Morphological Principles:

1. Compatibility with translation: Let the transformation l/J depend on the position of the origin 0 of the co-ordinate system, and denote such a transformation by l/J0 . If all points are translated by the vector −ℎ , it is expressed as l/J-h . The compatibility with translation principle is given by

If l/J does not depend on the position of the origin, then the compatibility with translation: principle reduces to invariance under translation

2. Compatibility with change of scale: Let AX represent the homothetic scaling of a point set X (i.e., the co-ordinates of each point of the set are multiplied by some positive constant A). This is equivalent to change of scale with respect to some origin. Let l/JA denote a transformation that depends on the positive parameter A (change of scale). Compatibility with change of scale is given by

If l/J does not depend on the scale A, then compatibility with change of scale reduces to invariance to change of scale

3. Local knowledge: The local knowledge principle considers the situation in which only a part of a larger structure can be examined—this is always the case in reality, due to the restricted size of the digital grid. The morphological transformation l/J satisfies the local knowledge principle if for any bounded point set Z, in the transformation l/J(X) there exists a bounded set Z , knowledge of which is sufficient to provide l/J. The local knowledge principle may be written symbolically as

4. Upper semi-continuity: The upper semi-continuity principle says that the morphological transformation does not exhibit any abrupt changes.

Binary Dilation and Erosion

• The sets of black and white pixels constitute a description of a binary image. Assume only black pixels is considered, and the others are treated as a background.

• The primary morphological operations are dilation and erosion, and from these two, more complex morphological operations such as opening, closing, and shape decomposition can be constituted.

Dilation

• The morphological transformation dilation ⊕ combines two sets using vector addition (e.g., (a, b) +(c, d) = (a+c, b+d)).

• The dilation X ⊕ B is the point set of all possible vector additions of pairs of elements, one from each of the sets X and B

• Fig 4 shows 256x256 original image on the left. A structuring element size 3x3 is used.

• The result of dilation is shown on the right side of Fig 4. In this case the dilation is an isotropic expansion (Fill or Grow).

Fig 4: Dilation as isotropic expansion

• Dilation with an isotropic 3x3 structuring element might be described as a transformation which changes all background pixels neighboring the object to object pixels.

• Dilation properties:

° Dilation operation is commutative

X ⊕ B = B ⊕ X

° Dilation operation is associative

X ⊕ (B ⊕ D) = (X ⊕ B) ⊕ D

° Dilation may also be expressed as a union of shifted point sets

° Invariant to translation

° Dilation is an increasing transformation

• Dilation is used to fill small holes and narrow gulfs in objects. It increases the object size if the original size needs to be preserved, and then dilation is combined with erosion.

Fig 5: Dilation where the representative point is not a member of the structuring element.

• Fig 5 illustrates the result of dilation if the representative point is not a member of the structuring element B, if this structuring element is used; the dilation result is substantially different from the input set.

Erosion

• Erosion ⊝ combines two sets using vector subtraction of set elements and is dual operator of dilation.

• This formula says that every point p from the image is tested; the result of the erosion is given by those points p for which all possible p + b are in X.

Example:

• The result of the erosion is shown in the right side of the Fig 7. Erosion with an isotropic structuring element is called as shrink or reduce.

Fig 7: Erosion as isotropic shrink

• Basic morphological transformations can be used to find the contours of objects in an image very quickly. This can be achieved, for instance, by subtraction from the original picture of its eroded version as in Fig 8.

Fig 8: Contours obtained by subtraction of an eroded image from the original (left).

• Erosion is used to simplify the structure of an object. It decomposes complicated object into several simple ones.

• The equivalent definition for erosion

X ⊝ B = {p ∈ E2: Bp ⊆ X}.

• The erosion might be interpreted by structuring element B sliding across the image X; then, if B translated by the vector p is contained in the image X, the point corresponding to the representative point B belongs to the erosion X ⊝ B.

• An implementation of erosion might be simplified by noting that an image X eroded by the structuring element B can be expressed as an intersection of all translations of the image X by the vector −b ∈ B

• If the representative point is a member of the structuring element, then erosion is an anti-extensive transformation; that is, if(0,0) ∈ B, then X ⊝ B ⊆ X.

• Erosion properties:

• Image intersection and dilation cannot be interchanged; the dilation of the intersection of two images is contained in the intersection of their dilations

(X ∩ Y) ⊕ B = B ⊕ (X ∩ Y) ⊆ (X ⊕ B) ∩ (Y ⊕ B).

• The order of erosion may be interchanged with set union. This fact enables the structuring element to be decomposed into a union of simpler structuring elements

• Successive dilation of the image X first by the structuring element B and then by the structuring element D is equivalent to the dilation of the image X by B ⊕ D