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In mathematics and applications, the signed distance function of a set Ω in a metric space, also called the oriented distance function, determines the distance of a given point x from the boundary of Ω, with the sign determined by whether x is in Ω. The function has positive values at points x inside Ω, it decreases in value as x approaches the boundary of Ω where the signed distance function is zero, and it takes negative values outside of Ω.

Definition

If (Ω, d) is a metric space, the signed distance function f is defined by

$$f(x) = \begin{cases} d(x, \Omega^c) & \mbox{ if } x\in\Omega \\ -d(x, \Omega)& \mbox{ if } x\in\Omega^c \end{cases}$$

where

$$d(x, \Omega) = \inf_{y\in \Omega}d(x, y)$$

and inf denotes the infimum.
Properties in Euclidean space

If Ω is a subset of the Euclidean space Rn with piecewise smooth boundary, then the signed distance function is differentiable almost everywhere, and its gradient satisfies the eikonal equation

$$|\nabla f|=1.$$

If the boundary of Ω is Ck for k≥2 (see differentiability classes) then d is Ck on points sufficiently close to the boundary of Ω.[1] In particular, on the boundary f satisfies

$$\nabla f(x) = N(x),$$

where N is the inward normal vector field. The signed distance function is thus a differentiable extension of the normal vector field. In particular, the Hessian of the signed distance function on the boundary of Ω gives the Weingarten map.

If, further, Γ is a region sufficiently close to the boundary of Ω that f is twice continuously differentiable on it, then there is an explicit formula involving the Weingarten map Wx for the Jacobian of changing variables in terms of the signed distance function and nearest boundary point. Specifically, if T(∂Ω,μ) is the set of points within distance μ of the boundary of Ω (i.e. the tubular neighbourhood of radius μ), and g is an absolutely integrable function on Γ, then

$$\int_{T(\partial\Omega,\mu)} g(x)\,dx = \int_{\partial\Omega}\int_{-\mu}^\mu g(u+\lambda N(u))\, \det(I-\lambda W_u) \,d\lambda \,dS_u,$$

where det indicates the determinant and dSu indicates that we are taking the surface integral.[2]
Algorithms

Algorithms for calculating the signed distance function include the efficient fast marching method , fast sweeping method[3] and the more general level set method.

Applications

Signed distance functions are applied for example in computer vision.

They have also recently been used in a method (advanced by Valve Software) to render smooth fonts at large sizes (or alternatively at high DPI) using GPU acceleration.[4] Valve's method computed signed distance fields in raster space in order to avoid of the computational complexity of solving the problem in the (continuous) vector space. More recently piece-wise approximation solutions have been proposed (which for example approximate a Bezier with arc splines), but even this way the computation can be too slow for real-time rendering, and it has to be assisted by grid-based discretization techniques to approximate (and cull from the computation) the distance to points that are too far away.[5]

Level set method
Eikonal equation
Parallel (aka offset) curve

Notes

Gilbarg 1983, Lemma 14.16.
Gilbarg 1983, Equation (14.98).
[ZHAO, Hongkai. A fast sweeping method for eikonal equations. Mathematics of computation, 2005, 74. Jg., Nr. 250, S. 603-627.
Green, Chris (2007). "Improved alpha-tested magnification for vector textures and special effects". ACM SIGGRAPH 2007 courses on - SIGGRAPH '07. doi:10.1145/1281500.1281665. edit