• Used to extract image components that are useful...

Digital Image Processing, 2nd ed.Digital Image Processing, 2nd ed.www.imageprocessingbook.com

Chapter 9Morphological Image Processing

• 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

• 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

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

{ }{ },

a Aa AA BC A BD A BA B

A w w A

A B w w A w B A B

• ∈• ∉• ⊆• =• =• =∅

• = ∉

• − = ∈ ∉ =

∪∪

• Operators by examples:

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

{ }( ) { }ˆ , for , Reflection

, for , Translationz

B w w b b B

A c c a z a A

• ∈ = − ∈

• ∈ = + ∈

• Reflection and Translation by examples:– Need for a reference point.

Reference point

• Basic Logic Operations:

• Examples

• Two Fundamental Morphological Operators:– Dilation:

• Set B: A structural elements.• Relation to Convolution mask:

– Flipping– Overlapping

• Other names: Grow, Expand

( ){ } ( ){ }ˆ ˆz z

A B z B A z B A A⊕ = ≠ ∅ = ⊆∩ ∩

• Dilation by example:

• Dilation by example:– B: a 3×3 mask.

• Application:– Gap filling

• Two Fundamental Morphological Operators:– Erosion:

• Set B: A structural elements.• Other banes: Shrink, Reduce

( ){ }zA B z B A= ⊆

• Erosion by example:– B: a 3×3 mask.

• Erosion by example:

11 pixels (Diameter)

• Dilation-Erosion Duality:

( ) ( ){ } ( ){ }( ){ }

( ){ }ˆ

ˆRemember:

ccc cz z

A B z B A z B A

z B A A B

A B z B A

= ⊆ = =∅

= ≠ ∅ = ⊕

⊕ = ≠ ∅

• Erosion Application:– Remove details

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

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

• Smooth contour• Break narrow isthmuses (Means: تنگه(• Remove thin protrusion

– Closing:• Smooth contour• Fuse narrow breaks, and long thin gulfs.• Remove small holes, and fill gaps.

• 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= Θ ⊕ = ⊆∪

( )A B A B B= ⊕ Θ

• Opening Illustration:

• Closing Illustration:

• Opening and Closing

• Medical Application:

Original Segmentation

Opening 5×5

Closing 5×5

• 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 A•B– 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.

( ) ( )ˆc cA B A B• =

Opening

Dilation of Opening Closing of Opening

• NoiseReduction

• Hit-or-Miss– Shape Detection– X-Y-X shape– X enclosed by W

– B1: Object related– B2: Background related

( ) ( )( )

( ) ( )1 2

A B A X A W X

A B A B A B

⎡ ⎤= Θ Θ −⎣ ⎦⎡ ⎤= Θ Θ⎣ ⎦

= Θ − ⊕

• Hit-or-Miss:– Another application:

• Corners

• Morphological Operator Applications:– Boundary Extraction:

( ) ( )A A A Bβ = − Θ

Eroded Difference

• Boundary Extraction:

• Region Filling:– Start form p inside boundary.

1Until:

X X B AX X+

• Region Filling Example:– Semi-automated to cancel reflection effect

• Connected components Extraction:– Start from p belong to desired region.

1Until: k k

X pX X B A

Original

Erosion (5×5)

Connected Components with size

• Convex Hull of S:– Smallest Convex set H, containing S

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

( )( )

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

i i ik k

i i iconverged

X X B A i k

C A D D X=

• = ∗ = =

• = =

Converged: C

• Shortcoming of previous algorithm:– Grow more than minimum required convex size.– Limit to vertical-horizontal expansion.

• Thinning:( ) ( )

{ } { }{ } ( )( )( )( )

* *Another approach:

Repeat until convergence

A B A A B A A B

A A B B B

⊗ = − =

⊗ = ⊗ ⊗ ⊗

Convert to m-connection

• Thickening:

• Structural elements are as before.

{ } ( )( )( )( )1 2

A B A A B

A A B B B

• 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.

• Skeleton by example:

• Formulation:( ) ( )

( ) ( ) ( )( ) ( )( )( )( )

( ){ }

( )( )

( ) ( )( )( )( )

, : Opening

: times

K=max k

Reconstruction:

: times

S A S A

S A A kB A kB B

A kB A B B B k

S A kB

A kB A B B B k

= Θ − Θ

Θ = Θ Θ Θ

Θ ≠∅

⊕ = ⊕ ⊕ ⊕

• Extension to Gray-Level images:– f(x,y): the input image – b(x,y): a structuring element (a subimage function)– (x,y) are 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)

• Extension to Gray-Level images:– Dilation:

• Df and Db are the domains of f and b, respectively.

• condition (s-x) and (t-y) have to be in the domain of fand (x,y) have to be in the domain of b is similar to the condition in binary morphological dilation where the two sets have to overlap by at least one element

( )( , ) max{ ( , ) ( , ) |( ), ( ) ; ( , ) }f b

f b s t f s x y t b x ys x t y D x y D

⊕ = − − +− − ∈ ∈

• Dilation Similarity with Convolution:– f(s-x) : f(-x) is simply f(x) mirrored with respect to the

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

– 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.

• 1D example:( )( ) max{ ( ) ( ) | ( ) ; }f bf b s f s x b x s x D x D⊕ = − + − ∈ ∈

• Erosion:

Condition (s+x) and (t+y) have to be in the domain of f and (x,y) have to be in the domainof b is similar to the condition in binary morphological erosion where the structuring element has to be completely contained by the set being eroded.

( )( , ) min{ ( , ) ( , ) |( ), ( ) ; ( , ) }f b

f b s t f s x y t b x ys x t y D x y D

Θ = + + −+ + ∈ ∈

• Similarity to 2D correlation– f(s+x) moves to the left for positive s and to the

right for negative s.• 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.

• 1D case:( )( ) min{ ( ) ( ) |

( ) ; }f b

f b s f s x b xs x D x D

Θ = + −+ ∈ ∈

• Erosion-Dilation Duality:

ˆ( ) ( , ) ( )( , )where

ˆ( , ) and ( , )

f b s t f b s t

f f x y b b x y

Θ = ⊕

= − = − −

• Dilation-Erosion

Original

Dilation

Erosion

• Opening-Closing:– Same (binary) relation with Dilation-Erosion:

( )( )

ˆ( )c c

f b f b bf b f b b

f b f b

= − ⊕• = ⊕ −

• 1D Example:One scan line

Rolling ball for opening

Opening results

Rolling ball for Closing

Closing Results

• Opening-Closing Properties:– Opening:

– Closing:

1 2 1 2

( ) ( )( ) if , then ( ) ( ),( ) ( ) ( )

i f b fii f f f b f biii f b b f b

↵↵ ↵

1 2 1 2

( ) ( )( ) if , then ( ) ( )( ) ( ) ( )

i f f bii f f f b f biii f b b f b

↵ •↵ • ↵ •• • = •

e↵r 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

• 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 theimage.

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

• Closing– The structuring element is rolled on top of the surface

of f– Peaks essentially are left in their original form (assume

that their separation at the narrowest points exceeds the diameter of the structuring element)

– So, closing is used to remove small dark details, while leaving bright features relatively undisturbed.

– The initial dilation removes the dark details and brightens the image

– The subsequent erosion darkens the image without reintroducing the details totally removed by dilation

• Opening Closing Example

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

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

• Gray Level Morphological Examples:– Smoothing:– Gradient:– Laplacian:

( ) ( )g f b f b= ⊕ − Θ

( ) ( ) 2g f b f b f= ⊕ + Θ −

( )( )g f b b= •

Dilation Erosion Smoothing Gradient Laplacian

• Smoothing

• Gradient:

• Top-hat:– Enhancing details in presence of shades

( )h f f b= −

• Used to extract image components that are useful...

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