Elliptical Adaptive AA (EAA)

• EAA is a frequency-aware rendering algorithm that replaces point sampling with geometry-aligned integration over elliptical pixel footprints to reduce aliasing.
• It computes pixel-integrated opacity by transforming projected 3D Gaussians into anisotropic, perspective-aware ellipses using adaptive quadrature and importance sampling.
• Its depth-ordered compositing and boundary-aware sampling stabilize photometric gradients, leading to sharper textures and improved real-time SLAM tracking.

Elliptical Adaptive Anti-aliasing (EAA) is a frequency-aware rendering algorithm for 3D Gaussian Splatting (3DGS) that replaces point sampling at pixel centers with fast, geometry-aligned numerical integration over the screen-space elliptical footprint of a projected 3D Gaussian. In MipSLAM, EAA is introduced to suppress aliasing across camera reconfigurations, including resolution, focal length, and zoom, while preserving the real-time characteristics required by a SLAM system. Its role is specifically to approximate pixel-integrated contributions of projected 3D Gaussians in image space, emphasize anisotropy and boundary structure, and stabilize the photometric gradients used for pose estimation (Li et al., 7 Mar 2026).

1. Problem setting and design objectives

Aliasing in 3D Gaussian Splatting arises when per-pixel contributions are evaluated by point sampling of the projected Gaussian density at the pixel center. Under changes in camera intrinsics or resolution, this violates Nyquist–Shannon sampling assumptions. The reported artifacts include high-frequency shimmer and moiré in textured regions such as blinds and keyboards, jagged edges and resolution-dependent blurring when zooming or downsampling, and instabilities in photometric gradients used for pose optimization, which can degrade SLAM tracking.

Naïve splatting or purely spatial map optimization does not resolve these effects because such methods do not integrate samples over the pixel footprint and do not account for the anisotropic, perspective-projected elliptical geometry of splats. EAA is therefore defined by three explicit goals: approximating frequency-aware, pixel-integrated contributions of projected 3D Gaussians in image space; suppressing aliasing by integrating over the correct ellipse footprint with emphasis on boundary regions and anisotropy; and maintaining computational efficiency suitable for real-time SLAM.

Within MipSLAM, these goals place EAA at the rendering side of a broader frequency-aware SLAM pipeline. The paper distinguishes this role from Spectral-Aware Pose Graph Optimization (SA-PGO), which addresses trajectory estimation and drift suppression in the frequency domain, and from the local frequency-domain perceptual loss, which is introduced to enhance fine-grained geometric detail recovery.

2. Rendering model and screen-space ellipse construction

EAA inherits the 3DGS primitive representation in which a Gaussian is parameterized by mean and covariance,

$$\boldsymbol\mu \in \mathbb{R}^3,\qquad \boldsymbol\Sigma \in \mathbb{R}^{3\times 3},$$

with

$$\boldsymbol{\Sigma}=\mathbf{O}\,\mathrm{diag}(\mathbf{s})^2\,\mathbf{O}^\top,$$

where $\mathbf{O}\in SO(3)$ is a rotation and $\mathbf{s}\in\mathbb{R}^3$ is a scale vector. The corresponding density is

$$G(\mathbf{x}) = \exp\!\Big(-\tfrac{1}{2}(\mathbf{x}-\boldsymbol{\mu})^\top \boldsymbol{\Sigma}^{-1} (\mathbf{x}-\boldsymbol{\mu})\Big).$$

The camera model uses intrinsics $\mathbf{K}$ and extrinsics $(\mathbf{R}, \mathbf{t})$. With camera-frame coordinates $\mathbf{X}_c=\mathbf{R}\mathbf{X}+\mathbf{t}$, the pinhole projection is

$\mathbf{u}=\boldsymbol\pi(\mathbf{X})= \begin{bmatrix}u\v\end{bmatrix} = \begin{bmatrix} f_x \frac{X_c}{Z_c} + c_x\[2pt] f_y \frac{Y_c}{Z_c} + c_y \end{bmatrix}.$

Standard 3DGS rendering uses front-to-back, depth-ordered alpha compositing. Given per-splat opacity $\alpha_i$ and view-dependent color $$\boldsymbol{\Sigma}=\mathbf{O}\,\mathrm{diag}(\mathbf{s})^2\,\mathbf{O}^\top,$$0, the pixel color at $$\boldsymbol{\Sigma}=\mathbf{O}\,\mathrm{diag}(\mathbf{s})^2\,\mathbf{O}^\top,$$1 is

$$\boldsymbol{\Sigma}=\mathbf{O}\,\mathrm{diag}(\mathbf{s})^2\,\mathbf{O}^\top,$$2

and the rendered depth is

$$\boldsymbol{\Sigma}=\mathbf{O}\,\mathrm{diag}(\mathbf{s})^2\,\mathbf{O}^\top,$$3

EAA replaces the point-sampled factor $$\boldsymbol{\Sigma}=\mathbf{O}\,\mathrm{diag}(\mathbf{s})^2\,\mathbf{O}^\top,$$4 by an integrated opacity over the pixel region.

The screen-space footprint is obtained by Jacobian linearization of projection at the Gaussian center. Let $$\boldsymbol{\Sigma}=\mathbf{O}\,\mathrm{diag}(\mathbf{s})^2\,\mathbf{O}^\top,$$5 and $$\boldsymbol{\Sigma}=\mathbf{O}\,\mathrm{diag}(\mathbf{s})^2\,\mathbf{O}^\top,$$6. The projection Jacobian with respect to world coordinates is

$$\boldsymbol{\Sigma}=\mathbf{O}\,\mathrm{diag}(\mathbf{s})^2\,\mathbf{O}^\top,$$7

where $$\boldsymbol{\Sigma}=\mathbf{O}\,\mathrm{diag}(\mathbf{s})^2\,\mathbf{O}^\top,$$8 are the rows of $$\boldsymbol{\Sigma}=\mathbf{O}\,\mathrm{diag}(\mathbf{s})^2\,\mathbf{O}^\top,$$9. First-order propagation gives the approximate 2D covariance

$\mathbf{O}\in SO(3)$0

This defines an elliptical Gaussian footprint centered at $\mathbf{O}\in SO(3)$1. Its eigendecomposition,

$\mathbf{O}\in SO(3)$2

provides the ellipse orientation $\mathbf{O}\in SO(3)$3 and principal variances $\mathbf{O}\in SO(3)$4. A $\mathbf{O}\in SO(3)$5-sigma footprint is then the level set

$\mathbf{O}\in SO(3)$6

This construction is central to EAA because it replaces isotropic or pixel-center reasoning by a perspective-aware elliptical domain whose area scales with $\mathbf{O}\in SO(3)$7 and can be compared to pixel area to determine sampling density.

3. Numerical integration over the pixel footprint

For a pixel region $\mathbf{O}\in SO(3)$8, the desired quantity for a single splat is the pixel-integrated contribution

$\mathbf{O}\in SO(3)$9

where

$\mathbf{s}\in\mathbb{R}^3$0

is the screen-space Gaussian and $\mathbf{s}\in\mathbb{R}^3$1 denotes shading, color, or opacity modulation. The formulation identifies direct analytic integration over a rotated, anisotropic Gaussian with perspective as expensive because it involves error functions and special functions and is ill-suited to real-time SLAM.

EAA therefore uses a geometry-aware numerical approximation. Samples are transformed into the ellipse principal-axis coordinates by

$\mathbf{s}\in\mathbb{R}^3$2

and the integration is adapted to footprint-to-pixel scale and anisotropy. In the principal frame, the integral for the opaque contribution is written as

$\mathbf{s}\in\mathbb{R}^3$3

The numerical quadrature is importance-weighted. Let $\mathbf{s}\in\mathbb{R}^3$4 and $\mathbf{s}\in\mathbb{R}^3$5. For samples $\mathbf{s}\in\mathbb{R}^3$6 drawn from a proposal $\mathbf{s}\in\mathbb{R}^3$7 over the pixel footprint $\mathbf{s}\in\mathbb{R}^3$8, EAA uses

$\mathbf{s}\in\mathbb{R}^3$9

with anisotropy-dependent enhancement

$$G(\mathbf{x}) = \exp\!\Big(-\tfrac{1}{2}(\mathbf{x}-\boldsymbol{\mu})^\top \boldsymbol{\Sigma}^{-1} (\mathbf{x}-\boldsymbol{\mu})\Big).$$0

and boundary proximity

$$G(\mathbf{x}) = \exp\!\Big(-\tfrac{1}{2}(\mathbf{x}-\boldsymbol{\mu})^\top \boldsymbol{\Sigma}^{-1} (\mathbf{x}-\boldsymbol{\mu})\Big).$$1

where $$G(\mathbf{x}) = \exp\!\Big(-\tfrac{1}{2}(\mathbf{x}-\boldsymbol{\mu})^\top \boldsymbol{\Sigma}^{-1} (\mathbf{x}-\boldsymbol{\mu})\Big).$$2 is the Euclidean distance to the nearest pixel edge. The stated intuition is that highly anisotropic splats and boundary-proximate samples receive higher weight to capture steep variations and reduce bleeding across discontinuities.

The resulting estimator for integrated per-splat opacity is

$$G(\mathbf{x}) = \exp\!\Big(-\tfrac{1}{2}(\mathbf{x}-\boldsymbol{\mu})^\top \boldsymbol{\Sigma}^{-1} (\mathbf{x}-\boldsymbol{\mu})\Big).$$3

where $$G(\mathbf{x}) = \exp\!\Big(-\tfrac{1}{2}(\mathbf{x}-\boldsymbol{\mu})^\top \boldsymbol{\Sigma}^{-1} (\mathbf{x}-\boldsymbol{\mu})\Big).$$4 is the adaptive number of quadrature samples. This integrated opacity is then used in front-to-back compositing in place of the point-evaluated term.

A notable aspect of the method is that sampling is performed in the pixel footprint clipped by the ellipse’s $$G(\mathbf{x}) = \exp\!\Big(-\tfrac{1}{2}(\mathbf{x}-\boldsymbol{\mu})^\top \boldsymbol{\Sigma}^{-1} (\mathbf{x}-\boldsymbol{\mu})\Big).$$5-sigma bounds to avoid negligible tails. Samples may be drawn either from a stratified grid aligned with $$G(\mathbf{x}) = \exp\!\Big(-\tfrac{1}{2}(\mathbf{x}-\boldsymbol{\mu})^\top \boldsymbol{\Sigma}^{-1} (\mathbf{x}-\boldsymbol{\mu})\Big).$$6 in the ellipse frame or from a pixel-aligned grid transformed into principal axes for weighting. The sampling density depends on the ellipse area relative to pixel area, the anisotropy $$G(\mathbf{x}) = \exp\!\Big(-\tfrac{1}{2}(\mathbf{x}-\boldsymbol{\mu})^\top \boldsymbol{\Sigma}^{-1} (\mathbf{x}-\boldsymbol{\mu})\Big).$$7, and local boundary proximity through $$G(\mathbf{x}) = \exp\!\Big(-\tfrac{1}{2}(\mathbf{x}-\boldsymbol{\mu})^\top \boldsymbol{\Sigma}^{-1} (\mathbf{x}-\boldsymbol{\mu})\Big).$$8. An optional approximation by pixel-footprint convolution is also described, in which a pixel covariance $$G(\mathbf{x}) = \exp\!\Big(-\tfrac{1}{2}(\mathbf{x}-\boldsymbol{\mu})^\top \boldsymbol{\Sigma}^{-1} (\mathbf{x}-\boldsymbol{\mu})\Big).$$9 is added so that $\mathbf{K}$0, but MipSLAM’s EAA focuses on direct numerical integration and does not require a separate $\mathbf{K}$1 (Li et al., 7 Mar 2026).

4. Geometry awareness, compositing, and implementation profile

EAA is geometry-aware in three explicit ways. First, it uses depth-ordered blending with front-to-back compositing, so farther splats contribute only through residual transmittance $\mathbf{K}$2. Second, the boundary-aware sampling and weighting term $\mathbf{K}$3 concentrates samples near pixel edges, where depth discontinuities and silhouette boundaries occur, reducing cross-boundary bleeding. Third, anisotropy-aware sampling through $\mathbf{K}$4 increases sampling support for thin or highly elongated splats, which are often aligned with geometric edges.

If available, z-buffer visibility checks per sample can be incorporated by rejecting samples whose estimated depth is behind previously accumulated opaque contributions. However, the reported implementation achieves robust occlusion handling primarily through depth-ordered alpha compositing and boundary-enhanced sampling, thereby avoiding the overhead of per-sample z-tests.

The per-splat/per-pixel computational structure is explicit. Computing $\mathbf{K}$5 and $\mathbf{K}$6 is $\mathbf{K}$7; the $\mathbf{K}$8 eigendecomposition is also $\mathbf{K}$9; adaptive sampling requires $(\mathbf{R}, \mathbf{t})$0 work per pixel per splat for weights and Gaussian factors; and the per-frame complexity is

$(\mathbf{R}, \mathbf{t})$1

Ellipse bounding-box culling limits evaluation to nearby pixels. The implementation strategy further includes tile-based rasterization, SIMD/GPU kernels for sample generation and evaluation, on-the-fly computation of $(\mathbf{R}, \mathbf{t})$2 and $(\mathbf{R}, \mathbf{t})$3 with shared memory usage, and early termination via a transmittance threshold to skip occluded contributions. The pseudocode also specifies early exit when $(\mathbf{R}, \mathbf{t})$4 falls below a threshold such as $(\mathbf{R}, \mathbf{t})$5 transmittance.

The reported runtime breakdown on Replica Room0 at $(\mathbf{R}, \mathbf{t})$6 is: $(\mathbf{R}, \mathbf{t})$7-blending at $(\mathbf{R}, \mathbf{t})$8 ms, tracking at $(\mathbf{R}, \mathbf{t})$9 ms, mapping at $\mathbf{X}_c=\mathbf{R}\mathbf{X}+\mathbf{t}$0 ms, SA-PGO at $\mathbf{X}_c=\mathbf{R}\mathbf{X}+\mathbf{t}$1 ms, for an overall $\mathbf{X}_c=\mathbf{R}\mathbf{X}+\mathbf{t}$2 FPS. This is described as matching the baseline while improving rendering quality and SLAM accuracy thanks to faster convergence after drift correction. A plausible implication is that EAA’s own rendering overhead is controlled sufficiently that the broader SLAM stack, rather than anti-aliasing alone, remains the dominant runtime determinant.

5. Relation to mip, elliptical filtering, and supersampling

EAA belongs to a family of anti-aliasing and prefiltering strategies, but its formulation is narrower and more geometry-specific than several neighboring approaches. Its relation to Mip-NeRF and mipmapping is stated in terms of frequency-aware prefiltering: both seek to prevent aliasing at appropriate scales, whereas EAA performs sampling directly in the ellipse domain of projected 3D Gaussians rather than through explicit levels of detail.

Its relation to EWA filtering is also explicit. EWA filters resample textures and point sets using anisotropic kernels aligned with projection geometry. EAA shares the elliptical alignment and anisotropy awareness, but its target is pixel-window integration of volumetric splats under perspective with front-to-back compositing. This distinction matters because EAA is embedded in the 3DGS rendering equation rather than acting as a generic texture resampler.

The comparison to Mip-Splatting is more operational. Mip-Splatting enforces minimum projected Gaussian size via 2D/3D filters and opacity clamping and is characterized as efficient, but biased for rotated, anisotropic ellipses and potentially disruptive to pose gradients during covariance inference when naively integrated into SLAM. EAA instead seeks anti-aliasing and gradient stability by integrating over the correct ellipse with adaptive sampling and boundary emphasis.

The contrast with MSAA is primarily about efficiency. Supersampling reduces aliasing, but it requires many uniform samples per pixel independent of geometry and anisotropy. EAA pursues a similar objective through importance sampling with far fewer samples.

These comparisons delimit what EAA is not. It is not analytic 2D integration; it is not equivalent to box filters or mip filters; and it is not simple oversampling. Its defining property is geometry-aligned quadrature over the perspective-induced elliptical footprint, coupled to depth-ordered alpha compositing (Li et al., 7 Mar 2026).

6. Empirical behavior, limitations, and significance in SLAM

The reported evaluations situate EAA within MipSLAM’s overall performance on Replica and TUM. On Replica at native $\mathbf{X}_c=\mathbf{R}\mathbf{X}+\mathbf{t}$3, MipSLAM at $\mathbf{X}_c=\mathbf{R}\mathbf{X}+\mathbf{t}$4 resolution exceeds MonoGS by $\mathbf{X}_c=\mathbf{R}\mathbf{X}+\mathbf{t}$5 dB PSNR and Scaffold-GS by $\mathbf{X}_c=\mathbf{R}\mathbf{X}+\mathbf{t}$6 dB. Averaged over eight sequences, it gains $\mathbf{X}_c=\mathbf{R}\mathbf{X}+\mathbf{t}$7 dB over SplaTAM, $\mathbf{X}_c=\mathbf{R}\mathbf{X}+\mathbf{t}$8 dB over MonoGS, and $\mathbf{X}_c=\mathbf{R}\mathbf{X}+\mathbf{t}$9 dB over GS-ICP. The overall averages are PSNR $\mathbf{u}=\boldsymbol\pi(\mathbf{X})= \begin{bmatrix}u\v\end{bmatrix} = \begin{bmatrix} f_x \frac{X_c}{Z_c} + c_x\[2pt] f_y \frac{Y_c}{Z_c} + c_y \end{bmatrix}.$0 dB, SSIM $\mathbf{u}=\boldsymbol\pi(\mathbf{X})= \begin{bmatrix}u\v\end{bmatrix} = \begin{bmatrix} f_x \frac{X_c}{Z_c} + c_x\[2pt] f_y \frac{Y_c}{Z_c} + c_y \end{bmatrix}.$1, and LPIPS $\mathbf{u}=\boldsymbol\pi(\mathbf{X})= \begin{bmatrix}u\v\end{bmatrix} = \begin{bmatrix} f_x \frac{X_c}{Z_c} + c_x\[2pt] f_y \frac{Y_c}{Z_c} + c_y \end{bmatrix}.$2. On TUM at native $\mathbf{u}=\boldsymbol\pi(\mathbf{X})= \begin{bmatrix}u\v\end{bmatrix} = \begin{bmatrix} f_x \frac{X_c}{Z_c} + c_x\[2pt] f_y \frac{Y_c}{Z_c} + c_y \end{bmatrix}.$3, the averages are PSNR $\mathbf{u}=\boldsymbol\pi(\mathbf{X})= \begin{bmatrix}u\v\end{bmatrix} = \begin{bmatrix} f_x \frac{X_c}{Z_c} + c_x\[2pt] f_y \frac{Y_c}{Z_c} + c_y \end{bmatrix}.$4 dB, SSIM $\mathbf{u}=\boldsymbol\pi(\mathbf{X})= \begin{bmatrix}u\v\end{bmatrix} = \begin{bmatrix} f_x \frac{X_c}{Z_c} + c_x\[2pt] f_y \frac{Y_c}{Z_c} + c_y \end{bmatrix}.$5, and LPIPS $\mathbf{u}=\boldsymbol\pi(\mathbf{X})= \begin{bmatrix}u\v\end{bmatrix} = \begin{bmatrix} f_x \frac{X_c}{Z_c} + c_x\[2pt] f_y \frac{Y_c}{Z_c} + c_y \end{bmatrix}.$6, with reported outperformance over SplaTAM, MonoGS, and GS-ICP across scales.

The visual evidence described for EAA is specifically multi-resolution. At low resolutions such as $\mathbf{u}=\boldsymbol\pi(\mathbf{X})= \begin{bmatrix}u\v\end{bmatrix} = \begin{bmatrix} f_x \frac{X_c}{Z_c} + c_x\[2pt] f_y \frac{Y_c}{Z_c} + c_y \end{bmatrix}.$7, it reduces moiré and shimmer on blinds and preserves object edges without dilation. At high resolutions such as $\mathbf{u}=\boldsymbol\pi(\mathbf{X})= \begin{bmatrix}u\v\end{bmatrix} = \begin{bmatrix} f_x \frac{X_c}{Z_c} + c_x\[2pt] f_y \frac{Y_c}{Z_c} + c_y \end{bmatrix}.$8, it yields crisper textures, including bottle caps and keyboards, and better structural fidelity than box-filter-based methods. For localization, the Replica ATE RMSE average is reported as $\mathbf{u}=\boldsymbol\pi(\mathbf{X})= \begin{bmatrix}u\v\end{bmatrix} = \begin{bmatrix} f_x \frac{X_c}{Z_c} + c_x\[2pt] f_y \frac{Y_c}{Z_c} + c_y \end{bmatrix}.$9 cm, compared with $\alpha_i$0 cm for MonoGS and larger errors for other 3DGS SLAM variants. The text attributes this to improved anti-aliasing stabilizing photometric gradients and operating in tandem with SA-PGO to reduce drift (Li et al., 7 Mar 2026).

The stated limitations are tied to the adaptive sample budget. Residual aliasing or blur can occur for extremely anisotropic splats with very large $\alpha_i$1, because thin geometry may demand more samples than real-time caps allow. Very large footprints spanning many pixels can over-smooth high-frequency details unless $\alpha_i$2 is increased. Complex occlusion layers with fine interpenetrations remain difficult: boundary emphasis reduces bleeding but cannot completely prevent cross-boundary contributions without per-sample visibility tests.

The proposed remedies remain within the paper’s own framework: adaptive $\alpha_i$3 escalation with local error estimates, hybrid schemes introducing pixel covariance for very large footprints, and optional z-buffer checks near strong depth discontinuities. This suggests that EAA’s principal trade-off is not conceptual but allocational: the method exchanges sample count against residual aliasing for extreme anisotropy or large projected footprints.

In the context of SLAM, EAA’s significance lies in coupling anti-aliased rendering to optimization stability. By replacing biased point sampling or box filtering with integrated opacity estimates, it suppresses moiré, shimmer, and jaggedness across resolutions while stabilizing the rendered signals that feed tracking and mapping. The result, as formulated in MipSLAM, is an anti-aliasing mechanism that is not merely visual post-processing but part of the estimation pipeline itself.