.PLY File Format – Splat properties

The vertex record

All properties are float (IEEE 754 single precision, 4 bytes). The offsets are for spherical harmonics of degree 3; for degree d there are 3 × ((d + 1)² − 1) f_rest properties: 0, 9, 24 or 45.

The file stores the values the optimiser works on, before the functions that keep them in range. The code names the functions in setup_functions: torch.exp for the scales, torch.sigmoid for the opacity and normalize for the rotation.

Position

x, y and z are the centre of the Gaussian, in the coordinate system of the camera poses the scene was trained from. Scenes made with COLMAP, the usual source of poses, have Y pointing down, which is why many viewers turn a scene over when they load it.

Base colour: f_dc

The colour of a splat is stored as spherical harmonics coefficients, one set for each of red, green and blue. f_dc_0, f_dc_1 and f_dc_2 are the degree 0 coefficients of red, green and blue. The colour seen from any direction starts from

colour = 0.5 + SH_C0 × f_dc          SH_C0 = 0.28209479177387814 (1 / (2√π))

The renderer adds the higher terms below, then clamps each channel at 0 from below.

Spherical harmonics: f_rest

The f_rest properties hold the coefficients above degree 0, for degree 1 (3 a channel), 2 (5 more) and 3 (7 more): K = 3, 8 or 15 a channel. The order is channel by channel: first all K coefficients of red, then those of green, then blue. Property f_rest_n is coefficient n mod K + 1 of channel ⌊n / K⌋.

The loader checks that the count is 3 × ((d + 1)² − 1) for the degree it was told to use, sorts the properties by their number, and reshapes them to three rows of K.

For a viewing direction (x, y, z), the unit vector from the camera to the splat’s centre, the renderer adds each coefficient times its basis function, in this order:

C1 is 0.4886025. These are the real spherical harmonics constants the Inria rasteriser uses, rounded here to seven places.

Opacity

opacity is a logit: the opacity is its sigmoid, 1 / (1 + e^−v), from 0 (invisible) to 1. A value of 0 means an opacity of 0.5; −5 is about 0.007. The training code resets opacities to a low value from time to time and removes splats that stay nearly transparent.

Scale

scale_0, scale_1 and scale_2 are the natural logarithms of the Gaussian’s standard deviations along its own three axes, before the rotation. The scale along axis i is e^scale_i. Typical values are negative: −5 is a scale of about 0.0067 scene units.

Rotation

rot_0 to rot_3 are a quaternion with the real part first: w, x, y, z. They are not stored at unit length, so a reader divides them by √(w² + x² + y² + z²) before use. A Gaussian with no rotation has rot_0 = 1 and the others 0.

The rotation matrix R of the normalised quaternion and the diagonal matrix S of the three scales give the Gaussian’s covariance:

Σ = R S Sᵀ Rᵀ

R = | 1 − 2(y² + z²)   2(xy − wz)       2(xz + wy)     |
    | 2(xy + wz)       1 − 2(x² + z²)   2(yz − wx)     |
    | 2(xz − wy)       2(yz + wx)       1 − 2(x² + y²) |

To draw a splat, a renderer projects the covariance onto the screen, which gives an ellipse, and blends the splats sorted by depth, each with an alpha of its opacity times the Gaussian’s value at the pixel.

A complete header

ply
format binary_little_endian 1.0
element vertex 1063091
property float x
property float y
property float z
property float nx
property float ny
property float nz
property float f_dc_0
property float f_dc_1
property float f_dc_2
property float f_rest_0
…
property float f_rest_44
property float opacity
property float scale_0
property float scale_1
property float scale_2
property float rot_0
property float rot_1
property float rot_2
property float rot_3
end_header