Signed distance function
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. However, the alternative convention is also sometimes taken instead (i.e., negative inside Ω and positive outside).
If Ω is a subset of a metric space, X, with metric, d, then the signed distance function, f, is defined by
where denotes the boundary of . For any ,
where inf denotes the infimum.
Properties in Euclidean space
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
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. 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.
- Chan, T.; Zhu, W. (2005). Level set based shape prior segmentation. IEEE Computer Society Conference on Computer Vision and Pattern Recognition.
- 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).
- 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: 9. doi:10.1145/1281500.1281665. ISBN 9781450318235.
- 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.)