Ordering Partial Partitions for Image Segmentation and Filtering: Merging, Creating and Inflating Blocks

Abstract

The refinement order on partitions corresponds to the operation of merging blocks in a partition, which is relevant to image segmentation and filtering methods. Its mathematical extension to partial partitions, that we call standard order, involves several operations, not only merging, but also creating new blocks or inflating existing ones, which are equally relevant to image segmentation and filtering techniques. These three operations correspond to three basic partial orders on partial partitions, the merging, inclusion and inflating orders. There are three possible combinations of these three basic orders, one of them is the standard order, the other two are the merging-inflating and inclusion-inflating orders. We study these orders in detail, giving in particular their minimal and maximal elements, covering relations and height functions. We interpret hierarchies of partitions and partial partitions in terms of an adjunction between (partial) partitions (possibly with connected blocks) and scalars. This gives a lattice-theoretical interpretation of edge saliency, hence a typology for the edges in partial partitions. The use of hierarchies in image filtering, in particular with component trees, is also discussed. Finally, we briefly mention further orders on partial partitions that can be useful for image segmentation.

Appendix:  Local Knowledge: Truncation and Restriction

We introduced numerous binary relations on Π ^∗(E), most of them being partial order relations, cf. Table 1. We will now consider how such relations are preserved by the following two operations on partial partitions, for any $A \in\mathcal{P}(E)$: truncation by A:

$$ \pi\mapsto\pi\wedge\mathbf{1}_A = \{ B \cap A \mid B \in\pi, \ B \cap A \ne\emptyset\} , $$

(42)

and restriction to A:

$$ \pi\mapsto\pi\cap\mathcal{P}(A) = \{ B \in\pi\mid B \subseteq A \} . $$

(43)

These operations are relevant to the problem of local knowledge, what Serra [32] calls class permanency: given a partial partition π and a restricted window A, viewing π through A means either truncating all blocks of π, that is, taking π∧1 _A , or restricting π to blocks inside A, that is, taking $\pi\cap\mathcal{P}(A)$; then it becomes interesting to know if these two operations of truncation and restriction preserve a given order on partial partitions.

We first consider compatibility with truncation. The following relations are preserved by truncation by A:

• The support inclusion, support containment and support equality relations, since supp(π _i ∧1 _A )=supp(π _i )∩A.

• The standard order: standard lattice theory gives π ₁≤π ₂ ⇒ π ₁∧1 _A ≤π ₂∧1 _A .

• The merging order, since it is the intersection of the standard order and the support equality relation.

• The inclusion order: for π ₁⊆π ₂, the blocks of π ₁∧1 _A are all non-void B∩A for B∈π ₁, so they belong to π ₂∧1 _A .

• The inclusion-inflating order, that is, the intersection of the standard order and of the singularity relation. Indeed, let π ₁≤π ₂ and π ₁⇚π ₂. Then π ₁∧1 _A ≤π ₂∧1 _A . The blocks of π ₁∧1 _A (resp., π ₂∧1 _A ) are the non-void B∩A for B∈π ₁ (resp., for B∈π ₂). If C∩A (C∈π ₂) contains B ₁∩A and B ₂∩A (B ₁,B ₂∈π ₁), then B ₁ and B ₂ intersect C, and as π ₁≤π ₂, B ₁,B ₂⊆C, but as π ₁⇚π ₂, we deduce that B ₁=B ₂, so B ₁∩A=B ₂∩A. Thus π ₁∧1 _A ⇚π ₂∧1 _A .

The following relations are not preserved by truncation by A:

• The singularity relation: take π ₁={B,C} and π ₂={A}, where B≬A≬C but B,C⊈A, then π ₁⇚π ₂ but $\pi_{1} \wedge\mathbf{1}_{A} = \{ A \cap B, A \cap C \} \not\Lleftarrow\pi_{2} = \pi_{2} \wedge\mathbf{1}_{A}$.

• The building order: take B⊃A⊃∅, π ₁=1 _B∖A and π ₂=1 _B , then π ₁⋐π ₂ but .

• The inflating order: take the counterexample given for the building order.

• The merging-inflating order: take the same counterexample.

We now consider compatibility with restriction. The following relations are preserved by restriction to A:

• The singularity relation: this is a special case of (12).

• The building order, see (7).

• The inclusion order: standard set theory gives $\pi_{1} \subseteq \pi_{2} \ \Rightarrow\ \pi_{1} \cap\mathcal{P}(A) \subseteq\pi_{2} \cap\mathcal{P}(A)$.

The following relations are not preserved by restriction to A:

• The support inclusion, support containment and support equality relations; indeed, the support does not tell anything about the existence of blocks included in A.

• The standard order.

• The merging order.

• The inflating order.

• The merging-inflating order.

• The inclusion-inflating order.

In fact, any order included in the standard order, that is compatible with restriction, must be included in the inclusion order. Indeed, if π ₁≤π ₂ but π ₁⊈π ₂, then there is B∈π ₁ and C∈π ₂ such that B⊂C, so $B \in\pi_{1} \cap\mathcal{P}(B)$ but $\pi_{2} \cap \mathcal{P}(B) \subseteq\pi_{2} \setminus\{C\}$, and B is not included in a block of π ₂∖{C}, thus $\pi_{1} \cap\mathcal{P}(B) \not\le \pi_{2} \cap\mathcal{P}(B)$.