DisC-GS: Discontinuity-aware Gaussian Splatting

Authors: haoxuan  qu
zhuoling  li
hossein  rahmani
yujun  cai
jun  liu
Venue: NeurIPS_cc_2024_Conference
Published: 2024-05-01 09:23:56

Abstract: Recently, Gaussian Splatting, a method that represents a 3D scene as a collection of Gaussian distributions, has gained significant attention in addressing the task of novel view synthesis. In this paper, we highlight a fundamental limitation of Gaussian Splatting: its inability to accurately render discontinuities and boundaries in images due to the continuous nature of Gaussian distributions. To address this issue, we propose a novel framework enabling Gaussian Splatting to perform discontinuity-aware image rendering. Additionally, we introduce a B\'ezier-boundary gradient approximation strategy within our framework to keep the ``differentiability'' of the proposed discontinuity-aware rendering process. Extensive experiments demonstrate the efficacy of our framework.

Main paper: 
 introduction: 
 Novel view synthesis aims to generate images accurately from novel viewpoints in a captured 3D scene. Its significance spans across diverse applications, such as autonomous driving [45] , virtual reality [14] , and 3D content generation [42] . Recently, for better tackling novel view synthesis, Neural Radiance Field (NeRF) [36] and a variety of NeRF-based methods [3, 4] have been proposed, which represent 3D scenes in an implicit manner as neural radiance fields. However, their general reliance on a heavy volume rendering mechanism often results in slow rendering speeds [30, 13] , limiting their practicality across real-world applications. While some methods [18, 19] have proposed to accelerate the rendering process of NeRF from different perspectives, they often achieve this at the expense of noticeably compromising the quality of the generated images [49] , which is evidently undesirable.More recently, Gaussian Splatting [30] , which explicitly represents the 3D scene as a collection of Gaussian distributions, has been proposed as an appealing alternative to NeRF. Specifically, rather than generating novel-view images through the time-consuming process of volume rendering, Gaussian Splatting enables images from novel viewpoints to be generated by simply splatting (projecting) [53, 54] these Gaussian distributions onto the image plane. By doing so, Gaussian Splatting achieves real-time rendering of novel-view images, while maintaining its rendered images to be of competitively high visual quality compared to NeRF-rendered ones. Due to its compelling capability, Gaussian Splatting has received lots of research attention [26, 13, 42, 49, 12, 21, 25] .Figure 1 : (a) Illustration of a ground truth image, containing numerous discontinuities and boundaries, that is expected to be rendered from a certain viewpoint of a 3D scene. We generate the boundary map in (a) utilizing the Canny algorithm [9] . (b) Illustration of Gaussian distributions projected onto the image plane. As shown, since Gaussian distributions are continuous, they can inevitably "pass over" the (hard) boundary represented by the curve. (c) Illustration of images rendered with and without applying DisC-GS. As shown, without DisC-GS, Gaussian Splatting can fail to accurately render boundaries. In contrast, applying DisC-GS ensures that boundaries and discontinuities in the image are properly rendered. More qualitative results are in Appendix B. (Best viewed in color.) However, in this paper, we argue that Gaussian Splatting may still be sub-optimal in accurately synthesizing novel views, due to its inherent weakness in representing (rendering) discontinuities and boundaries with its collection of continuous Gaussian distributions. Specifically, due to the general complexity of 3D scenes, the expected image to be rendered often contains numerous discontinuities and boundaries (as shown in Fig. 1(a) ). However, Gaussian Splatting represents each of its generated images using only continuous Gaussians projected onto the image plane. Considering this, as illustrated in Fig. 1 (b), the inherent continuity of Gaussian distributions can result in some parts of the distribution inevitably "passing over"("spilling over") the boundaries of sharp features in the image. This can lead to Gaussian Splatting rendering the sharp boundaries in the image with blurriness (as shown in Fig. 1(c )), which can significantly reduce the quality of the rendered image.Based on the above argument, in this paper, we aim to enable Gaussian Splatting to bypass its original intrinsic weakness, and render discontinuities and boundaries properly. However, this can be non-trivial owing to the following challenges: (1) Since 3D scenes generally can be complex, as illustrated in Fig. 1 (a), various different kinds of boundaries with diverse shapes can all exist in the image rendered from a certain 3D scene. Thus, it can be difficult to represent and render these diverse boundaries properly and seamlessly in a Gaussian-Splatting-based framework. (2) Meanwhile, recall that continuity serves as the prerequisite for a function to be differentiable. Thus, during the process of learning the 3D scene representation using Gaussian Splatting, how to maintain the "differentiability" of the process in existence of discontinuities, i.e., enabling the loss calculated over the rendered images that contain "discontinuous" (sharp) boundaries to properly guide the learning of the 3D scene representation, is also challenging. To handle the above challenges, in this work, we propose DisContinuity-aware Gaussian Splatting (DisC-GS), a novel framework that can for the first time, enable Gaussian Splatting to represent and render discontinuities properly in its image rendering process, which handles a key limitation of the original Gaussian Splatting technique. We illustrate the rendering process of our framework in Fig. 2 , and outline our framework as follows.Overall, to enable Gaussian Splatting to properly render discontinuities and boundaries, our framework introduces a "pre-scissoring" step. Specifically, for each Gaussian distribution representing the 3D scene, rather than directly rendering its entire 2D projection on the image plane, we first segment ("pre-scissor") the projected Gaussian distribution along the specified boundaries. However, achieving this requires representing boundaries with various shapes accurately. Here, we get inspiration from that, the cubic Bézier curve, conveniently represented by a group of four control points, has shown capable of efficiently parameterizing curves of various shapes with low computational complexity [16, 34] . Considering this, in our framework, we aim to use the cubic Bézier curves to represent boundaries. Specifically, we first introduce each Gaussian distribution representing the 3D scene with an additional attribute, which when projected onto the image plane, can serve as the control points of the cubic Bézier curves. After that, given a viewpoint, based on the control points projected onto the image plane corresponding to the viewpoint, we use the cubic Bézier curves formulated from these control points to represent the desired boundaries w.r.t. each projected Gaussian distribution. Finally, leveraging the derived boundaries, we can achieve discontinuity-aware image rendering through a modified α-blending function (as discussed in Sec. 4.1).Through the above process, we can render discontinuities and boundaries successfully (in the forward direction). However, the above process alone cannot be seamlessly integrated into the Gaussian Splatting pipeline. This is because, owing to the incorporation of the boundary information, the modified α-blending function now is no longer continuous everywhere. This can cause Gaussian Splatting, naively integrated with the above process, to become non-differentiable, and thus results in difficulties during the learning process of the 3D scene representation. To tackle this problem, in our framework, we further introduce a Bézier-boundary gradient approximation strategy, by which during backpropagation, we can enable gradients to properly pass through the modified α-blending function, and thus keep our framework to be still "differentiable". With the above designs properly involved, our DisC-GS framework can finally enable Gaussian Splatting to render discontinuities and boundaries properly, seamlessly addressing its original key intrinsic limitation.The contributions of our work are summarized as follows. 1) We proposed DisC-GS, an innovative framework for the novel view synthesis task. To the best of our knowledge, this is the first effort that enables Gaussian Splatting to represent and render boundaries and discontinuities properly in its image rendering pipeline, which tackles a key intrinsic limitation of Gaussian Splatting. 2) We introduce several designs in our framework to enable it to render images in a discontinuity-aware manner, while also to keep its "differentiability" in the presence of discontinuities. 3) DisC-GS achieves superior performance on the evaluated benchmarks.Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?Answer: [Yes] Justification: In both the abstract and the introduction, we have clearly mentioned the task (scope) and the contributions of this paper (e.g., in the last paragraph of the introduction).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. 
related work: 
 Novel View Synthesis. Owing to the wide range of applications, the task of novel view synthesis has received lots of research attention [23, 40, 39, 43, 24, 36, 3, 4, 5, 44, 50, 7, 11, 18, 19, 37, 30, 26, 13, 49, 20, 46, 22, 35, 33, 32, 48, 52, 17] . In the early days, with the emergence of CNN, different works have been proposed to leverage CNN in this task from different perspectives. Among them, Hedman et al. [23] proposed to use CNN to predict blending weights, and Sitzmann et al. [40] proposed to seek help from CNN in performing volumetric ray-marching. As time passed, NeRF tends to become a popular way in representing 3D scenes. Specifically, the original version of NeRF is first proposed by Mildenhall et al. in [36] and after it comes out, a variety of different NeRF-based methods have been further proposed, such as Mip-NeRF [3] , NeRF++ [50] , and Point-NeRF [44] . Despite the increased efforts, a weakness of NeRF-based methods can be that, to render novel-view images in high visual quality, they often still require a slow rendering process [30, 13] . This can negatively affect the usage of these methods in many real-world scenarios.Considering this, more recently, the Gaussian Splatting technique, which can render novel-view images in good quality while at the same time in real-time speed, as an attractive alternative to NeRF, has gained plenty of research attention. Specifically, Kerbl et al. [30] proposed to represent a 3D scene as a collection of 3D Gaussian distributions and made the first attempt to perform novel view synthesis using the Gaussian Splatting technique. After that, Huang et al. [26] pointed out that representing the 3D scene utilizing 3D Gaussian distributions can lead to a viewpoint inconsistency problem. To tackle this problem, they proposed to represent the 3D scene with 2D Gaussian distributions instead. Moreover, Cheng et al. [13] proposed to seek help from the classical patch matching technique to better guide the densification of Gaussian distributions, and Zhang et al. [49] formulated a new loss function in the frequency space to better regularize the learning process of Gaussian Splatting.Different from these existing Gaussian-Splatting-based methods that typically render complete Gaussian distributions during the image rendering process, we here argue that a key limitation of the original Gaussian Splatting technique lies in that, directly rendering the complete Gaussian distributions can lead boundaries and discontinuities in the image to be inaccurately rendered. Considering this, in this work, we propose to enable Gaussian distributions to be "pre-scissored" along desired boundaries before rendered. This for the first time, enables Gaussian Splatting to represent and render discontinuities and boundaries properly.Curve Representation. The idea of representing a curve in a parametric way has been studied in various tasks [34, 27, 38, 16, 28, 8] , such as lane detection [16] , trajectory prediction [27] , and text spotting [34] . Here in this work, we design a novel framework, which enables Gaussian Splatting to perform discontinuity-aware novel-view image rendering, via utilizing the cubic Bézier curves to parametrically contour the boundaries in the image plane.Question: Does the paper discuss the limitations of the work performed by the authors?Answer: [Yes] Justification: Following the reviewer's suggestion, we discuss the limitations at the end of the main paper.Guidelines:• The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper. • The authors are encouraged to create a separate "Limitations" section in their paper.• The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be. • The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated. • The authors should reflect on the factors that influence the performance of the approach.For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon. • The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size. • If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness. • While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations. 
preliminary: 
 Gaussian Splatting. Gaussian Splatting represents the 3D scene explicitly as a collection of anisotropic Gaussian distributions. In specific, in the collection, each Gaussian is defined with the following attributes: (1) its center µ ∈ R 3 , (2) its covariance matrixΣ ∈ R 3×3 , (3) its spherical harmonic (SH) coefficients c SH ∈ R 3×(k+1) 2representing its color from different viewpoints (where k denotes the order of SH), and (4) its opacity α ∈ R 1 . Regarding the covariance matrix Σ, it is important to ensure Σ remains positive semi-definite during the learning process of the 3D scene representation. To achieve this, Σ is expressed as Σ = RSS T R T , where R ∈ R 3×3 is the orthogonal rotation matrix of the Gaussian, and S ∈ R 3×3 denotes the diagonal scale matrix of the Gaussian.With the 3D scene represented as the collection of Gaussians defined in the above way, to render an image given a target viewpoint, inspired by [53] , each Gaussian in the collection is first projected onto the image plane corresponding to the viewpoint as:EQUATIONwhere µ 2D and Σ 2D respectively represent the center and the covariance matrix of the projected Gaussian distribution, W represents the viewing transformation matrix, P represents the projective transformation matrix, and J represents the Jacobian of the affine approximation of the projective transformation. After that, to perform image rendering on the image plane, for each pixel p of the image, its color C(p) is derived through an α-blending function as:C(p) = N i=1 c i β i i-1 j=1(1 -β j ), whereEQUATIONwhere N represents the number of projected Gaussians that overlap p, c i represents the color of the i-th Gaussian calculated from its corresponding SH coefficients, α i represents the opacity of the i-th Gaussian, µ 2D i represents the center of the i-th projected Gaussian, and Σ 2D i represents the covariance matrix of the i-th projected Gaussian. Note that, no matter whether Gaussian Splatting represents the 3D scene using 3D or 2D Gaussian distributions, the above equations can describe its rendering process consistently. In fact, as also mentioned in [26] , the difference between rendering images from 3D or 2D Gaussians can be reduced to that, when the scene is represented through 2D Gaussians, the scale matrix S of each of the 2D Gaussians should contain a zero column vector. In this work, we apply our framework to both 2D and 3D Gaussian Splattings, achieving performance improvements as shown in Tab. 2. Yet, as pointed out by [26] , using 3D Gaussians instead of 2D Gaussians to represent the scene can result in a viewpoint inconsistency problem. Thus, in Sec. 4, we first focus on explaining how our framework is applied to 2D Gaussian Splatting, in which we fix the last column of the scale matrix S of all the Gaussians to be a zero vector. We then discuss the application of our framework on 3D Gaussian Splatting in Sec. 4.3.A cubic Bézier curve is a parametric curve that can be formulated by leveraging a list of four ordered control points [ω 0 , ω 1 , ω 2 , ω 3 ] as:EQUATION) In the above equation, we can set t ∈ [0, 1] for B(t) to represent a segment of the curve that starts from ω 0 and ends at ω 3 . Alternatively, we can set t ∈ R to represent the entire curve. In this work, we set t ∈ R for B(t), as any segment of the curve may not be enough to represent the desired boundaries in the whole image plane. Note that when the four control points lie on the same straight line, the Bézier curve formulated by them would also be reduced to that straight line. Thus, besides representing smooth boundaries, the cubic Bézier curves, at their cross-interacting points, can also be used to represent the sharp corners (of human-made items) in the rendered image. 4 Proposed Method: DisC-GS Given a batch of source images of a 3D scene with their corresponding viewpoints, the goal of novel view synthesis is to generate novel-view images accurately. To handle this task, a common way is to first learn a 3D scene representation from the given source images. After that, the novel-view images can be rendered from the learned 3D scene. Recently, via representing the 3D scene through Gaussian distributions, Gaussian Splatting has enabled novel-view images to be generated both in real-time and with high rendering quality. It has thus attracted lots of research attention [30, 26, 13, 49 ].Yet, we here argue that Gaussian Splatting has a key intrinsic limitation: it may fail to render discontinuities and boundaries accurately. To tackle this problem, in this work, inspired by [54, 28] , we propose a novel framework named DisC-GS, which can seamlessly equip Gaussian Spatting with the discontinuity rendering ability. Specifically, during rendering images from the 3D scene, to render discontinuities properly, DisC-GS enables each Gaussian distribution projected onto the image plane to be first "pre-scissored" along certain desired boundaries before being rendered. However, such a "pre-scissoring" operation by itself can break the differentiability of the framework. Considering this, we further incorporate our framework with a Bézier-boundary gradient approximation strategy. Leveraging this strategy, during the learning process of the 3D scene, we can enable the gradient to properly backpropagate through the "pre-scissoring" operation. Below, we first describe the (forward) image rendering process of DisC-GS, and then explain the Bézier-boundary gradient approximation strategy.Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?Answer: [No] Justification: At this submission stage, we are sorry that we do not get enough approval to open-source our code.Guidelines:• The answer NA means that paper does not include experiments requiring code. • Providing as much information as possible in supplemental material (appended to the paper) is recommended, but including URLs to data and code is permitted. 
discontinuity-aware image rendering: 
 In the proposed DisC-GS, to perform discontinuity-aware rendering, we aim to preprocess Gaussian distributions projected onto the image plane by "scissoring" them along boundaries represented by cubic Bézier curves before rendering. To achieve this, we modify the conventional Gaussian Splatting technique through the following three steps.Introduction of an additional attribute. To facilitate the representation of cubic Bézier curves corresponding to the boundaries of each Gaussian distribution projected on the image plane, we introduce an additional attribute. We denote this attribute c curve ∈ R 4M ×2 , where M is a user-defined hyperparameter representing the number of Bézier curves. This attribute augments the original four attributes (discussed in Sec. 3) of each Gaussian distribution. Below, we introduce the physical interpretation of c curve . Specifically, for a certain 2D Gaussian distribution representing the 3D scene, denote the first column of its rotation matrix R to be r 1 and the second column of R to be r 2 . Over the 3D space, the 2D subspace that this Gaussian distribution lies in can be then described by a 2D coordinate system, which takes the center µ of the Gaussian as its origin, the direction of r 1 as the direction of its x-axis, and the direction of r 2 as the direction of its y-axis. Then for c curve of this Gaussian distribution, it can be understood as storing a total of 4M points in the above-defined coordinate system. Note that when these 4M points are projected onto the image plane (as discussed below), they can then serve as the control points of M cubic Bézier curves, which represent the desired boundary w.r.t. the current Gaussian distribution.Image plane projection of points in c curve . After introducing c curve to each Gaussian distribution that represents the 3D scene, given a viewpoint, we project points in c curve onto the image plane. Specifically, this is achieved in two steps: (1) We first transform each point (stored in c curve ) in the above-defined subspace coordinate system to the coordinate system of the 3D space as:EQUATION) where µ is the center of the Gaussian distribution. Note that here, since a column of a rotation matrix is already a unit vector, we can omit the normalization of r 1 and r 2 and directly transpose them. (2) After deriving c 3D curve ∈ R 4M ×3 storing the 4M points in the 3D space coordinate system, we can project each point in c 3D curve onto the image plane similar to what we have done in Eq. 1 as:c 2D curve [i] = P W c 3D curve [i], where i ∈ {0, ..., 4M -1}(5) where P represents the projective transformation matrix, and W represents the viewing transformation matrix. At this point, for each Gaussian projected onto the image plane via Eq. 1, we have gotten the control points of its desired cubic-Bézier-curves-represented boundary, stored in c 2D curve ∈ R 4M ×2 . Discontinuity-aware rendering. Finally, to perform discontinuity-aware image rendering, for each Gaussian distribution projected onto the image plane, we aim to first "scissor" the distribution along the M cubic Bézier curves formulated based on the 4M control points stored in c 2D curve . After that, we would like to only render the remaining parts of the distribution that are not "scissored out". To achieve this, assume that for each projected Gaussian distribution, we have built a binary indicator function g(•), which when passed with a pixel p on the image plane, can output 0 if the pixel is in the "scissored out" area of the distribution, and can output 1 otherwise. We can then perform discontinuity-aware rendering simply via modifying the α-blending function in Eq. 2 as:EQUATIONwhere g i (•) represents the indicator function w.r.t. the i-th projected Gaussian. Besides, same as in Eq. 2, N represents the number of projected Gaussians that overlap p, c i represents the color of the i-th Gaussian calculated from its corresponding SH coefficients, α i represents the opacity of the i-th Gaussian, µ 2D i represents the center of the i-th projected Gaussian, and Σ 2D i represents the covariance matrix of the i-th projected Gaussian. Note that via the above modified α-blending function, for Gaussians that no longer overlap with the pixel p due to the "scissoring" operation, we can zero out their contributions during calculating the color of p.Considering the above, the problem of enabling Gaussian Splatting to perform discontinuity-aware image rendering has now been reduced to building the indicator function g(•) for each projected Gaussian distribution based on its corresponding c 2D curve . Below, we discuss how we build g(•). For simplicity, we first consider the case where only one cubic Bézier curve exists per Gaussian distribution. In this case, denote the four control points of the curve ω 0 = (x 0 , y 0 ), ω 1 = (x 1 , y 1 ), ω 2 = (x 2 , y 2 ), and ω 3 = (x 3 , y 3 ). Then to build g(•), given a pixel p = (x p , y p ), we just need to determine (judge) whether p is on the inner side or the outer side of the curve. To achieve this, instead of directly leveraging the parametric representation of the cubic Bézier curve presented in Eq. 3, which may lead the judgment to be non-intuitive, we first leverage the implicitization technique in algebra [1] to represent the cubic Bézier curve in its implicit representation form as:EQUATIONwhere coefficients including γ xxx , γ xxy , γ xyy , γ yyy , γ xx , γ xy , γ yy , γ x , γ y , and γ 0 can all be obtained through basic arithmetic operations over the coordinates of the four control points of the curve in O(1) time complexity (more details are provided in Appendix C). Based on B imp (x, y), in the case where only one curve exists per Gaussian distribution, we can then build the single-curve indicator function g sc (•) intuitively and with O(1) time complexity as:EQUATIONAbove we introduce how we can build the indicator function g(•) as g sc (•) assuming that each projected Gaussian distribution is only "scissored" along one cubic Bézier curve. Here, in the case where M curves exist per Gaussian, for each Gaussian, we notice that a pixel can be regarded as in its "scissored out" area as long as the pixel is "scissored out" by at least one out of the M curves corresponding to the Gaussian. With this in mind, leveraging the g sc (•) function above, we can then define g(•) in cases where M > 1 as:EQUATIONLeveraging the indicator function g(•) defined in Eq. 9, along with the modified α-blending function in Eq. 6, we can then enable Gaussian Splatting to perform discontinuity-aware rendering.Above we discussed, how, in our framework, we perform discontinuity-aware rendering in the forward direction from the 3D scene representation to the 2D rendered image.Problems remain. Yet, this forward rendering process by itself cannot be seamlessly incorporated into the Gaussian Splatting pipeline. This is because of two reasons. Firstly, to enable the 3D scene representation to be properly learned from the source images of the 3D scene, Gaussian Splatting needs its rendering process to be (backward) differentiable. However, performing discontinuity-aware rendering leveraging the modified α-blending function in Eq. 6, with the discontinuous function g(•) in Eq. 9 incorporated, is no longer differentiable. Moreover, according to Eq. 8 and 9, g(•) is actually a piecewise constant function. Thus, even in its differentiable segments, the gradients of g(•) w.r.t. its inputs are always zero. In other words, even in segments of g(•) where its gradients are computable, these consistently zero gradients would fail to guide the update of the function g(•)'s inputs stored in c 2D curve , and consequently fail to guide the learning process of c curve introduced in Sec. 4.1.The big picture of our proposed strategy. To tackle the above problems and thus enable Gaussian Splatting to seamlessly render discontinuities, in our framework, we aim to further keep the "differentiability" of the whole discontinuity-aware rendering process. In other words, w.r.t. the discontinuous indicator function g(•) that is newly incorporated into the rendering process, we aim to approximate its gradients (partial derivatives) over the control point coordinates stored in c 2D curve , in a way that enables the approximated gradients to effectively guide the learning process of the 3D scene representation. To achieve this, inspired by [29] , we propose a Bézier-boundary gradient approximation strategy. Below, to ease our explanation of the strategy, we focus on discussing how we approximate ∂g ∂c 2D curve [0,0] , i.e., the partial derivative of the indicator function g(•) over the x coordinate of the first control point stored in c 2D curve . Note that the application of the strategy to the remaining coordinates stored in c 2D curve follows a similar process (more details are provided in Appendix D). Specifically, to approximate ∂g ∂c 2D curve [0,0] , based on the chain rule and according to Eq. 9, denoting [3] ; p) to be g 0 sc (p), we first have:g sc (c 2D curve [0], c 2D curve [1], c 2D curve [2], c 2D curveEQUATIONThen since g(•) is clearly differentiable over g 0 sc (•) based on its definition in Eq. 9, we can reduce our problem to approximate∂g 0 sc (p) ∂c 2Dcurve [0,0] , which is achieved through the following two steps. Determining if g 0 sc (p) is desired to be modified. Specifically, to approximate∂g 0 sc (p) ∂c 2Dcurve [0,0] , we first would like to determine, if the function g 0 sc (•) is desired to be modified at p or not. This is because, if g 0 sc (•) already outputs a satisfied value at p, we don't need to change c 2D curve [0, 0] to correspondingly modify g 0 sc (p). In other words, in such a case, we can simply set∂g 0 sc (p) ∂c 2Dcurve [0,0] to be zero. Denote the loss function used during the learning process to be L. Leveraging both ∂L ∂g 0 sc (p) and the current value of g 0 sc (p) as the conditions, below, we list the three situations in which g 0 sc (p) doesn't need to be further modified: (1) The first situation happens when ∂L ∂g 0 sc (p) = 0, which indicates that g 0 sc (•) given input p is already in an optimal state. (2) Besides, the second situation happens when ∂L ∂g 0 sc (p) > 0 and g 0 sc (p) = 0. Based on the gradient descent algorithm, this implies that, while we still hope the function g 0 sc (•) to output a smaller value at p, the function g 0 sc (•) already outputs its smallest allowed value. (3) Following the opposite logic of situation (2), the third situation happens when ∂L ∂g 0 sc (p) < 0 and g 0 sc (p) = 1. In this case, though we still want g 0 sc (p) to be larger, g 0 sc (•) at p already outputs its largest allowed value. In the above three situations, we can directly set∂g 0 sc (p) ∂c 2Dcurve [0,0] = 0 and omit the approximation performed in the next step.Approximating ∂g 0 sc (p) ∂c 2Dcurve [0,0] . Besides the above three situations, in the rest cases, for the value of function g 0 sc (•) at p to be properly modified based on the modification of the value of c 2D curve [0, 0], we aim to properly approximate the partial derivative∂g 0 sc (p) ∂c 2Dcurve [0,0] . To achieve this, recall that as a binary indicator function, g 0 sc (•) switches (modifies) its value at p between 0 and 1 only when its corresponding cubic Bézier curve passes through p. Considering this, below, we first identify: which value we should set (change) c 2D curve [0, 0] to be, so that the value switch of g 0 sc (•) at p can happen. To achieve this identification in an intuitive and analytical way, denoting p = (x p , y p ) and the desired value of c 2D curve [0, 0] to be ϕ, based on Eq. 3, we can first derive the following system of equations:EQUATION) In this system of equations, since x p , y p , and the coordinates in c 2D curve all have known values, we initially regard the second equation in the system as a cubic equation w.r.t. t, as t is now the only unknown variable in this equation. After solving this cubic equation and with t also known, we can then regard the first equation in the system as a cubic equation w.r.t. ϕ and solve it. Finally, by solving the above two equations (both in just O(1) time complexity), we obtain S ϕ as the set of all possible real number solutions for ϕ. Based on the solutions' scenarios within S ϕ , we approximate∂g 0 sc (p) ∂c 2Dcurve [0,0] in three different ways below.(1) The "no side" situation. The first situation happens when S ϕ = ∅. In this case, we simply set∂g 0 sc (p) ∂c 2Dcurve [0,0] = 0. This is because, the empty nature of S ϕ implies that, there exists no proper realnumber value that we can change c 2D curve [0, 0] to be, such that g 0 sc (p) can be desirably modified (i.e., either from 0 to 1 or from 1 to 0). We thus simply do not encourage c 2D curve [0, 0] to change. (2) The "single side" situation. The second situation occurs when all solutions in S ϕ lie on the same side of c 2D curve [0, 0] (i.e., all larger or all smaller than c 2D curve [0, 0]). In this situation, let ϕ denote the solution in S ϕ that is nearest to c 2D curve [0, 0]. Adjusting c 2D curve [0, 0] towards ϕ then implies the least-cost plan, facilitating the modification of g 0 sc (p) in a desired manner. With this in mind, to encourage c 2D curve [0, 0] to approach ϕ, inspired by previous studies [6, 10, 47, 29] , we approximate∂g 0 sc (p) ∂c 2Dcurve [0,0] via performing linear interpolation between c 2D curve [0, 0] and ϕ as:EQUATIONIn the above equation, we set ϵ = 10 -5 if ϕ -(c 2D curve [0, 0]) > 0 and we otherwise set ϵ = -10 -5 . ϵ here is a small number that is used to avoid the gradient exploding problem to happen when the distance between ϕ and c 2D curve [0, 0] is too short. (3) The "both sides" situation. The third situation happens when some solutions in S ϕ are on the left side of c 2D curve [0, 0], while other solutions are on the right side of c 2D curve [0, 0]. In this situation, we can achieve the desired modification of g 0 sc (p) via either moving c 2D curve [0, 0] to its left or right side. Thus, unlike the scenario described in the above situation (2) where we only consider ϕ from a single side of c 2D curve [0, 0], here, denoting ϕ 1 as the value that is nearest to c 2D curve [0, 0] from its left side, and ϕ 2 as the value that is nearest to c 2D curve [0, 0] from its right side, we approximateEQUATIONIn the above equation, we define g 0 sc (p) in the same way as in Eq. 12. Besides, both ϵ 1 and ϵ 2 are defined in the similar way as ϵ in Eq. 12.In summary, taking ∂g ∂c 2D curve [0,0] as an example, the above discussion explains how our proposed strategy approximates the gradient of g(•) with respect to the point coordinates stored in c 2D curve . With the incorporation of this strategy into our framework, we keep the "differentiability" of the whole rendering process, allowing Gaussian Splatting to seamlessly perform discontinuity-aware rendering.Above, we focus on describing how we use 2D Bézier curves in our DisC-GS framework and correspondingly apply DisC-GS on 2D Gaussian Splatting. Here in this subsection, we further describe how we use 3D Bézier curves in our DisC-GS framework and apply DisC-GS on 3D Gaussian Splatting. Specifically, the transition from 2D to 3D Bézier curves in DisC-GS requires only two minimal modifications. (1) Firstly, for each Gaussian representing the 3D scene, the control points of its Bézier curves are stored directly in the 3D spatial coordinate system rather than in a 2D subspace. Note that, this modification can be very simply made. Specifically, for each Gaussian in the 3D space in our DisC-GS framework, we only need to use c 3D curve ∈ R 4M ×3 instead of c curve ∈ R 4M ×2 to represent the control point coordinates of its 3D Bézier curves. In other words, for each 3D Gaussian, we only need to introduce it with c 3D curve instead of c curve as its new attribute. (2) Moreover, since we already directly introduce c 3D curve as the new attribute for each 3D Gaussian in our framework, during rendering, we omit Eq. 4 above in Sec. 4.1, which originally is used to acquire c 3D curve from c curve . Overall, the above two modifications are sufficient to incorporate DisC-GS with 3D instead of 2D Bézier curves.In DisC-GS, during training (i.e., learning the 3D scene representation from the source images), we follow a similar process as the typical Gaussian Splatting technique [30] . The involvement of the strategy introduced in Sec. 4.2 keeps the "differentiability" of our framework. During testing (i.e., image rendering), we use the discontinuity-aware image rendering process introduced in Sec. 4.1. 
experiments: 
 Datasets. To evaluate the efficacy of our proposed framework DisC-GS, following previous Gaussian Splatting works [30, 49] , we evaluate our framework on a total of 13 3D scenes, which include both outdoor scenes and indoor scenes. Specifically, among these 13 scenes, 9 of them are from the Mip-NeRF360 dataset [4] , 2 of them are from the Tanks&Temples dataset [31] , and 2 of them are from the Deep Blending dataset [23] . We also follow previous works [30, 49] in their train-test-split.Evaluation metrics. Following [30, 49] , we use the following three metrics for evaluation: Peak Signal-to-Noise Ratio (PSNR), Structural Similarity Index Measure (SSIM), and Learned Perceptual Image Patch Similarity (LPIPS) [51] .Implementation details. We conduct our experiments on an RTX 3090 GPU and develop our code mainly based on the GitHub repository [2] provided by Kerbl et al [30] . Moreover, we also get inspired by [35, 52, 46] during our code implementation, and make use of the LPIPS loss during our training process. Furthermore, for the newly introduced attribute c curve ∈ R 4M ×2 , we set its initial learning rate to 2e-4, and set the hyperparameter M to 3. Besides, in the densification procedure of our framework, when a Gaussian is cloned/splitted into two new Gaussians, we assign both the new Gaussians with the same attribute c curve as the original one.In Tab. 1, we compare our approach (applied on 2D Gaussian Splatting) with existing novel view synthesis methods evaluated on the same 13 3D scenes and report the PSNR, SSIM, and LPIPS results. Our framework consistently outperforms other methods on all three metrics and across various datasets, showing its effectiveness. We also show qualitative results in both Fig. 1(c ) and Appendix B. As shown, whether representing the 3D scene through 3D or 2D Gaussian distributions, the conventional Gaussian Splatting technique often struggles to render boundaries and discontinuities clearly and with high quality. In contrast, our framework can achieve good rendering quality, even in regions of the image containing numerous boundaries and discontinuities. This further underscores the efficacy of our approach.We conduct extensive ablation experiments on the Tanks&Temples dataset. More ablation studies w.r.t. the image areas with rich boundaries, the Bézier-boundary gradient approximation strategy, the hyperparameters, and the rendering speed of our framework are in Appendix A.  
conclusion: 
 In this paper, we have proposed an innovative novel view synthesis framework DisC-GS, which for the first time, enables Gaussian Splatting to properly represent and render discontinuities and boundaries in its image rendering process. Moreover, to keep the "differentiability" of our framework, we further introduce our framework with a Bézier-boundary gradient approximation strategy. Our framework consistently achieves superior performance across different evaluation benchmarks.Limitations. While our framework enables Gaussian Splatting to perform discontinuity-aware rendering, we acknowledge that same as existing Gaussian Splatting approaches, our framework still holds certain limitations, such as challenges in rendering large scenes.Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?Answer: [Yes] Justification: This paper has specified its data splits, its introduced hyperparameters, and other details in the Experiments section.Guidelines:• The answer NA means that the paper does not include experiments.• The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them. • The full details can be provided either with the code, in appendix, or as supplemental material. 
experiment statistical significance: 
 Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?Answer: [No] Justification: To make a fair comparison with the existing works, we follow their experimental settings which do not including any error bars.Guidelines:• The answer NA means that the paper does not include experiments.• The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper.• The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions). • The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.) • The assumptions made should be given (e.g., Normally distributed errors). • It should be clear whether the error bar is the standard deviation or the standard error of the mean. Guidelines:• The answer NA means that the authors have not reviewed the NeurIPS Code of Ethics.• If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics. • The authors should make sure to preserve anonymity (e.g., if there is a special consideration due to laws or regulations in their jurisdiction). 
broader impacts: 
 Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?Answer: [No] Justification: This paper focuses on addressing a key limitation of the Gaussian Splatting technique. To the best of our knowledge, it is not particularly tied to any particular deployments.Guidelines:• The answer NA means that there is no societal impact of the work performed.• If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact. • Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations. • The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster. • The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology. • If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML). 
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: To the best of our knowledge, this paper does not newly bring any risk for misuse and is thus not applicable to this question. 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 have discussed the licenses in our Appendix. 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. 
new assets: 
 Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?Answer: [NA] Justification: This paper does not release new assets.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. • At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file. 
crowdsourcing and research with human subjects: 
 Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?Answer: [NA] Justification: This paper does not involve crowdsourcing nor research with human subjects.Guidelines:• The answer NA means that the paper does not involve crowdsourcing nor research with human subjects. • Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper. • According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector. 
institutional review board (irb) approvals or equivalent for research with human subjects: 
 Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?Answer: [NA] Justification: This paper does not involve crowdsourcing nor research with human subjects.Guidelines:• The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.• Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper. • 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: In this section, we conduct more ablation experiments on the Tanks&Temples dataset. To achieve this, we evaluate our framework respectively over two parts of areas in each testing image. Specifically, for each testing image, we first pass the image over the Canny algorithm [9] followed by the dilation operation to highlight the areas with rich boundaries in the image (i.e., the areas that involve or surround the Canny-detected boundaries). After that, we let the first part (boundary-rich areas) include all the highlighted areas in each testing image, and let the second part (boundary-sparse areas) include all the rest areas in each testing image. To perform evaluation effectively over a part rather than the whole of each testing image, following [41] , we use MaskedSSIM as the evaluation metric. As shown in Tab. 4, compared to 2D Gaussian Splatting as a baseline, especially in the boundary-rich areas, our framework can achieve a significant performance improvement. This demonstrates the effectiveness of our method especially over image areas with rich boundaries. Impact of the "both sides" situation.To keep the "differentiability" of our framework, in this work, we propose a Bézier-boundary gradient approximation strategy. Specifically, in this strategy, when some solutions in S ϕ lie on the left side of c 2D curve [0, 0] while other solutions lie on its right side, we approximate∂g 0 sc (p) ∂c 2Dcurve [0,0] through Eq. 13 under the "both sides" situation, considering both the left and right sides of c 2D curve [0, 0] (with the "both sides" situation). To valid this design, we test a variant. In this variant (w/o the "both sides" situation), even when solutions in S ϕ exist on both the left and right sides of c 2D curve [0, 0], we still consider only the solution that is nearest to c 2D curve [0, 0], and approximate∂g 0 sc (p) ∂c 2Dcurve [0,0] through Eq. 12 under the "single side" situation. As shown in Tab. 7, our framework outperforms this variant. This shows the advantage of considering "both sides" when solutions in S ϕ lie on both the left and right sides of c 2D curve [0, 0]. Impact of the small numbers ϵ, ϵ 1 , and ϵ 2 . In our framework, during gradient approximation, to avoid the gradient exploding problem to happen, we add ϵ as a small number to the denominator part of Eq. 12, and we also add ϵ 1 and ϵ 2 to Eq. 13 in a similar way (with small numbers). To valid the efficacy of this design, we test a variant (w/o small numbers) in which we remove ϵ, ϵ 1 , and ϵ 2 from our gradient approximation process. As shown in Tab. 8, our framework involving these small numbers performs better than this variant, showing the efficacy of these small numbers. Impact of the number of Bézier curves per Gaussian M . In our framework, for each Gaussian distribution representing the 3D scene, we set the number of cubic Bézier curves M associate with the Gaussian to 3. As shown in Tab. 9, our framework achieves optimal performance when M is set to 3, and M = 3 is used in our experiments. Moreover, with different choices of M from 1 to 4, our framework consistently achieves good performance. This demonstrates the robustness of our proposed framework to this hyperparameter. Rendering time. In Tab. 11, we compare the rendering time of our framework with the existing NeRF-based method Mip-NeRF360 [4] , as well as two Gaussian Splatting baselines (i.e., 2D Gaussian Splatting and 3D Gaussian Splatting), on an RTX 3090 GPU in terms of seconds per image. As shown, our DisC-GS can achieve a competitive rendering time (speed) compared to the existing methods leveraging the conventional Gaussian Splatting technique, while obtaining much better performance. In this section, we present more qualitative results. Specifically, in Fig. 3 , we present images rendered by 2D Gaussian Splatting with and without applying our proposed framework DisC-GS; in Fig. 4 , we present images rendered by 3D Gaussian Splatting with and without DisC-GS. As shown, no matter representing the 3D scene through 2D or 3D Gaussian distributions, the typical Gaussian Splatting technique can fail to render boundaries and discontinuities clearly and with high quality. Yet, our DisC-GS framework, when applied, can achieve good rendering quality, even in regions of the image containing numerous boundaries and discontinuities. This further shows the efficacy of our approach.C Additional Details about Eq. 7 in the Main PaperIn Eq. 7 in the main paper, via leveraging the implicitization technique in algebra [1], we enable the cubic Bézier curve to be represented in its implicit representation form as:EQUATIONHere in this section, we discuss how we derive coefficients including γ xxx , γ xxy , γ xyy , γ yyy , γ xx , γ xy , γ yy , γ x , γ y , and γ 0 in Eq. 7 in the main paper (re-shown in Eq. 14 above). Specifically, denote the four control points of the cubic Bézier curve ω 0 = (x 0 , y 0 ), ω 1 = (x 1 , y 1 ), ω 2 = (x 2 , y 2 ), and ω 3 = (x 3 , y 3 ). To derive the coefficients in Eq. 14, following [1], we first define a set of intermediate coefficients as:ζ0 = x0; ζ1 = -3 × x0 + 3 × x1; ζ2 = -6 × x1 + 3 × x0 + 3 × x2; ζ3 = x3 -x0 -3 × x2 + 3 × x1; ζ4 = y0; ζ5 = -3 × y0 + 3 × y1; ζ6 = -6 × y1 + 3 × y0 + 3 × y2; ζ7 = y3 -y0 -3 × y2 + 3 × y1;After that, utilizing these intermediate coefficients, following [1], we can then compute coefficients in Eq. 14 as:γxxx = ζ7 × ζ7 × ζ7; γxxy = -3 × ζ3 × ζ7 × ζ7; γxyy = 3 × ζ7 × ζ3 × ζ3; γyyy = -ζ3 × ζ3 × ζ3; γxx = -3 × ζ3 × ζ5 × ζ6 × ζ7 + ζ1 × ζ6 × ζ7 × ζ7 -ζ2 × ζ7 × ζ6 × ζ6 + 2 × ζ2 × ζ5 × ζ7 × ζ7 + 3 × ζ3 × ζ4 × ζ7 × ζ7 + ζ3 × ζ6 × ζ6 × ζ6 -3 × ζ0 × ζ7 × ζ7 × ζ7; γxy = ζ1 × ζ3 × ζ6 × ζ7 -ζ2 × ζ3 × ζ5 × ζ7 -6 × ζ4 × ζ7 × ζ3 × ζ3 -3 × ζ1 × ζ2 × ζ7 × ζ7 -2 × ζ2 × ζ3 × ζ6 × ζ6 + 2 × ζ6 × ζ7 × ζ2 × ζ2 + 3 × ζ5 × ζ6 × ζ3 × ζ3 + 6 × ζ0 × ζ3 × ζ7 × ζ7; γyy = 3 × ζ1 × ζ2 × ζ3 × ζ7 + ζ3 × ζ6 × ζ2 × ζ2 -ζ2 × ζ5 × ζ3 × ζ3 -3 × ζ0 × ζ7 × ζ3 × ζ3 -2 × ζ1 × ζ6 × ζ3 × ζ3 -ζ7 × ζ2 × ζ2 × ζ2 + 3 × ζ4 × ζ3 × ζ3 × ζ3; γx = ζ2 × ζ3 × ζ4 × ζ5 × ζ7 -ζ1 × ζ2 × ζ5 × ζ6 × ζ7 -ζ1 × ζ3 × ζ4 × ζ6 × ζ7 + 6 × ζ0 × ζ3 × ζ5 × ζ6 × ζ7 + ζ5 × ζ1 × ζ1 × ζ7 × ζ7 + ζ7 × ζ2 × ζ2 × ζ5 × ζ5 + 3 × ζ7 × ζ3 × ζ3 × ζ4 × ζ4 + ζ1 × ζ3 × ζ5 × ζ6 × ζ6 -ζ2 × ζ3 × ζ6 × ζ5 × ζ5 -6 × ζ0 × ζ3 × ζ4 × ζ7 × ζ7 -4 × ζ0 × ζ2 × ζ5 × ζ7 × ζ7 -3 × ζ4 × ζ5 × ζ6 × ζ3 × ζ3 -2 × ζ0 × ζ1 × ζ6 × ζ7 × ζ7 -2 × ζ1 × ζ3 × ζ7 × ζ5 × ζ5 -2 × ζ4 × ζ6 × ζ7 × ζ2 × ζ2 + 2 × ζ0 × ζ2 × ζ7 × ζ6 × ζ6 + 2 × ζ2 × ζ3 × ζ4 × ζ6 × ζ6 + 3 × ζ1 × ζ2 × ζ4 × ζ7 × ζ7 + ζ3 × ζ3 × ζ5 × ζ5 × ζ5 + 3 × ζ0 × ζ0 × ζ7 × ζ7 × ζ7 -2 × ζ0 × ζ3 × ζ6 × ζ6 × ζ6; γy = ζ0 × ζ2 × ζ3 × ζ5 × ζ7 + ζ1 × ζ2 × ζ3 × ζ5 × ζ6 -ζ0 × ζ1 × ζ3 × ζ6 × ζ7 -6 × ζ1 × ζ2 × ζ3 × ζ4 × ζ7 -ζ1 × ζ1 × ζ1 × ζ7 × ζ7 -3 × ζ3 × ζ3 × ζ3 × ζ4 × ζ4 -ζ1 × ζ3 × ζ3 × ζ5 × ζ5 -ζ3 × ζ1 × ζ1 × ζ6 × ζ6 -3 × ζ3 × ζ0 × ζ0 × ζ7 × ζ7 + ζ2 × ζ6 × ζ7 × ζ1 × ζ1 -ζ1 × ζ5 × ζ7 × ζ2 × ζ2 -3 × ζ0 × ζ5 × ζ6 × ζ3 × ζ3 -2 × ζ0 × ζ6 × ζ7 × ζ2 × ζ2 -2 × ζ3 × ζ4 × ζ6 × ζ2 × ζ2 + 2 × ζ0 × ζ2 × ζ3 × ζ6 × ζ6 + 2 × ζ2 × ζ4 × ζ5 × ζ3 × ζ3 + 2 × ζ3 × ζ5 × ζ7 × ζ1 × ζ1 + 3 × ζ0 × ζ1 × ζ2 × ζ7 × ζ7 + 4 × ζ1 × ζ4 × ζ6 × ζ3 × ζ3 + 6 × ζ0 × ζ4 × ζ7 × ζ3 × ζ3 + 2 × ζ4 × ζ7 × ζ2 × ζ2 × ζ2; γ0 = ζ0 × ζ1 × ζ2 × ζ5 × ζ6 × ζ7 + ζ0 × ζ1 × ζ3 × ζ4 × ζ6 × ζ7 -ζ0 × ζ2 × ζ3 × ζ4 × ζ5 × ζ7 -ζ1 × ζ2 × ζ3 × ζ4 × ζ5 × ζ6 + ζ4 × ζ1 × ζ1 × ζ1 × ζ7 × ζ7 -ζ7 × ζ2 × ζ2 × ζ2 × ζ4 × ζ4 + ζ1 × ζ4 × ζ3 × ζ3 × ζ5 × ζ5 + ζ1 × ζ6 × ζ0 × ζ0 × ζ7 × ζ7 + ζ3 × ζ4 × ζ1 × ζ1 × ζ6 × ζ6 + ζ3 × ζ6 × ζ2 × ζ2 × ζ4 × ζ4 -ζ0 × ζ5 × ζ1 × ζ1 × ζ7 × ζ7 -ζ0 × ζ7 × ζ2 × ζ2 × ζ5 × ζ5 -ζ2 × ζ5 × ζ3 × ζ3 × ζ4 × ζ4 -ζ2 × ζ7 × ζ0 × ζ0 × ζ6 × ζ6 -3 × ζ0 × ζ7 × ζ3 × ζ3 × ζ4 × ζ4 -2 × ζ1 × ζ6 × ζ3 × ζ3 × ζ4 × ζ4 + 2 × ζ2 × ζ5 × ζ0 × ζ0 × ζ7 × ζ7 + 3 × ζ3 × ζ4 × ζ0 × ζ0 × ζ7 × ζ7 + ζ0 × ζ2 × ζ3 × ζ6 × ζ5 × ζ5 + ζ1 × ζ4 × ζ5 × ζ7 × ζ2 × ζ2 -ζ0 × ζ1 × ζ3 × ζ5 × ζ6 × ζ6 -ζ2 × ζ4 × ζ6 × ζ7 × ζ1 × ζ1 -3 × ζ0 × ζ1 × ζ2 × ζ4 × ζ7 × ζ7 -3 × ζ3 × ζ5 × ζ6 × ζ7 × ζ0 × ζ0 -2 × ζ0 × ζ2 × ζ3 × ζ4 × ζ6 × ζ6 -2 × ζ3 × ζ4 × ζ5 × ζ7 × ζ1 × ζ1 + 2 × ζ0 × ζ1 × ζ3 × ζ7 × ζ5 × ζ5 + 2 × ζ0 × ζ4 × ζ6 × ζ7 × ζ2 × ζ2 + 3 × ζ0 × ζ4 × ζ5 × ζ6 × ζ3 × ζ3 + 3 × ζ1 × ζ2 × ζ3 × ζ7 × ζ4 × ζ4 + ζ3 × ζ3 × ζ3 × ζ4 × ζ4 × ζ4 -ζ0 × ζ0 × ζ0 × ζ7 × ζ7 × ζ7 + ζ3 × ζ0 × ζ0 × ζ6 × ζ6 × ζ6 -ζ0 × ζ3 × ζ3 × ζ5 × ζ5 × ζ5;Note that, while the above calculation may look complicated, essentially, through the above, each of the coefficients in Eq. 14 is computed over the coordinates of the four control points of the curve (i.e. {x 0 , y 0 , x 1 , y 1 , x 2 , y 2 , x 3 , y 3 }) just through a group of basic arithmetic operations. In other words, the computation of the coefficients in Eq. 14 is just a task with O(1) time complexity.In our framework, we propose a Bézier-boundary gradient approximation strategy to keep the "differentiability" of the rendering process. In Sec. 4.2 in the main paper, we explain this strategy taking the approximation of ∂g ∂c 2D curve [0,0] as an example. Here in this section, with c curve ∈ R 4M ×2 , we further describe how we approximate ∂g ∂c 2D curve [i,j] , where i ∈ {0, ..., 4M -1} and j ∈ {0, 1}. Specifically, in our framework, ∂g ∂c 2D curve [i,j] is approximated in a similar manner as ∂g ∂c 2D curve [0,0] except in the following two places.(1) Definition change of g 0 sc (p). E Experiments on 11 3D Scenes.In Tab. 1 in the main paper, following [30, 49, 22] , we evaluate our method on a total of 13 3D scenes from the Mip-NeRF360 [4] , Tanks&Temples [31] , and Deep Blending [23] datasets. Here, following [35] , we also evaluate our method on another benchmark with 11 3D scenes from the above three datasets. As shown in Tab. 12, on this evaluation benchmark, our method can also achieve superior performance consistently, further demonstrating the efficacy of our proposed method. We use the Tanks&Temples dataset [31] by following the license of here. We use the Mip-NeRF360 dataset [4] by following the Apache-2.0 license. Moreover, we use the Deep Blending dataset [23] by following the license of here. Besides, we use part of the code owned by Kerbl et al. [30] by following the license of here.NeurIPS Paper Checklist 


Decision: This paper initially received mixed reviews. Reviewers raised concerns about the clarity of the manuscript and the role of the Bezier curve constraint on the quality of the results. The authors provided a rebuttal, and this paper was discussed at length by the reviewers and the AC. After the discussion, most reviewers agreed that the work is novel and that the additional results are sufficient to show the utility of the proposed Bezier curve constraint for improving reconstruction quality. The AC determined that the paper should be accepted. The authors should incorporate the results and clarifications from the rebuttal into the paper.

Reviews: review: paper_summary: paper_summary: The authors proposed an innovative framework DisC-GS, which enables Gaussian Splatting to represent and render boundaries and discontinuities in an image, they also introduce several designs to make the pipeline is discontinuity-aware and differentiability, and their method achieves superior performance on the evaluated benchmarks
strength_weakness: strength_weakness: The paper defines numerous labels, but they are presented in a noisy and unclear manner. It would be better to include a list of labels and their explanations. Additionally, I have to refer back to the main manuscript frequently as the lack of label explanations under each equation. 

The paper does not include an analysis of the computational complexity of the proposed method. 

There is no explanation or discussion on how the curves affect the densification procedure. For example, will the new cloned/splitted points are assigned the same curve attribute?
questions: questions: Is there any paneltation to optimize/adjust the positions of four contral points in a curve? 

As it will “scissored out” some part of a Gaussian point, will this affect the opacity of the whole point, and whether it will helps the point escape pruning.(pruning occurs when opacity < threshold)
limitations: limitations: The paper lacks of computational complexity of the proposed model. There is no explanation or discussion on the curves effect the densification procedure. Too many labels.
paper_summary: paper_summary: This paper proposes DisC-GS, a technique that enhances the boundary rendering quality for Gaussian splatting. DisC-GS takes into account the discontinuity of shapes and uses Bézier curves to model the boundaries. To enable differentiable rendering, the authors propose a novel discontinuity-aware rendering pipeline paired with a gradient approximation strategy. Experiments show that DisC-GS surpasses existing methods in rendering quality.
strength_weakness: strength_weakness: DisC-GS will increase the storage of the 3D scene representaion. The authors did not evaluate this in experiments.

The authors do not provide additional qualitative results, such as a video with continuous camera movements. It would be interesting to see if DisC-GS can preserve multi-view consistency. DisC-GS produces hard boundaries for Gaussians, which may cause artifacts (e.g., flickering) in such videos.

The methods compared in the paper are not very comprehensive; for example, there are no results compared with mip-splatting. The aliasing artifacts demonstrated in Figure 3 may not be entirely due to boundary issues, and more comparisons are needed to prove this.
questions: questions: If I understand correctly, the Bézier curves for each Gaussian are only defined in a 2D subspace. Is there a specific reason for this design? Can they be directly defined in a three-dimensional space? Is this design similar to 2D Gaussian splatting?
limitations: limitations: The authors do not provide any limitations.  There may be a limitation about the storage size.
paper_summary: paper_summary: This paper introduces DisC-GS, a method that utilizes Bezier curves for discontinuity-aware image rendering on 3DGS. By employing Bezier curves, this approach significantly enhances the rendering results of scene boundaries. A set of experiments and ablation studies substantiate the effectiveness of this proposed method.
strength_weakness: strength_weakness: 1. Introducing Bezier curves to refine Gaussian image rendering is indeed very clever. However, this paper only uses 2D parameters to control the position of the control points. Could this cause rendering problems in areas significantly affected by large viewpoint changes?
2. When additional properties such as control points and curve rendering are introduced, what impact does this have on training efficiency? While there is a time comparison for real-time rendering in the appendix, it would be interesting to know whether adding these parameters has any additional impact on training time and convergence (number of iterations).
3. Given that the authors changed the properties of each Gaussian, it would be interesting to see how these modifications affect the distribution of Gaussians in edge and surface regions. One aspect that could be explored is whether the number of Gaussians in the scene will differ significantly from the original 3DGS.
questions: questions: See the weakness part.
limitations: limitations: This paper lacks a discussion of limitations. It is recommended to provide a detailed description of the challenges associated with large scenes, the handling of huge parameters, posed images, and observation views that require improvement in reconstruction to ensure completeness.
paper_summary: paper_summary: This paper proposes a novel kernel function to model static scenes, addressing the difficulty of modeling high-frequency boundaries caused by the $r^2$ decay from center to edge of the Gaussian kernel. To tackle the smoothing decay of the Gaussian kernel, the authors first divide the Gaussian in screen space using M Bézier curves, introducing discontinuities in the kernel function. To address the gradient discontinuity issue caused by the segmented Gaussian, the authors propose the Bézier-boundary Gradient Approximation Strategy to approximate the gradients during backpropagation, ensuring stable optimization. Quantitative and qualitative experiments on real-world scenes demonstrate that DisC-GS can model high-frequency boundaries and achieve the best rendering quality.
strength_weakness: strength_weakness: 1. Too few qualitative comparisons and mismatched quantitative metrics. In Figs. 3 and 4, only two scenes from deep blending, two scenes from tanks&temples, and Mip360's room are shown. After carefully comparing the rendered images from corresponding views of Scaffold-GS, I found that although DisC-GS shows slightly better visual effects, the differences in rendering metrics, especially SSIM and LPIPS, should not be so significant. Hence, I have a few questions: Did DisC-GS use the same data as vanilla GS (without rerunning COLMAP)? What was the image resolution used for calculating the rendering metrics, and were the SSIM and LPIPS (VGG) calculations done in the same manner as for vanilla GS?
2. Lack of comparisons on synthetic scenes. In real-world scenes, there can be inaccuracies in camera poses, which may lead to improvements in metrics that are unrelated to the method itself. Therefore, the rendering quality on synthetic scenes would be more convincing. I am very much looking forward to seeing the rendering metrics and image quality of DisC-GS on the NeRF and NSVF datasets.
3. Lack of discussion on the number of Gaussians, training time and storage. I am very curious about the approximate number of Gaussians used in DisC-GS, as the number of Gaussians can greatly affect rendering quality.
4. I have some doubts about the optimization of DisC-GS. It seems that this optimization method could easily get stuck in local optima, and introducing polynomial computations in CUDA kernels might significantly reduce FPS. Did the authors use any special optimization strategies?
questions: questions: I have some doubts about evaluation in this paper. However, due to the strong results and sound methodology, if the authors can address my concerns, I am very open to raising my score to `accept`.
limitations: limitations: Please refer to the weakness part.

Comments: title: Response to reviewer ncnd [2/2]
Response to reviewer ncnd [1/2]
Feedback from reviewer ncnd
comment: Thanks for your timely response.

**(1)** Yes, in our framework, the 4M control points are first situated in a 3D space and then projected onto a 2D image plane.

**(2)** We're glad you think 2DGS appears to be compatible to this design. Below, we highlight that, with this design, when applying our framework on 3DGS, we can also achieve a good novel-view image rendering quality. This is because, as mentioned in existing works like [29], for Gaussian Splatting to perform good and detailed novel-view synthesis, in its (experimental) setting, Gaussian Splatting naturally expects the pose (angle) differences between the novel testing views and their surrounding training views to be not large. In this case with non-large view differences, even though we handle 3DGS in a quasi-2D manner (i.e., contouring scene boundaries on the image planes of the training views instead of directly in the 3D space), over the novel testing views, our framework can still acquire a good representation of scene boundaries. 

Meanwhile, we highlight that, rather than handling 3DGS in a quasi-2D manner, directly scissoring Gaussian kernels into non-Gaussian kernels in the 3D space and then projecting (splatting) these kernels onto the image plane can be a design that is not compatible to Gaussian Splatting. This is because, as mentioned in [52], while the splatting (projection) of Gaussian kernels approximately holds the closed-form solution (see Eq. 1 in paper) and can thus be easily performed, the splatting (projection) of non-Gaussian kernels can be a very difficult problem.

In summary, in our framework, from an innovative perspective, we propose to perform post-scissoring after Gaussian kernels have been projected onto the image plane. This can bypass the difficulty of splatting (projecting) non-Gaussian kernels, while at the same time can also enable scene boundaries to be effectively represented and rendered. As shown in Table 2 in paper, this also leads the applications of our framework on both 2DGS and 3DGS to consistently achieve good performance.

We will also discuss the above in paper.
Thank you to the author for providing additional explanation.

So, if we assume the presence of 4M control points, are these points situated within a 3D space and projected onto a 2D image plane? 
If this is the case, it appears that the impact on 2DGS might be minimal. However, for 3DGS, it seems that the method might inadvertently reduce 3DGS to a quasi-2D state.

Additionally, the MaskedSSIM comparison offers a clearer measure of improvement in the edge areas.
>*Q7: The paper does not show changes in the distribution of GS. DisC-GS theoretically reduces the number of GS required for modeling high-frequency boundaries, so the reduction in quantity and memory usage makes sense. However, the lack of visualization of the point cloud distribution is quite confusing.*

**A7:** Below, between our DisC-GS framework and 3D-GS, we compare the number (distribution) of Gaussian kernels, per unit volume ($1\times1\times1$) in the xyz 3D coordinate system of the 3D scene. Specifically, we perform comparisons in the edge regions of the 3D scene that are detected utilizing the Canny edge detection algorithm [9], as well as in the non-edge regions of the 3D scene, on the Tanks&Temples dataset.

|Method|Number of Gaussian kernels per unit volume in edge regions of the 3D scene|Number of Gaussian kernels per unit volume in non-edge regions of the 3D scene| 
|---|---|---|
| Original 3D-GS | 0.98 | 0.39 |
| Ours (on 3D-GS) | 0.31 | 0.30 |

As shown, compared to the original 3D-GS, our DisC-GS framework can represent the 3D scene with much fewer Gaussian kernels per unit volume, especially in the edge regions of the scene that contain a large number of high-frequency boundaries. This further experimentally shows that, DisC-GS can reduce the number of Gaussian kernels required for modeling high-frequency boundaries. Due to the format constraint here, we will also provide the visualization of the point cloud distribution in paper to further show changes in the distribution of Gaussian kernels.

>*Q8: While DisC-GS theoretically addresses the ability of 3D-GS to model high-frequency boundaries, is it really always better than 3D-GS? In many scenes (e.g., diffuse scenes), we actually need the smooth kernel provided by 3D-GS.*

**A8:** (1) **Yes, DisC-GS is always better than 3D-GS**. To investigate this, we measure the per-image metrics for all testing images in all 13 evaluated 3D scenes. We find that, in all testing images in different 3D scenes, DisC-GS can always perform better than 3D-GS. In specific, in all testing images, the [minimal per-image improvement, maximal per-image improvement] range of DisC-GS over 3D-GS is [+0.01, +9.64] for the PSNR metric, [+0.001, +0.059] for the SSIM metric, and [+0.002, +0.121] for the LPIPS metric. 

(2) We agree that, we still need smooth kernels in many scenes. Yet, we highlight that, **similar to 3D-GS, if needed, DisC-GS also can contain "smooth" kernels**. This is because, in DisC-GS, for each Gaussian kernel, the places where it is scissored over are learnable (see Lines 218-221 in paper). Thus, in DisC-GS, when a certain Gaussian kernel still needs to be a smooth kernel, DisC-GS is also able to learn to scissor this kernel only in places very far from its center. This leads the scissoring to have negligible effects on this kernel, and thus still keeps this kernel a "smooth" kernel. Yet, note that, different from 3D-GS, in DisC-GS, besides "smooth" kernels, DisC-GS also contains non-smooth kernels that can represent boundaries and discontinuities. This leads to the consistent performance improvements of DisC-GS over 3D-GS (see (1)).
Thank you for your time and effort. Below, we would like to answer your follow-up questions.

>*Q5: Fig. 3(b)(c) shows a significant reduction in floaters, but I have no idea about the relationship between this and DisC-GS itself. The essence of DisC-GS is to address the need for extensive boundary modeling in 3D-GS, which is unrelated to floater removal. The paper should clearly state what causes the removal of floaters instead of obscuring the components that actually contribute to this effect.*

**A5:** We carefully investigate Fig. 3(b)(c) and their corresponding 3D scene representations. We observe that, in the rendered images of existing methods in Fig. 3(b)(c), a large number of continuous Gaussian kernels are messily stacked in areas with rich boundaries and discontinuities. This stacking leads to floaters and blurriness, particularly in the red-boxed areas of these figures. We emphasize that this issue arises from a fundamental limitation of Gaussian Splatting, which our DisC-GS framework aims to address. Specifically, due to the continuous nature of Gaussian distributions, Gaussian Splatting struggles to accurately render discontinuities and boundaries in images (see Lines 3-5 in the paper). As a result, existing Gaussian Splatting methods often produce low-quality renderings in boundary-rich areas, with noticeable blurriness and floaters. The above points out that, **our floater reduction in Fig. 3(b)(c) is closely related to the essence of our DisC-GS framework**. We will also discuss this in paper.


>*Q6: Fig. 3(a) presents slightly better visual results compared to Scaffold-GS, but it does not sufficiently support the improvement of LPIPS (vgg) from 0.177 to 0.120 in the tanks scene.*

**A6:** (1) In **Fig. 3(a)**, our DisC-GS framework only slightly improves LPIPS over Scaffold-GS, i.e., from 0.188 to 0.179 by 0.009. This is consistent with the slight visual result improvement of DisC-GS over Scaffold-GS in Fig. 3(a).

(2) **Yet, we highlight that, the improvement of LPIPS from 0.177 to 0.120 in the tanks dataset is still sufficiently supported by its testing images**. This is because, in many other testing images in the tanks dataset, DisC-GS achieves much more significant LPIPS improvements over Scaffold-GS. For example, in the 10th testing image in the *Train* scene in this dataset, DisC-GS improves LPIPS over Scaffold-GS from 0.211 to 0.110 by 0.101; in the 13th testing image, DisC-GS improves LPIPS over Scaffold-GS from 0.203 to 0.101 by 0.102. Note that, the *Train* scene in the above sentence refers to the Train (railway) scene, but not the scene used for training. Meanwhile, we also observe that, in these testing images that DisC-GS achieves much more significant LPIPS improvements, DisC-GS can also achieve much more significant visual result improvements. So overall the improvement of LPIPS on the whole dataset is obvious. Thanks for this comment. Besides Fig. 3(a), we will also add these testing images in paper.
Thank you for your time and effort in reviewing our work. Below, we further answer your following questions.

>*Concerning A1: [...] the methodology for employing Bézier curves to confine a 3D Gaussian within three-dimensional space remains inadequately described.*

**Answer:** To clarify, in our framework, we do not scissor (confine) 3D Gaussians within 3D space. Instead, even for 3D Gaussian Splatting, as mentioned in Lines 240-242 in paper, our framework always scissors (confines) 2D Gaussians that have been projected on the image plane. This is because, in our framework, the final goal is to modify the $\alpha$-blending function and thus enable the function to perform discontinuity-aware image rendering (see Lines 85-87 in paper). Note that, even for 3D Gaussian Splatting, its $\alpha$-blending function is also calculated based on the 2D Gaussians that have been projected on the image plane (see Eq. 2 in paper). Thus, in our framework, to effectively and conveniently modify the $\alpha$-blending function, we directly scissor 2D Gaussians on the image plane. We will make the above clearer in paper to avoid confusion.

>*Regarding A2 and A3: My previous Q3 was about whether your method shows a significant performance difference between edge and non-edge areas. [...] The significant reduction in the number of Gaussians used in edge regions (approximately 1/3 of the original count) suggests a potential enhancement in edge regions' representation. I recommend that the authors conduct additional experiments to compare the number of Gaussians and the resulting image render quality, specifically in edge regions. This could further substantiate the benefits of your approach and strengthen its appeal to others.*

**Answer:** (1) Yes, our method shows a significant performance difference in the edge and non-edge areas of the testing images. As shown below on the Tanks&Temples dataset, in the edge areas of its testing images, our method can achieve a much more significant performance improvement compared to 3DGS. This further shows the effectiveness of our method over edge areas of the image. Note that here, to perform evaluation effectively over image sub-areas, following [40], we use the MaskedSSIM metric (the larger the better).

|Method|Number of Gaussians in edge areas of the scene|MaskedSSIM in edge areas of testing images|Number of Gaussians in non-edge areas of the scene|MaskedSSIM in non-edge areas of testing images|
|---|---|---|---|---|
| Original 3DGS | ~605K | 0.802 | ~1179K | 0.922 |
| Ours (on 3DGS) | ~196K | 0.865 | ~898K | 0.928 |
|Performance gain||+0.063||+0.006|

(2) As suggested, below, we also compare the number of Gaussians and the resulting image render quality, in edge areas on the Tanks&Temples dataset. As shown, in our framework, with the increase of the number of Bézier curves we equip for each Gaussian, the number of Gaussians in edge areas of the scene consistently decreases. At the same time, in the edge areas of the testing images, our framework's image render quality consistently improves. In paper, taking our framework's efficiency into consideration, we equip 3 Bézier curves for each Gaussian (see Line 371 in paper). 

|Method|Number of Gaussians in edge areas of the scene|MaskedSSIM in edge areas of testing images|Performance gain over 3DGS|
|---|---|---|---|
| 0 Bézier curve for each Gaussian (Original 3DGS)  | ~605K | 0.802 ||
| 1 Bézier curve for each Gaussian | ~324K | 0.838 |+0.036|
| 2 Bézier curves for each Gaussian | ~230K | 0.852 |+0.050| 
| 3 Bézier curves for each Gaussian | ~196K | 0.865 |+0.063|
| 4 Bézier curves for each Gaussian | ~191K | 0.866 |+0.064|

We will also add the above experiments to paper.
Thank you for your time and effort and also thanks for your suggestions.

**(1)** In our original paper, to ease readers' understanding, we first described the 2D version of Bézier curves as an example. After that, we described how the 3D version of Bézier curves can be similarly used in our framework (in Lines 388-392 in paper). Thank you for the suggestion, **to more sufficiently describe the 3D version of Bézier curves in our paper**, we will reorganize the methodology structure by adding a subsection at the end of the method section, as follows:

1. Specifically, in this subsection at the end of the method section, we will state that:

Above, we focus on describing how we use 2D Bézier curves in our DisC-GS framework and correspondingly apply DisC-GS on 2D Gaussian Splatting. Here in this subsection, we further describe how we use 3D Bézier curves in our DisC-GS framework. Specifically, the transition from 2D to 3D Bézier curves in DisC-GS requires only two minimal modifications. (1) Firstly, for each Gaussian representing the 3D scene, the control points of its Bézier curves are stored directly in the 3D spatial coordinate system rather than in a 2D subspace.  Note that, this modification can be very simply made. Specifically, for each Gaussian in the 3D space in our DisC-GS framework, we only need to use $c_{curve}^{3D} \in \mathbb{R}^{4M\times3}$ instead of $c_{curve} \in \mathbb{R}^{4M\times2}$ to represent the control point coordinates of its 3D Bézier curves. In other words, for each 3D Gaussian, we only need to introduce it with $c_{curve}^{3D}$ instead of $c_{curve}$ as its new attribute. (2) Moreover, since we already directly introduce $c_{curve}^{3D}$ as the new attribute for each 3D Gaussian in our framework, during rendering, we omit Eq. 4 above in Sec. 4.1, which originally is used to acquire $c_{curve}^{3D}$ from $c_{curve}$. Overall, the above two modifications are sufficient to incorporate DisC-GS with 3D instead of 2D Bézier curves.

2. Moreover, in this subsection, we will also draw a figure to better explain the usage of 3D Bézier curves in our framework. 

Specifically, we will draw this new figure in a similar way as the current Figure 2 in paper. That is, in this new figure, we will include three sub-figures. Among these three sub-figures, in sub-figure (a), we will draw a 3D coordinate system and demonstrate the control points of the 3D Bézier curves in 3D space. In sub-figure (b), we will draw an image plane, on which we will draw the control points of the Bezier curves that have been projected on that image plane. Finally, in sub-figure (c), similar to the sub-figure (c) in the current Figure 2, we will demonstrate how Bézier curves are used in our framework to scissor the Gaussian distribution.

We hope that this new subsection would facilitate the 3D version of Bézier curves to be more sufficiently described in our paper.

**(2)** Moreover, we agree that the visualization of the learned Bézier curves can help to better explain why our framework does not increase the storage size. We will follow your suggestion to add this to our paper.
Thanks to the authors for their response and the additional experimental results. Through their response, I realized that there are both 2D and 3D versions of Bézier curves. Even with this information, I believe that the description of the 3D version of Bézier curves in the paper is insufficient. It might be necessary to reorganize the methodology structure and provide some 3D illustrations to better explain it. Additionally, visualizing the learned Bézier curves could potentially offer a clearer explanation of why storage hasn't changed much, rather than vaguely attributing it to 'being aware of the discontinuity' and 'enhancing its representation ability.'
I have carefully reviewed the authors' responses and the supplementary material provided in the rebuttal.

**Concerning A1**: The authors have not fully addressed my concern regarding the utilization of Bezier curves to shape the Gaussian distributions in 3D space. While Figure 2 in the manuscript illustrates the use of two Bezier curves to maintain the sharpness of the Gaussian on a plane, the methodology for employing Bezier curves to confine a 3D Gaussian within three-dimensional space remains inadequately described, both in the original manuscript and in the rebuttal.

**Regarding A2 and A3**: The authors' method does enhance image rendering quality and reduces the number of Gaussians required, without a significant increase in training time.
My previous Q3 was about whether your method shows a significant performance difference between edge and non-edge areas.

|  | Edge | Non-edge | Total |
| --- | --- | --- | --- |
| Original 3DGS | ~605K | ~1179K | ~1784K |
| Proposed Method on 3DGS | ~196K | ~898K | ~1094K |

The significant reduction in the number of Gaussians used in edge regions (approximately 1/3 of the original count) suggests a potential enhancement in edge regions' representation. I recommend that the authors conduct additional experiments to compare the number of Gaussians and the resulting image render quality, specifically in edge regions. This could further substantiate the benefits of your approach and strengthen its appeal to others.
Thanks for authors' response and the additional experimental results based on the review. However, I still do not understand why there is such a significant improvement in the quantitative metrics (tanks from 0.178 to 0.120, db from 0.240 to 0.199) based on the authors' statement that they used the same resolution and rendering metrics as vanilla 3D-GS. I have the following questions:
- Fig. 3(b)(c) shows a significant reduction in floaters, but I have no idea about the relationship between this and DisC-GS itself. The essence of DisC-GS is to address the need for extensive boundary modeling in 3D-GS, which is unrelated to floater removal. The paper should clearly state what causes the removal of floaters instead of obscuring the components that actually contribute to this effect.
- Fig. 3(a) presents slightly better visual results compared to Scaffold-GS, but it does not sufficiently support the improvement of LPIPS (vgg) from 0.177 to 0.120 in the tanks scene. 
- The paper does not show changes in the distribution of GS. DisC-GS theoretically reduces the number of GS required for modeling high-frequency boundaries, so the reduction in quantity and memory usage makes sense. However, the lack of visualization of the point cloud distribution is quite confusing.
- While DisC-GS theoretically addresses the ability of 3D-GS to model high-frequency boundaries, is it really always better than 3D-GS? In many scenes (e.g., diffuse scenes), we actually need the smooth kernel provided by 3D-GS.

Overall, this paper still lacks sufficient evaluation to demonstrate the advantages of DisC-GS relative to vanilla 3D-GS (excluding improvements beyond changes in the kernel function). Therefore, I still maintain a negative evaluation of this work.