Orders on Partial Partitions Based on Block Apportioning

Abstract

Some problems in image analysis require considering an order on space partitions: in image filtering the partition of image zones must grow, while in image segmentation one maximizes the partition into image objects.In the classical refinement order, the growth of a partition is achieved through the merging of blocks. We generalize this operation into block apportioning: some blocks in a partial partition may be split, and their parts are merged with some remaining blocks; this includes the possibility of a block being merged with another without being split. We study this operation in detail; we obtain then the apportioning order, which extends the refinement order.For partial partitions, the standard order combines the merging of blocks with the growth of existing blocks and the creation of new blocks. Combining these operations with block apportioning leads to the apportioning-inflating and extended orders; the latter contains the standard order. Our analysis rests on a study of Serra’s building order on partial partitions, which intervenes in the above new orders.Except the building order, all these orders are graded, with a compound grading for the apportioning-inflating and extended orders; this gives the number of elementary operations involved in a growth of a partial partition.