Signed distance functions and implicit modelling explained

Conventional implicit modelling: RBF or kriging

To model a geological feature, data points get either distance values or indicator values. Points with a value of zero lie on the boundary, positive values on one side and negative on the other. Fitting an RBF works much like kriging: it estimates the distance to the boundary at points where it was not measured. An estimate of the distance to a boundary is, by definition, a signed distance function.

Using an SDF is implicit modelling

Seen this way, implicit modelling is a way to estimate distances from a boundary. For grade data you apply a threshold: values below the cut-off grade become negative and values above become positive. This is similar to indicator modelling but keeps the original values instead of converting them to indicators.

One caution: methods built for boundaries suit numeric interpolation, but the reverse is not true. Grade variation near a boundary is typically non-linear and differs from area to area.

Why it matters: speed

Points far from a boundary are largely redundant when estimating that boundary. Two points on opposite sides, equally far from a plane, define it precisely, and further distant points add nothing. Machine learning methods use this to cut the work. The source article reports speed-ups of about 100 times compared with conventional RBF interpolation and about 10 times compared with regression methods.

Distance values also give flexibility. A numeric problem can be turned into a signed distance problem with a threshold, and the boundary can then be moved by changing the threshold, without moving any data points. The value returned by the SDF determines where the boundary is generated.

A simple SDF: topography

A function that returns the elevation of a point relative to the topography is an SDF: positive above the surface, negative below, and zero on the surface. The same function can be locally modified, for example by adding building heights to the topography. Where a building stands, the function returns the roof elevation instead of the ground, which is the same as a Boolean operation in computational geometry.

Combining SDFs

Continuous SDFs handle gradual change well but struggle with sharp breaks. The fixed threshold, however, can be replaced by the values of another function. A fault plane can split a topography function so that each side has its own topography, and the fault itself is included, without building triangulated surfaces. Edits made with polylines or points can adjust an SDF too. The process becomes an exercise in adjusting function values, globally or locally, without committing to a full model at every step.

The same idea explains how modern graphics cards draw objects without triangles, using ray tracing against functions.

Further reading

• Jun Cowan (2011), ‘Wireframe-free Geological Modelling – An Oxymoron or a Value Proposition?’, AusIMM Publication Series.
• Inigo Quilez, work on signed distance functions at iquilezles.org.
• GEOREKA’s version 4 framework draws on these ideas and on Horizon 2020 ROBOMINERS research.