of the idea seems to make the method overly complex. NeuroGauss4D-PCI's pipeline contains 5 different components, where a mixture of representations are used. Besides more hand-tuning for hyperparameters and longer training time, this complexity could make it hard to interpret the fundamental principles and the key components behind the full model.
2. It seems that the method is not very efficient. For example, the "Iterable Gaussian Soft Clustering" procedure shown in Algorithm 1 is only capable of taking one timestamp of point cloud (N, 3) as input, which means that this has to be done for each timestamp of the point cloud sequence.
3. Similar to the point above, the method might be limited on the size of the point cloud sequence that it can handle. For example, line 218 says "we sample the input points to 1024 for object-level scenes and 8192 for autonomous driving scenes". A point cloud of 1024 points in autonomous driving scenes is very small, which could limit the method's applicability to real-world scenarios. The authors can provide more information on how the method will perform when we have a larger point cloud size or a longer sequence of point clouds.
4. The author provides rich information about the method, which is good, but the clarity of the writing can be improved. Some of these are mentioned in the "Questions" section, and it would be nice to include them in future versions of the paper. There are other inconsistencies in the paper, for example:
    - Equation 1 uses 0-based indexing, but line 29 uses 1-based indexing.
    - Line 109 has an extra double quote.
    - Line 117 says "Iterative Gaussian Cloud Soft Clustering" but the Figure 2 uses "**Iterable** Gaussian ...".
    - Equations shall be written as part of a sentence, not as separate sentences (Eq. 5, 8, 9, 10, 11; and also Eq. 4).
questions: questions: 1. It would be good to clarify in paper the exact meaning of "Frame-1, Frame-2, and Frame-3" in Table 2. Does it mean that we fit the scene with frame {0, 4, 8, 12, ...}, and then we evaluate on frame {1, 2, 3, 5, 6, 7, 9, 10, 11, ...}, and we call frame {1, 5, 9, ...} as Frame-1, frame {2, 6, 10, ...} as Frame-2, and frame {3, 7, 11, ...} as Frame-3?
2. In Table 5, when we say "4 frames of point clouds sampled at regular 3-frame intervals", does it mean that for each scene, we only train on 4 frames {0, 4, 8, 12} and we evaluate on 9 frames {1, 2, 3, 5, 6, 7, 9, 10, 11}? If this is the case, why do we only train on these 4 frames but not fit the full point cloud sequence of the scene in the raw dataset?
3. In line 456, when we say "initial frame at time step 0 and the final frame at time step 1", what do the "0" and "1" refer to? For example, in equation 1, it is clear that there can be time steps larger than 1, e.g. time step 4, 8, etc. Is the "0" and "1" in line 456 referring to some form of normalized time step? So, what exactly is the time step gap (in seconds) between the training frame and evaluation frames?
4. From equation 1, it seems that the method only fits one model for one whole scene, which is good. Can you confirm this?
5. In line 476 we report the "average training time is ∼ 1.31 seconds per iteration". Per my understanding, this "iteration time" means the training iteration. It is not clear if the time includes "Iterative Gaussian Cloud Soft Clustering" (which has to be done for each time step), and other feature extraction steps. It would be more clear to give a high-level timing number, for example, could you report "how long does it take to fit a full of ~X training frames each containing ~Y number of points"?
limitations: limitations: Yes, the authors have discussed the above.
paper_summary: paper_summary: This paper proposes a novel method for point cloud interpolation, which aims to address complex non-rigid scenarios. It turns point clouds into 3D Gaussians via iterative soft clustering, and then utilizes several 4D spatio-temporal modules to fuse latent features with neural fields. Specifically, this paper employs the temporal radial basis function for Gaussian interpolation, the graph convolutional network for feature extraction and the attention module for feature fusion. Comprehensive experiments demonstrate the leading performance across indoor and outdoor datasets, along with a thorough ablation study for the proposed components.
strength_weakness: strength_weakness: - The proposed method is overly complex and redundant, which shows a trivial combination of existing modules and limited contribution. Moreover, the proposed modules lack strong motivation and necessity, and the interpretability of the intermediate features is difficult to clarify.
- According to the Supplementary Material, nearly two hours of optimization time is required for just four frames of input, which is much more than the previous method. The problem may lie in the design of proposed method, which is not suitable for an optimization-based approach, but rather a learning-based approach. For example, some feature extraction or fusion may not be meaningful or necessary for a single sample fitting.
questions: questions: - This paper may not be related to the work of 3DGS series. It just uses 3D Gaussians but has nothing to do with "splatting". Also, the proposed method is more similar to the super-point concept or the Gaussian Mixture Model in clustering and should be discussed.
- Does DGCNN use pre-trained models or is it trained from scratch? Retraining DGCNN for each optimization can be time consuming and may not be meaningful for fitting individual samples.
- In the Temporal RBF-GR Module, the 3D Gaussian needs to be interpolated, so how to ensure the correspondence? Also, is it feasible to visualize the Residual Gaussian to prove its reasonableness and validity?
- Quantitative comparisons of efficiency with other methods. It would be better to analyze the training/inference time for each module. In addition, can the proposed method be extended to more frames or more points of input, and what is the corresponding efficiency?



typo: 

- L136 - "Table4.2"?

- missing arrows of ACC_S and ACC_R in Table 3
limitations: limitations: - The entire framework can be further simplified and improve its efficiency.
- Additional supervision or constraints can be introduced for 3D Gaussians rather than only points.
- Use more challenging datasets beyond DHB. In Table 1, the results for both Squat and Swing sequences are 0.00, which may need to retain higher precision or change to other datasets.
paper_summary: paper_summary: The paper introduces NeuroGauss4D-PCI, a model designed to address the challenges of point cloud interpolation (PCI) by using Gaussian soft clustering and a 4D neural field to model complex non-rigid deformations in dynamic scenes. The model excels in capturing spatial and temporal dynamics from sparse data, showing superior performance on object-level and large-scale datasets.
strength_weakness: strength_weakness: 1. The color scheme of the figures in the article is very poor, making many details difficult to discern.

2. Point cloud interpolation is a relatively niche direction in point cloud analysis. The author should more clearly emphasize the significance of studying this issue.

3. The most related work reported in this work might be the CVPR 23 paper Neuralpci.  Does that mean no more recent related works?
questions: questions: 1. How does NeuroGauss4D-PCI ensure robustness against varied data sparsity in real-world scenarios?

2. Can the model handle rapidly changing dynamic scenes, such as explosions or quick animal movements?

3. What are the computational costs associated with training and inference, and how do they compare to simpler models?

4. How does the model perform under conditions of significant occlusion or sensor noise?
limitations: limitations: 1. The complex interactions between Gaussian fields and neural networks may hinder understanding and tuning of the model.

2. The high computational demand may limit deployment in time-sensitive applications.
paper_summary: paper_summary: The paper presents a novel model, “NeuroGauss4D-PCI,” for point cloud frame interpolation, a popular but challenging 3D computer vision task in real-world scenarios such as Lidar point cloud densification. The model outperforms existing PCI methods on the most popular benchmark datasets, DHB and NL-Drive, showcasing scalability to tasks like auto-labeling and point cloud densification.
strength_weakness: strength_weakness: -	There are no details of the experiment platform or discussion of the computational efficiency of the proposed method.  
-	The potential of NeuroGauss4D-PCI seems significant. But as we all know, NeuralPCI was said to be the best model for PCI until now. A more detailed comparison with this model could be crucial in providing valuable insights and strengthening the paper's contribution to the field.
questions: questions: I don’t have additional detailed questions for this paper, as I may not be good enough to understand all the details of the proposed algorithm.
limitations: limitations: I wonder if it is convincing that all are testing their work on only two open datasets used in this paper.
paper_summary: paper_summary: Point Cloud Interpolation (PCI) is the task of predicting intermediate point cloud features from a sparser representation, often to construct point cloud representations for intermediate time frames. Contrary to interpolation in other fields, point clouds are largely unstructured and do not preserve consistent information across different sequence frames, and thus cannot be addressed adequately with classical interpolation functions such as linear interpolation. To properly represent point clouds in a latent space for learned approaches, a method that respects the volumetric nature of point distributions is crucial for minimising noisy artifacts caused by the unordered nature of point clouds.

This paper proposes three new components of PCI learning architectures that address this necessity. First, a Gaussian clustering function is used to group point sets into distinct regions, which are less prone to misrepresenting unintended variations in point cloud data, and facilitates a necessary structure for PCI. To my understanding, with the sparse latent representations, interpolation is achieved by first using Radial Basis Functions (RBF) to provide an initial smooth estimate of the interpolated Gaussian cluster sequence, followed by a learned graph-based network to determine a final interpolation based on the composition of RBF sequences from every cluster.
strength_weakness: strength_weakness: Section 1: Introduction
- The introduction writing structure is quite unconventional, and repeats itself word for word in the contributions statement. I suggest the introduction of the proposed method be explained in more detail with regards to its overall architectural structure, to incorporate a consistent flow with the presentation of the method's components.
- The mentioned challenges of PCI are sporadic and inconsistent with the way PCI is introduced as a research problem. The use of a mathematical formula to introduce PCI takes away from the intended motivation of PCI-centric research, and is therefore difficult to link to each challenge. e.g. If solving Eq.(1) is the goal, what does the representation of "multiple unordered point clouds" have to do with it? This is unclear with the current presentation.
- Figure 1 is a mess of graphs and qualitative examples that belong as separate (sub)figures in the Experiments section. The introduction of Figure 1 in the beginning of the work is inappropriate. What is the expected ground truth for the right side figures? It is unclear what the right side figure is trying to demonstrate.

Section 2: Related works
- Like the introduction, the presented explanation is centred around explaining mathematical formulae, rather than intuitively presenting technical problems and expressing it as formulae if necessary. It is generally unnecessary for the formula to be understood in order to analyse the current state of research for each subsection. For this reason, by linking the motives of past research to solving an otherwise arbitrary formula, the flow of the related works section is unclear.
- L81: "for efficient:" appears to be in error.

Section 3: Methodology
- 1), 2), 3) are not labelled in the diagram itself, and thus can be difficult to associate.
- Figure 3 fails to visualise the purpose of each component, i.e. RBF Activator, Gaussian/Feature Interpolators, in the proposed method. It is generally expected that a flowchart is used to both illustrate the information flow of the system, and provide labels for the purpose, i.e. I/O, of each component. Mathematical equations are highly ineffective for this.
questions: questions: 1. Figure 2: Is it correct to call the output of the soft-clustering process "spheres"? The figure appears to describe "ellipsoids" rather than consistent radius spheres.
2. Outlier removal in Section 4.2/Table 2 appears to improve the results of the method quite significantly. How does it affect other state-of-the-art methods?
limitations: limitations: Yes, limitations of the method are discussed briefly in the manuscript regarding its applicability in practical scenarios.

Comments: title: Response to Reviewer ddrR
Response to Reviewer 2Dkp
comment: Dear Reviewer Z54i,

We greatly appreciate your time and expertise in reviewing our work. Your insightful feedback is invaluable for improving our paper. In our rebuttal, we have:

- Addressed all your concerns
- Improved figure clarity
- Emphasized research significance
- Highlighted recent related work

Our response demonstrates NeuroGauss4D-PCI's effectiveness across various scenarios through:

- Visualizations
- Tables
- Detailed computational cost analysis

**With only about 20 hours left in the author-reviewer discussion period, we kindly request your feedback. Your thoughts on any remaining concerns are crucial for our paper's improvement.**

Thank you sincerely for your time and effort.

Best regards,

The Authors of Paper #7315
Dear Reviewer xq5Q,

Thank you for your consideration and the increased score. We appreciate your feedback and will incorporate your suggestions in our final paper.

Best regards,

Authors of Paper #7315
I thank the author for the rebuttal responses, where most of my concerns are covered. Score increase 4->5.
Thanks for your efforts during the rebuttal, which addressed most of my concerns. For the efficiency improvements that make a big difference from the original version, I hope that the authors will clearly modify the relevant content in the final version and include as much as possible the additional issues mentioned in the review and the corresponding experiments. Based on this, I would keep my positive rating.
Thank you very much for your valuable feedback and recognition of our work. We will do our best to improve the quality of our work based on your suggestions, ensuring that the final version meets your expectations.
Finally, thank you again for your time and insights.
Thanks for the detailed response and the additional experiment. I am satisfied with the current version. and my recommendation will not change from the previous one.
Thank you very much for your response and valuable feedback. We deeply appreciate your recognition of our revised paper. Your constructive comments have greatly improved the quality of our research.

We will do our utmost to further refine the presentation of our paper according to your suggestions, ensuring that the final version meets your expectations.

Once again, thank you for your time and insights.
Thank you for the response. The new figures, explanations, and results are much needed improvements to the original submission. I am inclined to increase my rating in order to reflect the technically sound and adept contributions of the paper, provided that its presentation is acceptable in the final version.