used to extract image components that are useful in the

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E. Fatemizadeh, Sharif University of Technology, 2011

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Digital Image Processing

Morphological Image processing

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• Used to extract image components that are useful in the representation and description of region shape, such as: – boundaries extraction

– skeletons

– convex hull

– morphological filtering

– thinning

– pruning

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• Sets in mathematic morphology represent objects in an image:

– binary image (0 = white, 1 = black): • The element of the set is the coordinates (x,y) of pixel belong to

the object Z2

– gray-scaled image: • The element of the set is the coordinates (x,y) of pixel belong to the object and the

gray levels Z3

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• Preliminary:

– A: A set in Z2 with elements of a=(a1,a2)

– Reflection:

– Translation:

– Used in Structuring Elements (SEs)

ˆ , for B w w b b B

, for z

B c c b z b B

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• Reflection and Translation by examples:

– Need for a reference point.

Reference point

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• Structuring Elements:

– Used to structure the objects: • 1 (or balck): Action

• 0 (or white): No Action

• × : Don’t Care

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• Example:

– An image

– A structural element

– A processing

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• Two Fundamental Morphological Operators:

– Erosion

– Dilation

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• Erosion: – Set B: A structural elements.

– Other Names: Shrink, Reduce

z

c

z

A B z B A

A B z B A

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• Erosion Example:

z

A B z B A

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• Erosion by example (1):

– B: a 3×3 mask (Full).

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• Erosion by example (2):

Disk: 11 pixels (Diameter)

Square: 9*9

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• Erosin by Example (3):

– Applied on white area

Original 11×11

45×45 15×15

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• Dilation:

– Set B: A structural elements.

– Other Names: Grow, Expanding

– Relation to Convolution mask: • Flipping

• Overlapping

ˆ

ˆ

z

z

A B z B A

A B z B A A

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• Dilation by example: ˆz

A B z B A

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• Dilation by example (1):

– B: a 3×3 mask (full).

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• Application:

– Gap filling

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• Dilation-Erosion Duality:

ˆ

ˆRemember:

ccc c

z z

c c

z

z

A B z B A z B A

z B A A B

A B z B A

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• Erosion Application:

– Remove details

1,3,5,7,9, and 15 Erode with 13 Dilate with 13

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• Opening and Closing: – Dilation expands and Erosion shrinks.

– Opening: • Smooth contour

• Break narrow isthmuses ( تنگه)

• Eliminates thin protrusion ( بیرون زدگی)

– Closing: • Smooth contour

• Fuse narrow breaks, and long thin gulfs.

• Eliminates small holes, and fill gaps.

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• Opening and Closing:

– Dilation expands and Erosion shrinks.

– Opening: • A erosion followed by a dilation using the same structuring

element for both operations.

– Closing: • A Dilation followed by a erosion using the same structuring

element for both operations.

z zA B A B B B B A

A B A B B

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• Opening Illustration:

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• Opening Example:

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• Closing Illustration:

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• Closing Example:

Disk

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• Closing Example:

Original Thresholded Closing

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• Opening and Closing

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• Medical Application:

Original Segmentation

Opening 5×5

Closing 5×5

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Opening and Closing Duality:

• Opening Properties: – A○B is a subset (subimage) of A

– If C is a subset of D, then C○B is a subset of D○B

– (A○B)○B = A○B ↔ Multiple apply has no effect.

• Closing Properties: – A is a subset (subimage) of AB

– If C is a subset of D, then CB is a subset of DB

– (AB)B = AB ↔ Multiple apply has no effect.

ˆc cA B A B

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Opening

Dilation of Opening Closing of Opening

• Noise

Reduction

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• Hit-or-Miss

– Shape Detection

– X-Y-X shape

– X enclosed by W

– B1: Object related

– B2: Background related

1 2

1 2ˆ

c

c

A B A D A W D

A B A B A B

A B A B A B

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• Hit-or-Miss:

– Another application: • Corners

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• Morphological Operator Applications:

– Boundary Extraction:

A A A B

Eroded Difference

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• Boundary Extraction:

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• Hole/Region Filling:

– Start form p inside boundary.

0

1

1

, 1, 2,3,

Until:

c

k k

k k

X p

X X B A k

X X

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• Region Filling Example:

– Semi-automated to cancel reflection effect

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• Connected components Extraction:

– Start from p belong to desired region.

0

1

1

1,2,3,

Until:

k k

k k

X p

X X B A k

X X

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• Example (1):

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Original

Thr

Erosion (5×5)

Connected Components with size

• Example (2)

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• Convex Hull:

– Convex Set:

– Convex Hull: Smallest Convex set H, containing S

• Define Four basic structural elements, Bi, i=1,2,3,4

0

1

4

1

1,2,3,4 and 1, 2,3,

,

i

i i i

k k

i i i

i

X A

X X B A i k

C A D D X

converged

, , 0,1 : 1s t s t S S

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C: Converged

C C

C C

• Example:

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• Shortcoming of previous algorithm:

– Grow more than minimum required convex size.

– Limit to vertical-horizontal expansion.

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• Thinning:

1 2

1 2

1

2

Another approach:

, , ,

One Pass with

One Pass with

...

One Pass with

Repeat until convergence

c

n

n

n

A B A A B A A B

B B B

A A B B B

B

B

B

B

B

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Convert to m-connection

• Example:

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• Thickening:

– Structural elements are as before (1 and 0 interchange).

– Usually thin the background, then complement the results.

1 2 n

A B A A B

A A B B B

B

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• Skeletonization A with notation S(A):

– For z belong to S(A) and (D)z, the largest disk centered at z and contained in A, one can not find a larger disk containing (D)z and included in A.

– Disk (D)z touches the boundary of A at two or more different points.

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• Skeleton by example:

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• Formulation:

0

k=0

, : Opening

: times

K=max k

Reconstruction:

A=

: times

k

k

k

k

k k

S A S A

S A A kB A kB B

A kB A B B B k

A kB

S A kB

S A kB S A B B B k

K

K

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• Example:

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• Pruning:

– Remove Parasitic components (clean-up)

1

8

2 1

1

3 2

4 1 3

k

k

X A

X X B

X X H A

X X X

B

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• Example:

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• Morphological Reconstruction:

– Until now: An Image and a structuring elements

– Now: Two images and one structuring elements • Marker Image: Starting points for transformation.

• Mask Image: Constrains the transformation

• Structuring element: Define connectivity

– Two main operator:

• Geodesic Dilation

• Geodesic Erosion

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• Geodesic Dilation:

– F: Marker Image

– G: Mask Image

– B: Structuring Elements

– Geodesic Dilation of size 1:

– Geodesic Dilation of size n:

– G: Limit the growing nature of dilation by B

1

GD F F B G

1 1 0,

n n

G G G GD F D F D F D F F

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• Illustration of Geodesic Dilation:

1

GD F F B G

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• Geodesic Erosion:

– F: Marker Image

– G: Mask Image

– B: Structuring Elements

– Geodesic Erosion of size 1:

– Geodesic Erosion of size n:

– G: Results is greater or equal to mask

1

GE F F B G

1 1 0,

n n

G G G GE F E F E F E F F

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• Illustration of Geodesic Erosion:

1

GE F F B G

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– By Dilation

1

, :k k kD

G G G GR F D F k D F D F

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– By Erosion

1

, :k k kE

G G G GR F E F k E F E F

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• Sample Application:

– Opening/Closing by Reconstruction: • Classical Opening:

– Erosion: Remove small objects

– Subsequent Dilation: Try to restore the shape of objects that remains.

– Restoration in strongly depend on similarity between SE and that objects.

• Restore exactly the shapes of the remained objects after erosion.!

• Opening Reconstruction of size n, we use F as mask

n D

R F

n E

R F

O F R F nB

C F R F nB

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• Example:

– Opening Reconstruction:

• Goal: Extract Characters with long vertical strokes

• B: 51×1

Original Erosion

Opening 1

FO F

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• Fully Automated Filling Hole Procedure:

– I(x,y): Binary Image

– F(x,y): Marker image that is 0 everywhere except borders

1 , , is on the border of ,

0 otherwise

c

cD

I

I x y x y IF x y

H R F

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• Example For Illustration:

– B: A full 3×3 mask

1

1, :

c

c c c c

c

c

I

k k kD

I I I I

cD

I

D F F B I

R F D F k D F D F

H R F

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• Practical Example:

I Ic

F H

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• Border Clearing:

– Goal: Remove objects that touch (connected to) the border

1 , , is on the border of ,

0 otherwise

D

I

I x y x y IF x y

X I R F

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• Example For Illustration:

– B: A full 3×3 mask

1

1, :

I

k k kD

I I I I

D

I

D F F B I

R F D F k D F D F

X I R F

D

IR F

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• Summary of Morphological Tools:

– Figure 9.33 and Table 9.1 (Pages 662-664)

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• Gray-Scale Morphoogy

– f(x,y): The input gray-scale image

– b(x,y): a structuring element (a subimage function)

– (x,y): Integers.

– f and b are functions that assign a gray-level value (real number or real integer) to each distinct pair of coordinate (x,y)

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• Two Types of Structuring Elements:

– Non-flat

– Flat

• Mostly used:

– Flat SE

– Symmetrical

Horizontal Intensity Profile Through the center

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• Erosion and Dilation:

– Erosion • Flat SE with origin in centered at (x,y)

• Similar to correlation procedure

– Dilation • Flat reflected SE with origin in centered at (x,y),

• Similar to Convolution procedure

,

, min ,s t b

f b x y f x s y t

,

, ,s t b

f b x y Max f x s y t

ˆ , ,b x y b x y

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• Example:

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• Erosion and Dilation:

– Erosion • Non-Flat SE with origin in centered at (x,y)

• Similar to correlation procedure

– Dilation • Non-Flat reflected SE with origin in centered at (x,y),

• Similar to Convolution procedure

,

, min , ,N

N Ns t b

f b x y f x s y t b s t

,

, max , ,N

N Ns t b

f b x y f x s y t b s t

ˆ , ,b x y b x y

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• Dilation Similarity with Convolution: – f (x-s,y-t): is simply mirrored version of f (x,y) with respect

to the original of the x-y axis. the function f (x-s,y-t) moves to the right for positive s-t, and to the left for negative s-t.

– Max operation replaces the sums of convolution – Addition operation replaces with the products of

convolution.

• General effect – If all the values of the structuring element are positive, the

output image tends to be brighter than the input – Dark details either are reduced or eliminated, depending

on how their values and shapes relate to the structuring element used for dilation.

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• 1D example:

max ( ) ( )N

Ns b

f b x f x s b x

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• Erosion similarity to 2D correlation

– f (x+s,y+t) moves to the left for positive s-t and to the right for negative s-t.

• General effect – If all the elements of the structuring element are positive,

the output image tends to be darker than the input

– The effect of bright details in the input image that are smaller in area than the structuring element is reduced, with the degree of reduction being determined by the gray-level values surrounding the bright detail and by the shape and amplitude values of the structuring element itself.

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• 1D Example:

minN

N Ns b

f b x f x s b s

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• Erosion-Dilation Duality:

– For flat and non-flat SE:

ˆ , ,

ˆ , ,

ˆ( , ) and ( , )

c c

c c

c

f b x y f b x y

f b x y f b x y

f f x y b b x y

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• Dilation-Erosion by Example:

Original Dilation

Erosion

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• Opening-Closing:

– Same (binary) relation with Dilation-Erosion:

ˆ

ˆ

c c

c c

f b f b b

f b f b b

f b f b

f b f b

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• Opening/Closing 1D Example:

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• Opening-Closing Properties:

– Opening:

– Closing:

1 2 1 2

( )

( ) if , then ,

( )

i f b f

ii f f f b f b

iii f b b f b

1 2 1 2

( )

( ) if , then

( )

i f f b

ii f f f b f b

iii f b b f b

er indicates that the domain of e is a subset of the domain of r, and also that e(x,y) ≤ r(x,y) for any (x,y) in the domain of e

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• Opening – The structuring element is rolled underside the surface of f

– All the peaks that are narrow with respect to the diameter of the structuring element will be reduced in amplitude and sharpness

– So, opening is used to remove small light details, while leaving the overall gray levels and larger bright features relatively undisturbed.

– The initial erosion removes the details, but it also darkens the image.

– The subsequent dilation again increases the overall intensity of the image without reintroducing the details totally removed by erosion.

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• Opening Closing Example (1)

Opening: Decreased size of small bright details. No changes to dark region

Closing: Decreased size of small dark details. No changes to bright region

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• Opening and Closing Example (2):

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• Gray Level Morphological Examples:

– Smoothing:

– Gradient:

– Laplacian:

g f b f b

2g f b f b f

g f b b

Dilation Erosion Smoothing Gradient Laplacian

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• Morphological Smoothing:

– Opening

– Closing

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• Morphological Gradient:

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• Top-hat and Bottom-hat Transforms:

hat

hat

T f f f b

B f f b f

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• Example:

Original Thresholding

Opening

Top-hat

Thresholding

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• Granulometry:

– Determine size distribution of particles in an image.

– Apply opening with SE’s on increasing size.

– Calculate sum of pixel values in the opening.

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• Example:

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• Example (Cont.):

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• Textural Segmentation:

– Two different texture in an image

– Goal: Find Boundary between them.

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• Gray-Scale Morphological Reconstruction:

– f: Marker Image

– g: Mask Image

– Same size and f ≤ g

– ˄: Point-wise minimum operator

– ˅: Point-wise maximum operator

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• Geodesic Dilation of size 1:

• Geodesic Dilation of size n:

1

gD f f b g

1 1 0,

n n

g g g gD f D f D f D f f

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• Geodesic Erosion of size 1:

• Geodesic Srosion of size n:

1

gE f f b g

1 1 0,

n n

g g g gE f E f E f E f f

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– By Erosion

– By Dilation

1, :

k k kE

g g g gR f E f k E f E f

1, :

k k kD

g g G GR f D f k D f D f

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• Opening/Closing by Reconstruction:

n D

R f

n E

R f

O f R f nb

C f R f nb

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• Example:

used to extract image components that are useful in the

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