  In the deep metric learning approach to image segmentation, a convolutional
net densely generates feature vectors at the pixels of an image. Pairs of
feature vectors are trained to be similar or different, depending on whether
the corresponding pixels belong to same or different ground truth segments. To
segment a new image, the feature vectors are computed and clustered. Both
empirically and theoretically, it is unclear whether or when deep metric
learning is superior to the more conventional approach of directly predicting
an affinity graph with a convolutional net. We compare the two approaches using
brain images from serial section electron microscopy images, which constitute
an especially challenging example of instance segmentation. We first show that
seed-based postprocessing of the feature vectors, as originally proposed,
produces inferior accuracy because it is difficult for the convolutional net to
predict feature vectors that remain uniform across large objects. Then we
consider postprocessing by thresholding a nearest neighbor graph followed by
connected components. In this case, segmentations from a "metric graph" turn
out to be competitive or even superior to segmentations from a directly
predicted affinity graph. To explain these findings theoretically, we invoke
the property that the metric function satisfies the triangle inequality. Then
we show with an example where this constraint suppresses noise, causing
connected components to more robustly segment a metric graph than an
unconstrained affinity graph.
