Morphology

• Morphology deals with form and structure

• Mathematical morphology is a tool for extracting image components useful in:– representation and description of region shape

(e.g. boundaries)– pre- or post-processing (filtering, thinning,

etc.)• Based on set theory

Morphology

• Sets represent objects in images• Sets in binary images (x,y)• Sets in gray scale images (x,y,g)• Some morphological operations:

Dilation & ErosionOpening & Closing

Hit-or-Miss TransformBasic Algorithms

Basic Concepts of Set Theory• A is a set in , a=(a1,a2) an element of A, aA• If not, then aA : null (empty) set• Typical set specification: C={w|w=-d, for d D}• A subset of B: AB• Union of A and B: C=AB• Intersection of A and B: D=AB• Disjoint sets: AB= • Complement of A:• Difference of A and B: A-B={w|w A, w B}=• Reflection of B: • Translation of A by z=(z1,z2):

€

Z 2

€

Ac = {w | w ∉ A}

€

A ∩ Bc

€

ˆ B = {w | w = −b,b∈ B}

€

(A)z = {c | c = a + z,a∈ A}

Morphological Image Processing

Morphological Image Processing

Morphological Image Processing

Dilation & Erosion

• Basic definitions:

– A,B: sets in Z2 with components a=(a1,a2) and b=(b1,b2)

– Translation of A by x=(x1,x2), denoted by (A)x is defined as:

(A)x = {c| c=a+x, for a∈A}

Dilation & Erosion

• More definitions:

Reflection of B: = {x|x=-b, for b∈B}

Complement of A: Ac = {x|xA}

Difference of A & B: A-B = {x|x∈A, x B} = A∩Bc

B̂

Dilation & Erosion

• Dilation: : empty set; A,B: sets in Z2

– Dilation of A by B:

€

A ⊕B = {x | ( ˆ B )x ∩ A ≠ ∅}

Dilation & Erosion

• Dilation:– Obtaining the reflection of B about its origin

and then shifting this reflection by x

– The dilation of A by B then is the set of all x displacements such that and A overlap by at least one nonzero element…

B̂

Dilation & Erosion

• Dilation:

}])ˆ[(|{ AABxBA x ⊆∩=⊕

B is the structuring element in dilation.

Morphological Image Processing

Morphological Image Processing

Dilation & Erosion

• Erosion:

i.e. the erosion of A by B is the set of all points xsuch that B, translated by x, is contained in A.

In general:BABA cc ˆ) ( ⊕=

})(|{ ABxBA x ⊆=

Morphological Image Processing

Morphological Image Processing

Opening & Closing

• In essence, dilation expands an image and erosion shrinks it.

• Opening:– generally smoothes the contour of an image,

breaks isthmuses, eliminates protrusions.

• Closing:– smoothes sections of contours, but it

generally fuses breaks, holes, gaps, etc.

Opening & Closing

• Opening of A by structuring element B:

BBABA ⊕= ) (o

• Closing:

BBABA )( ⊕=•

Morphological Image Processing

Morphological Image Processing

Morphological Image Processing

Morphological Image Processing