Mathematical Morphology - CMPcmp.felk.cvut.cz/.../lecture_morphology_sara.pdf · Mathematical Morphology A mathematical tool for the extraction and analysis of discrete quantized

1

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

2

Binary Mathematical Morphology

Motivation example: pre-processing for hand-writtencharacter recognition

Mathematical Morphology R. Sara

3

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)

Mathematical Morphology R. Sara

4

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

Mathematical Morphology R. Sara

5

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

Mathematical Morphology R. Sara

6

Translation Xh

Xh(p) = X(p− h), p ∈ X ⊂ Z2

Example:

h = (1, 2)

X Xh

�

Mathematical Morphology R. Sara

7

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.

Mathematical Morphology R. Sara

8

Binary Dilation Example

X

B

X ⊕B

Mathematical Morphology R. Sara

9

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. · · ·

Mathematical Morphology R. Sara

10

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

Mathematical Morphology R. Sara

11

Binary Erosion Example

X

B

X B

(X B)⊕ B

• Observation: (X B)⊕B 6= X

Mathematical Morphology R. Sara

12

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=

Mathematical Morphology R. Sara

13

6. (X B)⊕B 6= X

=

⊕ = 6=

7. · · ·

Mathematical Morphology R. Sara

14

Morphological Opening and Closing

X ◦B = (X B)⊕B opening

X •B = (X ⊕B)B closing

X

X ◦B

X •B

B

Mathematical Morphology R. Sara

15

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. · · ·

Mathematical Morphology R. Sara

16

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

Mathematical Morphology R. Sara

17

Morphological Gradient Example

X

β(X)

β∗(X)

Mathematical Morphology R. Sara

18

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

Mathematical Morphology R. Sara

19

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

Mathematical Morphology R. Sara

20

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)

Mathematical Morphology R. Sara

21

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.

Mathematical Morphology R. Sara

22

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

Mathematical Morphology R. Sara

23

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)

Mathematical Morphology R. Sara

24

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})

Mathematical Morphology R. Sara

25

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

Mathematical Morphology R. Sara

26

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.

Mathematical Morphology R. Sara