Dilation as proximity: Gonzalez & Woods show that morphological dilation by a disk of radius r is equivalent to the set of all pixels within distance r of the object. Proximity Areas generalize this: a zone from distance r₁ to r₂ is the dilation by r₂ minus the dilation by r₁. But computing this via distance maps is more efficient — a single distance transform computation provides all possible zone boundaries simultaneously, rather than requiring separate dilation operations at each distance.
Neighborhood analysis (MIT Statistical Models): In spatial statistics, the behavior of a process often varies as a function of distance from a boundary or reference point. Proximity Areas operationalize this by creating discrete distance bins in which spatial statistics can be computed. The resulting distance-dependent profiles are the empirical analog of the theoretical spatial correlation functions used in geostatistics and spatial point process analysis.
The invasive margin — a biologically defined proximity zone: The tumor invasive margin is clinically defined as the region within ~500 µm of the tumor border. This is a proximity area by definition. The Immunoscore for colorectal cancer separately quantifies immune cell density in the tumor center and at the invasive margin — two proximity areas defined relative to the tumor boundary. The prognostic power of the Immunoscore validates that proximity-based spatial analysis captures biologically and clinically meaningful information.
Proximity Areas are generalized dilation — instead of expanding objects by one fixed distance, they create multiple concentric zones that partition the space around reference objects. This enables distance-dependent analysis: how immune cell density decreases with distance from the tumor, or how PD-L1 expression changes near blood vessels. The concept directly supports clinical spatial biomarkers like the Immunoscore.
