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);