# Dual-Dimensional Gaussian Splatting Integrating 2D and 3D Gaussians for Surface Reconstruction
## 1 Introduction  
Novel view synthesis (NVS) and surface reconstruction from multi-view images are fundamental research fields in computer vision. These techniques aim to render photorealistic images from novel viewpoints or reconstruct accurate 3D meshes. They are essential for applications including 3D content creation [1, 2], autonomous driving [3, 4], virtual reality [5, 6], and the metaverse [7, 8].  
Recently, Three-Dimensional Gaussian Splatting (3 DGS) [9] has been proposed to represent a scene as a collection of 3D Gaussians, enabling efficient and high-quality real-time rendering. This is enabled by the differentiable nature of 3D Gaussians, along with efficient tile-based rasterization and optimization techniques. However, their volumetric nature makes it difficult to represent flat surfaces accurately, limiting their effectiveness in surface reconstruction. Moreover, affine transformations induced by each Gaussian's covariance matrix can introduce geometric distortions that lead to projection errors.  
Two-Dimensional Gaussian Splatting (2 DGS) [10] replaces ellipsoid-shaped 3D Gaussians with oriented, elliptical disk-shaped 2D Gaussians, enhancing geometric detail in surface reconstruction. In addition, 2 DGS replaces the covariance matrix with a homogeneous transformation matrix, allowing precise 2D-to-2D mapping. Depth maps are estimated to facilitate mesh extraction through truncated signed distance fusion (TSDF) using Open3D [11]. This depth information functions as a regularization term, encouraging the concentration of 2D Gaussians and aligning rendered normal vectors to achievesmoother surface reconstruction. However, these regularization constraints often lead to overly smooth surface reconstruction [10], further exacerbated by the limited capacity of 2D Gaussians to capture volumetric details.  
In this paper, we introduce Dual-Dimensional Gaussian Splatting (DDGS), a novel hybrid representation that strategically integrates the complementary advantages of 2D and 3D Gaussians, to enhance flexibility and accuracy in surface reconstruction. Specifically, volumetric structures are modeled using 3D Gaussians, while flat surfaces are represented by 2D Gaussians, allowing adaptive selection of the optimal representation for different regions within a scene. This approach improves geometric accuracy across diverse environments by leveraging the complementary strengths of both representations.  
Furthermore, we propose gradient-based regularization terms to improve mesh quality. Since the normal consistency term in 2 DGS leads to excessively smoothed surfaces, we modulate it using a gradient-aware approach. This term aligns the rendered normal map with the normal map generated from the depth map. However, sharp depth changes can introduce errors, resulting in over-smoothed surfaces. To address this, we introduce a gradient-aware normal consistency term that applies stronger penalties in proportion to image gradients. Additionally, we incorporate regularization terms that minimize both the gradient and second-order gradient differences between the rendered and ground truth images, encouraging Gaussians to better fit the object surfaces.  
In summary, the contributions of this paper are as follows:  
* We propose DDGS, which integrates both 2D and 3D Gaussians to enable flexible and precise surface reconstruction by adaptively selecting the optimal Gaussian representation based on the characteristics of the object surfaces.
* We introduce gradient-based regularization terms to improve the alignment accuracy between rendered Gaussian normals and the underlying ground truth surfaces, thus enhancing geometric fidelity.