2D Gaussian Splattering

The splattering of 2D Gaussians can be seen as a simpler alternative to 3D Gaussians, and from a historical perspective, they had already been introduced earlier.

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

The orientation of the 2D Gaussian can be organized in a rotation matrix $3 \times 3$ (parameterizable as a classical rotation in 3D) after defining $t_w = t_u \times t_v$ and the scaling factors in a diagonal matrix $\mathbf{S}=\diag (s_u, s_v, 0)$.

The 2D Gaussian is therefore defined on a tangent plane described by the equation

$$P(u,v) = \mathbf{p}_k + s_u \mathbf{t}_u u + s_v \mathbf{t}_v v = \mathbf{H} (u,v,1,1)^{\top}$$ (9.101)

after defining the homographic matrix $\mathbf{H}=\begin{bmatrix} s_u \mathbf{t}_u & s_v \mathbf{t}_v & 0 & \mathbf{p... ...begin{bmatrix} \mathbf{R} \mathbf{S} & \mathbf{p}_k \\ 0 & 1 \end{bmatrix}$. To each point $(u,v)$ in the plane coordinates, there is clearly associated a Gaussian described by the equation $e^{\frac{u^2 + v^2}{2}}$.