Analytic-Splatting: Anti-Aliased 3D Gaussian Splatting via Analytic Integration

Abstract

3DGS gained its popularity recently by combining the advantages of both primitive-based and volumetric 3D representations , resulting in improved quality and efficiency for 3D scene rendering. However, 3DGS is not alias-free , and its rendering at varying resolutions could producesevere blurring or jaggies. This is because 3DGS treats each pixel as an isolated , single point rather than as an area , causing insensitivity to changes in the footprints of pixels. Consequently, this discrete sampling scheme inevitably results in** aliasing**** ,** owing to the restricted sampling bandwidth.

In this paper, we derive an analytical solution to address this issue. More specifically, we use a conditioned logistic function as the analytic approximation of the cumulative distribution function (CDF) in a one-dimensional Gaussian signal andcalculate the Gaussian integral by subtracting the CDFs.

We then introduce this approximation in the two-dimensional pixel shading , and present Analytic-Splatting, which analytically approximates the Gaussian integral within the 2D-pixel window area to better capture the intensity response of each pixel.

Moreover, we use the approximated response of the pixel window integral area to participate in the transmittance calculation of volume rendering, making Analytic-Splatting sensitive to the changes in pixel footprint at different resolutions.

Experiments on various datasets validate that our approach has better anti-aliasing capability that gives more details and better fidelity.

Figure

Figure 1

For shading a pixel by a Gaussian signal, 3DGS only treats the Gaussian signal value corresponding to the pixel center as the intensity response.

Analytic-Splatting instead considers an analytic approximation of the integral over the pixel window area as the intensity response.

Compared to 3DGS, Analytic-Splatting has anti-aliasing capability and better detail fidelity.

Figure 2

Example diagram of the signal integration within window area and the approximation schemes used in different methods.

Figure 3

Example diagram of the pixel integration domain and the domain after rotation.

(a) the yellow lines are thecoordinate axes of 2D screen space ;

(b) the yellow lines are theeigenvectors scaled by the eigenvalues.

Figure 4

Error Analysis of using our conditioned logistic function to approximate CDF of Gaussian signals and window integration. Note that the scaling factor of the error is .

Figure 5

Error Analysis of approximating the window integral using different schemes.

Figure 6

Qualitative comparison of full-resolution and low-resolution (1/8) on Multi-Scale Blender.

All methods are trained on images with downsampling rates covering**[1, 2, 4, 8]**. Our method can better overcome the artifacts in 3DGS with better fidelity of details.

Figure 7

Qualitative comparisons of full-resolution and low-resolution on Multi-Scale Mip-NeRF 360.

All methods are trained on images with downsampling rates covering [1, 2, 4, 8]. Our method can better overcome the artifacts with better fidelity of details.

Figure 8

Qualitative comparison of super-resolution (2×) on Multi-Scale MipNeRF 360.

All methods are trained on images with downsampling rates covering [1, 2, 4, 8].

Conclusion

In this paper, we first revisit the window response of one-dimensional Gaussian signals and reason about an analytical and accurate approximation using a conditioned logistic function.

We then introduce this approximation in the two-dimensional pixel shading and present Analytic-Splatting, which approximates the pixel area integral response to achieve anti-aliasing capability and better detail fidelity.

Our extensive experiments demonstrate the efficacy of AnalyticSplatting in achieving state-of-the-art novel view synthesis results under multiscale and super-resolution settings.

Limitations

Compared with 3DGS and Mip-Splatting, our shading implementation introducesmore root and exponential operations , which inevitably increases the computational burden and reduces the frame rate. Despite this, our frame rate is only 10% lower than Mip-Splatting, which is also an anti-aliasing approach.