2D Gaussian Splatting

Instead of representing the scene using three-dimensional volumetric Gaussians, it is possible to use Gaussian primitives defined on local two-dimensional elements. This formulation, known as 2D Gaussian Splatting, predates modern techniques based on three-dimensional Gaussians and represents a different trade-off between geometric modeling and computational cost.

2D Gaussians are represented by a center point $\mathbf{p}_k$, two unit tangent vectors ($\mathbf{t}_u$, $\mathbf{t}_v$), and a scale factor $\mathbf{S}=(s_u,s_v)$ that controls the two-dimensional variance of the Gaussian.

The orientation of the 2D Gaussian can be organized into a rotation matrix $3 \times 3$ of rotation $\mathbf{R} = [t_u, t_v, t_w]$ (parameterizable as a standard 3D rotation), with $t_w = t_u \times t_v$ defined and the scale factors arranged in a diagonal matrix $\mathbf{S}=\diag (s_u, s_v, 0)$.

The plane parameterization can be written as an affine transformation of the local coordinates $(u,v)$:

$$P(u,v) = \mathbf{p}_k + s_u\mathbf{t}_u u + s_v\mathbf{t}_v v = \mathbf{H} \begin{bmatrix} u\\ v\\ 1 \end{bmatrix},$$ (10.123)

and

$$\mathbf{H} = \begin{bmatrix} s_u\mathbf{t}_u & s_v\mathbf{t}_v & \mathbf{p}_k \end{bmatrix}.$$ (10.124)

where $\mathbf{H}$ is the matrix of the affine embedding.

Each point $(u,v)$ in plane coordinates is clearly associated with a Gaussian defined by the equation $e^{-\frac{u^2 + v^2}{2}}$.