Metropolis-Hastings Sampling for 3D Gaussian Reconstruction
Authors: Hyunjin Kim
Haebeom Jung
Jaesik Park
Venue: NeurIPS 2025 poster ()
Keywords: 3D Gaussian Splatting
Metropolis-Hastings Sampling
Primary area: applications
TL;DR: A Metropolis–Hastings driven adaptive sampling framework for 3D Gaussian Splatting that unifies densification and pruning to achieve more efficient, high-quality real-time scene reconstructions with fewer Gaussians.

Abstract: We propose an adaptive sampling framework for 3D Gaussian Splatting (3DGS) that leverages comprehensive multi-view photometric error signals within a unified Metropolis-Hastings approach. Vanilla 3DGS heavily relies on heuristic-based density-control mechanisms (e.g., cloning, splitting, and pruning), which can lead to redundant computations or premature removal of beneficial Gaussians. Our framework overcomes these limitations by reformulating densification and pruning as a probabilistic sampling process, dynamically inserting and relocating Gaussians based on aggregated multi-view errors and opacity scores. Guided by Bayesian acceptance tests derived from these error-based importance scores, our method substantially reduces reliance on heuristics, offers greater flexibility, and adaptively infers Gaussian distributions without requiring predefined scene complexity. Experiments on benchmark datasets, including Mip-NeRF360, Tanks and Temples and Deep Blending, show that our approach reduces the number of Gaussians needed, achieving faster convergence while matching or modestly surpassing the view-synthesis quality of state-of-the-art models.

Metropolis-Hastings Sampling for 3D Gaussian Reconstruction
Hyunjin Kim!
Haebeom Jung"
Jaesik Park?
IC San Diego
Seoul National University
hyk071Qucad.edi fhaebeom. jaesik.pan parkk ac.
Abstract
We propose an adaptive sampling framework for 3D Gaussian Splatting (3DGS) that leverages comprehensive multi-view photometric error signals within unified Metropolis- Hasttnngg approach. Vanilla 3DGS heavily relies on heuristic-based density-contre mechanisms (e.g., cloning, splitting, and pruning), which can lead to redundant computations or premature removal of beneficial Gaussians. Our framework overcomes these limitations by reformulating densification and pruning as probabilistic sampling process, dynamically inserting and relocating Gaussians based on aggregated multi-view errors and opacity scores. Guided by Bayesian acceptance tests derived from these error-based importance scores, our method substantially reduces reliance on heuristics, offers greater flcxibility, and adaptively infers Gaussian distributions without requiring predefined scene complexity. Experiments on benchmark datasets, including Mip-NeRF360, Tanks and Temples and Deep Blending, show that our approach reduces the number Gaussians needed, achieving faster convergence while matching or modestly surpassing the fiew-synthesis quality of state-of ootte--f--hh-arr models. Our project page is available at https: / j/j.hunjjjji. inkim. gi thub. 1o/MH-3DD 3DGS.
Introduction
Novel view synthesis (NVS) focuses creating photo-realistic images of 3D scene from unseen viewpoints. is significant due to its broad applicability across various real-world scenarios. While Neural Radiance Fields (NeRF) [36] has largely advanced Novel View Synthesis by mapping 3D loca- tions and view directions to view-dependent colors and volumetric densities, its slow rendering speed hinders real-world deployment. To address these limitations, 3D Gaussian Splatting (3DGS) [20] recently emerged, enabling real-time photo-realistic rendering by representing complex scenes with explicit 3D Gaussians, each which is projected the screen and defined by its position, anisotropic covariance, and opacity.
Despite these advantages, 3DGS and its extensions [18, 20, 56] rely heavily heuristic-based adaptive density control mechanisms, such as cloning, splitting, and pruning by introducing fixed thresholds opacity, size, and positional gradients in the view space. This leads inflation of memory with redundant Gaussians when thresholds are loose, or to a sacrifice of fidelity when they are tight. To tackle this problem, Kheradmand al. [22] recast 3DGS updates Stochastic Gradient Langevin Dynamics (SGLD), framing Gaussian updates as Markov Chain Monte Carlo (MCMC) samples from an underlying probability distribution representing scene fidelity. As SGLD updates Gaussians in the direction of the reonstrrconsttucttion gradient and adds small noise term, the procedure behaves like gradient-basee optimizer. Although replaces heuristics with state transitions, it still fixes the total number of Gaussians upfront, limiting adaptability to scenes of varying complexity.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Motivated by these considerations, we propose an adaptive sampling framework that uniquely leverages comprehensive multi-view error signals within a unified Satropolis-Hastimng (MH) [6. 35] approach. We reformulate 31D Gaussian Splatting (3DGS) newly derive a concise, rigorously proven acceptance probability grounded in the classical MH rule. By casting the densification and pruning stages of 3DGS as a probabilistic sampling problem, our method selectively adds Gaussians in regions with high visual or geometric significance as indicated by aggregated view-consistent photometric discrepancies. By deriving Gaussian-level importance scores from opacity and multi-view error metrics, we generate Gaussian candidates through coarse-to sor--to-fiiin fine proposals, which subsequently accepted rejected via Bayesian acceptance tests. This probabilistic process adaptively converges to a well-balanced Gaussian distribution with only minimal reliance on heuristics. It requires no prior knowledge of scene complexity while seamlessly integrating with existing 3DGS workflows to achieve better reconstruction quality, capturing both coarse structures and fine details.
MH sampling process and
We demonstrate the effectiveness of our approach on real-world scenes from the Mip-NeRF360 [1], Tanks and Temples [23], and Deep Blending [17] datasets. Our method reduces the number of Gaussians needed to represent each scene solely through an improved sampling mechanism, while consistently matching or surpassing the view-synthesis quality of 3DGS across diverse scenes. Additionally, our approach converges faster than 3DGS-MCMC [22], achieving target PSNR values in fewer training iterations and reduced wall-clock time. By coupling principled probabilistic inference with dense multi-view error signals, our method matches the rendering accuracy of 3DGS- MCMC [22] with fewer Gaussians, demonstrating that comparable quality can be achieved with leaner, more generalizable representation that converges more rapidly.
We summarize our contributions as follows:
We introduce an adaptive Metropolis- Hastingss (MH) framework that replaces densification heuris- tics in 3DGS with a closed-form Bayesian posterior for each Gaussian and a lightweight photometric surrogate that turns the MH acceptance ratio into a single logistic-voxel product.
To this end, we derive Metropolis-Hastinge transition equations that are specialized to 3DGS and prove that this MH sampler is mathematically sound, demonstrating that the heuristic densification rule in 3DGS can be recast as principled MH update.
Our experiments show that our framework achieves compelling view synthesis quality while converging faster.
2 Related Work
2.1 Neural 3D Scene Representation
Novel View Synthesis generates realistic images from novel viewpoints that differ from the original captures [56]. Neural Radiance Fields (NeRF) [36] pioneered this field by using MLPs to model 3D scenes from multi-view 2D images, inspiring extensions for large scenes [13, 46, 52], dynamic content [8, 14, 40, 41, 49], and pose-free reconstruction [2. 19, 27, 33, 34, 57]. However, NeRF's extensive training and rendering time prompted numerous acceleration efforts [4, 10, 11, 37, 42, 45].
3D Gaussian Splatting (3DGS) [20] emerged as a compelling alternative, using differentiable ras- terization of 3D Gaussian primitives initialized from sparse point clouds generated from Structure- from-Motion [44]. This approach enables efficient optimization and high-resolution output, spurring extensions in dynamic scenes [25, 26, 31, 51, 53], 3D generation [5, 29, 47, 48, 55]. and large-scale reconstruction [21, 28, 30J. Yet, 3DGS suffers from a significant memory burden due to covari- ance matrices and spherical harmonics, leading memory-efficiee variants that rely on heuristic density-control rules [9, 15, 16, 24, 38, 28,33,555 39,50].
Recent work has explored probabilistic sampling in 3D reconstruction: Goli et al. [12] and Bortolon et al. [3] apply Metropolis- for sample refinement and surface point selection in NeRF While these methods use Metropolis-Hasting for specific components, our work reformulates the entire 3D reconstruction problem as a Metropolis-Hastings sampling process, replacing heuristic density controls in 3DGS with unified probabilistic framework that enhances both efficiency and generality.

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2.2 Adaptive Density Control for 3D Gaussian Splatting
Kheradmand et al. [22] recently introduced the use of Stochastic Gradient Langevin Dynamics (SGLD) for sampling and optimization in 3DGS, combined with an ppacity-driven relocation strategy to shift low-opacity Gaussians towards higher-opacity regions, while enforcing fixed limit on the total number of Gaussians. Additionally, Rota Bulò et al. [43] introduces error- error-based densification with perceptual error prioritization and sets a global limit on the number of Gaussians. Similarly, Ye et al. [54] addresses the limitations of gradient collision in adaptive density control by proposing homodirectional gradient as a criterion for densification. Moreover, Mallick et al. [32] presents a score-basee guided densification approach that uses training-time time priors to restrict growth. More recently, Deng et al. [7] introduced deterministic geometric splitting rules for efficient density control. As 3DGS-MCMC [22] aligns most closely with our objectives and delivers the best overall reconstruction quality, it serves as our primary baseline.
In contrast, our proposed framework leverages a unified Metropolis-Hasting sampling strategy grounded in comprehensive multi-view error signals. Rather than relying on thresholds, our method computes Gaussian importance scores from aggregated multi-view photometric errors and dynami- cally adapts the number and distribution of Gaussians via probabilistic inference. We derive the MH apportance-weigheee proposals preserve
acceptance rule in the context of 3DGS and prove that detailed balance with respect to the multi-view photometric likelihood. This rigorous formulation distinguishes our framework from prior methods and enables natural scaling with scene complexity.
Preliminary
3.1
3D Gaussian Splatting
3D Gaussian Splatting (3DGS) [20] represents the 3D scene as set of anisotropic 3D Gaussians, further optimized by differentiable tile rasterization. Each 3D Gaussian is characterized by position vector and covariance matrix Σ:
(xx) e (x Σ RSS R
where the covariance matrix is factorized into a scaling matrix S and rotation matrix R. The transformed 2D covariance matrix E' is calculated by the viewing transform W and the Jacobian the affine transformation of the projective transformation J as >' JWEW J". The color C' of pixel can be computed by blending N ordered points overlapping the pixel C =iEN C;0; [1 (1 a,), where e and a; represent the view-dependent color and opacity computed from 2D Gaussian distribution.
To achieve high-quality representation, 3DGS employs adaptive density control that clones Gaus- sians in under-reconstructed areas and splits Gaussians in over-reconstructee regions. During den- sification, 3DGS evaluates the positional gradient v,C of each Gaussian, refining further if the gradient exceeds predefined threshold Tp. If a Gaussian surpasses the size threshold т., indicating over-reconstrrucionn is split into smaller Gaussians. Alternatively, under-reconttruuete areas prompt cloning by shifting Gaussians along the gradient direction for better scene representation.
3.2 Metropolis- Hastings Algorithm
The Metsonolis-Hastings algorithm [6, 35] is a widely used Markov Chain Monte Carlo (MCMC) method for sampling from complex, nigh-dimmnsional probability distributions. The objective is to construct Markov chain whose stationary distribution i (x) matches the target distribution, enabling efficient sampling where direct methods are impractical.
Given a target distribution x xxx) defined over a state space X, the algorithm generates a sequence (x(1) ) N of samples by iteratively proposing candidates from a proposal distribution g y) and accepting or rejecting them based on an acceptance probability. At each iteration i, new candidate y is proposed from the distribution (xx(1)..)
.).. The acceptance probability is further computed as
p(x(0.,) = min T(v)g(r,(()
( (1=xx(0)((((),)))
(2)
3

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The candidate is accepted with probability p(x("),) otherwise, the chain remains at the current state, x(++11 = xx) The acceptance probability is designed to ensure that the Markov chain satisfies
the detailed balance condition:
क (x) y)) = (()))((x,)p(),)>>
(3)
which guarantees that (x) is the stationary distribution of the Markov chain. The distribution the generated samples converges the target distribution 7xx) as the number of iterations N increases.
Method
Our method begins with key design choice: unlike 3DGS, which densifies points at fixed thresholds, employ a lean point-based sampler that keeps the most informative Gaussians, sharply reducing storage while preserving fidelity. Section 4. I lays the theoretical groundwork for this: we extend the MetrooolissHHasting algorithm to 3DGS and rigorously derive acene-adaptive equations that generalize the traditional formulation. Section 4.2 describes how the sampler proposes candidates guided by multi-view photometric and opacity errors, and Sec. 4.3 presents the MH acceptance test with a newly derived practical expression for the MH acceptance probability. The complete algorithmic flow in Algorithm 1 and 2 in Appendix D.
4.1 Formulation of the Metropolis- Framework
Our main objective is eliminate heuristic dependence in adaptive density control by recasting densification and pruning as a principled sampling problem. Specifically, we interpret each modifi- cation of the Gaussian chain perturbs the current scene representation θ by (i) inserting new Gaussians near pixels that still exhibit high multi-view error (ii) relocating Gaussians whose opacity has collapsed. The proposed scene O' is then accepted or rejected according to a probabilistic rule that, in expectation, favors configurations that reproduce the captured images better while remaining spatially sparse.
proposal in Metropolis- Hastings Markov Chain. Each step in the
of Gaussian splats, where each splat contains parameters (x,, Σ, C₁2 0;2) ). representing the position, covariance, color, and opacity of a Gaussian. We define the overall loss function as:
In our formulation, scene is described as a collection = {
(()) = (1 A)C1 + ACD-SSI + Aopacity σ + Ascale Σ
where L1 and SSMS are the loss terms from the original 3DGS [20] method. The last two terms are regularizers on the mean opacity and covariance, used to penalize large or widespread opacities and large Gaussian spreads, respectively, as advocated by 3DGS-MCMC [22]. We use
0.2, Aopacity = 0.01 and Ascale = 0.01 in our framework.
From likelihood viewpoint, we treat the minimization of L(O) equivalent to maximizing the likelihood of the observed image set D = (T()). Thus, we can interpret this as:
In (D () = - L(e)
(5)
Since likelihood terms are usually expressed with base e, we adopt the natural logarithm (In) throughout to keep the formulas clean and consistent.
Furthermore, photometric fidelity alone does not stop the optimizer from piling Gaussians onto the same patch once converges locally. Thus, we discourage such redundancies by introducing a voxel-wise prior:
In p() = ~ Σ In(1+ce(oo) NEV
where ce(v) (v) is the number of Gaussians in voxel v. This prior encourages at most one or two Gaussians per voxel. Since the cost grows logarithmically, empty cells get almost no penalty, but the penalty rises quickly once the voxel becomes crowded. Combining Eq. 4 and 6, we are able to define a negative log-posterior:
E(0) (())= + A Σ In(1 1n(1+cen cc
VEV

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Based on this derivation, we can define the posterior density based on Bayesian statistics:
1 E(() Z
t ((+)
(8)
where Z is normalization constant (See Appendix A for details). Our framework samples Gaussian configurations in proportion to (+) and adaptively refines the 3D Gaussian representation to cover undersampled regions. To facilitate this sampling. we construct Markov Chain whose states are successive scene representations Og I,.... O₁ From a given state O, we draw a proposal O' by adding Gaussians in high-error areas or relocating low opacity Gaussians, with probability density q(e' (t). The construction of g( (e,) is detailed in Sec. 4.2.
We accept or reject O' according to the Metropolis- Heatopolis-Hastingss (MH) rule:
p(O' (.) = min (1, ((,)) ).)
((0)) ((O,
p( E
(9)
where p(O') (t) indicates the standard MH acceptance probability. At each step we draw u U(0, 1); if < p. the proposal is accepted and we set 0:+1 O', otherwise we keep 0 = θ We explain how the MH rule accepts or rejects O' again in Sec. 4.3. Over many iterations, this Markov Chain converges toward our target distribution (()), systematically densifying the Gaussian set, reducing loss, and relocating Gaussians that fail to improve image fidelity.
4.2 Proposal Generation
Building on the Metropolis-Hattings framework, we now specify how a candidate configuration O' is generated from the current state Ô. The guiding principle is to concentrate proposals in problematic regions where è either (1) lacks sufficient opacity, indicating missing or overly sparse geometry, or (2) exhibits large photometric error.
4.2.1 Per-pixel importance field
At every densification step, we first choose a working view subset C2 = I Cr J from the full training set ensure balanced use of all viewpoints over time and broad viewing-angle coverage. Note that this selected subset C₁ is used only for computing the importance field. The subset size is annealed according
k = max (I, (1, |cl]),
Emin
Te
(10)
fmax
Emin
so that early iterations (M) s 0) ensure broad coverage across diverse views, whereas we reduce the subset size focus on specific vantage points requiring finer coverage. We use a round-robin schedule that walks through C in contiguous blocks of length k2 as this allows every viewpoint to contribute equally to the eиror signal after a few iterations. For each viewpoint
Ct. we compare the current prediction f(() the ground truth image ((e) and record per-pixel SSIM and Lı errors. To construct a unified importance field for Gaussian proposal generation, we aggregate these error maps and construct view-averaged importance maps:
SSIMage (p)
- 2[ (p), (( (p)). Clagg()) Ke CECs
-1) ((c
(11)
he oEC.
where i is the number of selected viewpoints, and p indicated image pixels. This aggregation produces spatial field that captures the difficulty reconstruction across the current viewing subset. We then feed SSIMagg (p) and L1 Clags (p) along with the opacity (p)), which is directly obtained by projecting the per-Gaussian opacity on pixel p. into our importance score. After applying robust normalization to each quantity (to mitigate outliers and match dynamic ranges), the three cues are fused through a logistic function:
spp) (a0(p) BSSIM agg (p)
aallag (p))
a(2) Itc-x
(12)
where a, 8, control the relative emphasis opacity, structural similarity, and photometric fidelity, respectively. We use a = 0.8,3 3 = 0.5, 0.5 in our tests. The scalar field E (0, 1) highlights the pixels that are simultancously under-covered and photometrically inaccurate across the chosen views.
5

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4.2.2 Mapping Pixels to Gaussian Importance
Given the 3D position calibrated projection II: IR + R2. Rather than integrating the importance field over the full elliptical footprint of gi, which requires evaluating hundreds of pixels per splat, we adopt lightweight surrogate that uses the floor operator to sample only the pixel lying directly beneath the center of gi:
R of Gaussian Ji. we obtain its image-plane coordinates via the
( = ((I(())))
(13)
where s(-) is the logistic importance map Eq. 12. The importance weight T(i) then normalized once per iteration to give a categorical distribution Pizc(i) = T(t)/ Σ3 T()).
4.2.3 Proposal Distribution
Having established Bayesian viewpoint on our 3D Gaussian representation and a per-pixel impor- tance score sp)), we now detail the proposal mechanism in Metropolis-Hastings (MH) sampler. We maintain a Markov Chain θg - Q - .... each O, comprising the current Gaussian set. From state , we form a proposal O' in two phases.
of size B. from pick- For every During the Coarse Phase, we sample a batch Ie = (i,,.... BB.) index i Ie, we create new Gaussian:
gi¹ x; + 5;, Σ,, C₁ a.), 5 N2 connse T)
(14)
where Coarse is chosen to be large enough to enable the sampler to explore broad coverage gaps. In the Fine Phase, second batch Ij { ( Similar to the coarse phase, we create new Gaussian for every index in i Ir:
'n } is also drawn from the distribution Ppick-
gft (x, 8% 2,, C, at), 8 N(0,omme
(15)
where Oine < Ocuarse. The smaller perturbation radius focuses on residual high- high-importance regions that still yield large weights
I(i) (and hence large Ppicx(t) after the coarse phase fills the major coverage gaps. This allows our method to refine the geometry further using locally placed Gaussians. See Fig. for visualization of the coarse-fine proposal phase.
Finally, all the newly spawned Gaussians are combined to form the birth component O'in of the proposal. This is combined the current state θ to yield full proposal O':
Oiith = fo," E U (of i Zz). O'= = O, U Ohinh
(16)
Formally, the probability of drawing the birth component is:
q ())
II iET,UT
(1)
(17)
4.2.4 Relocation of Low Contributing Gaussians
In addition to densification, our framework incorporates a relocation mechanism inspired by 3DGS- MCMC [22] to reuse Gaussians with persistently low opacity. Let D {i a; S] denote the set of Gaussians with low opacity, and A = {j a, > r} the set of high opacity Gaussians. We consider T 0.005. For each 9: E D, we sample replacement index j E AA with probability
ag
p(j)
KEEA
and update g.'s parameters by transferring those from gj. optionally modulated by a ratio reflecting the sampling frequency. The relocation step preserves the representational fidelity of our framework by continuously reinforcing effective Gaussians while reassigning mnder-contiibutin ones.
4.3 Metropolis- Proposal Acceptance
Now, the Metropolis-Hastion sampler must decide whether the proposal O' is kept or rejected. Recall the MH rule in Eq. 9. Considering the posterior density in Eq. 8, we can rewrite the rule as:
( O')
Psm = min 1,
(18)
e θ

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PSNR over training iterations for our method and -MCGC averaged across all scenes. Dotted vertical lines indicate when each method reaches 98 % of its eventual PSNR, high- lighting near-optimal convergence speed. While both methods ultimately attain comparable final PSNR, ours converges faster.
Figure 1:
Table I: Time equal PSNR comparison be-
tween our method and 3DGS-MCMC, averaged across all benchmark scenes. Our method con- sistently reaches target PSNR thresholds faster.
Target PSNR (dB)
Time Ours (s / mins)
Time 3DGS-MCMC
(s/ mins)
21
16,30 / 0.27
17.08 / 0.28
24
61.34 / 1.02
98.38 / 1.64
3DGG-MONC
27
287.01 / 4.78
341.64 / 5.69
30
851.52 / 14.19
983.05 / 16.38
Figure 2: 3D visualization of (a) coarse and (b)
fine proposal stages (see Sec. 4.2.3 for details).
10
Ours
DDGGS-MMMC
31.9 min
13.4 min
1005
10000
15000
20000
25000
30000
Iteration
(b)
where AE ((e')-E(()) The proposal O' is batch of new Gaussians, but since the perturbations are independent, we can evaluate the acceptance separately. Considering one candidate Gaussian whose center falls in voxel v', the insertion of that particular Gaussian leads to AE as:
1+ (
+ +)( - AC - In D(V') 1 (nce(T")
AE = AC Au In
(19)
where AL is the photometric change and c(v') is the voxel occupancy before the insertion. (See Appendix B for details.) However, evaluating L requires rendering every view in the set C. which is computationally expensive. Thus, we use the tight empirical correlation between the importance map I(i) and the loss reduction, -AC Rv I ((i), as a surrogate. (See Appendix for details.)
Moreover, the calculation of q() O') in Eq. 18 requires the enumeration of the reverse action, which is deleting the particular newborn Gaussian. As our framework does not generate such move, we absorb the unavailable eeverse-propote density into the voxel factor D(r') = (1 + X, (c'))= which already penalises insertions in crowded cells. Substituting both surrogates into Eq. 18 gives an upper bound min(1, (1, (0) D(u'), and mapping the exponential through the logistic a=2)
(1 + +)= yields a practical version of the MH rule:
p(1) = (T()) D(o')
(20)
Following the Metropolis-Hasting algorithm, we draw a uniform random number u. ~ (/(0,) 1) for each candidate Gaussian. The proposal is accepted and added to the current set O when < p(). otherwise is rejected. Because p(i) = o( (I()) (t") is large only when the per-Gaussiag importance [()) is high and the voxel factor D(o') is close one, Gaussians that target significant photometric errors in sparsely populated regions are accepted with high probability, whereas candidates in already crowded voxels or low-error areas are mostly discarded.
4.4 Comparison with 3DGS-MCMC [22]
Although both methods fall under the broad scope of Monte-Carlo sampling, they embody funda- mentally different MCMC philosophies. 3DGS-MCMC treats every Gaussian parameter as latent variable and applies SGLD: it perturbs all dimensions with small, isotropic noise and accepts every proposed move, so the chain drifts locally through the existing cloud.
In contrast, our method runs Metsopolis-Hastinggs birth sampler that targets only positions. We draw global proposals straight in image-space voids highlighted by a multi-view error map, then pass each
7

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Table 2: Quantitative results of our method and state-of-the art models evaluated on Mip-NeRF 360 [1]. Tanks&Templee [23] and Deep Blending [17] datasets. We present the baseline results from 3DGS [20]. For a fair comparison, we also report the results for 3DGS obtained from our experiments (Denoted as 3DGS*), as well as those for 3DGS and 4DGS-MCMC trained with the same number of Gaussians as our method (denoted as 3DGS-S and
respectively). The unit of the number of Gaussians is million. The results are colored as best second-best, and third-best. To
avoid redundancy, only 3DGS results are considered instead of 3DGS.
Dataset
Mip-NeRF 360
Tanks & Temples
Deep Blending
Method
PSNR
SSIM
LPIPS)
# GS.
PSNR1
SSIM
LPIPS|
# GSL
PSNR
SSIM
LPIPS|
# GSL
Plenoxels [10]
23.08
0.626
0.463
21.08
0.719
0.379
23.06
0.795
0.510
INGP base [37]
25.30
0.671
0.371
21.72
0.723
0.330
23.62
0.797
01.423
INGP-big [37]
25.59
0.699
0.331
21.92
0.745
0,305
24.96
0.817
0.390
Mip-NeRF 360 [1]
27.69
0.792
0.237
22.22
0.759
0.257
29.40
0,901
0.245
3DGS [20]
27.21
0.815
0.214
23.14
0.841
0.183
29.41
0.903
0.243
3DGS [20]
27.44
0.811
0.223
3.176
23.63
0.848
0.177
1.831
29.53
0.904
0.244
2.815
3DGS-S [20]
26.62
0.761
0.299
0.723
23,38
0.828
0,214
0.773
29.27
0.897
0.272
0.751
3DGS- MCMC [22]
27.69
0.816
0.234
0.723
24.30
0.862
0.170
0.773
29.84
0.906
0.254
0.751
Ours
27.34
0.798
0.241
0.723
23.99
0.852
0.166
0.773
30,12
0.909
0.245
0.751
candidate through a Metropolis- Hastinggs accept-reject test that weighs its predicted photometric gain against sparsity prior. This decouples exploration ("where to propose") from retention ("what to keep"), enabling long jumps to uncovered regions and the principled dismissal of redundant splats, which Langevin-noise walks with unconditional acceptance cannot achieve. Moreover, we analytically derive and rigorously prove that the 3DGS densification process can be recast as a Metsopolis-Hastings sampler, establishing a principled bridge between 3DGS and our formulation.
In principle, 3DGS-MCMC is a local, always-accept SGLD chain, whereas our approach is global, importance-driven MH chain focused on the scene's 's unexplored areas. This leads to our method converging faster, as shown in Fig. 1. Mcasuring time to equal PSNR across all benchmark scenes, our method reaches 30 dB approximately 2.2 minutes faster than 3DGS-MCMC with consistently faster convergence at intermediate quality thresholds, as shown in Tab. 1.
Experiments
5.1 Experimental Settings
Dataset and Metrics. We perform comprehensive experiments using the same real-world datasets employed in 3DGS. Specifically, utilize the scene-scale view synthesis dataset from Mip- NeRF360 [1], which includes nine large-scale real-world scenes comprising five outdoor and four indoor environments. Additionally, we selected two scenes each from the Tanks and Temples [23] and the Deep Blending [17] datasets, using the same scenes as in the original 3DGS. We evaluate each method using the standard evaluation metrics: peak signal-to-ne -o-ise ratio (PSNR), structural similarity index measure (SSIM), and learned perceptual image patch similarity (LPIPS).
Implementation Details. Our framework is based on 3DGS [20] implementation. We also use the differentiable tile rasterizer implementation provided by 3DGS-MCMC [22]. We conducted experiments on an NVIDIA RTX 3090 GPU. Averaged across all datasets, our method runs at 0.0528 seconds per iteration and 26.4 minutes for 30K iterations. During densification, a normalized progress value (E [0, 1) linearly transitions parameters from coarse (larger offsets, voxel sizes, and batch sizes) to fine (smaller offsets and voxel sizes) to shift focus from broad to precise refinements. Coarse proposal scales range from 10.0 to 5,0, fine proposal scales from 2.0 to 1.0, and voxel sizes from 0.02 to 0.005. Batch sizes for coarse and fine proposals are set to 4,500 and 16,000, respectively.
Comparison with Baselines. We compare against methodologies that model large-scale scenes. We compare our approach against state-of-the-arr baselines, such as Plenoxels [10], Instant-NGP [37], Mip-NeRF 360 [1], and 3DGS [20]. For the baselines aside from 3DGS, we present the original results as reported 3DGS. Moreover, we include a comparison with 3DGS-MCMC [22], which has similar motivation and shows state-of-the- performance.
8

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Figure 3: Qualitative results of method compared 3DGS and the corresponding ground truth image from held-out test views. As shown the red bounding box, our method better captures the fine and coarse details that 3DGS [20] and 3DGS-MCMC [22] miss.
Room
Trechill
Playroom Deep Blending
Train
Mip-NeRF 360
Mip-NeRF 360
Tanks & Temples
306
Ours
5.2 Results
Table 2 presents the quantitative evaluations, and Fig. 3 demonstrates the qualitative evaluations on real-world scenes. Our method consistently surpasses 3DGS performance with the same number of Gaussians and achieves or exceeds 3DGS performance with fewer points. Unlike 3DGS, which relies on fixed thresholds for densification, our framework leverages a streamlined point-based sampling strategy that drastically reduces storage requirements while preserving high-fidelity reconstructions. In particular, Tab. C in Appendix F demonstrates our approach's 's consistent efficiency across varied scene complexities. We also provide the full per-scene evaluations in Appendix E.
Although 3DGS-MCMC can yield slightly higher metrics, updates and optimizes every Gaussian parameter, which increases its inefficiency and necessitates knowing the scene's 's complexity (the number of Gaussians) in advance. In contrast, our method focuses exclusively on sampling and optimizing point positions, providing a simpler yet equally accurate alternative. As evidenced by Fig. 1, our method reaches the 98 % PSNR threshold faster than 3DGS-MCME both in iteration and wall-clock time, demonstrating faster convergence. This shows that our framework can still achieve state-of-the-arn reconstruction quality even without exhaustive parameter updates.
5.3 Ablation Study
Components for the Importance Scores. We conducted an ablation study on the key Gaussian components to identify the most influential factors in importance score calculation for accurate sam- pling. The results, shown in Tab. 3. support the effectiveness of our Importance Score Computation design. Specifically, combining SSIM, LI loss, and opacity values yields higher PSNR scores and reduces the required number of Gaussians compared to both the baseline and variants that rely solely on photometric losses. This study underscores the need for careful factor selection.
Effectiveness Relocation. We explore whether relocating Gaussians was effective Tab. 5. Our experiments confirm that the relocation strategy is essential for maintaining high representational fidelity. shows that dynamically reallocating underperforming Gaussians areas where they contribute more effectively improves performance in both indoor and outdoor scenes.

---

Table 3: Ablations on the importance score components.
Method Dataset
Mip-NeRF360 [1]
Tanks&Templer [23]
Deep Blending [17]
SSIM Opacity
PSNR1
SSIM
LPIPS.
PSNR1
SSIMT
LPIPS]
PSNR1
SSIMI
LPIPS]
27.22
0.796
0.243
23.89
0.851
0.168
29.87
0.907
0.247
27.30
0.796
0.242
23.92
0.851
0.169
29.85
0.907
0.246
27.30
0.796
0.243
23.87
0.852
0.167
30.02
0.908
0.247
27.34
0.798
0.241
23.99
0.852
0.166
30.12
0.909
0.245
Table 4: Ablations on the core sampling strategy: Hassopsssss (MH) vs. Stochastic Gradient Langevin Dynamics (SGLD). We denote the number of Gaussians next to the method.
Mip-NeRF360
Tanks& Temples
Deep Blending
PSNR 1
SSIM
LPIPS
PSNR f SSIM
LPIPS
PSNR t SSIM
LPIPS
3DGS MH (0.85M)
26.87
0.782
0.260
23.66
0.843
0.173
29.46
0.898
0.258
3DGS SGLD (0.85M)
26.80
0.789
0.250
23.47
0.841
0.178
29.45
0.898
0.259
Table 5:
Ablations relocation and the multi-view importance score. Results presented for 3DGS use the same number of Gaussians as ours.
Mip-NeRF360
TempleT
Deep Blending
PSNR t SSIM
LPIPS
PSNR f SSIM
LPIPS
PSNR 1
SSIMI LPIPS
3DGS
26.62
0.761
0.299
23.38
0.828
0.214
29.27
0.897
0.272
Ours w/o Relocation
26.94
0.778
0.267
23.69
0.844
0.184
29.57
0.901
0.267
Ours w/o Importance Score
27.31
0.800
0.234
23.79
0.851
0.167
29.97
0.906
0.245
Ours
27.34
0.798
0.241
23.99
0.852
0.166
30.12
0.909
0.245
Contribution of Metropolis-Hasting Sampling. To measure the exact contribution of our core Metropolis-Hastings (MH) sampling strategy, conducted an ablation study comparing our method directly against the Stochastic Gradient Langevin Dynamics (SGLD) approach from 3DGS- MCMC [22]. We implemented only the core sampling algorithms from each method while maintain- ing matched Gaussian counts to ensure fair comparison. The results shown in Tab. 4 demonstrate that MH sampling consistently outperforms SGL.D across all benchmark datasets, confirming the effectiveness of our sampling strategy.
Furthermore, to understand the source of our method's effectiveness, conducted ablation study in Tab. 5 that isolates the contributions of the sampling process and the multí-view importance score. Our experiments demonstrate that the method retains strong performance even when the importance scores are removed. This indicates that the underlying sampling mechanism is the primary driver of our approach, while the multi view importance scores act an intelligent guidance mechanism.
Conclusion
We present an adaptive sampling framework that unifies densification and pruning for 3D Gaussian Splatting via probabilistic Metropolis- Hastings scheme. The importance-weighte sampler auto- matically targets visually or geometrically critical regions while avoiding redundancy. We derive new, concise probabilistic formulation that links proposals to scene importance, enabling faster con- vergence without sacrificing accuracy. Our experiments show that our method matches or surpasses state-of-the-ar quality with fewer Gaussians, and our method's simplicity invites further advances in 3D scene representation.
Limitations and Future Work. Our Metropolis-Hastine sampler, though aware of low-density regions, still prioritizes high-density areas, leading to outdoor scenes with sparser data performing worse than indoor scenes. Moreover, using multi-view photometric errors when proposing points adds roughly 5 minutes of extra training time per scene. Crafting multi-scale representations or dual-stream foreground/background samplers, and introducing faster optimization to reduce training time, would be promising directions for future research.
10

---

Acknowledgments
This work was supported by LITP grant (RS-2021-1211343 AI Graduate School Program at Scoul Detailed 3D reconstruction for urban areas from unstruc-
National Univ. (5%), RS- SSSS2023-00222990 tured images (35%). and RS-2025-0022 2025-02303703: Realworld multi-space fusion and 6DoF free- viewpiit immersive visualization for extended reality (60%) funded by the Korea government (MSIT).
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14

---

A Extended Derivation for Eq. 8
In Bayesian statistics, the posterior distribution of the parameters θ given data is defined as:
p(D ( p(e)
p(0 D)
(21)
( p)) de
Z
where p(D ()) is the likelihood, p(O) is the prior, and Z is the marginal likelihood.
Eq. 7, we introduced a negative log-posterior
E()) [()) + X
Σ
In(1 + ce()))
(22)
VEV
In Eq. 5, we found out that In p(D
() = -()). Combining these two, we can derive:
(() = p(D θ) ln)
(23)
If we exponentiate -E(O):
- p(D (
(24)
we can derive the unnormalized posterior density. To turn this into proper probability distribution, we normalize by Z .e -- E()) dO, yielding:
1
p(D ())
(
Z e [())
p( D))
(25)
Z
As π (()) matches the posterior distribution form Eq. 21, we can define it as the posterior density.
B Extended Derivation for Eq. 19
Below is the derivation for Eq. 19. Recall Eq. 7, which defines the negative log-posterior, also known as the energy function:
[(O) (()) + X Σ
(26)
VEV
Thus AE = E(O¹) E()) can be calcuated as:
A E(e') E())
L(0') + Ae Σ
In(1 + (o)) (C(O) +
X Σ In(1+co(u)))
"EY
"EV
=CC(O' - L(9)) + A Σ DEV
- In co
As we considered the case of having added exactly one Gaussian into voxel v', we can define cu), the number of Gaussians in of after the move,
cer(v') (o +
(27)
Moreover, the sum collapses to a single term for voxel v'. Thus, continuing our derivation:
AΣ = (L(e') L())) + Ao Σ
[In(1 +1(1+ce) co(o') + 1) In(1 + ce())))
DEV
1 ++ co') +1
AC+ X, In
1 co((")
1 co('') An
= AC - In
1 co(')) +1
Setting
1+ca(e") [+oo[1))1]
as D(V).. we can derive:
AE AC - In D(v')
(28)
15

---

Now set I-ce(v") We can use the first-order Taylor series expansion (also known as the first-order Maclaurin expansion) to approximate
A X In + = A 1 1+
(29)
We can further approximate as follows:
In D(v') -o In 1 1 + ces(")
- In(1 + x) ~ A,t
D(v') 2
(30)
Using the Maclaurin expansion, we can approximate:
1
(31)
Ca
1 + 2
+0()3)
1+a
Thus can approximate Aut
I+h
D(
(32)
Au
1 )cce(1))
[+]
C Extended Derivation for Surrogate
This section explains why we approximate - -AC to I(i).
Let's say insert tiny splat {e gi) to the current state O. We'l We'll call this state O'. By the first-order term in the multivariate Taylor expansion in the direction of g::
AC = -
(V,, [()), Eg,
(33)
where V 9. (()) is the gradient of the loss w.r.t. the Gaussian's parameters and the inner product measures the first-order change when adding E gr.
Since only affects p; [[l(x)),, we can show:
AC x a O(p:) + SSIMagg (pc) + 7 (Pi) = 2i
(34)
As stated in Eq. 12 and 13, we set:
T(i) a(z)) Ite
(35)
The Maclaurin series for (2)) I+e-= around 0 is:
o²(0)
(3)
(2) = (0) + +0) +
22
+
3
2!
3!
1
=
2
3
2 +
48
4
Thus, (=x,) + 12z Since our Metropolis-Hasee
rule uses only relative magnitudes of a (z.). the constant and the scale are absorbed, yielding ozz)) x 2. Finally, if we put everything together:
AL R z; R 0(2)) 7(x)
(36)
D Algorithms
Our algorithms for our coarse-fine Metropolis- Hastings sampling framework and overall pipeline are summarized in Alg. and 2.
16

---

Algorithm Coarse-Fine Meiropolis-Hastings Sampling for 3DGS
Require:
Scene = [ where 9i = (xi, Ei, a.)
View set C; iteration total steps Lmax
Voxel size u; density penalty A_ Proposal stds (a,, ot): batch sizes (Be, B:)
Importance weights (ar, r)) opacity threshold
// Select diverse view subset for aggregation
1: +
VIEWSUBSET(CC 1, fmax)
I/ Compute multi-view error maps (SSIM, L1)
2: SSIMags: Clags
|e,1 Exec, [ISSSI ((o),, (160) - |600
// Relocate Gaussians with low opacity
3: for gi E ② where oş S r do
4:
REI.OCATE(gS)
5: end for
// Compute importance scores per Gaussian
6: for g: É θ do
7: Pi I1(x:)
Project center image plane
O(p.) + O,
Use projected Gaussian opacity
9:
S(1) + --a O(p.) + (P. + (Pa)s(P))
10: I(i) + a(S(i))
11: end for
12: c(-) VOXELCOUNTS((0.0)
// Metsopolis-Hastings insertion (Coarse - Fine)
13: for (o, B) {(a,, Be), (o,, B.) do
14: Sample I of size B from θ with Pr(i) (()
15: for I. do
+ x, + NN ITI) v' + VOXELINDEX(X', v) D(e') t ap
17:
18:
19:
T(1) D(v')
b Acceptance probability
20: if UNIFORM(() 1) then
((x', a,)))
22: end
23: end for
24:
end for
25:
return
Algorithm 2 Overall pipeline of Metropolis- Hastings 3DGS
1: Input: M,S, C, A (SfM Points, Covariances, Colors, Opacities); image dimensions w, h.
2: Output: Optimized and densified attributes M, S, C.
3: i-0 0
Iteration count
4: while not converged do
5:
{V, V] + SAMPLETRAININGVVENN
> Camera V and image
6:
+ M,
7:
L LOs
8:
SSSIM,: + SSIM(I,
9:
SUL. LI(I, S.S.C,AN +- AVL) then
Update attributes
11:
12:
for Current Scene do METROPOLIS SETROPOLIS_HASTINGS HASTINGS SAMAIIGSSSAMLIIIG/IAOOI MAM A,
13:
14:
end for end if
15:
16:
i ++1
17:
end while
18:
return M,

---

E Per-Scene Results
In this section, we provide the full per-scene evaluations. We evaluated method, 3DGS [20] and 3DGS-MCME [22] on various datasets and scenes. We provide the per-scene results for Mip-NeRF 360 [1], Tanks & Temples [23] and Deep Blending [17] datasets in Tab. A and B. All evaluations have been conducted on a NVIDIA Gelorce RTX 3090.
Table A: Full per-scene results evaluated on Mip-NeRF 360 dataset.
Type Datavet
Indoor
Ouidoor
Seone
Bunsai
Counter
Kitchen
Room
Bicycle
Flowers
Garden
Stump
Trechill
Avergee
PSNR
32.14
28.98
31.25
31.65
25.10
21.33
27.31
26,64
22.54
27.44
0.931
0.635
0.197
0.216
3DGS (20)
Train (mmsss)
28:04
43:25
31:05
26:29
42:41
31:21
43:16
34:00
30:53
34:34
# of Caussians
1249672
1174274
1763905
1478813
5728256
3480107
5667876
4494105
3522801
3175479
Sturage (MB)
296.56 56
277.73
419.18
349.76
1351,80
823.09
13400.2
1062.91
833.18
750.86
SSIM LPIPS
0,09
0.924
0.918
0.616
0.488
0.794
0.677
0.540
0,761
0.055
0.196
0.134
0.214
0.410
0.478
0.230
0.381
0.444
0.299
3DGS-8 [30]
Train (mmsss)
26:42
2X:04
33:59
25:22
25:54
23:58
28:17
23:02
28:00
27:02
A of Gauscians
797782
696371
623992
764539
741748
811695
697792
690517
712492
Sturage
164.49
PSNR
29.27
31.41
32.15
25.13
21.64
26.71
27.21
23.09
27.69
SSIM
0,949
0,920
0.930
0.933
0.745
0.595
0.832
0.791
0.648
0.816
LPIPS
0.176
0.177
0.126
0.186
0.275
0.377
0.170
0,247
0.370
0.234
3DGS-MCMC
(mmass)
Cuussians
777782
764539
695495
811605
690517
Storage (MB)
183.96
165
147.58
181.02
175.43
164.49
191.98
165.
163.32
170.90
PSNR
32.52
29.33
31.35
32.15
24.53
20.99
26.49
25.95
22,80
27.34
SSIM
0.950
0.918
0.930
0.934
0.711
0.556
0.819
0.743
0.621
0.798
Ours
Train (mm:ss)
41:05
16:59
11:20
16:31
34:58
35:13
31:36
32:16
37:05
38:37
A of Cousssians
777782
696871
623992
764539
741748
695495
8116895
697792
690511
722492
Storage (MB)
183.96
165.29
147.58
181.02
175.43
164.49
191.99
165.04
163.32
170.96
Table B: Full per-scene results evaluated on Tanks Temples and Deep Blending datasets.
Dataset
Tanks& Temples
Deep Blending
Scene
Train
Truck
Average
Johnson
Playroom
Average
PSNR (1)
21.89
25.37
23.63
29.08
29.97
29.53
SSIM (t)
0.814
0.882
0.848
0.901
0.907
0.904
LPIPS (L)
0.207
0.147
0.177
0.244
0.244
0.244
3DGS [20]
Train (mm:ss)
13:22
19:09
16:15
29:26
26:20
27:53
of Gaussians
1085904
2575516
1830710
3308886
2320688
2814787
Storage (MB)
256.83
609.14
432.99
782.59
548. 87
665.73
PSNR
21.80
24.95
23.38
28.99
29.54
29.27
SSIM (1)
0.791
0.865
0,828
0.892
0.901
0.897
LPIPS (L)
0.245
0.183
0.214
0.280
0.263
0.272
3DGS-S [20]
Train (mm:ss)
19:16
18:05
18:41
20:18
20:23
20:21
of Gaussians
788942
756104
772523
766469
735409
750939
Storage (MB)
186.60
178.83
182.72
181.28
173.93
177.61
PSNR (t)
22.58
26.02
24.30
29.37
30.30
29.84
SSIM (t)
0.834
0.889
0.862
0.902
0.910
0.906
LPIPS (L)
0.196
0.143
0.170
0.260
0.248
0.254
3DGS-MCMC [22]
Train (mm:ss)
15:50
15:45
15:48
26:28
23:39
25:03
of Gaussians
788942
756104
772523
766469
735409
750939
Storage (MB)
186.60
178.83
182.72
181.28
173.93
177.61
PSNR
22.44
25.54
23.99
29.77
30.47
30.12
SSIM (t)
0.821
0.883
0.852
0.904
0.913
0.909
LPIPS ())
0.198
0.134
0.166
0.249
0.240
0.245
Ours
Train (mm:ss)
20:38
19:40
20:09
32:27
29:54
31:10
of Gaussians
788942
756104
772523
766469
735409
750939
Storage (MB)
186.60
178.83
182.72
181.28
173.93
177.61
18

---

Comparison of the Number of Gaussians
Table C contrasts the number of Gaussians needed by our approach and by 3DGS for each scene. In every case, our method attains equal superior rendering quality while requiring substantially fewer Gaussians.
Table C: Comparison of our method and 3DGS on the number of Gaussian center points used to represent scenes in the Mip-NeRF 360 [1]. Tanks & Temples [23], and Deep Blending [17] datasets.
Temple
Bleoding
Method
Bonial
Counter
Kinchen
Room
Bayyle
Flowers
Garden
Stump
Treehill
Tizin
Trick
Dr Johnson
Plyyoom
SDGSS [20]
1,249.622
1,174,274
1,763,905
1,297,736
4,585.552
2,887,135 695,495
5,667,876
4.324,009 697,793
3,224,115
L085,904
2.550,408
3,308,86 766,469
2.320,458
Duss
777,782
698,871
623,992
765.370
741,748
811,695
696,517
788,942
756,184
735,409
G Broader Impacts
As our method focuses on the core and theoretical problem of 3D Scene Reconstruction, our work has the potential for positive societal impact on various downstream applications using 3D Gaussian Splatting [20]. As we reduce the reliance on heuristics and the number of Gaussians used work to enhance many downstream applications that require less memory usage. However, as we did not introduce any fundamental new content generation capabilities through our work, there is little potential for work to have negative societal impact beyond the ethical considerations already present in the field of 3D reconstruction.
scene, expect
H Dataset Licenses
We used the following datasets:
Mip-NeRF360 [1]: The license term is unavailable. Available at https: info/ mipnerf360/
Tanks and Temples [23]: Published under the Creative Commons Attribution 4.0 International (CC
BY 4.0) License. Available at h/ww..
e Gandssmttemmpee..rg//lcmmmml
Deep Blending [17]: The license term is unavailable. Available at http: //wis///vlaaaaaa//e cs ac. uk/ hta/pubbs/deepbbenddng
19

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Reviews:
[{"review_id": "ypkS3NDOmR", "reviewer": "Reviewer_U3jc", "summary": "This paper re-casts the splitting and pruning operations of 3DGS training into the framework of Metropolis-Hastings (MH) Monte Carlo methods, similar in spirit to the recent MCMC-3DGS paper (Both are Markov Chain methods, except that MH adds in an accept probability for proposals). The method also introduces a number of heuristics, such as a voxel occupancy prior to discourage \"overpopulation\" of solved areas, a per-pixel importance field based on a combination of photometric error and accuracy, and new heuristics for \"coarse\" and \"fine\" gaussian proposals. Performance metrics are provided on benchmark datasets, which do not consistently outperform MCMC-3DGS. A limited series of ablations justify some of their design choices.", "questions": "How much _exactly_ does MH contribute to performance _gains_ in a direct \"apples-to-apples\" comparison between your method and 3DGS-MCMC. I realize a direct comparison may not be strictly possible, but it doesn't appear as though every pains were taken to make this comparison possible. Furthermore, how much do all your design heuristics advance reconstruction metrics, if at all? A strong indication that MH improves performance in all cases would shift me to \"accept\" or a thorough analysis of why MH alone fails.\n\nWhat is the point of aggregating the same pixel from different images in Eq. 9? presumably the same pixel in different images represent rays passing through entirely different parts of the 3D scene, so It's unclear how this could be useful information. Presumably you are computing the importance per-Gaussian, then averaging views, so it seems as though the notation is wrong?", "limitations": "Yes.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 1, "originality": 2, "strengths_and_weaknesses": "The paper is weak in it's quantitative metrics, just barely producing a few SotA results on the Deep Blending datasets, but is only two scenes. On the larger set of scenes in MipNerf-360 and Tanks and Temples, 3DGS-MCMC still achieves superior metrics. It's also difficult to draw conclusions from the qualitative results - the inset images do show superior quality with their method, but without access to the high-res images or error map visualizations, there could concievably be worse areas in their images as well.\n\nAny performance gains in timings are also unclear. Figure 1 overlays wall-time vertical lines on an iteration axis, which are entirely different measures. This would seem to imply that the relative performance of each method is equivalent, but is not directly discussed?. I'm also not sure what the point of a \"95% of final PSNR\" metric is - it almost seems like metric only constructed demonstrate the method in a good light. Why is the more typical \"time to equal PSNR\" metric not appropriate in this setting?\n\nAt a high level, I really want to like this paper, as I'm a fan of more mathematically justifiable techniques such as Metropolis-Hastings. However, computer vision is as much \"art\" as it is \"science\", and this paper does not _fully_ disentangle the gains made by heuristics (Eq. 4, 10, 12, 13, round-robin scheduling of sec 4.2.1), and those from the addition of MH. MH could very well be a consistent win, but this paper does not make that clear, making it of limited interest to the NeurIPS audience, hence my reccommendation of reject.", "was_revised": true, "final_justification": "The authors have provided evidence that their method converges faster than 3DGS-MCMC. I really would have preferred to see a graph for each dataset, as there is concievable ambiguity in how these numbers were arrived at (average all scenes training PSNR first, then timing, or each scene individually, then average). Regardless, I'm trusting the authors that there are genuine performance gains, and am revising my reccommendation to \"borderline accept\".", "rebuttal": "We sincerely appreciate your insightful comments and questions. We have tried to respond to all of your concerns as follows.\n\n## [W1] Weakness in quantitative results\n\nWe would like to first highlight that our core Metropolis-Hastings (MH) sampling strategy demonstrates improvements over 3DGS-MCMC's SGLD approach in a direct comparison (please see our detailed analysis in Q1/W3). However, we acknowledge that our overall method shows mixed quantitative results on MipNeRF-360 and Tanks & Temples, which is a point we also acknowledge in our paper's limitations section.\n\nQualitatively, our method is on par with or even superior to baselines in capturing the intricate geometry and fine details of the primary subjects, as can be seen in Fig. 2. However, the low quantitative metrics are predominantly caused by our model's difficulty in representing vast, low-frequency backgrounds, such as the sky. Since these metrics are averaged over the entire images, background imperfections can disproportionately penalize scores despite the high-quality foreground reconstruction. \n\nThe results on the indoor scenes with manageable scale variance demonstrate our method's strong potential. Our method outperforms the baselines in these scenes, suggesting that core algorithmic contributions are sound. This challenge stems from the extreme scale variance (exceeding 170,000x) inherent to unbounded outdoor scenes, where our model struggles to simultaneously optimize for the compact, detailed foreground Gaussians and the massive, smooth background Gaussians. Adapting our model to robustly handle both ends of this scale spectrum is a clear direction for future work, and this can be addressed by exploring techniques such as multi-scale representation. \n\n## [W2] Clarity on performance gains on timing \nIn Fig. 1, the horizontal axis indeed counts training iterations. We imposed vertical markers that correspond to actual wall-clock time at which each method reaches a chosen quality threshold. We intended to give a single, compact view of how fast along the iteration trajectory each method would be experienced in real time. \n\nRegarding the reviewer's concern with the \"95% of final PSNR\" metric, we would like to clarify that we were not intending to demonstrate our method in a good light; we were rather unaware of the typical “time to equal PSNR” metric. We initially chose “95% of the final PSNR” as the metric, as a confidence-interval-like view of near-final quality that is directly comparable across methods, especially as absolute PSNR ceilings differ across scenes. Moreover, using 95% is adopted as a practical convergence metric in recent vision works; for instance, [1] uses 95% to compare optimization speed. \n\nFollowing the reviewer's suggestion, we provide the “time to equal PSNR” metrics averaged across all scenes for both our method and 3DGS-MCMC. As the approximate highest average PSNR value was 30 dB, we selected our final target PSNR as 30 dB. The results show that our method converges approximately 2.18 minutes faster than 3DGS-MCMC. We also show lower target PSNR values to show that our method consistently converges faster. We will include these metrics in our final manuscript. \n\nNote that we used the official implementation provided by 3DGS-MCMC and conducted experiments on NVIDIA GeForce RTX 3090 for both methods to ensure a fair comparison.\n\n| Target PSNR  | Time - Ours           | Time - 3DGS-MCMC       |\n|--------------|----------------------|------------------------|\n| 21           | 16.30s / 0.27 mins   | 17.08s / 0.28 mins     |\n| 24           | 61.34s / 1.02 mins   | 98.38s / 1.64 mins     |\n| 27           | 287.01s / 4.78 mins  | 341.64s / 5.69 mins    |\n| 30           | 851.52s / 14.19 mins | 983.05s / 16.38 mins   |\n\n[1] Zhang, Yunxiang, et al. \"Image-gs: Content-adaptive image representation via 2d gaussians.\" *arXiv preprint arXiv:2407.01866* (2024)\n\n## [Q1/W3] Direct Comparison of MH & 3DGS-MCMC and impact of design heuristics\n\nThank you for this important question. The concern about disentangling the gains from our core Metropolis-Hastings (MH) framework versus our other design choices is crucial. To address this directly, we have conducted experiments to measure the contribution of each component.  \n\n**1. Quantifying the Contribution of Metropolis-Hastings (MH) in an \"Apples-to-Apples\" Comparison with 3DGS-MCMC**\n\nTo isolate the exact contribution of our core MH sampling strategy, we ran experiments with both our method and 3DGS-MCMC, both methods with only their key algorithmic methodology implemented. We believe that this setup creates the most direct \"apples-to-apples\" comparison between our Metropolis-Hastings (MH) proposal strategy and 3DGS-MCMC’s Stochastic Gradient Langevin Dynamics (SGLD) approach. \n\n| Method | Mip-NeRF360 | Tanks&Temples | Deep Blending |\n|--------|-------------|---------------|---------------|\n|        | PSNR$\\uparrow$/SSIM$\\uparrow$/LPIPS$\\downarrow$ | PSNR $\\uparrow$/SSIM$\\uparrow$/LPIPS$\\downarrow$ | PSNR $\\uparrow$/SSIM$\\uparrow$/LPIPS$\\downarrow$ |\n| 3DGS + MH (0.85M) | 26.87 / 0.782 / 0.260 | 23.66 / 0.843 / 0.173 | 29.46 / 0.898 / 0.258 |\n| 3DGS + SGLD (0.85M) | 26.80 / 0.789 / 0.250 | 23.47 / 0.841 / 0.178 | 29.45 / 0.898 / 0.259 |\n\nThe results demonstrate that our Metropolis-Hastings sampling strategy provides consistent improvements over 3DGS-MCMC's SGLD approach. This result indicates that the MH sampling strategy provides a reliable advantage in reconstruction quality.\n\n**2. Disentangling Design Heuristics**\n\nTo address the reviewers' concern regarding this, we wish to clarify that our model’s components are theoretically grounded designs that function as interdependent parts of our Metropolis-Hastings framework. Our paper theoretically derived how these components must work together. As mentioned in Eq. 18, the first term $\\sigma\\bigl(I(i)\\bigr)$ represents the photometric gain of adding a new Gaussian, which is calculated from Eq. 10 and the second term $D(v’)$ represents the sparsity penalty, calculated from Eq. 4. A proposal is accepted only if the product of these two terms is high, and this ensures that our method prioritizes Gaussians that are photometrically useful and spatially efficient. \n\nTo further disentangle our design components, we present a detailed ablation study breaking down our method. We start with pure MH Sampling, which is the MH sampling algorithm with the regularizers. We incrementally added our key design components, and all the experiments were conducted in the same setting as our original experiments, with no separate parameter tuning for individual components. The results are summarized in the table below. \n\n| Method | Mip-NeRF360 | Tanks&Temples | Deep Blending |\n|--------|-------------|---------------|---------------|\n|        | PSNR$\\uparrow$/SSIM$\\uparrow$/LPIPS$\\downarrow$ | PSNR $\\uparrow$/SSIM$\\uparrow$/LPIPS$\\downarrow$ | PSNR $\\uparrow$/SSIM$\\uparrow$/LPIPS$\\downarrow$ |\n| MH (0.85M) | 27.42 / 0.797 / 0.237 | 23.87 / 0.852 / 0.167 | 29.96 / 0.907 / 0.245 |\n| MH + Eq. 4 (0.77M) | 27.31 / 0.800 / 0.234 | 23.79 / 0.851 / 0.166 | 29.97 / 0.906 / 0.243 |\n| MH + Eq. 10 (0.85M) | 27.37 / 0.797 / 0.236 | 23.81 / 0.853 / 0.163 | 29.94 / 0.905 / 0.245 |\n| MH + Eq.4, Eq.10 (1.45M) | 27.52 / 0.809 / 0.214 | 23.94 / 0.858 / 0.148 | 29.98 / 0.907 / 0.233 |\n| MH + Eq. 12, Eq. 13 (1.45M) | 27.53 / 0.807 / 0.223 | 23.97 / 0.853 / 0.164 | 29.99 / 0.903 / 0.243 |\n| MH-3DGS (0.75M) | 27.34 / 0.798 / 0.241 | 23.99 / 0.852 / 0.166 | 30.12 / 0.909 / 0.245 |\n\nThe results show that naively combining our design components without the full, synergistic framework (e.g., Pure MH + Eq.4, Eq.10) leads to an inefficient model. While it achieves a high PSNR, it does so by nearly doubling the number of Gaussians to 1.45M. This demonstrates that the components, when not properly balanced by the complete acceptance rule, can lead to uncontrolled growth, chasing marginal gains at a significant computational cost.\n\nWhile Pure MH achieves a marginally higher PSNR on the Mip-NeRF360 dataset, our final MH-3DGS model achieves this competitive result using 12% fewer Gaussians (0.75M vs. 0.85M). With all the design components together, we are able to create a much more compact and robust performance across all scenes.\n\n## [Q2] Equation 9\n\nTo clarify the notation, when we refer to pixel p in Eq. 9, we mean the pixel coordinate where a specific Gaussian projects in each camera's image plane. Since the same 3D Gaussian will project to different pixel coordinates in different camera views, the aggregation is performed per-Gaussian across its projected pixel locations in multiple views, not at the same pixel coordinate across different images. \n\nThe process works as follows: we compute residuals locally for each camera $c$ at the pixel location where each Gaussian projects, then average these residuals across the $k_t$ cameras in $C_t$. This aggregation pools error statistics across multiple perspectives of the scene for stable error estimates. The per-Gaussian importance is then evaluated based on where each Gaussian projects in the aggregated error maps.\n\nWe agree with the reviewer that the current notation can be misleading, and we will revise the notation for the final manuscript. Thank you for pointing out this important issue."}, {"review_id": "a3sCeeZ5D9", "reviewer": "Reviewer_gybH", "summary": "This paper targets the stage of densification and sparsification in 3D gaussian splitting (3DGS). In the 3DGS work this stage is conducted with a heuristic rule based on opacity and photometric loss gradients on each Gaussian primitive. This paper shows that the process can be viewed as a sampling process, where a new configuration of Gaussian sets are sampled and accepted/rejected. The authors propose to use Metroplis Hasting as the sampling framework, with proposal generation composed of Gaussian densification and relocation. An method to calculate importance scores is introduced to conduct the sampling.  \n\nThe authors tested the proposed method on common 3DGS benchmarks. The method can lead to on-par reconstruction quality with vanilla 3DGS but with much fewer number of Gaussians, saving compute and storage. Authors also compare the method with 3DGS-MCMC, another work formulating the specific stage as sampling problem. The propose method yields faster convergence than this baseline.", "questions": "One interesting question I have is whether the method's effectiveness comes from the sampling process or the multi-view importance score. The original 3DGS uses gradients in a single-view batch as a cue for the densification, while in this work, a large set of views are used to evaluate whether a Gaussian needs to be \"cloned.\" It would be great if the authors could provide insight on this matter.", "limitations": "Yes.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "+ The formulation of densification and sparsification as point configuration sampling provides a systematic view of the originally heuristics based stage which is empirically found to be critical to 3DGS reconstruction quality. \n+ The proposed important scored based on multi-view reconstruction error seems to provide a better cue for densification and sparsification over the original 3DGS's loss gradients which is dependent on mini-batch sampling. \n+ In experiment, the method yield both fast convergence and smaller number of Gaussian than the baselines, showing its effectiveness. \n+ The authors have made good effort to make the theoretical part of this work easy to understand.\n\nWeakness\n- As the authors also pointed out, the method requires evaluating multiple views for the importance score in each sampling step, which adds up to the total training cost. \n- In measuring the per Gaussian importance score, a simple surrogate is used to reduce complexity. Not much discussion is provided on what is the quality impact of this choice.", "was_revised": true, "final_justification": "The authors have addressed my questions regarding this work. The merit of this work is as described in the reviews. I would like to keep my rating as borderline accept.", "rebuttal": "We sincerely appreciate your insightful comments and questions. We have tried to respond to all of your concerns as follows.\n\n## [W1] Multiple-Views for Importance Score - Training Cost\n\nWhile our method does require multi-view evaluation for importance scoring, hence leading to an increase in the total training cost, the faster convergence shown in Fig. 1 demonstrates that this overhead is worthwhile in practice. Our method reaches 95% of the final PSNR faster than 3DGS-MCMC in both iteration count and wall-clock time, meaning the improved sampling efficiency more than compensates for the additional per-iteration cost. The key insight is that spending slightly more time per iteration to make better sampling decisions results in needing fewer total iterations to converge, and this is achieved even with our current unoptimized implementation. \n\nAdditionally, our approach uses only a camera subset (annealed from broad to focused coverage) rather than the full training set, which helps control the computational burden. Moreover, our method achieves comparable or better reconstruction quality while using dramatically fewer Gaussians (as shown in Table 4), leading to significant memory savings. Nevertheless, we hope to explore more efficient importance score computation or adaptive view selection strategies to further reduce this overhead.\n\n## [W2] Discussion on the surrogate\n\nComputing the exact loss reduction ($-\\Delta\\mathcal L$) for each candidate Gaussian would require re-rendering the entire scene from all viewpoints in the selected camera subset for every proposed splat. This would increase computational complexity from O(N) to O(N×K×M), where N is the number of candidates, K is the number of cameras, and M is the scene complexity, making the method computationally impractical. To solve this problem, we derived a surrogate by analyzing the first-order multivariate Taylor expansion based on inserting a small Gaussian in the scene. (Mentioned in our paper in Appendix C)\n\nIndeed, the best way to measure the quality impact would be a direct comparison, but a direct empirical comparison between the surrogate and exact $-\\Delta\\mathcal L$ would require computing full multi-view rendering errors for thousands of candidate Gaussians, which was precisely the computational burden our surrogate is designed to avoid. This creates a paradox where validating the surrogate's accuracy would negate its computational benefits. \n\nNevertheless, we believe that the quality impact of selecting a surrogate to calculate $-\\Delta\\mathcal L$ is minimal, as the surrogate maintains the relative ordering of candidate importance. Since our surrogate is derived from a rigorous first-order approximation, and the logistic function $\\sigma(z)$ is strictly monotonic, the relative ordering is mathematically preserved. For example, if candidate A truly provides greater improvement than candidate B, it would be $I(A) > I(B)$ in our surrogate. This ordering preservation is sufficient for quality maintenance because the MH acceptance test ensures that better candidates receive higher acceptance probabilities, guiding the sampler toward high-impact regions. Over many iterations, this statistical bias toward high-importance areas drives convergence to optimal Gaussian distributions, as evidenced by our results showing maintained or improved reconstruction quality. \n\n## [Q1] Model's effectiveness coming from sampling or importance score\n\nWe conducted an ablation study by removing the multi-view importance score and other components in our method. We also reported the average number of Gaussians across all scenes in the parentheses. \n\n| Method | Mip-NeRF360 | Tanks&Temples | Deep Blending |\n|--------|-------------|---------------|---------------|\n|        | PSNR$\\uparrow$/SSIM$\\uparrow$/LPIPS$\\downarrow$ | PSNR $\\uparrow$/SSIM$\\uparrow$/LPIPS$\\downarrow$ | PSNR $\\uparrow$/SSIM$\\uparrow$/LPIPS$\\downarrow$ |\n| No Importance Score (0.75M) | 27.31 / 0.800 / 0.234 | 23.79 / 0.851 / 0.167 | 29.97 / 0.906 / 0.245 |\n| MH-3DGS (0.75M) | 27.34 / 0.798 / 0.241 | 23.99 / 0.852 / 0.166 | 30.12 / 0.909 / 0.245 |\n\nBased on the study, our method without the multi-view importance scores still achieves strong performance, outperforming 3DGS-MCMC on all three metrics on Deep Blending and LPIPS in Tanks & Temples. This shows that replacing heuristic-based densification with principled Metropolis-Hastings sampling is effective. \n\nTo further clarify, our multi-view component is strategically integrated into the sampling framework, not as the primary driver of performance, but as an intelligent guidance mechanism that makes the probabilistic sampling more informed and consistent. Overall, our sampling methodology demonstrates that principled sampling with robust guidance is effective."}, {"review_id": "KBIM20j3Tx", "reviewer": "Reviewer_PUCv", "summary": "The authors propose a novel approach to improve the 3DGS sampling strategy. They represent the importance of gaussians through measuring their multi-view pixel errors, and model the density control process in 3DGS based on Metropolis-Hastings sampling, such that densification process can be performed by generating new gaussian proposals with probability according to their importance. Subsequently, they can reduce the heuristic dependence in the density control process. Experiments show that the new approach can outperform vanilla 3DGS with fewer numbers of gaussians, and achieve faster convergence comparing to 3DGS-MCMC.", "questions": "My main concern is about the advantages comparing to 3DGS-MCMC, could the authors illustrate more on the significance of the proposed method comparing to 3DGS-MCMC? And it would be helpful to provide some failure cases on the Mip-NeRF 360 and TNT datasets and the corresponding insights to help future research, since in Figure 2 the proposed method successfully reconstruct the distant details comparing to 3DGS-MCMC, while underperforming by 0.3dB PSNR in Table 1.", "limitations": "Please see the weaknesses and questions above.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n1. Novelty: Bringing the Metropolis-Hastings sampling process into 3DGS density control is novel and reasonable, which models the probability of densification through measuring the multi-view pixel errors to focus on the areas with larger errors, while avoiding introducing strongly heuristic strategies.\n2. Soundness: The paper provides strict mathematical derivation of applying the Metropolis-Hastings sampling to 3DGS density control.\n\nWeaknesses:\n1. Performance: The proposed method seems to slightly underperform comparing to 3DGS-MCMC after convergence. The major significance of the proposed method comparing to 3DGS-MCMC is a bit unclear.", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely appreciate your insightful comments and questions. We have tried to respond to all of your concerns as follows.\n\n## [W1/Q1] Performance - Significance of MH and Failure Cases \n\nIn order to directly compare our methodology with 3DGS-MCMC, we conducted an ablation study where both methods had only their key algorithm implemented. We removed our design components (voxel-wise prior, importance score) and also took out opacity and scaling regularizers from both methods. We believe that this setup creates the most direct comparison between our MH sampling and the SGLD approach.\n\n| Method | Mip-NeRF360 | Tanks&Temples | Deep Blending |\n|--------|-------------|---------------|---------------|\n|        | PSNR$\\uparrow$/SSIM$\\uparrow$/LPIPS$\\downarrow$ | PSNR $\\uparrow$/SSIM$\\uparrow$/LPIPS$\\downarrow$ | PSNR $\\uparrow$/SSIM$\\uparrow$/LPIPS$\\downarrow$ |\n| 3DGS + MH (0.85M) | 26.87 / 0.782 / 0.260 | 23.66 / 0.843 / 0.173 | 29.46 / 0.898 / 0.258 |\n| 3DGS + SGLD (0.85M) | 26.80 / 0.789 / 0.250 | 23.47 / 0.841 / 0.178 | 29.45 / 0.898 / 0.259 |\n\nThe results demonstrate that our Metropolis-Hastings sampling strategy itself provides consistent improvements over 3DGS-MCMC's SGLD approach. Furthermore, this result indicates that the MH sampling strategy provides a reliable advantage in reconstruction quality.\n\nMoreover, beyond the significance highlighted in our paper, key significances of our MH framework are as follows:\n\n1. Automatic scene-complexity adaptation. As mentioned in the paper, 3DGS-MCMC must commit to a fixed budget of Gaussians before training, which can under-fit simple scenes or waste memory on complex ones. By contrast, our accept/reject chain keeps inserting points only while the posterior still yields photometric gain, so the optimization automatically stops at the scene-appropriate count with no prior complexity. \n\n2. Dimension-selective updates that accelerate convergence. 3DGS-MCMC injects Langevin noise into every parameter of every Gaussian each step, which inflates stochastic gradient variance and forces smaller learning rates. On the other hand, we restrict stochastic moves to positions only. This trims the update dimensionality and lets each accepted proposal deliver a larger effective step. Thus, the PSNR curve for our method is smoother and achieves convergence earlier than 3DGS-MCMC (Fig. 1), confirming that the dimensionality cut translates into faster practical convergence. \n\nHowever, as mentioned in our paper, in sparse outdoor scenes such as Flowers, Stump, etc. (Mip-NeRF 360) or Train, Truck (T&T), our PSNR trails that of 3DGS-MCMC. These gaps arise not from missed high-frequency details, which we capture well as can be seen in Fig. 2 and the supplementary video, but from faint background strata that carry little image gradient yet contribute to pixel-wise error. Our MH sampler weighs proposals by multi-view photometric discrepancy; once high-error foreground patches are saturated, the residual background error is dispersed across many low-density voxels, so their individual acceptance probabilities fall below the threshold, leaving some regions undersampled. To improve these failure cases, adopting adaptive temperature annealing to MH acceptance or adopting multi-scale representations or dual-stream foreground/background samplers could be viable solutions, and open grounds for future work."}, {"review_id": "2lgFsgTkMI", "reviewer": "Reviewer_6AXt", "summary": "This work proposes an improved Gaussian Splatting framework based on Metropolis-Hastings Sampling. Following the formulation and overall framework of 3DGS-MCMC, this work adopts Metropolis-Hastings Sampling to accept or reject newly added Gaussians. Besides, it considers the number of Guassians in each subvoxel (as a voxel-wise prior) to constraint the overall model size, and includes photometric error as long as opacity to construct Gaussian importance. Experimental results on several datasets show the proposed method is comparable with baseline algorithms.", "questions": "1. It would be better to provide results of Mip-NeRF360 and Tanks&Temples in Figure 1.\n2. It would be better to provide models with varying number of Gaussians.\n3. It would be better to provide ablation study to show that Metropolis-Hastings Sampling is better than Stochastic Gradient Langevin Dynamics in 3DGS-MCMC with a fair comparison (i.e., without the voxel-wise prior and photometric error as importance).", "limitations": "yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 2, "originality": 3, "strengths_and_weaknesses": "### Paper Strengths\n\n\nIt is reasonable and intuitive to incorporate the Metropolis-Hastings sampling method for the MCMC framework.\n\n\n\n\n### Major Weaknesses\n\nAbstract  \n1. Misleading clarification. This work claims that 'Traditional 3DGS methods heavily rely on heuristic based density-control mechanisms ... Our framework overcomes these limitations...' (line 3 to 8). Note that this limitation is dealed by 3DGS-MCMC with their state transition, and this work follows the overall pipeline of 3DGS-MCMC for this problem.\n\nRelated Work\n1. Lack of related work for Sec. 2.3. Considering that Gaussians importance has been discussed and adopted by a great number of works, many studies on Gaussian densification have been overlooked. Besides, considering that this work highlights their contribution for efficient reconstruction (' a reduced number of Gaussians'), many studies on efficient Gaussian models are also ignored.\n\nPreliminary\n1. It would be better to include the introduction of 3DGS-MCMC and Metropolis-Hastings Sampling in this part.\n\nMethod \n1. Confusing description (line 176). What is 'voxel' in the context of Guassian Splatting and how to construct voxels.\n2. Cherry picking (Figure 1). In Figure 1., the proposed method shows better performance and converges faster than 3DGS-MCMC. However, according to Table 1, the proposed method only performs better than 3DGS-MCMC on deepblending. What about Mip-NeRF360 and Tanks&Temples?\n\nExperiments\n1. Limited improvement (Table 1). The proposed method shows limited improvements on most scans. \n2. Performance with varying number of Gaussians. Although this work focuses on the case with limited number of Guassians, it would be better to provide models with varying number of Gaussians.\n3. Ablation study. Considering that the key contribution of this work is to leverage Metropolis-Hastings Sampling for MCMC framework, it would be better to provide ablation study to show that Metropolis-Hastings Sampling is better than Stochastic Gradient Langevin Dynamics in 3DGS-MCMC with a fair comparison (i.e., without the voxel-wise prior and photometric error as importance).\n\n\n\n### Minor Weaknesses\n\ntypo\n1. line 30: samples->sampling", "was_revised": true, "final_justification": "After reading the comments from other reviewers and the author response, Reviewer 6AXt acknowledges the technical contribution of MH sampling presented in this work. However, the limited performance improvement somewhat constrains the overall impact of the contribution. Therefore, the reviewer maintains the rating as a borderline accept.", "rebuttal": "We sincerely appreciate your insightful comments and questions. We have tried to respond to all of your concerns as follows. Moreover, thank you for pointing out the typo; we will revise it for our final manuscript. \n\n## [W1] Misleading Clarification \n\nWe acknowledge the reviewer's concern about our positioning. While we follow a similar overall pipeline (relocation and loss formulation) and address the same fundamental problem of replacing heuristic density control, we believe our characterization is accurate, as we propose a fundamentally different solution approach to the same problem. Specifically, 3DGS-MCMC employs SGLD (Stochastic Gradient Langevin Dynamics) that perturbs all Gaussian parameters with noise and accepts every move, whereas our method uses Metropolis-Hastings sampling that targets specific locations based on multi-view error aggregation and employs principled accept/reject decisions. We will revise our final manuscript to better acknowledge 3DGS-MCMC's pioneering role while more clearly stating how our MH-based approach with multi-view importance scoring represents a distinct methodological contribution to the same problem space. Our intent was not to diminish prior work, but rather to demonstrate an alternative probabilistic framework that achieves faster convergence. \n\n## [W2] Lack of Related Work\n\n**Efficient Reconstruction**\nWe have included the following works on efficient reconstruction in Sec. 2.2 of our paper [9, 22, 36, 38, 48], but we will further add additional more recent works on efficient reconstruction [e1, e2, e3] to Sec. 2.2. These works focus on deterministic geometric analysis or post-hoc uncertainty pruning on converged models. Unlike these methods, our probabilistic framework reduces the number of Gaussians through principled Bayesian sampling that dynamically adapts to scene-specific reconstruction signals.\n\n[e1] Papantonakis, Panagiotis, et al. \"Reducing the memory footprint of 3d gaussian splatting.\" Proceedings of the ACM on Computer Graphics and Interactive Techniques 7.1 (2024): 1-1\n\n[e2] Hanson, Alex, et al. \"Speedy-splat: Fast 3d gaussian splatting with sparse pixels and sparse primitives.\" Proceedings of the Computer Vision and Pattern Recognition Conference. 2025.\n\n[e3] Hanson, Alex, et al. \"Pup 3d-gs: Principled uncertainty pruning for 3d gaussian splatting.\" Proceedings of the Computer Vision and Pattern Recognition Conference. 2025. \n\n**Densification**\nWe have included related work regarding densification in Sec. 2.3 [4, 20, 30, 52], but as densification was a rapidly growing field of interest during the past year, we may have unintentionally overlooked recent studies. We will add recent densification works using neural prediction [d1] or deterministic geometric splitting [d2]. These methods can face limitations, such as generative approaches may not generalize well to novel scene types [d1], and geometric methods use fixed rules that may not adapt to diverse scene complexity [d2]. In contrast, our framework provides theoretical convergence guarantees via stochastic exploration of the solution space.\n\n[d1] Nam, Seungtae, et al. \"Generative Densification: Learning to Densify Gaussians for High-Fidelity Generalizable 3D Reconstruction.\" Proceedings of the Computer Vision and Pattern Recognition Conference. 2025.\n\n[d2] Deng, Xiaobin, et al. \"Efficient density control for 3d gaussian splatting.\" arXiv preprint arXiv:2411.10133 (2024).\n\n## [W3] Preliminary\nWe will move the introduction of 3DGS-MCMC and Metropolis-Hastings Sampling to the preliminary in the final manuscript. \n\n## [W4] Confusing Description \nIn the context of our paper, a voxel is the axis-aligned cube of edge length v, which is used purely for counting how many Gaussians occupy each bit of world space. To construct the voxel grid, we first lay a regular cubic lattice of size v across the scene’s coordinate frame. After that, the centre of every Gaussian is mapped to one of these voxels by dividing its 3D position by v and taking the integer part. We then record only how many Gaussians fall inside each voxel, $c_\\Theta(v)$. These occupancy counts feed the sparsity voxel-wise prior as denoted in Eq. 4, which penalizes voxels that hold more than one or two Gaussians. Note that as we only use the voxels for counts, and never for rendering, we don’t need to store a full 3D grid. \n\n## [W5/Q1] Results in Figure 1.\n\nAs we are not allowed to provide images, we will provide the statistics regarding timing. Our method shows faster convergence for both Mip-NeRF360 and Tanks and Temples datasets, and shows faster convergence even when measured by the average of all the datasets. We provide the statistics of the convergence time when averaged across all 13 scenes below, and include the \"98% of Final PSNR\" as another reference.\n\n|                     | Ours     | 3DGS-MCMC |\n|---------------------|----------|-----------|\n| 95% of Final PSNR | 8.7 mins | 9.0mins   |\n| 98% of Final PSNR | 11.9 mins| 13.4mins  |\n\nWe also provide a “time-to-equal-PSNR” metric, as suggested by reviewer U3jc, below. \n\n| Target PSNR  | Time - Ours           | Time - 3DGS-MCMC       |\n|--------------|----------------------|------------------------|\n| 21           | 16.30s / 0.27 mins   | 17.08s / 0.28 mins     |\n| 24           | 61.34s / 1.02 mins   | 98.38s / 1.64 mins     |\n| 27           | 287.01s / 4.78 mins  | 341.64s / 5.69 mins    |\n| 30           | 851.52s / 14.19 mins | 983.05s / 16.38 mins   |\n\nOur method consistently shows faster convergence compared to 3DGS-MCMC. We will update the figure to reflect all datasets in our final manuscript. \n\n## [W6] Limited Improvement\nOur method shows mixed quantitative results on MipNeRF-360 and Tanks & Temples, which is a point we acknowledge in our paper's limitations section. \n\nQualitatively, our method is on par with or even superior to baselines in capturing the fine details of the primary subjects, as can be seen in Fig. 2 and our supplementary video. However, as discussed in our limitations, the lower quantitative metrics are predominantly caused by our model's difficulty in representing vast, low-frequency backgrounds, such as the sky. Since these metrics are averaged over the entire image, imperfections in the large background regions can penalize the overall score.\n\nThe high-quality results of reconstructing indoor scenes demonstrate our method's strong potential when scale variance is more manageable. On these scenes, our method achieves superior metrics across all measures, solidifying that the core algorithmic contributions are sound. This trade-off stems from the extreme scale variance (exceeding 170,000x) inherent to unbounded outdoor scenes, where our model struggles to simultaneously optimize for the compact, detailed Gaussians of the foreground and the massive, smooth Gaussians required for the background. As we state in our limitations, adapting our model to robustly handle both ends of this scale spectrum is a clear direction for future work, and this can be addressed by exploring techniques regarding multi-scale representation. \n\n## [W7/Q2] Varying Number of Gaussians \n\nWe provide results of our method, 3DGS-MCMC and 3DGS, with varying numbers of Gaussians across all scenes. Note that the performance of models is similar across varying numbers of Gaussians. \n\n**Average Number of Gaussians across all scenes = 0.53M**\n\n| Method | Mip-NeRF360 | Tanks&Temples | Deep Blending |\n|--------|-------------|---------------|---------------|\n|        | PSNR$\\uparrow$/SSIM$\\uparrow$/LPIPS$\\downarrow$ | PSNR $\\uparrow$/SSIM$\\uparrow$/LPIPS$\\downarrow$ | PSNR $\\uparrow$/SSIM$\\uparrow$/LPIPS$\\downarrow$ |\n| MH-3DGS    | 27.14 / 0.790 / 0.253              | 23.75 / 0.847 / 0.179              | 29.99 / 0.906 / 0.251              |\n| 3DGS       | 26.46 / 0.756 / 0.301              | 23.30 / 0.824 / 0.221              | 29.27 / 0.897 / 0.277              |\n| 3DGS-MCMC  | 27.39 / 0.808 / 0.246              | 24.02 / 0.854 / 0.181              | 29.84 / 0.905 / 0.264              |\n\n**Average Number of Gaussians across all scenes = 1.40M**\n\n| Method | Mip-NeRF360 | Tanks&Temples | Deep Blending |\n|--------|-------------|---------------|---------------|\n|        | PSNR$\\uparrow$/SSIM$\\uparrow$/LPIPS$\\downarrow$ | PSNR $\\uparrow$/SSIM$\\uparrow$/LPIPS$\\downarrow$ | PSNR $\\uparrow$/SSIM$\\uparrow$/LPIPS$\\downarrow$ |\n| MH-3DGS    | 27.58 / 0.810 / 0.212              | 24.06 / 0.857 / 0.153              | 30.04 / 0.911 / 0.233              |\n| 3DGS       | 27.08 / 0.786 / 0.261              | 23.52 / 0.842 / 0.192              | 29.60 / 0.903 / 0.255              |\n| 3DGS-MCMC  | 27.88 / 0.829 / 0.207              | 24.46 / 0.867 / 0.157              | 29.90 / 0.908 / 0.246              |\n\n## [W8/Q3] Ablation Study \n\nThank you for the valuable feedback. To directly compare Metropolis-Hastings (MH) Sampling with SGLD, we conducted an ablation study where both methods had only their key algorithm implemented. We removed our design components (voxel-wise prior, importance score) and also took out opacity and scaling regularizers from both methods. This setup creates the most direct comparison between our MH sampling and the SGLD approach.\n\n| Method | Mip-NeRF360 | Tanks&Temples | Deep Blending |\n|--------|-------------|---------------|---------------|\n|        | PSNR$\\uparrow$/SSIM$\\uparrow$/LPIPS$\\downarrow$ | PSNR $\\uparrow$/SSIM$\\uparrow$/LPIPS$\\downarrow$ | PSNR $\\uparrow$/SSIM$\\uparrow$/LPIPS$\\downarrow$ |\n| 3DGS + MH (0.85M) | 26.87 / 0.782 / 0.260 | 23.66 / 0.843 / 0.173 | 29.46 / 0.898 / 0.258 |\n| 3DGS + SGLD (0.85M) | 26.80 / 0.789 / 0.250 | 23.47 / 0.841 / 0.178 | 29.45 / 0.898 / 0.259 |\n\nThe results demonstrate that our Metropolis-Hastings sampling strategy provides consistent improvements over 3DGS-MCMC's SGLD approach. This result indicates that the MH sampling strategy provides a reliable advantage in reconstruction quality."}]

Author rebuttal:

Meta-review: This paper proposes a densification and sparsification process in 3D Gaussian splatting (3DGS). The proposed method uses importance values of each Gaussian and uses Metropolis-Hastings sampling to achieve probability-based densification. Experiments show that the proposed method outperforms 3DGS with fewer numbers of gaussians, and achieves faster convergence compared to 3DGS-MCMC.

The reviewers valued the novelty of the paper for the new sampling process for 3DGS. Since the original 3DGS relies on the heuristics of the densification-sparsification processes, the sampling method would be beneficial. As reviewers mentioned, the 3DGS-MCMC aims at a similar goal, while the proposed method using Metropolis-Hastings sampling achieves faster convergence.

Overall, the reviewers' assessment weakly converges to the acceptance of the paper. Meanwhile, the AC suggests clarifying and emphasizing the technical difference and merit of using the proposed method over 3DGS-MCMC in the final version.
Decision: Accept (poster)