Mathematical Morphology - CMPcmp.felk.cvut.cz/.../lecture_morphology_sara.pdf · Mathematical...

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Mathematical Morphology

A mathematical tool for the extraction and analysis ofdiscrete quantized image structure.

• Does not change image representation.(It is a system of transformations from the space ofdiscrete quantized images onto itself.)

• Implemented as set-theoretic operations withstructuring elements.

• Fast algorithms (quasi-paralel processing), manyapplications, mainly microscopy image processing.

Mathematical Morphology R. Sara

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Binary Mathematical Morphology

Motivation example: pre-processing for hand-writtencharacter recognition


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Literature

Serra, J. Image Analysis and MathematicalMorphology. Academic Press, London 1982

Hlav, V. and onka, M. Potaov vidn. Grada Praha,1992 (str. 76–96)

onka, M. Hlav, V., Boyle, R. Image Processing,Analysis, and Machine Vision. Thomson, 1998. (str.559–598).

Haralick, R. M, Shapiro, Linda G. Computer andRobot Vision. Addison Wesley, 1992 (str. 157–262)

Gonzalez, R. C., Woods, R. E. Digital ImageProcessing. Addison Wesley, 1992 (str. 518–566)


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Representation of imageand the structuring element

• image = a set of labeled vertices (pixels)

• a regular rectangular (or hexagonal) grid in thespace of dimension n (here n = 2)

• binary (integer) values

0 0 0000

1000 0 0 0

01 10

00

origin unspecified

• we will assume zero values behind image edges

• binary morphology is based on set-theoreticoperations with images


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Mathematical Morphology: Notation

X, Y – discrete quantized image

B, E, L – structural element

D(B) – domain of structural element B

Xc – complement of set X

Xh – translation of set X by vector h

X ⊕B – dilation (of X by B)

X B – erosion

X ◦B – opening

X •B – closing

β(X) – morphological gradient of X

X ⊗B – Serra transform (hit-or-miss)

X �B – thinning


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Translation Xh

Xh(p) = X(p− h), p ∈ X ⊂ Z2

Example:

h = (1, 2)

X Xh

�


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Binary Dilation X ⊕B

X ⊕B =⋃

{y, B(y)=1}

Xy

Example:

X B X ⊕B

X ⊕B = X(0,0) ∪X(1,0)

�

• Locus of all non-zero image pixels translated by theset of vectors defined by the structuring element.


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Binary Dilation Example

X

B

X ⊕B


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Dilation Properties

Observation: decomposability of element B

= ⊕ = ⊕ ⊕

B = D3 ⊕ S3,3 = D3 ⊕ S1,3 ⊕ S3,1

1. X ⊕B = B ⊕X

2. X ⊕ (B ⊕D) = (X ⊕B)⊕D decomposable element

3. X ⊕ (B ∪D) = (X ⊕B) ∪ (X ⊕D)

= ∪

4. X ⊕B 6⊇ X dilation is not ‘inflation’

⊕ =

5. · · ·


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Binary Erosion X B

X B =⋂

{y, B(y)=1}

X−y

Example:

X B X B

X B = X(0,0) ∩X(−1,0)

�

• Erosion: Locus of (non-zero) image pixels to whichstructuring element B can be inserted


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Binary Erosion Example

X

B

X B

(X B)⊕ B

• Observation: (X B)⊕B 6= X


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Erosion Properties

1. X B 6= B X

=

=

2. X (B ⊕D) = (X B)D decomposable element

3. X (B ∪D) = (B X) ∩ (D Y )

4. If (0, 0) ∈ B then X B ⊆ X

= counter-example

5. (X ⊕B)B 6= X

⊕ =

= 6=


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6. (X B)⊕B 6= X

=

⊕ = 6=

7. · · ·


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Morphological Opening and Closing

X ◦B = (X B)⊕B opening

X •B = (X ⊕B)B closing

X

X ◦B

X •B

B


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Properies of Opening and Closing

1. idempotence

(X ◦B) ◦B = X ◦B

(X •B) •B = X •B

2. antiextensivity of opening

X ◦B ⊆ X

3. extensivity of closing

X ⊆ X •B

4. · · ·


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Morphological Gradient β(X)

β(X) = X \ (X S3,3)

Example:

X X S3,3 β(X)

S3,3 =

�

Remarks

• interior boundary

• 4-connectivity

• exterior boundary: β∗(X) = (X ⊕ S3,3) \X


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Morphological Gradient Example

X

β(X)

β∗(X)


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Serra Transform (Hit-or-Miss)

X ⊗B = (X B1) ∩ (Xc B2)

correlation with two constraints

B = {B1, B2}, B1 ∩B2 = ∅:

1. X B1 – locus of object pixels similar to B1

2. Xc B2 – locus of background pixels similar to B2

• not every pair B gives X ⊗B 6= ∅

Example:

detection of “endpoints” from the left:

B = : B1 = , B2 =

X X ⊗B


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Sequential Thinning

X �B = X \ (X ⊗B)

X � {Bi}ni=1 = X �B1 �B2 � · · · �Bn

• the result is order-dependent!

Example:

X L1 X � L1 �

Golay alphabet:

· · ·L1 L2 L3 L4 · · · L8

E1 E2 E3 E4


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Sequential Thinning Example

image I

X = (I < 245)

Y = X ◦ S3,3

Y � {Li}8i=1, repeat till convergence

our goal: smoothed skeleton (see later)


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Skeleton1 Smoothing

Input: skeleton X

1. shortening of endings n-times (n = 8)

X1 =⟨X � {Ei}4

i=1

⟩n

k=1

2. ending point detection

X2 =4⋃

i=1

(X ⊗ Ei)

3. conditional dilation n-times (n = 8)

X3 =⟨(X2 ⊕ S3,3) ∩X

⟩n

k=1

4. smoothed skeleton

Y = X1 ∪X3

1I will call the result of sequential thinning a skeleton even if this is incorrect.


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Analogy with Convolution and Correlation

(X ⊕B)(x) =⋃

{y, B(y)=1}

Xy =⋃

{y, B(y)=1}

X(x− y)

(X B)(x) =⋂

{y, B(y)=1}

X−y =⋂

{y, B(y)=1}

X(x + y)

(X ⊕B)(x) = maxy∈D(B)

(X(x− y) + B(y)

)(X B)(x) = min

y∈D(B)

(X(x + y)−B(y)

)(f ∗ g)(x) =

∑y∈D(f)

f(y) g(x− y)

(f ~ g)(x) =∑

y∈D(f)

f∗(y) g(x + y)

convolution binary dilation gray-scale dilation

× ++ ∪ max

correlation binary erosion gray-scale erosion

× −+ ∩ min


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Gray-Scale Morphology

XC = C −X, C – maximum element (e.g. 255)

Xh(p) = X(p− h)

X ⊕B = maxy∈D(B)

X(x− y) + B(y)

X B = miny∈D(B)

X(x + y)−B(y)

X ◦B = (X B)⊕B

X •B = (X ⊕B)B

β(X) = X − (X S3,3)

X ⊗B = min(X B1, Xc B2)

X �B = X − (X ⊗B)


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Gray-Scale Morphology Examples

Example 1: Local maxima/minima detection in image:application of morphological gradient.

Example 2: Segmentation of cell boundaries in theimages of human cornea: application of morphologicalwatershed.

X {p} W (X, {p})


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Example 3: 100% visual quality inspection of amaximal thermometer capillary application of top hattransform.

X X ◦ S20,1 X −X ◦ S20,1

GRAY SCALE IMAGE OPENED IMAGE

THRESHOLDTOP HAT


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Example 4: Granulometry2

X largest square probes

0 10 20 30 40 50 60 70 80 90 1000

1000

2000

3000

4000

5000

6000

7000

granulometric spectre2Sorry for this example.


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