Negative Alpha Blending in Gaussian Splatting

• The paper introduces a negative alpha blending modification to 3D Gaussian Splatting, enabling both additive and subtractive residual correction in CT super-resolution.
• It replaces the softplus function with a leaky ReLU activation to permit signed density coefficients, critical for modeling residual corrections.
• Empirical evaluations demonstrate improved PSNR and SSIM, achieved through a careful balance of residual learning, controlled densification, and regularization.

Searching arXiv for the cited NAB-GS and related Gaussian Splatting papers. Negative Alpha Blending (NAB-GS) is a rendering and optimization modification for 3D Gaussian Splatting in which Gaussian density representation is allowed to take negative values, so that per-splat opacity contributions can be signed rather than restricted to the standard non-negative regime. It was introduced in the context of zero-shot volumetric CT super-resolution using 3D Gaussian Splatting with upsampled 2D X-ray projection priors, where the goal is to learn residuals between upsampled low-resolution projections and diffusion-generated high-resolution projections (Noh et al., 21 Aug 2025). In this formulation, signed opacity enables both additive and subtractive residual correction in projection space: positive splats restore or sharpen structure, while negative splats suppress blurred or over-smoothed attenuation. NAB-GS is closely related, but not identical, to the earlier idea of negative contributions in Gaussian Splatting developed in "NegGS: Negative Gaussian Splatting," which realizes subtraction through negative colors while keeping alpha non-negative (Kasymov et al., 2024).

1. Definition and conceptual scope

NAB-GS denotes a variant of Gaussian Splatting in which the usual non-negativity constraint on Gaussian density is relaxed. In the standard 3D Gaussian Splatting pipeline for radiative or tomographic rendering, each Gaussian contributes non-negative attenuation or opacity, and front-to-back compositing assumes $\alpha_i \in [0,1]$ with monotone decreasing transmittance. NAB-GS preserves the same compositing equations structurally, but permits $\alpha_i \in \mathbb{R}$ by allowing signed density coefficients $\rho_i$ (Noh et al., 21 Aug 2025).

The method was introduced for zero-shot 3D CT super-resolution, where only a single low-resolution volume is available and internal information alone is insufficient to recover high-frequency anatomical structure. In that setting, the reconstruction problem is reframed as residual learning in projection space relative to an upsampled low-resolution baseline, with high-resolution guidance supplied by diffusion-generated 2D X-ray projections. Because such residuals are signed, a model limited to non-negative opacity cannot represent them directly; NAB-GS addresses that restriction by permitting negative opacity contributions (Noh et al., 21 Aug 2025).

A central distinction from NegGS is that NegGS does not implement negative alpha. Instead, NegGS allows color vectors $c_i$ to have negative components while keeping opacity standard, thereby preserving conventional transmittance semantics. NAB-GS is therefore a stronger intervention into the Gaussian Splatting renderer, because it modifies not only the contribution term but also the multiplicative transmittance chain through signed $\alpha_i$ (Kasymov et al., 2024).

2. Standard Gaussian Splatting formulation and the NAB-GS modification

In the radiative Gaussian field formulation used for tomographic reconstruction, each 3D Gaussian is parameterized by center $\mathbf{p}_i \in \mathbb{R}^3$, covariance $\boldsymbol{\Sigma}_i \in \mathbb{R}^{3\times 3}$, and density coefficient $\rho_i$. The 3D Gaussian kernel is

$$G_i^3(\mathbf{x}\mid\rho_i,\mathbf{p}_i,\boldsymbol{\Sigma}_i) = \rho_i \cdot \exp\!\left(-\frac{1}{2}(\mathbf{x}-\mathbf{p}_i)^\top \boldsymbol{\Sigma}_i^{-1}(\mathbf{x}-\mathbf{p}_i)\right).$$

The continuous density field is

$$\sigma(\mathbf{x}) = \sum_{i=1}^{M} G_i^3(\mathbf{x}\mid\rho_i,\mathbf{p}_i,\boldsymbol{\Sigma}_i).$$

After projection to the image plane, each Gaussian induces a 2D splat with projected center $\alpha_i \in \mathbb{R}$0, projected covariance $\alpha_i \in \mathbb{R}$1, and per-pixel opacity

$\alpha_i \in \mathbb{R}$2

where $\alpha_i \in \mathbb{R}$3 is a covariance-related normalization term. In the CT setting described for NAB-GS, emission is assumed zero and the local attenuation contribution is set as $\alpha_i \in \mathbb{R}$4 (Noh et al., 21 Aug 2025).

Standard front-to-back accumulation is

$\alpha_i \in \mathbb{R}$5

Equivalently, under a piecewise-constant volumetric interpretation,

$\alpha_i \in \mathbb{R}$6

The standard assumptions are $\alpha_i \in \mathbb{R}$7, $\alpha_i \in \mathbb{R}$8, and non-increasing transmittance. NAB-GS keeps the same rendering and sorting structure but removes the non-negativity bottleneck by replacing softplus with a leaky ReLU parameterization of density:

$\alpha_i \in \mathbb{R}$9

This yields signed $\rho_i$0, and therefore signed $\rho_i$1, while retaining the same front-to-back depth ordering and compositing equations (Noh et al., 21 Aug 2025).

3. Residual learning interpretation in zero-shot volumetric CT super-resolution

The immediate motivation for NAB-GS is residual learning between an upsampled low-resolution projection and a diffusion-based high-resolution projection prior. Let $\rho_i$2 denote the upsampled low-resolution projection and $\rho_i$3 the target high-resolution projection generated from the diffusion prior. NAB-GS models the residual as

$\rho_i$4

Because $\rho_i$5 is signed, negative opacity is used to cancel erroneous or over-smoothed attenuation, while positive opacity adds missing high-frequency structure (Noh et al., 21 Aug 2025).

This residual interpretation is the main reason NAB-GS departs from standard physical attenuation constraints. The paper explicitly notes that physical attenuation and density in CT are non-negative, and does not claim physical interpretability for the signed densities. Instead, signed contributions operate in residual space relative to low-resolution projections. This suggests that the method should be understood as a reconstruction device for correcting low-resolution blur under projection consistency, rather than as a literal model of material attenuation (Noh et al., 21 Aug 2025).

A plausible implication is that NAB-GS is best viewed as a hybrid between volumetric rendering and residual field estimation. Its signed splats do not merely encode scene occupancy or attenuation; they encode correction terms conditioned on an external prior. That differentiates it from standard radiance-field-style compositing and also from NegGS, where subtraction is introduced through signed color but the opacity chain remains physically conventional (Kasymov et al., 2024).

4. Optimization, regularization, and implementation

The optimization objective combines reconstruction, residual, structural, and total-variation terms:

$\rho_i$6

$\rho_i$7

where $\rho_i$8 is the ground-truth high-resolution projection, $\rho_i$9 is the predicted projection, $c_i$0 is the residual ground truth, and $c_i$1 is the residual prediction. The loss weights are $c_i$2 and $c_i$3 (Noh et al., 21 Aug 2025).

The renderer itself is minimally modified. The GS pipeline is unchanged except for the density activation: softplus is replaced by leaky ReLU, while the rasterizer, depth sorting, over-operator, and X-ray accumulation remain the same. For a pixel $c_i$4, the front-to-back procedure is:

• Initialize $c_i$5 and $c_i$6.
• For splats sorted by depth:
  • Compute the 2D Gaussian weight

  $c_i$7 - Compute signed density $c_i$8. - Compute signed alpha $c_i$9. - Accumulate $\alpha_i$0 with $\alpha_i$1. - Update $\alpha_i$2.

Training uses PyTorch on a single NVIDIA A6000 with Adam for 5k iterations. Initial learning rates are position $\alpha_i$3, density $\alpha_i$4, scale $\alpha_i$5, and rotation $\alpha_i$6, all exponentially decayed by a factor of $\alpha_i$7 by the end. Initialization uses 50k Gaussians, density threshold $\alpha_i$8, and scaling term $\alpha_i$9. Adaptive densification is performed between iterations 500 and 5000 with gradient threshold $\mathbf{p}_i \in \mathbb{R}^3$0 every 100 iterations, and Gaussians are pruned when density lies in $\mathbf{p}_i \in \mathbb{R}^3$1 (Noh et al., 21 Aug 2025).

The paper emphasizes that numerical stability is not enforced through explicit clamping of $\mathbf{p}_i \in \mathbb{R}^3$2 or $\mathbf{p}_i \in \mathbb{R}^3$3. Instead, stability is achieved empirically through the leaky-ReLU negative slope $\mathbf{p}_i \in \mathbb{R}^3$4, total variation regularization, and gradient-controlled densification and pruning. The gradients of the density activation are piecewise constant: $\mathbf{p}_i \in \mathbb{R}^3$5 for $\mathbf{p}_i \in \mathbb{R}^3$6 and $\mathbf{p}_i \in \mathbb{R}^3$7 for $\mathbf{p}_i \in \mathbb{R}^3$8 (Noh et al., 21 Aug 2025).

5. Diffusion priors, PAS, and the broader reconstruction pipeline

NAB-GS is not presented as an isolated renderer modification; it is embedded in a zero-shot CT super-resolution framework that uses diffusion-generated upsampled 2D X-ray projection priors. The diffusion prior is constructed using DDNM and DDNM+, with degradation model $\mathbf{p}_i \in \mathbb{R}^3$9. The DDNM estimate at diffusion step $\boldsymbol{\Sigma}_i \in \mathbb{R}^{3\times 3}$0 is

$\boldsymbol{\Sigma}_i \in \mathbb{R}^{3\times 3}$1

and with noise correction in DDNM+,

$\boldsymbol{\Sigma}_i \in \mathbb{R}^{3\times 3}$2

The framework also introduces per-projection adaptive sampling strategy (PAS) through DDIM initialization

$\boldsymbol{\Sigma}_i \in \mathbb{R}^{3\times 3}$3

followed by a per-projection change criterion

$\boldsymbol{\Sigma}_i \in \mathbb{R}^{3\times 3}$4

and adaptive step selection

$\boldsymbol{\Sigma}_i \in \mathbb{R}^{3\times 3}$5

These high-resolution projections become the targets $\boldsymbol{\Sigma}_i \in \mathbb{R}^{3\times 3}$6 and residuals $\boldsymbol{\Sigma}_i \in \mathbb{R}^{3\times 3}$7 used for NAB-GS training (Noh et al., 21 Aug 2025).

The diffusion model is trained unconditionally on ChestX-ray14 with 112,120 images and CheXpert with 80,845 images at $\boldsymbol{\Sigma}_i \in \mathbb{R}^{3\times 3}$8, batch size 12 over 620k iterations, with attention at $\boldsymbol{\Sigma}_i \in \mathbb{R}^{3\times 3}$9, $\rho_i$0, and $\rho_i$1. Inference uses DDIM with 50 steps and DDNM+ with noise $\rho_i$2. The PAS thresholds are $\rho_i$3 for $\rho_i$4 and $\rho_i$5 for $\rho_i$6. Projections are acquired at 100 uniformly spaced angles from $\rho_i$7 to $\rho_i$8 using TIGRE (Noh et al., 21 Aug 2025).

This design places NAB-GS within a specific computational role: it is the 3D volumetric residual integrator that reconciles low-resolution consistency and external high-frequency priors. A plausible implication is that the signed-opacity formulation derives much of its practical value from this coupling with projection-space priors; the paper does not claim that signed alpha alone is sufficient independent of the diffusion stage.

6. Empirical behavior, ablations, and comparison with related negative-contribution methods

The reported evaluation covers MELA and UHRCT at super-resolution scales of $\rho_i$9 and $$G_i^3(\mathbf{x}\mid\rho_i,\mathbf{p}_i,\boldsymbol{\Sigma}_i) = \rho_i \cdot \exp\!\left(-\frac{1}{2}(\mathbf{x}-\mathbf{p}_i)^\top \boldsymbol{\Sigma}_i^{-1}(\mathbf{x}-\mathbf{p}_i)\right).$$0, measured by PSNR and SSIM. Quantitatively, the method reports:

Dataset/scale	Ours (NAB-GS)	Selected comparisons
MELA 4×	34.13 / 0.9518	CuNeRF 33.76 / 0.9096; ArSSR 32.92 / 0.9589
MELA 8×	30.74 / 0.9153	CuNeRF 30.11 / 0.8535; ArSSR 30.38 / 0.9294
UHRCT 4×	25.41 / 0.8963	CuNeRF 25.25 / 0.8459; ArSSR 24.81 / 0.8900
UHRCT 8×	21.94 / 0.8255	CuNeRF 21.04 / 0.7572; ArSSR 21.72 / 0.8458

Against R$$G_i^3(\mathbf{x}\mid\rho_i,\mathbf{p}_i,\boldsymbol{\Sigma}_i) = \rho_i \cdot \exp\!\left(-\frac{1}{2}(\mathbf{x}-\mathbf{p}_i)^\top \boldsymbol{\Sigma}_i^{-1}(\mathbf{x}-\mathbf{p}_i)\right).$$1-GS on MELA, the reported gains are $$G_i^3(\mathbf{x}\mid\rho_i,\mathbf{p}_i,\boldsymbol{\Sigma}_i) = \rho_i \cdot \exp\!\left(-\frac{1}{2}(\mathbf{x}-\mathbf{p}_i)^\top \boldsymbol{\Sigma}_i^{-1}(\mathbf{x}-\mathbf{p}_i)\right).$$2 dB PSNR at $$G_i^3(\mathbf{x}\mid\rho_i,\mathbf{p}_i,\boldsymbol{\Sigma}_i) = \rho_i \cdot \exp\!\left(-\frac{1}{2}(\mathbf{x}-\mathbf{p}_i)^\top \boldsymbol{\Sigma}_i^{-1}(\mathbf{x}-\mathbf{p}_i)\right).$$3 and $$G_i^3(\mathbf{x}\mid\rho_i,\mathbf{p}_i,\boldsymbol{\Sigma}_i) = \rho_i \cdot \exp\!\left(-\frac{1}{2}(\mathbf{x}-\mathbf{p}_i)^\top \boldsymbol{\Sigma}_i^{-1}(\mathbf{x}-\mathbf{p}_i)\right).$$4 dB PSNR at $$G_i^3(\mathbf{x}\mid\rho_i,\mathbf{p}_i,\boldsymbol{\Sigma}_i) = \rho_i \cdot \exp\!\left(-\frac{1}{2}(\mathbf{x}-\mathbf{p}_i)^\top \boldsymbol{\Sigma}_i^{-1}(\mathbf{x}-\mathbf{p}_i)\right).$$5 (Noh et al., 21 Aug 2025). For direct ground-truth reconstruction, where residual learning is still used, the paper reports R$$G_i^3(\mathbf{x}\mid\rho_i,\mathbf{p}_i,\boldsymbol{\Sigma}_i) = \rho_i \cdot \exp\!\left(-\frac{1}{2}(\mathbf{x}-\mathbf{p}_i)^\top \boldsymbol{\Sigma}_i^{-1}(\mathbf{x}-\mathbf{p}_i)\right).$$6-GS at 40.88 / 0.9865 and NAB-GS at 43.14 / 0.9902 (Noh et al., 21 Aug 2025).

Ablation on activation functions is especially important because it isolates the signed-density mechanism. On MELA, softplus gives 33.65 / 0.9433 with 49k Gaussians at $$G_i^3(\mathbf{x}\mid\rho_i,\mathbf{p}_i,\boldsymbol{\Sigma}_i) = \rho_i \cdot \exp\!\left(-\frac{1}{2}(\mathbf{x}-\mathbf{p}_i)^\top \boldsymbol{\Sigma}_i^{-1}(\mathbf{x}-\mathbf{p}_i)\right).$$7 and 30.60 / 0.9009 with 50k Gaussians at $$G_i^3(\mathbf{x}\mid\rho_i,\mathbf{p}_i,\boldsymbol{\Sigma}_i) = \rho_i \cdot \exp\!\left(-\frac{1}{2}(\mathbf{x}-\mathbf{p}_i)^\top \boldsymbol{\Sigma}_i^{-1}(\mathbf{x}-\mathbf{p}_i)\right).$$8; tanh gives 33.95 / 0.9481 with 224k Gaussians at $$G_i^3(\mathbf{x}\mid\rho_i,\mathbf{p}_i,\boldsymbol{\Sigma}_i) = \rho_i \cdot \exp\!\left(-\frac{1}{2}(\mathbf{x}-\mathbf{p}_i)^\top \boldsymbol{\Sigma}_i^{-1}(\mathbf{x}-\mathbf{p}_i)\right).$$9 and 30.05 / 0.8953 with 540k Gaussians at $$\sigma(\mathbf{x}) = \sum_{i=1}^{M} G_i^3(\mathbf{x}\mid\rho_i,\mathbf{p}_i,\boldsymbol{\Sigma}_i).$$0 and is described as grainy/noisy; tanh† with a 100k cap gives 34.05 / 0.9499 with 92k Gaussians at $$\sigma(\mathbf{x}) = \sum_{i=1}^{M} G_i^3(\mathbf{x}\mid\rho_i,\mathbf{p}_i,\boldsymbol{\Sigma}_i).$$1 and 30.45 / 0.9042 with 134k at $$\sigma(\mathbf{x}) = \sum_{i=1}^{M} G_i^3(\mathbf{x}\mid\rho_i,\mathbf{p}_i,\boldsymbol{\Sigma}_i).$$2; leaky ReLU, i.e. NAB-GS, gives 34.13 / 0.9518 with 68k Gaussians at $$\sigma(\mathbf{x}) = \sum_{i=1}^{M} G_i^3(\mathbf{x}\mid\rho_i,\mathbf{p}_i,\boldsymbol{\Sigma}_i).$$3 and 30.74 / 0.9153 with 75k at $$\sigma(\mathbf{x}) = \sum_{i=1}^{M} G_i^3(\mathbf{x}\mid\rho_i,\mathbf{p}_i,\boldsymbol{\Sigma}_i).$$4 (Noh et al., 21 Aug 2025). The paper therefore attributes improved fidelity and fewer artifacts specifically to the leaky-ReLU signed-density parameterization.

The negative slope $$\sigma(\mathbf{x}) = \sum_{i=1}^{M} G_i^3(\mathbf{x}\mid\rho_i,\mathbf{p}_i,\boldsymbol{\Sigma}_i).$$5 is dataset-dependent: $$\sigma(\mathbf{x}) = \sum_{i=1}^{M} G_i^3(\mathbf{x}\mid\rho_i,\mathbf{p}_i,\boldsymbol{\Sigma}_i).$$6 for MELA $$\sigma(\mathbf{x}) = \sum_{i=1}^{M} G_i^3(\mathbf{x}\mid\rho_i,\mathbf{p}_i,\boldsymbol{\Sigma}_i).$$7, $$\sigma(\mathbf{x}) = \sum_{i=1}^{M} G_i^3(\mathbf{x}\mid\rho_i,\mathbf{p}_i,\boldsymbol{\Sigma}_i).$$8 for MELA $$\sigma(\mathbf{x}) = \sum_{i=1}^{M} G_i^3(\mathbf{x}\mid\rho_i,\mathbf{p}_i,\boldsymbol{\Sigma}_i).$$9, $\alpha_i \in \mathbb{R}$00 for UHRCT $\alpha_i \in \mathbb{R}$01, and $\alpha_i \in \mathbb{R}$02 for UHRCT $\alpha_i \in \mathbb{R}$03 (Noh et al., 21 Aug 2025). The paper states that ablations show sensitivity to $\alpha_i \in \mathbb{R}$04, indicating that signed blending is beneficial but not plug-and-play.

NegGS provides the closest related precedent for negative contributions in Gaussian Splatting. That work introduces negative Gaussians interpreted as items with negative colors, motivated by a difference-based density family built from two Gaussian PDFs and analyzed through the ratio of Gaussian PDFs. It reports improved modeling of high-frequency elements with rapid color transitions and improved representation of shadows, while preserving standard alpha semantics and CUDA kernels (Kasymov et al., 2024). NegGS explicitly states that it does not introduce negative alpha, and frames "Negative Alpha Blending (NAB-GS)" as a hypothetical, riskier alternative that can break transmittance semantics and stability unless constrained (Kasymov et al., 2024). NAB-GS, by contrast, operationalizes exactly that stronger alternative in the residual CT setting (Noh et al., 21 Aug 2025).

7. Interpretation, limitations, and technical significance

NAB-GS introduces signed alpha into a front-to-back renderer whose conventional semantics assume non-negative extinction. The resulting transmittance chain no longer guarantees $\alpha_i \in \mathbb{R}$05 or monotonic decrease, because negative $\alpha_i \in \mathbb{R}$06 can increase later transmittance factors. The paper acknowledges this indirectly by noting that negative $\alpha_i \in \mathbb{R}$07 increases $\alpha_i \in \mathbb{R}$08 and subtracts from accumulated attenuation, and by emphasizing that training stability must be maintained through activation design, pruning, densification control, and TV regularization rather than through explicit theoretical constraints (Noh et al., 21 Aug 2025).

From a reconstruction perspective, the method is justified not as a physically interpretable attenuation model but as a residual corrector. This suggests that its admissibility depends on the task formulation. In zero-shot CT super-resolution, where the objective is to recover high-frequency structure absent from the low-resolution volume but hinted by diffusion priors, signed opacity is presented as the mechanism that makes residual volumetric correction feasible (Noh et al., 21 Aug 2025).

The contrast with NegGS clarifies an important conceptual point. Negative contributions in Gaussian Splatting can be introduced at least two different ways: through signed color with conventional alpha, or through signed alpha itself. NegGS chooses the former because it preserves transmittance semantics and implementation stability, while NAB-GS adopts the latter because residual learning in projection space requires signed attenuation rather than merely signed radiance or color (Kasymov et al., 2024, Noh et al., 21 Aug 2025). This division also resolves a potential misconception: NAB-GS is not simply another name for NegGS. The two methods are related by the broader goal of subtractive modeling, but they intervene at different points in the rendering equation.

In practical terms, NAB-GS is a minimal renderer modification with nontrivial modeling consequences. The paper reports that the density activation change adds no separate overhead in the pipeline, training proceeds for 5k iterations on a single A6000 GPU, and the number of active Gaussians remains moderate due to controlled densification, for example approximately 68–75k for MELA (Noh et al., 21 Aug 2025). A plausible implication is that the principal cost of NAB-GS is not computational but statistical and numerical: careful tuning of $\alpha_i \in \mathbb{R}$09, regularization, and pruning is required to exploit signed blending without destabilizing the renderer.