  Semantic segmentation consists of predicting a semantic label for each image
pixel. Conventional deep learning models do not take advantage of ordinal
relations that might exist in the domain at hand. For example, it is known that
the pupil is inside the iris, and the lane markings are inside the road. Such
domain knowledge can be employed as constraints to make the model more robust.
The current literature on this topic has explored pixel-wise ordinal
segmentation methods, which treat each pixel as an independent observation and
promote ordinality in its representation. This paper proposes novel spatial
ordinal segmentation methods, which take advantage of the structured image
space by considering each pixel as an observation dependent on its neighborhood
context to also promote ordinal spatial consistency. When evaluated with five
biomedical datasets and multiple configurations of autonomous driving datasets,
ordinal methods resulted in more ordinally-consistent models, with substantial
improvements in ordinal metrics and some increase in the Dice coefficient. It
was also shown that the incorporation of ordinal consistency results in models
with better generalization abilities.
