Signed Distance Functions (SDF)

The signed distance function

The signed distance function (SDF) is a lesser-known but elegant 3D data representation. While PyTorch3D doesn’t directly implement signed distance functions, they’re becoming more popular due to a few interesting properties.

What is a signed distance function?

A signed distance function (SDF), is a mathematical expression of a 3D scene that computes the distance $d$ between a given point $p$ and a surface in a given space. The SDF is a function that takes a position $p$ as an input and returns a scalar value $d$, with positive values representing distances outside the surface and negative values representing distances inside the function. This is where it gets the name signed distance function.

Where:

• $d < 0$ for points inside of the surface.

• $d = 0$ for points along the surface.

• $d > 0$ for points outside of the surface.

Many primitive shapes can be defined with closed-form mathematical expressions as an SDF. For example, the SDF of a sphere of radius $R$ centered at point $(0, 0, 0)$ can be expressed as:

where ...