DisC-GS: Discontinuity-aware Gaussian Splatting
Authors: Haoxuan Qu
Zhuoling Li
Hossein Rahmani
Yujun Cai
Jun Liu
Venue: NeurIPS 2024 poster (2024)
Keywords: Gaussian Splatting
discontinuity awareness
gradient approximation
Primary area: machine_vision

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.

DisC-GS: Discontinuity-aware Gaussian Splatting
Haoxuan Qu Lancaster University U.K. h. w..uullaaasttrr.cmmmm ac.
Zhuoling Li Lancaster University U.K.
Hossein Rahmani Lancaster University
U.K.
z. w..1810lancaster... ac uk
@echani@@lncasseeno ac.
Yujun Cai University of Queensland
Jun Liu Lancaster University
Australia vanora. cat.yggaaaijjjmmill..... com
U.K.
j 1.1iu810lancaster ac.uk
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 novel view synthesis. this paper, we highlight fundamental limitation Gaussian Splatting: its inability to accurately render discontinuities and boundaries in images due to the continuous nature Gaussian distributions. To address this issue. propose novel framework enabling Gaussian Splatting perform discontinuity- -ware image rendering. Additionally, we introduce Bézier-boute boundary gradient approximation strategy within framework to keep the "differentiability" the proposed discontinuity-awal rendering process. Extensive experiments demonstrate the efficacy of our framework.
Introduction
Novel view synthesis aims to generate images accurately from novel viewpoints in a captured 3D) scene. Its significance spans across diverse applications, such autonomous driving [45), virtual reality [14, and 3D content generation
[42] Recently, for better tackling novel view synthesis, Neural Radiance Field (NeRF) (36] and variety of NeRF-based methods 3 have been proposed, which represent 3D scenes in an implicit manner as neural radiance fields. However, their general reliance on heavy volume rendering mechanism often results in slow rendering speeds 30
limiting their practicality across real-world applications. While some methods [18 19] have proposed accelerate the rendering process of NeRF from different perspectives, they often achieve this 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 appealing alternative NeRF. Specifically, rather than generating novel-view images through the time-consuming process of volume rendering, Gaussian Splatting enables images from novel viewpoints be generated by simply splatting 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 126 13
of Gaussian distributions, has been proposed
(projecting) 153 54
49
12 21 25).
BBoth authors contributed equally the work. Conresponding Author
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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Ground truth imagee From as ceriain viewspoint
Boundary map of
(b)
Parts of the two Gaussian distributions that are "Inside" 'the boundary
the left-hand side image
Perts of the two Geussian distributions that "pass over" boundary
Image rendered by Gaussion Splettingwio wio Disc-GS
Image rendered by Geussian Splatting with DiaC-OS
Figure 1: (a) Illustration of 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 f'il 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) discontinuiries and boundaries with its collection of contimuous Gaussian distributions. Specifically, due to the general complexity of 3D scenes, the expected image to be rendered often contains numerous discontinaities and boundaries (as shown in Fig. [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. (Tb). the inherent continuity of Gaussian distributions can result in some parts of the distribution inevitably "passing s"pillng over") the boundaries sharp features the image. This can lead to Gaussian Splatting rendering the sharp boundaries in the image with blurriness (as shown in Fig. (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. [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 Gaussian Caussian-Splatting based framework. (2) Mcanwhile, recall that continuity serves as the prerequisite for function to be differentiable. Thus, during the process of learning the 3D scene representation using Gaussian Splatting, how to maintain the "diffcrentiability" of the process in existence of discontinuities, i.c., 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-ave 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. [ and outline our framework as follows.
Overall, 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 group of four control points, has shown capable of efficiently parameterizing curves of various shapes with low computational complexity 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 the control points of
16 34

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the cubic Bézier curves, After that, given 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-awan image rendering through a modified a-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 a- a-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 bézier-boundary gradient approximation strategy, by which during backpropagation, we can enable gradients to properly pass through the modified a-blending function, and thus keep our framework to 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 key intrinsic limitation of Gaussian Splatting. 2) We introduce several designs in our framework to enable it to render images in a discontinuity-waan manner, while also to keep its "differentiability"" in the presence of discontinuities. 3) DisC-GS achieves superior performance on the evaluated benchmarks.
2
Related Work
Novel View Synthesis. Owing to the wide range of applications, the task of novel view synthesis has
received lots of research attention [3
40 39 43 24 36
4 50 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 popular way in representing 3D scenes. Specifically, the original version of NeRF is first proposed
by Mildenhall et al. in (36) and after comes out, a variety of different -baed methods have been further proposed, such as Mip-NeRF 3. NeRF++ [50], and Point-NeRF [44). Despite the increased efforts, weakness of NeRF-based methods can be that, to render novel-view images in
high visual quality, they often still require 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. 300 proposed to represent 3D scene as 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 new loss function in the frequency space to better regularize the learning process of Gaussian Splatting.
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.
Different from these existing Gaussian-Splattee
3

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Curve Representation. The idea of representing a curve in parametric way has been studied in
various tasks [34
28 8 such as lane detection [16], trajectory prediction [27]. and text Here in this work, we design a novel framework, which enables Gaussian Splatting to perform discontinuity-wwal novel-view image rendering, via utilizing the cubic Bézier curves to
27 38
16
spotting [34]
parametrically contour the boundaries in the image plane.
3 Preliminary
Gaussian Splatting. Gaussian Splatting represents the 3D scene explicitly as collection of anisotropic Gaussian distributions. In specific, in the collection, each Gaussian is defined with the following attributes: (1) its center μa R$, (2) its covariance matrix E R (3) its spherical harmonic (SH) coefficients сSH E 3xx(+++1* representing its color from different viewpoints (where k denotes the order of SH), and (4) its opacity a E RR. Regarding the covariance matrix E, 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 Rtt where I& R3x3 is the orthogonal rotation matrix of the Gaussian, and S E R3x3 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 the collection is first projected onto the image plane corresponding to the viewpoint as:
2220_ PW,, p,, 5:2 JWEWT yf
(1)
where 1.2D and D2D 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 represents the Jacobian of the affine approximation of the projective transformation. After that, to perform image rendering the image plane, for each pixel p of the image, its color C(p) is derived through an a-blending function as:
N
i-1
c(p)= = Σ C.S.
II Bj), where 3, = a,e -x (x-pr? j=1
(2)
where N represents the number of projected Gaussians that overlap p. c, represents the color of the i-th Gaussian calculated from its corresponding SH coefficients, a; represents the opacity of the i-th Gaussian, 1T? represents the center of the i-th projected Gaussian, and E?? represents the covariance matrix of the i-th projected Gaussian. Note that, 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, 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 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. | we first focus on explaining how our framework is applied 2D Gaussian Splatting, in which we fix the last column of the scale matrix 5 of all the Gaussians to be zero vector. We then discuss the application of our framework on 3D Gaussian Splatting in Sec. 4.3
Cubic Bézier curve. A cubic Bézier curve is a parametric curve that can be formulated by leveraging list of four ordered control points (wo w1, w2, wa] as:
(tt) (1 -1-18 3) + 3(1 t) + 3(1 - t)t? (01-t)tPuu + t2ws
In the above equation, we can set E [0, 1] for B(t) to represent a segment of the curve that starts from W/ and ends w3. Alternatively, we can set t E IR to represent the entire curve. this work. we set E IR for B(t), any segment of the curve may not be enough to represent the desired boundaries 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.

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Gaussian distribution
Control points of the first Bézier
Control paints of second Bézier
Renclared part Gaussian distribution
x
(a)
(b)
(c)
Figure 2: Illustration of the discontinuty--wwar rendering process over single Gaussian distribution. Specifically, over each 2D Gaussian distribution representing the 3D scene, we first introduce it with new attribute Ceurve E R4Mx2 (represented by the red and purple points (a)). Here we set M = 2. After that, given a viewpoint, as shown in (b), we project both the Gaussian distribution and the points stored in Ceurve onto the image plane corresponding to the viewpoint. Finally, leveraging the modified a-blending function in Eq. 6 we can perform discontinuity-
rendering and render
only the part of the Gaussian distribution masked with the dotted lines in (c). (Best viewed in color.)
4 Proposed Method: DisC-GS
Given batch of source images 3D scene with their corresponding viewpoints, the goal of novel view synthesis is to generate novel-view images accurately. To handle this task, 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
49)
Yet, we here argue that Gaussian Splatting has a key intrinsic limitation: it may fail render discontinuities and boundaries accurately. To tackle this problem, in this work, inspired by [84, we propose 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 Bézier-boundaty 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, first describe the (forward) image rendering process of DisC-GS, and then explain the Bézier- boundary gradient approximation strategy.
4.1 Discontinuity-wwars 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 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 additional attribute. We denote this attribute Ceurve 422 where A is 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 Ceurve- Specifically, for a certain 2D Gaussian distribution representing the 3D scene, denote the first column of its rotation matrix R be 71 and the second column of R be r2- Over the 3D space, the 2D subspace that this Gaussian distribution lies in can be then described by 2D coordinate system, which takes the center μ of the Gaussian as its origin, the direction of ri as the direction of its x-axis, and the direction of as the direction of its y-axis. Then for Ceurve of this Gaussian distribution, it can understood as storing a total 4M points the above-defined coordinate system. Note that when these 4M points are projected onto the image plane (as discussed
5

---

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 Courve- After introducing Ceurve to each Gaussian distribution that represents the 3D) scene, given a viewpoint, we project points in Ceurve Onto the image plane. Specifically, this is achieved in two steps: (I) We first transform each point (stored in Ceurve) in the above-defined subspace coordinate system to the coordinate system of the 3D space as:
3D Ceurvel []] = μ + Ceurve ( 0] x r + Ceurve [ē, 1] x r where i E {0, ..., 4M 1
(4)
where ju the center of the Gaussian distribution. Note that here, since a column of rotation matrix is already a unit vector, we can omit the normalization of 11 and 12 and directly transpose them. (2) After deriving Ceurve E RR44M x3 storing the 4. points in the 3D space coordinate system. we can c3D
project each point in Courve onto the image plane similar to what we have done in Eq. as:
Ceurve [i] = PW ceurvelt], where i E (0.....444 - 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. [ we have gotten the control points of its desired Sebbee-Bziiee-curees-rrrrreessse
boundary, stored in Cuuree 1Mx.
Discontinuit--wval rendering. Finally, to perform discontinuity-aaan image rendering, for each Gaussian distribution projected onto the image plane, aim to first "scissor" the distribution along the . cubic Bézier curves formulated based on the 4M control points stored in 2D 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 otherwise. We can then perform discontinuity-awer awwrr rendering simply via modifying the a-blending function in Eq. 2 as:
C(p) Σ cB II (1 (1),,,, where B, =a:g(p)e =00-n (""")"">>>
(6)
where g:(-) 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 c; represents the color of the i-th Gaussian calculated from its corresponding SH coefficients, a represents the opacity of the i-th Gaussian, 14? represents the center of the i-th projected Gaussian, and Σ represents the covariance matrix of the i-th projected Gaussian. Note that via the above modified a-blending function, for Gaussians that no longer overlap with the pixel 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 image rendering has now been reduced to building the indicator function g(-) for each projected
2D Gaussian distribution based on its corresponding Couree. 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 wn = (To, Uo).
(I1, y)),
We (z), and W3 = (123, y/). Then build g()) given a pixel p = (Ipz: Up), 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- intuitivee we first leverage the implicitization technique i algebra I to represent the cubic Bézier curve in its implicit representation form as:
Bamp(a. v) = T 3
+ 7 y T + Twvel 3 + + Tayay + Tuyy" + 7:x2 + T5y +
(7)
where coefficients including Tozzz Tezy. Tawy. Tygus Year Yay. Tsy Tax Ty. and To can all be obtained through basic arithmetic operations over the coordinates of the four control points of the curve in (11 time complexity (more details are provided in Appendix C. Based on B (2. y), in the case where only one curve exists per Gaussian distribution, we can then build the single-curve indicator function (-e(-) intuitively and with (11) time complexity as:
1, if Bamp Up) > O,
goo (wo, W1, w2, W3: = 0, otherwise
(8)
Above we introduce how we can build the indicator function g(-) as g-c(-) 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

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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 Isc(-) function above, we can then define g(-) in cases where M > as:
M-1
((p)
II
Bec(Ceuree [4i], Ceurve [4 + 11, C [4 + 2),ccor [4i + 3]:p)
(9)
x=
Leveraging the indicator function g(-) defined in Eq. 9 along with the modified a-blending function in Eq. 6 we can then enable Gaussian Splatting to perform discontinuity-awan rendering.
4.2 Bézier-boundary Gradient Approximation Strategy
Above we discussed, how, in our framework, we perform discontinuity- aware rendering in the forward direction from the 3D scene representation 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, enable the 3D scene representation to be properly learned from the source images of the 3D scene, Gaussian Splatting needs its rendering process be (backward) differentiable. However, performing discontinuity-ave -wwar rendering leveraging the modified a-blending function in Eq. [6] with the discontinuous function g(-) in Eq. / incorporated. no longer differentiable. Morcover, according to Eq. and 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(.) inputs stored in Courne: and consequently fail guide the learning process of Course introduced in 4.1.
9 g is actually
The big picture of our proposed strategy. To tackle the above problems and thus enable Gaus- sian Splatting to seamlessly render discontinuities, in our framework, we aim to further keep the "differentiability"" of the whole discontinuity-awar -wwar 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 2DD Currve 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 Bézier-boundan gradient approximation strategy. Below, to ease our explanation of the strategy, we focus on discussing how we Bg of the first control point stored in Ceurve- Note that the application of the strategy to the remaining coordinates stored in 2D
approximate 10.0 i.e., the partial derivative of the indicator function g(.) over the x coordinate
2DD
Ceurve follows a similar process (more details are provided in Appendix D)
Specifically, to approximate Curve [0,0] based on the chain rule and according to Eq. 9 denoting
0g
Ise( (2 [0], cabroe [1], 2e [2], courve (3::)) to be (p),,,, we first have:
0g
Ög X De2D 0,0
(g)
(10)
De2D 0..0 0
(p)
Then since g(-) is clearly differentiable over dee (-) based on its definition in Eq. 9 we can reduce our ay..(p)
problem to approximate Be [.0
which is achieved through the following two steps.
Determining if (p) is desired to be modified. Specifically, to approximate
0e20 0.0 we first
0g
would like to determine, if the function gla (-) is desired to modified at p or not. This is because,
(.) already outputs a satisfied value at p. we don't 't need to change Ceurve [0, 0) to correspondingly
2D
modify gd (p). In other words, in such a case, we can simply set
1..0 to be zero
Be20
Denote the loss function used during the learning process to Leveraging both 0g (p) and the current value of ghe as the conditions, below, we list the three situations in which go (p) doesn't 't need to be further modified: (1) The first situation happens when aL (ø) 0. which indicates that
L given input is already in an optimal state. (2) Besides, the second situation happens when Tgh > 0 and gd (p) = O. Based on the gradient descent algorithm, this implies that, while we still hope the function 9 (.) to output smaller value at p. the function glo(-) already outputs its smallest allowed value. (3) Following the opposite logic of situation (2), the third situation happens when
7

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AL 80% (p)
< 0 and (p) = 1. In this case, though we still want (p) to be larger, (.c(.) at already 0g
outputs its largest allowed value. In the above three situations, we can directly set 8020 0..0 and omit the approximation performed in the next step.
Approximating
0e [0.0] Besides the above three situations, in the rest cases, for the value of
function (-) at p to be properly modified based the modification of the value of c [0, 0j,
we aim to properly approximate the partial derivative ag.c(p) To achieve this, recall that as
0ct0r.
binary indicator function, (.c(.) switches (modifies) its value at p between and 1 only when its corresponding cubic Bézier curve passes through p. Considering this, below, first identify: which value we should set (change) 2DD Courvel 0) to be, so that the value switch of (.) at can happen.
To achieve this identification in an intuitive and analytical way, denoting p (Ixp Up) and the desired value of Cuurve [0, 0] to be σ) based on Eq. 3 we can first derive the following system of equations:
xp (1 1) + 3(1 U (1 (c [0, 1]) + 3(1 t) [1, 1)) + 3(1 t)t? (c (2, 1)) 8 (c22
curvel [1, 0)) + 3(1 - ()( 2D
curve 0]) + (122 [3. 01)
3. 11)
(11)
In this system of equations, since Tp. Bp,, and the coordinates in Courve all have known values, we initially regard the second equation in the system cubic equation w.r.t. 4, as / is now the only unknown variable in this equation. After solving this cubic equation and with & also known, we can then regard the first equation in the system solving the above two equations (both in just ((1) time complexity), we obtain Sy as the set of all possible real number solutions for 6. Based the solutions' scenarios within Sg. approximate (g)n(p)
cubic equation w.r.t. & and solve it. Finally, by
in three different ways below.
DeaD
Ganne [0,0
(1) The "no side" situation. The first situation happens when S. Ø. In this case, we simply set 3g. = This is because, the empty nature of Sy implies that, there exists no proper real- 2D change Ceurse 10, 0] to be, such that (P) can be desirably modified (i.e., either from 0 to 1 or from 1 to 0). We thus simply do not encourage 2D eeurve [0, 0J to change.
3c2aree 0000
number value that
(2) The "single side" situation. The second situation occurs when all solutions in S. lie on the same side of 2D [0, 0] (i.e., all larger or all smaller than ceurve [0, 0). In this situation, let ø denote the solution in S' that is nearest to ce [0, Oj. Adjusting <ewrve [0, 0] towards á then implies the least-cost plan, facilitating the modification of (p)(p) in desired manner. With this in mind, to
encourage 2D [0, 0] to approach ő, inspired by previous studies 66 10 47 29].
we approximate
00% (p) via performing linear interpolation between c2D [0, 0) and Φ as:
Beaane 0,00
DeaD
0g
Lhe - (p)
where (2) {1.
if (2) =, 0, otherwise
(12)
0.0 - (c220re 01) +
In the above equation, we set 10-5 if (36 - (220 J0, 0))) > and we otherwise set e = 110- e here is a small number that is used to avoid the gradient exploding problem to happen when the distance between o and e2p [0, 0] is too short.
(3) The "both sides" situation. The third situation happens when some solutions in S. are on the left side of 2D (0, 0). while other solutions are on the right side of 2D [0, 0). In this situation, we 20ree
can achieve the desired modification of go (p) via either moving 20 [0, 0] to its left or right side.
Thus, unlike the scenario described in the above situation (2) where we only consider á from a single
side of 2D [0, 0J. here, denoting Φi as the value that is nearest to 2D [0, 0] from its left side, and 22 as the value that is nearest to
2D 2200ee [0, 0] from its right side, we approximate Be20 (p)
Perers [0.0] as:
0g (p) a De2D currue [0.0]
9 (p) (o((p)
(2) (d
+ (02 (c2D 10, 00)))) + E2
(13)
(Ø1
(c 2D
(c2000(0,0) 0]) +
8

---

In the above equation, we define Peo (p) the same way as in Eq. Besides, both 61 and 62 are defined in the similar way as in Eq. 12
In summary, taking strategy approximates the gradient of g(-) with respect to the point coordinates stored in 2D With the incorporation of this strategy into framework, we keep the "differentiability" of the whole rendering process, allowing Gaussian Splatting to seamlessly perform discontinuity- aware rendering.
Bg
as an example, the above discussion explains how our proposed
Seree [0.0]
4.3 DisC-GS on 3D Gaussian Splatting
Above, we focus on describing how 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 Cuuree E R R X3 instead of Ceurve E g4M 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 cap instead of Course its new attribute. (2) Moreover, since we already directly introduce Ceuree as the new attribute for each 3D Gaussian in our framework, during rendering,
3D
eDD
omit Eq. 4 above in Sec. [4.] which originally is used to acquire Caurpe from Ceurve. Overall, the above two modifications are sufficient to incorporate DisC-GS with 3D instead of 2D Bézier curves.
3D
4.4 Overall Training and Testing
In DisC-GS, during training (i.e., learning the 3D scene representation from the source images), we follow similar process as the typical Gaussian Splatting technique [30]. The involvement of the strategy introduced in Sec. 4.2 keeps the "diffcrentiability" of our framework. During testing (i.c., image rendering), we use the discontinuity-wwar image rendering process introduced in Sec.
Experiments
Datasets. To evaluate the efficacy our proposed framework DisC-GS, following previous Gaussian Splatting works 30 49,, we evaluate our framework on total of 13 3D scenes, which include both outdoor scenes and indoor scenes. Specifically, among these 13 scenes, of them are from the M30-NeRF360 dataset [4] 2 of them are from the T Temples dataset [31] and 2 of them are from the Deep Blending dataset 23
We also follow previous works (30
their train-test-split.
44), we use the following three metrics for evaluation: Peak Signal-to-Noiss Ratio (PSNR), Structural Similarity Index Measure (SSIM), and Learned Perceptual Image Patch Similarity (LPIPS) (51)
Evaluation metrics. Following 30
Implementation details. We conduct our experiments on an RTX 3090 GPU and develop code mainly based on the GitHub repository (2) provided by Kerbl et 30 Moreover, we also get inspired by 35 52 46 during code implementation, and make use of the LPIPS loss during our training process. Furthermore, for the newly introduced attribute Ceurve E R-4x2 we set its initial learning rate to 2e-4, and the hyperparameter M 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 Courve the original one.
5.1 Experimental Results
In Tab. I 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 effectiveness. We also show qualitative results both Fig. (Tc) and Appendix B As shown, whether representing the 3D scene through 3D or 2D Gaussian distributions,

---

Table 1: Performance comparison the Tanks& Tanks&Temples Temples, Mip-NeRF360, and Deep Blending
datasets.
Tanks& Temples
Mip-NcRF360
Dataset
Deep Blending
Method
SSIM
PSNR
LPIPS
SSIM
PSNR1
LPIPS!
SSIM
PSNR
LPIPS_
Plenoxels 18
0.719
21.08
0.379
0.626
23.08
0.463
0.795
23.06
0.510
INGP-Base 37
0.723
21.72
0.330
0.671
25.30
0.371
0.797
23.62
0.423
INGP-Big R7
0.745
21.92
0.305
0.699
25.59
0.331
0.817
24.96
0.390
336 4
0.759
22.22
0.257
0.792
27.69
0.237
0.901
29.40
0.245
3D-GS 30
0.841
23.14
0.183
0.815
27.21
0.214
0.903
29.41
0.243
Surfsplatting 26
0.837
23.42
0.202
0,804
27.03
0.239
0.895
28.89
0.261
FreGS 49
0.849
23.96
0.178
0.826
27.85
0.209
0,904
29.93
0.240
GES 22]
0.836
23.35
0.198
0.794
26.91
0,250
0.901
29.68
0.252
Mip-Splatting
48
0.851
23.78
0.178
0.827
27.79
0,203
0,904
29.69
0.248
Ours
0.866
24.96
0.120
0.833
28.01
0.189
0.907
30.42
0,199
the conventional Gaussian Splatting technique often struggles 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.
5.2 Ablation Studies
We conduct extensive ablation experiments on the Tanks& Temples dataset. More ablation studies w.r.t. the image areas with rich boundaries, the strategy, the hyperparameters, and the rendering speed of our framework are in Appendix A
gradient approximation
Impact of representing the scene with 2D or 3D Gaussians in DisC-GS. In Sec. 4.1 and
Table 2: Evaluation of our framework on both 2D)
and 3D Gaussian Splattings.
Sec. 3 we focus introducing how we apply DisC-GS on 2D Gaussian Splatting. After that,
Method
SSIMI
PSNRT
LPIPS]
21 Gaussian Splatting
0.836
23.30
0.205
in Sec. 4.3 we introduce how DisC-GS can be applied on 3D Gaussian Splatting in similar
2D Gaussian Splatting Ours
0.866
24.96
0.120
3D Gaussian Splatting
0.841
23.14
0.183
3D Gaussian Splatting Ours
0.863
24.67
0.123
way. Here verify the generality of our frame-
work, we test applying our framework on both 2D and 3D Gaussian Splatting. As shown in Tab. 2 our framework, when applied on both 2D and 3D Gaussian Splattings, can consistently achieve performance improvements, demonstrating the generality our framework.
Impact of the number of control points per Bézier curve. In our framework, inspired by we represent boundaries in the image with the cubic Bézier curve, each of which is formulated by leveraging 4 control points. Here
Table 3: Evaluation on the number of control points per Bézier curve. Method
[16 34
SSIM
PSNRT LPIPS.
control points per curve
0.853
24.14
control points per
0.861
24.58
0.127
we evaluate formulating each Bézier curve by other numbers of control points, and report the results in Tab. 3 As shown, our framework gets optimal performance when the number of control points per Bézier curve is set to 4, and we thus formulate each Bézier curve by utilizing control points in our experiments. Besides, with different choices of the number of control points per Bézier curve from 2 to 5, our framework outperforms the previous state-of-the-art method consistently. This shows the robustness of our framework to the number of control points per Bézier curve.
control points per
0.866
24.96
0.120
control points curve
0.863
24.68
0.126
6 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 "differentiaiility" of our framework, further introduce framework with Bézier- beundaryb gradient approximation strategy. Our framework consistently achieves superior performance across different evaluation benchmarks.
Limitations. While our framework enables Gaussian Splatting to perform discontinuity rendering, we acknowledge that same as existing Gaussian Splatting approaches, our framework still holds certain limitations, such as challenges in rendering large scenes.
10

---

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A Additional Ablation Studies
In this section, we conduct more ablation experiments on the Tanks& Temples dataset.
Evaluation especially over areas with rich boundaries in the testing images. In this work, we point out that Gaussian Splatting holds an
Table 4: Evaluation especially over areas with rich boundaries in the testing images.
MaskedSSIM
Method
Boundary-nch areas Boundary- sparse areas
Baseline(2 Gaussian Splatting)
0.819
0.922
Ours
0.855
0.934
inherent weakness in rendering discontinuities and boundaries, and we propose DisC-GS framework for Gaussian Splatting to render boundaries in the image more accurately. Here, we aim to test our framework, particularly over image areas with rich boundaries. To achieve this, 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.c., the areas that involve or surround the Canny-detecter 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 2D Gaussian Splatting as a baseline, especially in the boundary-rice areas, our framework can achieve significant performance improvement. This demonstrates the effectiveness of our method especially over image areas with rich boundaries.
Impact of introduc- ing different Gaus- sians with different numbers of Bézier curves. In our frame- work, for each Gaus-
Table Evaluation on introducing different Gaussians with different num- bers of Bézier curves.
Method
SSIM 0.863
PSNRT
LPIPS
Different numbers of curves for different Gaussians
24.79
0.122
A curves for each Gaussian
0.866
24.96
0.120
sian distribution representing the 3D scene, we introduce it with M cubic Bézier curves (M curves for each Gaussian). Here, to investigate whether introducing different Gaussians with different numbers of curves can further benefit our framework, we test another variant (different numbers of curves for different Gaussians). Specifically, in this variant, before the start of the training process w.r.t. a certain 3D scene, for each of the source images, we first identifyy its boundary-rich areas similarly as in the above ablation study. After that, during training. when a Gaussian distribution is newly created through the adaptive control process of Gaussian Splatting, instead of directly introducing it with M curves, we introduce the Gaussian with M + Bézier curves if the center فن of its corresponding projected Gaussian lies in the highlighted boundary introduce the newly created Gaussian with M does not result in better performance compared to our framework. This might be because, during training, for each Gaussian representing the 3D scene, its center is learnable, and the Gaussian is thus movable. In other words, for Gaussians that initialized with more curves and thus may be able to represent boundaries more accurately, they can move to areas with fewer (or no) boundaries in the 3D scene. At the same time, for Gaussians that are initialized with fewer curves, they can also move to areas with rich boundaries in the 3D scene. With the above in mind, in our experiments, we introduce each Gaussian representing the 3D scene consistently with the same number of M Bézier curves, equipping each Gaussian with the same level of power in boundary representation.
areas. Otherwise, Bézier curves. As shown in Tab. [5] this variant
Image sharpness evaluation. In this work, we propose DisC-GS, which enables Gaussian Splatting render the sharp boundaries in the image more accurately. Considering this, to per- form further evaluation of our framework, we here also evaluate our framework from the im- age sharpness perspective. Specifically, following [15, we measure image sharpness leveraging the
Table 6: Evaluation on the sharpness of the ren- dered images.
Method
Image sharpness
Baseline(2D Gaussian Splatting)
$1.50 % 57.72 %
Ours
energy gradient function. As shown in Tab. 6 with our framework applied, we can render images more sharply. This further shows the efficacy of our proposed framework.
14

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Impact of the "both sides" situation.
Table Evaluation on the "both sides" situation.
To keep the "differentiability"" of our framework, in this work, we propose a Bézier-boundary gradient approxi- mation strategy. Specifically, in this strategy, when some solutions in S lie on the left side of Ceuree 0] while other solutions lie on its dghclp)
Method
SSIMT
PSNRT
LPIPS
w/o the "both sides" situation
0.854
24.38
0.135
with the "both sides" situation
0.866
24.96
0.120
right side, we approximate Be (0.0 through Eq. under the "both sides" situation, considering (eu Ceurve [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 So exist on both the left and right sides of 22D
both the left and right sides of
0g. ce,rvel0, O], we still consider only the solution that is nearest to Corree (0, 0). and approximate Test [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 So lie on both the left and right sides of 2 [0, 0].
Impact of the small numbers 6, E1, and Eg. In our framework, during gradient approximation, to avoid the gradient exploding problem to happen, we add e small number to the denominator part of Eq. [2] and we also add 61 and 62 to Eq. 1 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 e, é1, and 62 from our gradient approximation process. As shown in Tab. g our framework involving these small numbers performs better than this variant, showing the efficacy of these small numbers.
Table 8: Evaluation on the small numbers 6, E1, and E2.
Method
SSIM 0.858
PSNR1
LPIPS_
w/o small numbers with small numbers
24.37
0.130
0.866
24.96
0.120
Impact of the number of Bézier curves per Gaussian M. In our framework, for each Gaussian distribution rep- resenting 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 A is set to 3, and M 3 is used in our experiments. Moreover, with different choices of M from I to 4, our framework consistently achieves good performance. This demonstrates the robustness of our proposed framework this hyperparameter.
Table 9: Evaluation on the number of
Bézier curves per Gaussian M.
Method
SSIM
PSNRT
LPIPSI
M =
0.853
24.36
0.139
M =2
0.863
24.76
0.124
==3
0.866
24.96
0.120
M
0,862
24.81
0.125
Impact of the initial learning rate set to Ceurve- In our framework, we introduce a new attribute Ceurves for which in our experiments, we set its initial learn- ing rate (Ireurve) to 2e-4. Here also assess the other choices of Ircurve from le-4 to le-3 and re- port the results in Tab. I0 As shown, with different choices of Ireurves the performance of our frame- work is consistent, which shows the robustness of our framework to Ir Tourve
Table 10: Evaluation on the initial learning
rate (Ireurve) set to Ceurve-
Method
SSIM
PSNRT
LPIPS.
Fourre le-4
0,861
24.60
0.126
Dourre 22-4
0,866
24.96
0.120
Treuree 5c-4
0,861
24.64
0.125
Drawren le-3
0,857
24.30
0.132
Rendering time. In Tab. pare 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 Splat- ting and 3D Gaussian Splatting), on an RT'X 3090 GPU in terms of seconds per im- age. As shown, our DisC-GS can achieve competitive rendering time (speed) com- pared to the existing methods leveraging the conventional Gaussian Splatting technique, while obtaining much better performance.
com-
Table 11: Analysis of rendering time in terms of seconds per image. Our framework can run efficiently and sat- isfy most real-time requirements, yet achieves superior performance.
Method
PSNR
Rendering time
Mip-NeRF360 [
22.22
7.143s
2D Gaussian Splatting
23.30
0.007s
3D Gaussian Splatting
23.14
0.007s
Ours (on 2D Gaussian Splatting)
24.96
0.008s
Ours (on 3D Gaussian Splatting)
24.67
0.008s
15

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2D Gaussian Splatting w/o DisC-GS
2D Gaussian Splatting with DisC-GS
Ground Truth
213
Figure 3: Qualitative results of 2D Gaussian Splatting with and without DisC-GS.
B Additional Qualitative Results
In this section, present more qualitative results. Specifically, in Fig. 3 present images rendered by 2D Gaussian Splatting with and without applying our proposed framework DisC-GS; in Fig. 4 we present images rendered by 3DD Gaussian Splatting with and without DisC-GS. As shown, matter representing the 3D scene through 2D or 3D Gaussian distributions, the typical Gaussian Splatting technique fail to render boundaries and discontinuities clearly and with high quality. Yet, 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 Paper
In Eq. in the main paper, via leveraging the implicitization technique in algebra [ enable the cubie Bezier curve be represented in its implicit representation form as:
Bemp (x. y) = " 3 " 2
"wwpy"
2 2 Neyyy + Touy 7a0 + Yoy + 0
(14)
Here in this section, we diseuss how we derive coefficients including Yanns Trzys Yayys Tyuys Tazz Tays Tags Tex Nyo and 7o in Eq. in the main paper (re-shown in Eq. (4lbbve).
16

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3D Gaussian Splatting w/o DisC-GS
3D Gaussian Splatting with DisC-GS
Ground Truth
Figure 4: Qualitative results of 3D Gaussian Splatting with and without DisC-GS.
Specifically, denote the four control points of the cubic Bézier curve wip = (0o, Jo), (I), 31). w2 = (I2, 1/2), and w3 (I3, Vs). To derive the coefficients in Eq, 14 following (1. we first define intermediate coefficients as:
So Toi
(1 3 x 2u +3 Z1;
(2 6 x +3 n0 +3x 22
G3 Fз 20 3x + 3x
Ç4 Mo:
G -3 x 30 +3 x V1;
Çe x 91 +3 +3
G 23 9o
3 02 3 yr
After that, utilizing these intermediate coefficients, following 1 can then compute coefficients in Eg. 14as: as:
Yeur Ç7 x Çr x Çr:
"aaz -3 x Ça x Çr G7; Tayy 3 Ç X Ča X Ca;
Tapy -3 X Ç3 Ç3;
17

---

7ex = -<<_<//wK<1113333333333333333333
Jay [1 75555555855530000000666666333333 775g5376606777773 377777<66<6643
3
799 = 3 [uXXoxxXceccaaaa         766<<<n X
T2 &5 X5534411555533111117777764333333
- "XXX5//4555000XXXXXXXXX006343343
= C 0U233344555711N113333X3377766764
+ "x735577(kkkc7333377733333333333
70
+3 x X
X Xx Cs X Go
C3N x <s:
5o x
Note that, while the above calculation may look complicated, essentially, through the above, each of the coefficients Eq. 14 is computed over the coordinates of the four control points of the curve (i.e. (zo, yo. 31, V/. T2. 1/2. v3. ys just through group of basic arithmetic operations. In other words, the computation of the coefficients Eq. 14 is just a task with (1) time complexity.
D Additional Details about the Bézier-boundary Gradient Approximation Strategy
In our framework, we propose a Bézier-boundary gradient approximation strategy to keep the "differentiability"" of the rendering process. In Sec. Sec. in the main paper, we explain this strategy taking the approximation Bea Be [0.0] as an example. Here in this section, with Courve E 124Xx2
åg we further describe how we approximate ae2p [t.f) where i E {O..., 4M - 1} and j E {0.1), 1).
Specifically, in our framework,
8g is approximated in a similar manner as De7t Vecuree 0.0 except
89
Jeam oeeeee
in the following two places.
18

---

(1) Definition change of (p). To approximate Bq
ducurve (.5] we first need to redefine (e) as Ceurnal [4m 2 2D
Ise (Ggarne [4m], 2D for Ize (p) to accurately represent the Bézier curve w.r.t. curvee [..
Ceurnel [4m + 1], 2DD
Curoen 4m + 3|: p). where m = 4. This done 2D
(2) Reformulation of Eq. 11. Moreover, during approximating
0g (urome [5)) for S to be corrected derived, we also need to reformulate Eq. 11 according the Bézier curve w.r.t. 20
Veurveli. [i Note that
the reformulated equation, should used place of c2D cerrvel..]]
With the above two changes made, we can seamlessly use the strategy introduced in Sec. 4.2 in the
main paper to approximate de [.3]
0g
E Experiments on 11 3D Scenes.
we evaluate our method on total of 13 scenes from the Mip-NeRF360 [4) Tanks& Temples 3 and Deep Blending [23 datasets. Here, following 35 we also evaluate our method on another benchmark with 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.
In Tab. iab.[in the main paper, following 30
40
Table 12: Performance comparison following the evaluation benchmark
[35
Mip NeRF360 Dataset
Tanks& Temples
Deep Blending
Method
SSIM
PSNRT
LPIPS
SSIM
PSNKT
LPIPS]
SSIM
PSNRT
LPIPS.
Scatfold-GS
35
0.848
28.84
0.220
0.853
23.96
0.177
0.906
30.21
0.254
Ours
0.885
29.58
0.158
0.866
24.96
0.120
0.907
30.42
0.199
Licenses
We the Lanks&Templee dataset [31] by following the license here We use the Mip-NeRF360 dataset [4] by following the Apache-2.0 license, Moreover, we use the Deep Blending dataset [2 by following the license of here Besides, we use part of the code owned by Kerbl et al. by following the license of here
19

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26


Reviews:
[{"review_id": "smygbJTvOW", "reviewer": "Reviewer_TofM", "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", "questions": "Is there any paneltation to optimize/adjust the positions of four contral points in a curve? \n\nAs 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": "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.", "rating": 5, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "1. The authors propose a robust and sound pipeline that effectively emphasizes discontinuities and boundary situations while maintaining differentiability. \n\n2. The proposed method demonstrates superior performance on the evaluated benchmarks \n\n3. The paper presents a well-reasoned argumentation and reasoning process", "weaknesses": "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. \n\nThe paper does not include an analysis of the computational complexity of the proposed method. \n\nThere 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?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": ">*Q1: Defines numerous labels. [...] It would be better to include a list of labels and their explanations. [...] label explanations under each equation.*\n\n**A1:** Thanks for your suggestion. Following it, (1) below, we formulate a list of labels (symbols) and their explanations. (2) Meanwhile, under each equation in our paper, we will fully explain all the labels used in that equation. For example, we will re-explain $\\mu$, $r_1$, and $r_2$ under Eq. 4, and re-explain $P$ and $W$ under Eq. 5. We will also include a list of all labels and their explanations in the Appendix of our paper.\n\n\n|Label|Explanation|\n|---|---|\n|$\\mu$|Center of the Gaussian|\n|$\\Sigma$|Covariance matrix of the Gaussian|\n|$c_{SH}$|Spherical harmonic coefficients of the Gaussian|\n|$\\alpha$|Opacity of the Gaussian|\n|$R$|Rotation matrix of the Gaussian|\n|$r_1$, $r_2$|First column and second column of the rotation matrix $R$|\n|$S$|Scale matrix of the Gaussian|\n|$\\mu^{2D}$|Center of the projected Gaussian|\n|$\\Sigma^{2D}$|Covariance matrix of the projected Gaussian|\n|$W$|Viewing transformation matrix|\n|$P$|Projective transformation matrix|\n|$J$|Jacobian of the affine approximation of the projective transformation|\n|$p$|Pixel|\n|$C(p)$|Color at pixel $p$|\n|$\\omega_0$, $\\omega_1$, $\\omega_2$, $\\omega_3$|Four control points of the cubic Bézier curve|\n|$B(t)$|Cubic Bézier curve function where $t$ is a random real number|\n|$B_{imp}(x, y)$|Implicit representation form of $B(t)$|\n|$M$|Hyperparameter representing the number of Bézier curve|\n|$c_{curve}$|Fifth attribute of the Gaussian|\n|$c_{curve}^{3D}$|$c_{curve}$ in 3D space coordinate system|\n|$c_{curve}^{2D}$|$c_{curve}$ in image plane coordinate system|\n|$g(p)$|Indicator function w.r.t. all the $M$ Bézier curves|\n|$g_{sc}(\\omega_0, \\omega_1, \\omega_2, \\omega_3; p)$|Single-curve indicator function|\n|$g_{sc}^0(p)$|Single-curve indicator function w.r.t. the first Bézier curve|\n|$L$|Loss function|\n|$\\phi$|Desired value of $c_{curve}^{2D}[0, 0]$|\n|$S_{\\phi}$|Set of all possible real number solutions for $\\phi$|\n|$\\widetilde{\\phi}$|Solution in $S_{\\phi}$ that is nearest to $c_{curve}^{2D}[0, 0]$|\n|$\\widetilde{\\phi_1}$|Solution in $S_{\\phi}$ that is nearest to $c_{curve}^{2D}[0, 0]$ from its left side|\n|$\\widetilde{\\phi_2}$|Solution in $S_{\\phi}$ that is nearest to $c_{curve}^{2D}[0, 0]$ from its right side|\n|$\\epsilon$, $\\epsilon_1$, $\\epsilon_2$|Small numbers used to avoid the gradient exploding problem|\n\n>*Q2: Analysis of the computational complexity of the proposed method.*\n\n**A2:** Below, we show both the training time and the rendering time (inference time) of our proposed method, on the Tanks&Temples dataset on an RTX 3090 GPU.\n\n| Method | PSNR↑ | Training time | Rendering time (per image) |\n|---|---|---|---|\n| Mip-NeRF360 | 22.22 | 48 hours | 7.143s |\n| 2D Gaussian Splatting | 23.30 | 0.27 hour | 0.007s |\n| 3D Gaussian Splatting | 23.14 | 0.27 hour | 0.007s |\n| Ours (on 2D Gaussian Splatting) | 24.96 | 0.32 hour | 0.008s |\n| Ours (on 3D Gaussian Splatting) | 24.67 | 0.33 hour | 0.008s |\n\nAs shown, though our framework achieves obviously better performance, it only brings a relatively small increase of training time. Besides, during inference, our framework can also achieve a competitive rendering time (speed) compared to the existing methods leveraging the conventional Gaussian Splatting technique and satisfy most real-time requirements. We will extend Tab. 11 in the paper for the table to also include the analysis of training time (besides the analysis of rendering time).\n\n>*Q3: How the curves affect the densification procedure. For example, will the new cloned/splitted points be assigned the same curve attribute?*\n\n**A3:** Yes. In the densification procedure of our framework, when a point (Gaussian) is cloned/splitted into two new points (Gaussians), similar to how other attributes such as the spherical harmonic coefficients and the opacity are assigned to the new points, we simply assign both the new points with the same curve attribute as the original point. We will discuss this in the paper.\n\n>*Q4: Is there any penalty to optimize/adjust the positions of four control points in a curve?*\n\n**A4:** Yes. In our framework for which we have maintained its differentiability, similar to the other attributes of the Gaussian, the positions of the four control points stored in the curve attribute are also learnable parameters. This means that, during training, based on the loss function as the penalty, the gradients would first backpropagate from the loss function to the positions of control points stored in the curve attributes. Leveraging such backpropagated gradients, the positions of control points would then be correspondingly optimized/adjusted to facilitate the accurate representation of the 3D scene.\n\n>*Q5: Will this affect the opacity of the whole point, and whether it will help the point escape pruning.*\n\n**A5:** (1) In Gaussian Splatting, the opacity of the whole point (Gaussian) that would be used during point (Gaussian) pruning is the opacity attribute $\\alpha$. In our framework, to perform \"scissoring out\", we add a binary indicator function $g$ to the color blending function and this does not edit the opacity attribute $\\alpha$ (as can be seen in Eq. 6 in the paper). (2) In other words, as long as \"opacity $\\alpha$ < threshold\", the pruning process of a point (Gaussian) would always still occur in our framework, and our framework would not interfere this process. Thus, our \"scissoring out\" process would not help the point escape pruning.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "oQSwqG1rB4", "reviewer": "Reviewer_F99C", "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.", "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": "The authors do not provide any limitations.  There may be a limitation about the storage size.", "rating": 5, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 4, "strengths": "This paper addresses a very important problem regarding the representability of 3DGS. It introduces a reasonable pipeline to tackle this issue. The idea of using Bézier curves to define the shape is novel and sound.\n\nDisC-GS encodes Bézier curves as an additional attribute of Gaussians, and the rendering and training schemes are compatible with the original 3DGS. This compatibility means that this method can be easily adopted by most 3DGS-based methods.\n\nFurthermore, this paper proposes an effective gradient approximation strategy, making the training of this representation feasible.", "weaknesses": "DisC-GS will increase the storage of the 3D scene representaion. The authors did not evaluate this in experiments.\n\nThe 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.\n\nThe 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.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": ">*Q1: Evaluation of the storage.*\n\n**A1:** Below, we show the storage of our framework on the Tanks&Temples dataset on an RTX 3090 GPU. As shown, either applied on 2D or 3D Gaussian Splattings, our framework **does not increase the storage size**. \n\nThis can be because, admittedly, for each Gaussian representing the 3D scene, our framework equips it with a fifth attribute to represent the Bézier curves and thus makes it have slightly more parameters. Yet, on the other hand, with this additional attribute incorporated, our framework also enables each Gaussian representing the 3D scene to be aware of the discontinuity of shapes and thus enhance its representation ability. As shown below, this allows our framework to ultimately represent the 3D scene with much fewer Gaussians, thus not incurring the increase of memory storage. We'll discuss this more in paper.\n|Method|Storage|Total number of Gaussians|\n|-|-|-|\n|2D Gaussian Splatting|376MB|~1585K|\n|Ours (on 2D Gaussian Splatting)|299MB|~909K|\n|3D Gaussian Splatting|423MB|~1784K|\n|Ours (on 3D Gaussian Splatting)|410MB|~1094K|\n\n>*Q2: Additional qualitative results.*\n\n**A2:** (1) As suggested, we have rendered additional qualitative results (i.e., videos over different 3D scenes each with continuous camera movements). Among these videos, we observe that DisC-GS can consistently preserve multi-view consistency. Meanwhile, in these videos, artifacts such as flickering are also not observed. This can be because, while DisC-GS produces hard boundaries, the boundaries produced over different viewpoints are all based on the projection of the same sets of control points stored in the Gaussians representing the 3D scene. Thus, (multi-view) consistency of boundaries across different views can be kept. Meanwhile, with boundaries rendered in a multi-view consistent manner, artifacts such as flickering are also not observed. (2) During rebuttal, due to its format and space constraint, we provide a sample of such video qualitative results in GIF format in the PDF at the bottom of the \"Author Rebuttal by Authors\" comment. To view those results in animation mode, please use a computer and Adobe Acrobat Reader Version 2024.002 (downloadable from Adobe Reader's official website). We'll also include more results in paper. \n\n>*Q3: Comparison with mip-splatting.*\n\n**A3:** Below, we also compare our method with mip-splatting. As shown, on all three metrics and across various datasets, our framework consistently outperforms mip-splatting. This further shows the efficacy of our framework. We'll include mip-splatting in Tab. 1 in paper. \n\n|Method|Tanks&Temples-PSNR↑|Tanks&Temples-SSIM↑|Tanks&Temples-LPIPS↓|Mip-NeRF360-PSNR↑|Mip-NeRF360-SSIM↑|Mip-NeRF360-LPIPS↓|Deep Blending-PSNR↑|Deep Blending-SSIM↑|Deep Blending-LPIPS↓| \n|-|-|-|-|-|-|-|-|-|-|\n| Mip-splatting | 23.78 | 0.851 | 0.178 | 27.79 | 0.827 | 0.203 | 29.69 | 0.904 |  0.248 |\n| Ours  | **24.96** | **0.866** |  **0.120** | **28.01** | **0.833** |  **0.189** | **30.42** | **0.907** |  **0.199** |\n\n>*Q4: More comparisons w.r.t. boundary issues.*\n\n**A4:** In Figs 3 and 4 in paper, we demonstrate that, compared to both 2D and 3D Gaussian Splattings, our method can mitigate aliasing artifacts and achieve better rendering quality, especially in image regions containing numerous boundaries. To more clearly show that such advantages of our framework are due to its ability in handling boundary issues, we also perform quantitative comparisons.\n\nIn Tab. 4 in paper (re-shown below), we quantitatively test our framework, particularly over the boundary-rich areas of the testing images. Specifically, for each testing image, to acquire its boundary-rich areas, we first use the Canny algorithm [9] to detect its boundaries. We then define its areas that involve or surround its Canny-detected boundaries as its boundary-rich areas, and define its rest areas as its boundary-sparse areas. To enable evaluation over image sub-areas, following [40], we use the MaskedSSIM metric (the larger the better). As shown, especially in the boundary-rich areas of the testing images, our framework can achieve a significant performance improvement. This shows our framework's ability to handle boundary issues from one perspective.\n|Method|Boundary-rich areas|Boundary-sparse areas|\n|-|-|-|\n|Baseline(2D Gaussian Splatting)|0.819|0.922|\n|Ours|0.855|0.934|\n\nMeanwhile, in Tab. 6 in paper (re-shown below), to evaluate if our framework can render sharp boundaries accurately instead of rendering them with blurriness, we also test our framework from the image sharpness perspective. Following [15], we measure image sharpness using the energy gradient function. As shown, our framework can increase the sharpness of its rendered images. This also from another perspective implies our framework's ability in accurately rendering the sharp boundaries in the image.\n|Method|Image sharpness|\n|-|-|\n|Baseline(2D Gaussian Splatting)|51.50%|\n|Ours|57.72%|\n\nWe will further clarify the above in paper.\n\n>*Q5: Bézier curves in a 2D subspace.*\n\n**A5:** (1) In DisC-GS, we can define the Bézier curves in a 2D subspace, as well as directly in a three-dimensional space (3D space). (2) In the method section, we define the Bézier curves in a 2D subspace since we there take 2D Gaussian Splatting as an example and explain how we apply our framework on 2D Gaussian Splatting (as mentioned in lines 168-169 in paper). (3) We can also similarly define the Bézier curves in a 3D space for 3D Gaussian Splatting. As mentioned in lines 388-392 in paper, to achieve this, we just need to modify a single place of our framework (i.e., defining the control points of the Bézier curves for each Gaussian in the 3D space coordinate system instead). (4) As shown in Tab. 2 in paper, no matter when defining the Bézier curves in the 2D subspace or the 3D space, our framework can consistently achieve performance improvements. We will further clarify this in paper.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "mapS07HXbv", "reviewer": "Reviewer_Kjj1", "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.", "questions": "See the weakness part.", "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.", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The author thoroughly discusses the discontinuous characteristics of GS, which lead to inaccurate drawing of boundary curves in images. Innovatively, they propose using Bezier curves to draw continuous boundaries.\n2. The experimental results are highly convincing, demonstrating the superiority of the proposed method over state-of-the-art algorithms, and are supported by abundant ablation experiments.\n3. The authors’ clear writing, well-structured discussion of the problem, detailed methods, and comprehensive presentation of experimental results make the paper easy to understand.", "weaknesses": "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?\n2. 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).\n3. 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.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": ">*Q1: 2D parameters to control the position of the control points.*\n\n**A1:** As mentioned in lines 121-125 in the paper, in existing works, both 2D and 3D Gaussian Splattings have been utilized to represent the 3D scene. In this work, we proposed a framework that can be applied to both 2D and 3D Gaussian Splattings. Specifically, as mentioned in lines 218-220 and lines 388-392 in the paper, **when applying our framework on 2D Gaussian Splatting, we use 2D parameters to control the positions of the control points, while when applying our framework on 3D Gaussian Splatting, we instead use 3D parameters**. As shown in Tab. 2 in the paper, no matter when applied to 2D or 3D Gaussian Splattings, our framework can consistently achieve performance improvements. Meanwhile, in both cases, we do not observe rendering problems in areas significantly affected by large viewpoint changes. We will further clarify the above in the paper to avoid confusion. \n\n>*Q2: When additional properties such as control points and curve rendering are introduced, what impact does this have on training efficiency? [...] impact on training time and convergence(number of iterations).*\n\n**A2:** **(1) Training time.** Below, similarly to the rendering time shown in the Appendix, we also show the training time of our framework, on the Tanks&Temples dataset on an RTX 3090 GPU.\n\n| Method | PSNR↑ | Training time |\n|-|-|-|\n| Mip-NeRF360 | 22.22 | 48 hours |\n| 2D Gaussian Splatting | 23.30 | 0.27 hour |\n| 3D Gaussian Splatting | 23.14 | 0.27 hour |\n| Ours (on 2D Gaussian Splatting) | 24.96 | 0.32 hour |\n| Ours (on 3D Gaussian Splatting) | 24.67 | 0.33 hour |\n\nAs shown, though our framework achieves obviously better performance, no matter when applied to 2D or 3D Gaussian Splattings, our framework only brings a relatively small increase of training time. \n\n**(2) Convergence (Number of iterations).** Meanwhile, in the PDF uploaded during rebuttal (at the bottom of the \"Author Rebuttal by Authors\" comment), we also provide loss curves over the 3D scene Train in the Tanks&Temples dataset. As shown, our framework consistently achieves a similar convergence rate compared to the baseline without parameters such as the control points added. Moreover, for a fair comparison, in the experiments in the paper, we train our framework for the same number of iterations (i.e., 30k iterations) as the existing Gaussian Splatting works. \n\nWe will add the above discussion on training time and convergence to the paper.\n\n>*Q3: Affects on the distribution of Gaussians in edge and surface regions. [...] whether the number of Gaussians in the scene will differ significantly from the original 3DGS*\n\n**A3:** Below, we aim to compare the total number of Gaussians in the whole 3D scene, as well as the number (distribution) of Gaussians in the edge and plane regions of the 3D scene, of our framework with the original 3DGS. Yet, both the edge regions and the plane regions of the 3D scene are not annotated in the dataset. Thus, to enable the latter of the above comparisons, below, we first describe how we estimate the edge regions and the plane regions of each 3D scene.\n\nSpecifically, given a 3D scene, to estimate its edge regions, we here conduct the following 4 steps: (E.1) for each testing image of the scene, we first pass the image over the Canny algorithm [9] for detecting the pixels that are on the edge regions of the image. (E.2) After that, each testing image is further passed over the Depth Anything model [a] to acquire the depth values of its edge-region pixels detected in (E.1). (E.3) Then for each edge-region pixel detected in (E.1), leveraging its 2D coordinate, its depth value, and the viewpoint of its corresponding testing image, we can map this pixel back to the 3D space of the 3D scene. (E.4) Finally, we can define the union of the 3D positions of all the edge-region pixels as the estimated edge regions of the given 3D scene.\n\nMeanwhile, to estimate the plane regions of a 3D scene, we here conduct the following 3 steps: (P.1) for each testing image of the scene, we first pass it over the Depth Anything model [a] to acquire its depth map. (P.2) Given the depth map of a testing image, following [b], using Hough Transform, we can detect the pixels that are on the plane regions of the image. (P.3) Then similar to (E.3-E.4) above, we can transform all the plane-region pixels to the space of the 3D scene, and we can then finally define the union of the 3D positions of all the plane-region pixels as the estimated plane regions of the 3D scene.\n\nWith the edges and the plane regions of the 3D scenes estimated in the above way, we can then estimate the number of Gaussians that cover (overlap with) these regions and thus perform comparisons over the number of Gaussians in the edge and plane regions of the 3D scene. Below, we compare the total number of Gaussians in the whole 3D scene, as well as the number of Gaussians in the edge and plane regions of the 3D scene, between our framework and the original 3DGS, on the Tanks&Temples dataset.\n\n|Method|Number of Gaussians in edge regions|Number of Gaussians in plane regions|Total number of Gaussians in the whole scene| \n|-|-|-|-|\n| Original 3DGS | ~605K | ~647K | ~1784K |\n| Ours (on 3DGS) | ~196K | ~433K | ~1094K |\n\nAs shown, compared to the original 3DGS, our framework can represent the 3D scene with much fewer Gaussians, especially over its edge and plane regions. This further demonstrates our framework's ability in performing discontinuity-aware rendering and handling boundary issues. We will also discuss the above more in paper.\n\n[a] Depth Anything: Unleashing the Power of Large-Scale Unlabeled Data.\n\n[b] Hough Transform for real-time plane detection in depth images.\n\n>*Q4: Description of the challenges.*\n\n**A4:** Thanks for your suggestion. We agree that the Gaussian Splatting technique still faces challenges. As suggested, we will provide a detailed description of the mentioned challenges in the paper.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "JmYKmyt6Rs", "reviewer": "Reviewer_ncnd", "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.", "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": "Please refer to the weakness part.", "rating": 4, "confidence": 4, "soundness": 4, "presentation": 2, "contribution": 2, "strengths": "I think the ideas proposed in this paper are sound and easy to follow. Using two Bézier curves to segment the Gaussian, making it better at modeling high-frequency boundaries, is very reasonable and much needed by the community. The comparison in Tab. 1 with numerous compelling baselines and the excellent rendering metrics are greatly appreciated. While achieving the best rendering quality, a balance with FPS was also maintained, as shown in Tab. 11, where the rendering speed did not significantly decrease. The ablations are also appreciated.", "weaknesses": "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?\n2. 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.\n3. 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.\n4. 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?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": ">*Q1: Qualitative comparisons and quantitative metrics.*\n\n**A1:** **(1) Qualitative comparisons.** In the PDF uploaded during rebuttal (at the bottom of the \"Author Rebuttal by Authors\" comment), besides in the 3D scenes in Figs 3 and 4 in paper, we also provide qualitative comparisons in more 3D scenes. Moreover, Scaffold-GS is also added in the comparisons. As shown, across different 3D scenes, our framework consistently achieves better visual effects than existing methods such as Scaffold-GS, further showing the efficacy of our framework. Due to the space limitation of the PDF, we'll also add qualitative comparisons in more 3D scenes to paper. \n\n**(2) Quantitative metrics.** Moreover, for the mentioned mismatch between the difference in rendering metrics and the difference in visual effects, this can be because, different testing views are of different complexities (i.e., containing different numbers and scales of discontinuities). During measuring the rendering metrics, all different testing views (including those very complex ones) are utilized and they all contribute to the measurement. Yet, in Figs 3 and 4 in paper which serve as the qualitative results of our method in general, we did not dedicatedly show all those very complex testing views that suffer from very severe boundary modeling issue.\n\nAs shown in the PDF uploaded during rebuttal, while good visual effects can be consistently achieved by our framework across different viewpoints and different 3D scenes, especially over those testing views that are more complex (richer in boundaries), the difference in visual effects between our method and existing methods such as Scaffold-GS tends to be more significant. These complex testing views, during quantitative evaluation, also tends to more significantly contribute to the superior performance of our method in metrics such as SSIM and LPIPS. We will also add more qualitative comparisons over those complex testing views across different 3D scenes to paper.\n\n**(3) Questions w.r.t. evaluation details.** (3.1) Yes. We use the same data as vanilla GS without rerunning COLMAP. (3.2) Following the vanilla GS, during calculating the rendering metrics, for images with width exceeding 1600, we would rescale them for their width to be 1600. For rest images we would use their original resolution. (3.3) Yes, SSIM and LPIPS calculations are both done in the same manner as vanilla GS. \n\n>*Q2: Synthetic scenes.*\n\n**A2:** As suggested, we also evaluate our method on the synthetic NSVF and NeRF datasets, on an RTX 3090 GPU. \n\nFor rendering metrics, below, we report the SSIM, PSNR, and LPIPS metrics. Moreover, we also show images rendered over these two datasets in the PDF uploaded during rebuttal.\n|Method|NSVF-PSNR↑|NSVF-SSIM↑|NSVF-LPIPS↓|NeRF-PSNR↑|NeRF-SSIM↑|NeRF-LPIPS↓|\n|-|-|-|-|-|-|-|\n|2D Gaussian Splatting|37.59|0.984|0.014|33.96|0.969|0.032| \n|Ours (on 2D Gaussian Splatting)|**38.37**|**0.988**|**0.010**|**34.18**|**0.973**|**0.025**| \n|3D Gaussian Splatting|37.07|0.987|0.015|33.32|0.970|0.031|\n|Ours (on 3D Gaussian Splatting)|**38.32**|**0.988**|**0.012**|**34.01**|**0.972**|**0.029**|\n\nAs shown above, on synthetic scenes, our framework, when applied on both 2D and 3D Gaussian Splattings, can consistently achieve performance improvements. Moreover, as shown in the PDF, on synthetic scenes, our framework can also render images with high quality. This further shows our framework's efficacy.\n\n>*Q3: Discussion on the number of Gaussians, training time and storage.*\n\n**A3:** Below, we show the number of Gaussians, storage, training time, and inference time of our framework, on the Tanks&Temples dataset on an RTX 3090 GPU.\n|Method|PSNR↑|Number of Gaussians|Storage|Training time|Inference time (per image)|\n|-|-|-|-|-|-|\n|2D Gaussian Splatting|23.30|~1585K|376MB|0.27 hour|0.007s|\n|Ours (on 2D Gaussian Splatting)|24.96|~909K|299MB|0.32 hour|0.008s|\n|3D Gaussian Splatting|23.14|~1784K|423MB|0.27 hour|0.007s|\n|Ours (on 3D Gaussian Splatting)|24.67|~1094K|410MB|0.33 hour|0.008s|\n\nAs shown, though our framework achieves obviously better performance, it uses less number of Gaussians, does not increase the memory storage, only brings a relatively small increase of training time, and can perform rendering efficiently during inference. We'll discuss the above in paper.\n\n>*Q4: Optimization of DisC-GS.*\n\n**A4:** **(1) Computations.** No, we do not use special optimization strategies. Instead, we would like to highlight that computations introduced by DisC-GS are not mathematically complex. Specifically, in the forward rendering process, as mentioned in lines 261-265 in paper, for each Bézier curve, the binary indicator function that DisC-GS builds for it is with $O(1)$ time complexity. Meanwhile, in the gradient backpropagation process, the cubic functions that DisC-GS introduces (Eq. 11 in paper) also have closed-form solutions, and can thus be very efficiently solved also in $O(1)$ time complexity. Thus, overall, as shown above in **A3**, our framework would only bring a small increase of training time. Meanwhile, during inference, our framework can also perform rendering efficiently. \n\n**(2) Local optima.** As mentioned above, the computations our framework introduce are not complex. It thus would not add significant complexity to the typical Gaussian Splatting technique. Specifically, across different 3D scenes among different datasets, our framework consistently outperforms both 2D and 3D Gaussian Splattings leveraging the typical Gaussian Splatting technique. This also implies that our framework would not easily get stuck in local optima. Meanwhile, in the PDF uploaded during rebuttal, we also provide loss curves over the 3D scene Train in the Tanks&Temples dataset. As shown, no matter when applied to 2D or 3D Gaussian Splattings, our framework consistently achieves a similar loss reduction trend. This further implies that our framework would not raise the risk of easily getting stuck in local optima.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:
We thank all reviewers for recognition of our contributions (Reviewer TofM: "an innovative framework", "a robust and sound pipeline"; Reviewer F99C: "a novel discontinuity-aware rendering pipeline", "addresses a very important problem", "the idea of using Bézier curves to define the shape is novel and sound"; Reviewer Kjj1: "significantly enhances the rendering results of scene boundaries", "innovatively, they propose using Bézier curves to draw continuous boundaries"; Reviewer ncnd: "a novel kernel function", "very reasonable and much needed by the community").

Meta-review: 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.
Decision: Accept (poster)