The digital morphological sampling theorem

Morphological sampling reduces processing time and cost and yet produces results sufficiently close to the result of full processing. A morphological sampling theorem is described which states: (1) how a digital image must be morphologically filtered before sampling in order to preserve the relevant information after sampling; (2) to what precision an appropriate morphologically filtered image can be reconstructed after sampling; and (3) the relationship between morphologically operating before sampling and the more computationally efficient scheme of morphologically operating on the sampled image with a sampled structuring element. The digital sampling theorem is developed first for the case of binary morphology, and then it is extended to gray-scale morphology through the use of the umbra homomorphism theorems