Principle of Superposition

What does the superposition principle for binary data mean? For gray value images it is defi ned as

H (a G + b G ') = aH G + bH G '.                                  (18.7)

The factors a and b make no sense for binary images; the addition of images corresponds to the union or logical or of images. If the superpo- sition principle is valid for morphological operations on binary images, it has the form

M( G ∪ G ') = (M G ) ∪ (M G ') or M( G ∨ G ') = (M G ) ∨ (M G '). (18.8)

∨ '

The operation G G ' means a pointwise logical or of the elements of the matrices G and G . Generally, morphological operators are not additive in the sense of Eq. (18.8). While the dilation operation conforms to the superposition principle, erosion does not. The erosion of the union of two objects is generally a superset of the union of two eroded objects:

( G ∪ G ') ∅ M    ⊇ ( G ∅ M ) ∪ ( G ' ∅ M )

( G ∪ G ') ⊕ M   = ( G ⊕ M ) ∪ ( G ' ⊕ M ).

Commutativity and Associativity

Morphological operators are generally not commutative:

 

(18.9)

M ₁ ⊕ M ₂ = M ₂ ⊕ M ₁, but M ₁ ∅ M ₂ ≠ M ₂ ∅ M ₁.                           (18.10)

⊃

We can see that erosion is not commutative if we take the special case that M ₁ M ₂. Then the erosion of M ₂ by M ₁ yields the empty set. However, both erosion and dilation masks consecutively applied in a cascade to the same image G are commutative:

( G ∅ M ₁) ∅ M ₂ = G ∅ ( M ₁ ⊕ M ₂) = ( G ∅ M ₂) ∅ M ₁

( G ⊕ M ₁) ⊕ M ₂ = G ⊕ ( M ₁ ⊕ M ₂) = ( G ⊕ M ₂) ⊕ M ₁.

 

(18.11)

These equations are important for the implementation of morphological operations. Generally, the cascade operation with k structure elements

18.3 General Properties                                                                485

 

=   ⊕

=  ⊕ ⊕ ⊕

M ₁, M ₂,..., M _k is equivalent to the operation with the structure element M M ₁ M ₂... M _k. In conclusion, we can decompose large structure elements in the very same way as we decomposed linear shift-invariant operators. An important example is the composition of separable struc- ture elements by the horizontal and vertical elements M M _x M _y. Another less trivial example is the construction of large one-dimensional structure elements from structure elements including many zeros:

[11 1] ⊕ [10 1] = [1111 1]

[1111 1] ⊕ [1000 1] = [11111111 1].                          (18.12)

[11111111 1] ⊕ [10000000 1]

= [1111111111111111 1]

In this way, we can build up large exponentially growing structure ele- ments with a minimum number of logical operations just as we built up large convolution masks by cascading in Section 11.6. It is more diffi cult to obtain isotropic, i. e., circular-shaped, structure elements. The prob- lem is that the dilation of horizontal and vertical structure elements always results in a rectangular-shaped structure element, but not in a circular mask. A circular mask can be approximated, however, with one- dimensional structure elements running in more directions than only along the axes. As with smoothing convolution masks, large structure elements can be built effi ciently by cascading multistep masks.

 

Monotony

Erosion and dilation are monotonic operations

G ₁ ⊆ G ₂ ~ G ₁ ⊕ M ⊆ G ₂ ⊕ M G ₁ ⊆ G ₂ ~ G ₁ ∅ M ⊆ G ₂ ∅ M.

 

 

(18.13)

Monotony means that the subset relations are invariant with respect to erosion and dilation.

 

Distributivity

Linear shift-invariant operators are distributive with regard to addition. The corresponding distributivities for erosion and dilation with respect to the union and intersection of two images G ₁ and G ₂ are more complex:

 

and

( G ₁ ∩ G ₂) ⊕ M    ⊆ ( G ₁ ⊕ M ) ∩ ( G ₂ ⊕ M ) ( G ₁ ∩ G ₂) ∅ M = ( G ₁ ∅ M ) ∩ ( G ₂ ∅ M )

( G ₁ ∪ G ₂) ⊕ M    = ( G ₁ ⊕ M ) ∪ ( G ₂ ⊕ M ) ( G ₁ ∪ G ₂) ∅ M        ⊇ ( G ₁ ∅ M ) ∪ ( G ₂ ∅ M ).

 

(18.14)

 

 

(18.15)

Erosion is distributive over the intersection operation, while dilation is distributive over the union operation.

486                                                                                                        18 Morphology

 

Duality

Erosion and dilation are dual operators. By negating the binary image, erosion is converted to dilation and vice versa:

 

                                                          

G ∅ M = G ⊕ M G ⊕ M      = G ∅ M.

 

(18.16)

The mathematical foundation of morphological operations including complete proofs for all the properties stated in this section can be found in the classic book by Serra [168].

 

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