Signed distance function
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.
If (Ω, d) is a metric space, the signed distance function f is defined by
and 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 Software) 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 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.
- 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.
- 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.)
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