Delta Metric
Computes the discrepancy between two sets A and B according to Baddeley's delta-metric.
deltametric(A, B, p = 2, c = Inf, ...)
A,B |
The two sets which will be compared.
Windows (objects of class |
p |
Index of the L^p metric.
Either a positive numeric value, or |
c |
Distance threshold.
Either a positive numeric value, or |
... |
Arguments passed to |
Baddeley (1992a, 1992b) defined a distance between two sets A and B contained in a space W by
Δ(A,B) = [ (1/|W|) * integral of |min(c, d(x,A))-min(c, d(x,B))|^p dx ]^(1/p)
where c ≥ 0 is a distance threshold parameter,
0 < p ≤ Inf is the exponent parameter,
and d(x,A) denotes the
shortest distance from a point x to the set A.
Also |W| denotes the area or volume of the containing space W.
This is defined so that it is a metric, i.e.
Δ(A,B)=0 if and only if A=B
Δ(A,B)=Δ(B,A)
Δ(A,C) ≤ Δ(A,B) + Δ(B,C)
It is topologically equivalent to the Hausdorff metric (Baddeley, 1992a) but has better stability properties in practical applications (Baddeley, 1992b).
If p=Inf and c=Inf the Delta metric is equal to the Hausdorff metric.
A numeric value.
Adrian Baddeley Adrian.Baddeley@curtin.edu.au
and Rolf Turner r.turner@auckland.ac.nz
Baddeley, A.J. (1992a) Errors in binary images and an L^p version of the Hausdorff metric. Nieuw Archief voor Wiskunde 10, 157–183.
Baddeley, A.J. (1992b) An error metric for binary images. In W. Foerstner and S. Ruwiedel (eds) Robust Computer Vision. Karlsruhe: Wichmann. Pages 59–78.
X <- runifrect(20) Y <- runifrect(10) deltametric(X, Y, p=1,c=0.1)
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