Signed distance function

A disk (top, in grey) and its signed distance function (bottom, in red). The x-y plane is shown in blue.

A more complicated set (top) and its signed distance function (bottom, in red).

In mathematics and its applications, the signed distance function (or oriented distance function) of a set Ω in a metric space 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.^[1] However, the alternative convention is also sometimes taken instead (i.e., negative inside Ω and positive outside).^[2]

Definition

If Ω is a subset of a metric space, X, with metric, d, then the signed distance function, f, is defined by

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

where $\partial \Omega$ denotes the boundary of $\Omega$. For any $x\in X$,

${\displaystyle d(x,\partial \Omega ):=\inf _{y\in \partial \Omega }d(x,y)}$

where 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 C^k for k≥2 (see differentiability classes) then d is C^k on points sufficiently close to the boundary of Ω.^[3] 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 W_x 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 dS_u indicates that we are taking the surface integral.^[4]

Algorithms

Algorithms for calculating the signed distance function include the efficient fast marching method , fast sweeping method^[5] 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 Corporation) to render smooth fonts at large sizes (or alternatively at high DPI) using GPU acceleration.^[6] Valve's method computed signed distance fields in raster space in order to avoid 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 Bézier 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.^[7]

See also

• Distance function
• Level set method
• Eikonal equation
• Parallel (aka offset) curve

Notes

1. Chan, T.; Zhu, W. (2005). Level set based shape prior segmentation. IEEE Computer Society Conference on Computer Vision and Pattern Recognition.
2. Malladi, R.; Sethian, J.A.; Vemuri, B.C. (1995). "Shape modeling with front propagation: a level set approach". IEEE Transactions on Pattern Analysis and Machine Intelligence. 17 (2).
3. Gilbarg 1983, Lemma 14.16.
4. Gilbarg 1983, Equation (14.98).
5. [ZHAO, Hongkai. A fast sweeping method for eikonal equations. Mathematics of computation, 2005, 74. Jg., Nr. 250, S. 603-627.
6. Green, Chris (2007). "Improved alpha-tested magnification for vector textures and special effects". ACM SIGGRAPH 2007 courses on - SIGGRAPH '07: 9. doi:10.1145/1281500.1281665. ISBN 9781450318235.
7. https://www.youtube.com/watch?v=7tHv6mcIIeo

References

• Stanley J. Osher and Ronald P. Fedkiw (2003). Level Set Methods and Dynamic Implicit Surfaces. Springer.
• Gilbarg, D.; Trudinger, N. S. (1983). Elliptic Partial Differential Equations of Second Order. Grundlehren der mathematischen Wissenschaften. 224 (2nd ed.). Springer-Verlag. (or the Appendix of the 1977 1st ed.)