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Rand error

The Rand index is a well-known measure of the similarity between two data clusterings[1]. Recently, it has been proposed as a measure of segmentation performance, since a segmentation can be regarded as a clustering of pixels[2]. More formally, define a segmentation as an integer-valued labeling of an image. Each object in a segmentation consists of a set of pixels sharing a common label.

The Rand index is defined as a measure of agreement:

Given two segmentations \(S_1\) and \(S_2\) of an image \(I\) with \(n\) pixels, we define:

  • \(a\), the number of pairs of pixels in \(I\) that are in the same object in \(S_1\) and in the same object in \(S_2\) (i.e., they have the same label)
  • \(b\), the number of pairs of pixels in \(I\) that are in different objects in \(S_1\) and in different objects in \(S_2\) (i.e., they have different labels)

The Rand index, \(RI\), is: \(RI = \frac{a+b}{n \choose 2 }\)

Here we instead define the closely related Rand error, which is a measure of disagreement. The Rand error (RE) is the frequency with which the two segmentations disagree over whether a pair of pixels belongs to same or different objects:

\[RE = 1 - RI\]

Implementation in Fiji

The Rand error metric is implemented in the Trainable Weka Segmentation library. Here is an example of how to use it in a Beanshell script:

import trainableSegmentation.metrics.RandError;
import ij.IJ;

// original labels
originalLabels = IJ.openImage("/path/original-labels.tif");

// proposed (new) labels
proposedLabels = IJ.openImage("/path/proposed-labels.tif");

// threshold to binarize labels
threshold = 0.5;

metric = new RandError( originalLabels, proposedLabels );
randError = metric.getMetricValue( threshold );

IJ.log("Rand error between source image " + originalLabels.getTitle() + " and target image "
+ proposedLabels.getTitle() + " = " + randError);

See also

References

[1]

William M. Rand (1971), “Objective criteria for the evaluation of clustering methods”, Journal of the American Statistical Association 66: 846–850 , doi:10.2307/2284239.

[2]

R. Unnikrishnan, C. Pantofaru, and M. Hebert (2007), “Toward objective evaluation of image segmentation algorithms”, IEEE Transactions on Pattern Analysis and Machine Intelligence 29: 929-944 , doi:10.1109/TPAMI.2007.1046.