The location, size, and number of the regions of change are generally unknown. However, we might expect that these properties will remain fairly stable over a wide range of threshold values, whereas down at the noise level small changes in the threshold value can substantially alter the number of regions. Such an observation suggests that if a range of threshold values is found that leads to a stable number of regions, then these regions are unlikely to come from noise, and so a value from this range will provide a suitable threshold. This approach was suggested by O'Gorman [12] for intensity image thresholding (and was recently proposed again by Pikaz and Averbuch [13]), and was applied to difference images by Rosin and Ellis [15]. Rather than counting the number of regions the image's Euler number can be used, and was found to give almost identical results [15]. The advantage of calculating connectivity over region counting is that the Euler number is locally countable [5], and can be determined efficiently in a single raster scan of the image by just a few lines of code.
A stable threshold range will correspond to a plateau in the graph of the Euler numbers against thresholds. Initially we assumed that the plateaus were perfectly flat, and detected them by looking for the longest such range in the graph [15]. However, given the noisy, fragmented nature of images this was not found to be reliable as sometimes the Euler number varied slowly within the stable range. An alternative procedure that we have found more effective is to model the shape of the graph as a decaying exponential. At low threshold values there will be many regions and holes caused primarily by the noise, and the Euler number will change rapidly with threshold. At high threshold values there will be few regions, and the Euler number will be stable. (We only consider thresholds up until there are no regions remaining, and the Euler number becomes zero). Therefore a suitable partition point between the signal and noise is the ``corner'' of the curve. which we find as the point on the curve with maximum deviation from the straight line drawn between the end points of the curve.
If we count regions then the curve at low thresholds which is generated primarily by noise will appear roughly Gaussian [13] and the number of regions eventually drops to one at a threshold of zero. When using Euler numbers the shape is different since the number can become negative if there are more holes than connected components. To avoid difficulties for the corner detection we start the straight line from the first positive peak in the curve (see for example figure 1l).