this lecture is about mathematical...

This lecture is about

• Mathematical Morphology

Lecture 10 c©2002 Fredrik Georgsson, Umeå University

Mathematical Morphology

• Within biology, the term morphology is

used for the study of the shape and

structure of animals and plants

• In image processing mathematical

morphology is theoretical framework for

representation, description and

pre-processing.

• The mathematical morphology is based on

set-theory

• The mathematical morphology is an

example of non-linear operations



Linear filtering



Morphological filtering



Median filtering



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Morphological filtered background

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Let A and B be sets of points from Z2 with

objects a = (a1, a2) and b = (b1, b2).

The Translation of A with x = (x1, x2) is

defined as:

(A)x = {c | c = a + x, ∀a ∈ A}

The Reflection of A is defined as:

A = {x | x = −a, ∀a ∈ A}

The Complement of A is defined as:

Ac = {x | x /∈ A}

The Difference between two sets is defined as:

A−B = {x | x ∈ A, x /∈ B} = A ∩Bc

The intersection and union of sets is supposed

to be familiar


Mathematical Morphology The Basics

• A set represents a shape in mathematical

morphology

• Binary images is a subset of Z2 where each

pixel is a tuple of coordinates (x, y) for

either the black or white pixels.

• Intensity images is a subset of Z3 where

each element is a three-dimensional points

who’s first two elements describe the

spatial locality and the last element

describes the intensity at that point.


Mathematical Morphology Dilation

The Dilation of A ⊂ Z2 and B ⊂ Z

2 is

defined as

A⊕B = {x | (B)x ∩A 6= Ø}

The dilation is all translations x that yields an

intersection between A and the reflected B that

is not the empty set.

∴ dilation ensures that the intersection between

A and B is a subset of A or

A⊕B = {x | [(B)x ∩A] ⊆ A}

B is called the structuring element of the

dilation (as in all other morphological

operations).



1 1 1

1 1 1

1 1 1

Original image Dilated image



1 0 0

0 1 0

0 0 1

Original image Dilated image



A more intuitive definition of the dilation is

(Minkowsky addition)

A⊕B =⋃

b∈B

(A)b

This is, of course, equivalent to the previous

definition

Proof

→: Suppose that x ∈ A⊕B, then there is an

a ∈ A and b ∈ B so that x = a + b which mean

that x ∈ (A)b

←: Suppose that x ∈⋃

b∈B(A)b then there is a

b ∈ B, x ∈ (A)b. This implies that it exists

ana ∈ A so that x = a + b implying that

x ∈ A⊕B



The Minkowsky addition allows us to visualize

the dilation as follows: A copy of the structure

element B is placed at each point x ∈ A.

B

A A⊕B


Mathematical Morphology Erosion

The Erosion of A ⊂ Z2 and B ⊂ Z

2 is defined

as:

AB = {x | (B)x ⊆ A}

In other words: The Erosion of A with B are

the translation x in which B is completely

covered by A.



1 1 1

1 1 1

1 1 1

Original image



1 0 0

0 1 0

0 0 1

Original image Eroded image


Mathematical Morphology Duality

Erosion and dilation are dual with with respect

to complement and reflection:

(AB)c = Ac ⊕ B

Since (starting from definitions)

(AB)c = {x | (B)x ⊆ A}c (1)

= {x | (B)x ∩Ac = ∅}c (2)

= {x | (B)x ∩Ac 6= ∅} (3)

= Ac ⊕ B (4)


Mathematical Morphology Opening

• By combining erosion with dilation

according to

A ◦B = (AB)⊕B

we have an operation that is called

opening.

• The Opening smooths objects by

removing pixels

• An other definition is

A ◦B =⋃

{(B)x | (B)x ⊂ A}


Mathematical Morphology Closing

• By reversing the erosion and dilation in the

opening, a closing operation is defined:

A •B = (A⊕B)B

• Closing smooths objects by adding pixels.

• All smoothing (both for closing and

opening) is in relation to the size of the

structuring element B.


Mathematical Morphology Closing

Original image Opened image

Original image Closed image


Mathematical Morphology Dualitet

Opening and closing are dual with respect to

complement and reflection

(A •B)c = ((A⊕B)B)c

= (A⊕B)c ⊕ B

= {x | (B)x ∩A 6= ∅}c ⊕ B

= {x | (B)x ∩A = ∅} ⊕ B

= {x | (B)x ∩Ac 6= ∅} ⊕ B

= {x | (B)x ⊆ Ac} ⊕ B

= (Ac B)⊕ B

= (Ac ◦ B)


Mathematical Morphology Dualitet

For opening the following holds

1. A ◦B ⊂ A

2. If C ⊂ D then C ◦B ⊂ D ◦B

3. (A ◦B) ◦B = A ◦B

and for closing

1. A ⊂ A •B

2. If C ⊂ D then C •B ⊂ D •B

3. (A •B) •B = A •B

Operators for which property (3) holds are said

to be idempotent. That is, it does not matter

how many times the operation is applied, the

result is the same as if it only had been applied

once.


Mathematical Morphology (A ◦B) •B

Original image

Opened image

Opened+Closed image


Mathematical Morphology (A •B) ◦B

Original image

Closed image

Closed+Opened image


Mathematical Morphology (A •B) ◦B


Mathematical Morphology Hit & Miss

If the aim is to locate objects in an image we

can do the following:

1. By calculating AX all objects smaller

than X will vanish. X is reduced to a

point.

2. By calculating Ac (W −X), where W is a

frame to the object, all objects larger than

X will vanish, X is reduced to a point.

The intersection between (1) and (2) will be

the mid-point for each X in the image.



Or slightly more formal:

(AX) ∩ [Ac (W −X)]

If we let B = (B1, B2) and B1 = X,

B2 = W −X we have

A ~ B = (AB1) ∩ (Ac B2)

Which can be written as

A ~ B = (AB1) ∩ (Ac B2) (5)

= (AB1)− (Ac B2)c (6)

= (AB1)− (A⊕ B2) (7)



A R1 = AX

W −X R2 = Ac (W −X)

R1 ∩R2 (R1 ∩R2)⊕X


Mathematical Morphology Edge

extraction

The pixels that constitutes the edge of an

object A is defined as the points that belong to

the object and has at least one neighbor that

belongs to the background.

These pixels are denoted β(A), and are

precisely the pixels that are removed by

performing an erosion B,

β(A) = A− (AB)

Original image, A A−(A eroded with B)


Mathematical Morphology Region

growing

Given that we know a point p in side a region

defining with a contour A we can let the point

grow until it fills the entire region by iterating

with

Xk = (Xk−1 ⊕B) ∩Ac

until Xk = Xk+1. X0 is sat to be p.

If the region is 8-connected the complement

will be 4-connected and B must have the

following appearance

0 1 0

1 1 1

0 1 0


Mathematical Morphology Region

growing

Original Regionfill after 90 iterations

Regionfill after 180 iterations Regionfill after 253 iterations


Mathematical Morphology Connected

components

Given a point p inside an object we can use the

same principle as in region-growing with a

slight modification. The difference is that the

since object is now defined by ones (and not

zeros surrounded by a contour) we do not have

to invert A.

Xk = (Xk−1 ⊕B) ∩A

where X0 = p.

If A is 8-connected, the object will be

8-connected and we have to use the following B

1 1 1

1 1 1

1 1 1


Mathematical Morphology Connected

components

Original Connected comp. after 15 iterations

Connected comp. after 30 iterations Connected comp. after 41 iterations


Mathematical Morphology Convex hull

• Given a set A we can find an

approximation to the convex hull C(A)

by repeated application of

Xik = (X ~ Bi) ∪A

where i = 1, 2, 3, 4 and k = 1, 2, 3, . . ..

Bi are different structuring elements.

• For each i, {X ik} will converge to a set Di.

• The convex hull is the union of all Di

C(A) =

4⋃

i=1

Di



• The following Bi are used in the algorithm

B1

1 × ×

1 0 ×

1 × ×

B2

1 1 1

× 0 ×

× × ×

B3

× × 1

× 0 1

× × 1

B4

× × ×

× 0 ×

1 1 1

× indicates don’t care.

• This definition (with don’t care) makes it

impossible to use MatLabs implementation

of dilation and erosion since they only use

binary structuring elements.

• The convex hull is implemented in MatLab

by using look-up tables (LUT), see help

makelut and help applylut for more

information.



Result X416

Result X126

Result X236

Result X320



Approximation of convex hull


Math

em

atic

alM

orpholo

gy

Convex

hull

Result X126

Result X236

Result X320

Result X416

Result X531

Result X630

Result X763

Result X830

Lecture

10

c©2002

Fredrik

Georgsson,U

meå

Univ

ersity

Page

43


Approximation of convex hull


Mathematical Morphology The Skeleton

The skeleton S(A) of an object A can be

expressed as

S(A) =K⋃

k=0

Sk(A)

where

Sk(A) = (A kB)−((A kB) ◦B

)

and

K = max{k | (A kB) 6= Ø}

and finally

(A kB) = ((. . . ((AB)B) . . .)B︸︷︷︸

k times

Observe: There is nothing in the algorithm

that guarantees that S(A) will be a connected

component.



An object and its skeleton

K = 15



• A nice property of S(A) is the A can be

reconstructed if we know K.

• This is not possible with a normal skeleton

• The reconstruction is made according to

A =K⋃

k=0

(Sk(A)⊕ kB)


Mathematical Morphology Pruning

• After finding the skeleton, it is often

desirable to remove short branches from

the graph that constitutes the skeleton.

This is done by removing end-points from

the graph without breaking connectivity.

• Given knowledge of the object (that is how

long branches that can be expected) we

can remove just enough of the graph.


Mathematical Morphology Pruning

Image, skeleton and graph


Mathematical Morphology Gray level

• To use mathematical morphology on

gray-scale images the crisp set theory must

be replaced by fuzzy set theory.

• This means that each point (x, y) has a

membership function f(x, y) describing

the degree of membership that the point

has to the foreground and background.

• f(x, y) is the image, b(x, y) is the

structuring element. f c(x, y) is defined as

−f(x, y) and b(x, y) as b(−x,−y).



Dilation A dilation of f with b is defined as

(f ⊕ b)(s, t) = max{f(s− x, t− y) + b(x, y) |

(s− x), (t− y) ∈ Df , (x, y) ∈ Db}

Erosion The erosion of f with b is defined as

(f ⊕ b)(s, t) = min{f(s− x, t− y)− b(x, y) |

(s− x), (t− y) ∈ Df , (x, y) ∈ Db}

Df and Db are the domain for f and b

respectively. Since (x, y) ∈ Db only the points

in the support of f and b can be members of

the final set.

Dilation and erosion are still dual :

(f b)c(x, y) = (f c ⊕ b)(x, y)



Eroded image

Dilated image



• The Opening of f with b is defined as

(f ◦ b)(s, t) = ((f ⊕ b) b)(s, t)

• The opening suppress bright objects

smaller than the structuring element b

• The Closing of f with b is defined as

(f • b)(s, t) = ((f b)⊕ b)(s, t)

• The closing suppress dark objects smaller

than the structuring element b.



Opened image

Closed image



• A morphological low-pass filtering can

be calculated as

l = (f ◦ b) • b

• A morphological gradient filtering can

be calculated as

g = (f ⊕ b)− (f b)

• A ’top-hat’ transform is defined as

h = f − (f ◦ b)



Opened and closed image

Morphological gradient

Top−Hat



Gray-level morphology is implemented in

MatLab (see imdilate, imerode,. . . )

However, these implementations require b to be

a binary function. A more general

implementation (as the one used in the

Gonzales book) is given by:

function F=grayerode(f, b)

B=im2col(b,size(b),’sliding’);

fun=inline(’min(x-P1*ones(1,length(x)))’,1);

F=colfilt(f,size(b),’sliding’,fun,B);

function F=graydilate(f, b)

B=im2col(b,size(b),’sliding’);

fun=inline(’max(x+P1*ones(1,length(x)))’,1);

F=colfilt(f,size(b),’sliding’,fun,B);


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