Some localization performance (distance) measures for binary images.
locperf(X, Y, which.stats = c("qdmapdiff", "med", "msd", "ph", "fom", "minsep"),
alpha = 0.1, k = 4, distfun = "distmapfun", a=NULL, ...)distob(X, Y, distfun = "distmapfun", ...)
distmapfun(x, ...)
locperf returns a list object with components depending on which.stats: one or more of the following, each of which is a single numeric, except as indicated.
matrix or numeric depending on p and number of thresholds.
numeric giving the Hausdorff distances for each threshold.
matrix or numeric, depending on k and number of thresholds, giving the value of the quantile (or k-th highest value) of the difference in distance maps for each threshold.
numeric giving the value of the mean error/square error distance for each threshold.
matrix or numeric, depending on alpha and number of thresholds, giving the value of Pratt his Figure of Merit for each threshold.
numeric giving the value of the minimum boundary separation distance for each threshold.
distob returns a single numeric.
distmapfun returns a matrix of same dimension as the input argument's field.
list object giving a pixel image as output from solutionset from package spatstat. This corresponds to the set B in the Details section below.
list object giving a pixel image as output from solutionset from package spatstat. This corresponds to the set A in the Details section below.
list object of class “owin” as returned by solutionset from package spatstat.
character vector stating which localization performance measure to calculate.
numeric giving the scaling constant for Pratt's figure of merit (FOM). Only used for which.stat method “fom”.
single numeric giving the order for the rank/quantile of the difference in distance maps. If 0 <= k < 1, this is assumed to be a quantile for use with the quantile function. Otherwise, k should be a whole number such that 1 <= k <= Nxy, where nxy is the total number of grid points in the set.
character specifying a distance metric that returns a matrix of same dimension as X yielding, at each point x, the shortest distances from x to the set of events in the field. Default is distmapfun, which returns the Euclidean distances.
Not used. For compatibility with locmeasures2d.
Optional arguments to the distfun function. In the case of distmapfun, these are the optional arguments to distmap from package spatstat.
Eric Gilleland
This function computes localization performance (or distance) measures detailed in Peli and Malah (1982) and Baddeley (1992), as well as a modification of one of these distances detailed in Zhu et al. (2011); distob.
First, it is helpful to establish some notation. Suppose a distance rho(x,y) is defined between any two pixels x and y in the entire raster of pixels/grid (If distfun is distmapfun (default), then rho is the Euclidean distance) that satisfies the formal mathematical axioms of a metric. Let d(x,A) denote the shortest distance (smallest value of rho) from the point x in the entire raster to the the set A contained in the raster. That is, d(x,A) = min(rho(x,a): a in A contained in the raster) [formally, the minimum should be the infimum], with d(x, empty set) defined to be infinity. Note that the distfun argument is a function that returns d(x,A) for all x in the raster.
The mean error distance (“med”) is the mean of d(x,A) over the points in B. That is e.bar = mean( d(x,A)), over all x in B. Because it is not symmetric (i.e., MED(A, B) != MED(B, A)), it is given as medMiss = MED(Forecast, Observation) and medFalseAlarm = MED(Observation, Forecast).
The mean square error distance (“msd”) is the mean of the squared d(x,A) over the points in B. That is, e2.bar = mean( d(x,A)^2), over all x in B. Similarly to MED, it is given as msdMiss or msdFalseAlarm.
Pratt's figure of merit (“fom”) is given by: FOM(A,B) = sum( 1/(1+alpha*d(x,A)^2))/max(N(A),N(B)), where x in B, and N(A) (N(B)) is the number of points in the set A (B) and alpha is a scaling constant (see, e.g., Pratt, 1977; Abdou and Pratt, 1979). The scaling constant is typically set to 1/9 when rho is normalized so that the smallest nonzero distance between pixel neighbors is 1. The default (0.1) here is approximately 1/9. If both A and B are empty, the value returned for max(N(A), N(B)) is 1e16 and for d(x,A) for x in B is given a value of zero so that the returned value should be close to zero.
Minimum separation distance between boundaries (“minsep”) is just the smallest value of the distance map of one field over the subset where events occur in the other. This is mainly for when single features within the fields are being compared.
distob is a modification of the mean error distance where if there are no events in either field, the value is 0, and if there are no events in one field only, the value is something large (in this case the length of the longest side of the grid).
The Hausdorff distance for a finite grid is given by max( max( d(x,B); x in A), max( d(x,A); x in B)), and can be written as max( abs(d(x,A) - d(x,B)), over all x in the raster). The quantile of the difference in distance mapse (“qdmapdiff”) is also potentially useful, and replaces the maximum in the latter equation with a k-th order statistic (or quantile). The modified Hausdorff distance is no longer given from this function, but can easily be computed using output from this function as it is given by mhd(A,B) = max( e.bar(A,B), e.bar(B,A)), and in some literature the maximum is replaced by the minimum. See, e.g., Baddeley, (1992) and Schwedler and Baldwin (2011).
For computational efficiency, the distance transform method is used via distmap from package spatstat for calculating d(x,A) x in the raster.
Abdou, I. E. and Pratt, W. K. (1979) Quantitative design and evaluation of enhancement/thresholding edge detectors. Proc. IEEE, 67, 753--763.
Baddeley, A. (1992) An error metric for binary images. In Robust Computer Vision Algorithms, W. Forstner and S. Ruwiedel, Eds., Wichmann, 59--78.
Peli, T. and Malah, D. (1982) A study on edge detection algorithms. Computer Graphics and Image Processing, 20, 1--21.
Pratt, W. K. (1977) Digital Image Processing. John Wiley and Sons, New York.
Schwedler, B. R. J. and Baldwin, M. E. (2011) Diagnosing the sensitivity of binary image measures to bias, location, and event frequency within a forecast verification framework. Wea. Forecasting, 26, 1032--1044, doi:10.1175/WAF-D-11-00032.1.
Zhu, M., Lakshmanan, V. Zhang, P. Hong, Y. Cheng, K. and Chen, S. (2011) Spatial verification using a true metric. Atmos. Res., 102, 408--419, doi:10.1016/j.atmosres.2011.09.004.
x <- y <- matrix( 0, 10, 12)
x[2,3] <- 1
y[4,7] <- 1
x <- im( x)
y <- im( y)
x <- solutionset( x > 0)
y <- solutionset( y > 0)
locperf( x, y)
# Note that ph is NA because there is only 1 event.
# need to have at least k events if k > 1.
par( mfrow=c(1,2))
image.plot( distmapfun(x))
image.plot( distmapfun(y))
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