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**:

## 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:<a href={{doi}}></a>

[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:<a href={{doi}}></a>