huberFilter

smooth grain boundaries but keep genuine corners sharp

Description

Same variational idea as curvatureFilter, but the curvature is penalized by a Huber function instead of a square

|V - V0|^2 + alpha * sum rho(|L V|)

with rho quadratic below a threshold and linear above it. Quadratic means least squares, which spreads a large deviation over many vertices and so rounds a corner off. Linear - an l^1 penalty - is minimized by concentrating the deviation in as few vertices as possible, which leaves a corner standing.

So the filter smooths gentle undulations exactly like curvatureFilter, while a boundary that is genuinely faceted keeps its facets and its corners. The threshold says how sharp a turn has to be to count as a corner rather than as noise.

The Huber problem is solved by iteratively reweighted least squares: the l^1 part is replaced by a quadratic with weight threshold/|L V, the resulting linear system is solved, and the weights are recomputed. This is an iteration, but it runs until it stops moving - |iterMax is a safety limit, not a tuning knob.

threshold is the turning angle at a single vertex of the result, not the total angle of the corner. The smoothing spreads a corner over roughly smoothingLength/h vertices, and the reweighting then sharpens it back, so the total angle that survives comes out several times larger than the threshold. On a hexagon smoothed with smoothingLength = 8*h the 60 degree corners come back as

threshold    30    15     8     4     2   degree
corner       22    28    49    68    80   degree

A boundary with no corner at all is unaffected: a circle of radius R turns by h/R per vertex, far below any sensible threshold, so the filter returns exactly what curvatureFilter would.

Syntax

grains = smoothBoundary(grains,huberFilter)

F = huberFilter;
F.smoothingLength = 3;
F.threshold = 3*degree;
grains = smoothBoundary(grains,F)

Class Properties

See also

grain2d.smoothBoundary boundaryFilter curvatureFilter taubinFilter