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The time and space complexities of Markov random field (MRF) algorithms for image segmentation increase with the number of edges that represent statistical dependencies between adjacent pixels. This has made MRFs too computationally complex for cutting-edge applications such as joint segmentation of longitudinal sequences of many high-resolution magnetic resonance images (MRIs). Here, we show that simply removing edges from full MRFs can reduce the computational complexity of MRF parameter estimation and inference with no notable decrease in segmentation performance. In particular, we show that for segmentation of white matter hyperintensities in 88 brain MRI scans of elderly individuals, as many as 66% of MRF edges can be removed without substantially degrading segmentation accuracy. We then show that removing edges from MRFs makes MRF parameter estimation and inference computationally tractable enough to enable modeling statistical dependencies within and across a larger number of brain MRI scans in a longitudinal series; this improves segmentation performance compared to separate segmentations of each individual scan in the series.