2D Gaussian Splatting

2D Gaussian Splatting can be viewed as a simpler alternative to 3D Gaussians and, historically, 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 scale factor $\mathbf{S}=(s_u,s_v)$ that controls the Gaussian's variance in two dimensions.

The orientation of the 2D Gaussian can be organized in a rotation matrix $3 \times 3$ $\mathbf{R} = [t_u, t_v, t_w]$, parameterizable as a standard 3D rotation, by defining $t_w = t_u \times t_v$ and the scale factors 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 $(u,v)$:

$$P(u,v) = \mathbf{p}_k + s_u\mathbf{t}_u u + s_v\mathbf{t}_v v = \mathbf{H} \begin{bmatrix} u\\ v\\ 1 \end{bmatrix},$$ (10.120)

and

$$\mathbf{H} = \begin{bmatrix} s_u\mathbf{t}_u & s_v\mathbf{t}_v & \mathbf{p}_k \end{bmatrix}.$$ (10.121)

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

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