Suvarna Garge (Editor)

Morphological skeleton

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In digital image processing, morphological skeleton is a skeleton (or medial axis) representation of a shape or binary image, computed by means of morphological operators.

Contents

Morphological skeletons are of two kinds:

  • Those defined by means of morphological openings, from which the original shape can be reconstructed,
  • Those computed by means of the hit-or-miss transform, which preserve the shape's topology.

    • Continuous images

    In (Lantuéjoul 1977), Lantuéjoul derived the following morphological formula for the skeleton of a continuous binary image X ⊂ R 2 :

    S ( X ) = ⋃ ρ > 0 ⋂ μ > 0 [ ( X ⊖ ρ B ) − ( X ⊖ ρ B ) ∘ μ B ¯ ] ,

    where ⊖ and ∘ are the morphological erosion and opening, respectively, ρ B is an open ball of radius ρ , and B ¯ is the closure of B .

    Discrete images

    Let { n B } , n = 0 , 1 , … , be a family of shapes, where B is a structuring element,

    n B = B ⊕ ⋯ ⊕ B ⏟ n  times , and 0 B = { o } , where o denotes the origin.

    The variable n is called the size of the structuring element.

    Lantuéjoul's formula has been discretized as follows. For a discrete binary image X ⊂ Z 2 , the skeleton S(X) is the union of the skeleton subsets { S n ( X ) } , n = 0 , 1 , … , N , where:

    S n ( X ) = ( X ⊖ n B ) − ( X ⊖ n B ) ∘ B .

    Reconstruction from the skeleton

    The original shape X can be reconstructed from the set of skeleton subsets { S n ( X ) } as follows:

    X = ⋃ n ( S n ( X ) ⊕ n B ) .

    Partial reconstructions can also be performed, leading to opened versions of the original shape:

    ⋃ n ≥ m ( S n ( X ) ⊕ n B ) = X ∘ m B .

    The skeleton as the centers of the maximal disks

    Let n B z be the translated version of n B to the point z, that is, n B z = { x ∈ E | x − z ∈ n B } .

    A shape n B z centered at z is called a maximal disk in a set A when:

  • n B z ∈ A , and
  • if, for some integer m and some point y, n B z ⊆ m B y , then m B y ⊈ A .

    • Each skeleton subset S n ( X ) consists of the centers of all maximal disks of size n.

    References

    Morphological skeleton Wikipedia
    (Text) CC BY-SA