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A well-known heuristic for segmenting an image into gray level subpopulations is to select thresholds at the bottoms of valleys on the image's histogram. When the subpopulations overlap, valleys may not exist, but it is often still possible to define good thresholds at the `shoulders' of histogram peaks. Both valleys and shoulders correspond to concavities on the histogram, and this suggests that it should be possible to find good candidate thresholds by analyzing the histogram's concavity structure. Histogram concavity analysis as an approach to threshold selection is investigated and its performance on a set of histograms of infrared images of tanks is illustrated.