morphological image processing - ele – puc rioraul/imageanalysis/morphology.pdf · 9/3/2019...

Morphological Image Processing

Raul Queiroz Feitosa

9/3/2019 Morphological Image Processing 2

Objetive

To introduce basic morphological tools that are

useful :

in the representation and description of region shape

In pre- and post-processing to improve the

segmentation outcome .


Introduction

In mathematical morphology Objects in the image are represented by sets.

In binary images sets Z2, where each element is a tuple (x,y)

with the coordinates of the black (or white) pixels.

In gray-scale digital images sets Z3, where each element is a

tuple (x,y,f) with the coordinates and the discrete gray level of the

pixels.

pre-

processing

segmen-

tation

feature

extraction

feature

selection

classifi-

cation

post-

processing data label


Basic Concepts

Let A be a set in Z2. If a=(a1,a2) is an element of A then

we write

a A

Similarly if a is not an element of A, we write

a A

The set with no elements is called null or empty set,

denoted with .

The elements we are concerned are the coordinates of a

pixel belonging to objects or features of interest.


Basic Concepts

If every element of a set A is also an element of another set B, then we write

A B

The union of two sets A and B, is denoted as

C=AB

The intersection of two sets A and B, is denoted as

D = A B

Two sets A and B, are said to be disjoint if

A B =


Basic Concepts

A complement of a set A, is the set of elements not contained in A

Ac={w | w A}

The difference of two sets A and B, the set of elements of A that do not belong to B

A-B={ w | w A, w B }= ABc


Basic Concepts


Basic Concepts

A reflection of a set B, denoted is defined as

The translation of a set A by point z=(z1,z2), denoted (A)z is

defined as

BbbwwB para,ˆ

B̂

AazaccA z for ,


Logic Operations

Some logic operations

A B NOT(A)

A OR B A AND B A XOR B


Dilation

Definition:

Examples:

A B

ABxBA x ˆ|

A BAStructuring

Element

(SE)

B

B

.

B̂

B̂

A


Dilation

Example:


Dilation

Application Example:

SE


Erosion

Definition:

Examples:

B

A BA

BAA

B

ABxBA x |

Example:


Erosion


Erosion

Application example: eliminating irrelevant details

image of squares of

size 1, 3, 5 ,7, 9 e 15

after with dilation with

13 13 square SE

after erosion with

13 13 square SE


Erosion

Application example: eliminating lines using square SE

15×15 45×45

input image

486×486

11×11


Erosion-Dilation Duality

Erosion and Dilation are duals, formally

and

in other words, the erosion of A by B is the complement of

the dilation of A by , and vice-versa.

BABA c ˆ)(

BABA cc ˆ)(

B̂


Opening and Closing

Opening

Closing

BBABA )()(

BBABA )()(

)( BA BBABA )()(

BBABA )()()( BA

original image SE

Geometric Interpretation of Opening

The “rolling ball” rolls around the inside of A’s boundary.

The points covered by B is A○B


Opening and Closing

ABBBA zz


Opening and Closing

Geometric Interpretation of Closing

The “rolling ball” rolls on the outer boundary of A.

The points covered by B is the complement of AB.


Opening-Closing Duality

Opening and Closing are duals, formally

and

in other words, the complement of the closing of A by B is

the opening of the complement of A by , and vice-versa.

BABA cc ˆ

BABA cc ˆ

B̂


Opening and Closing

Application Example


Hit or Miss

Definition: for

A

Ac

1B

2B

× ×

×

21 BB

A 1B

cA 2B

BA Ο

hit

miss

𝐴⊛𝐵1,2 = 𝐴⊖𝐵1 ⋂ 𝐴𝑐 ⊖𝐵2

Hit or Misssim


single pixel

detection

upper-right

corner

detection

vertical-right

border

detection


Basic Morphological Algorithms

Boundary Extraction

)( BAAA



Boundary Extraction

)( BAAA



Hole Filling

Problem formulation Let A be the set of pixels in the

contour of a region Y

Let p a pixel in the interior of the region Y

Compute Y

Solution

where

X0=p

B is a proper SE

Y=Xk=Xk-1

,...3,2,1,)( 1 kABXX c

kk



Extracting Connected Components

X- ray of a chicken breast

with bone fragments

Thresholded image

Image eroded with a

5×5 SE



Thinning: Reduces binary objects or shapes to strokes that

are a single pixel wide

input image after many thinnings



Thickening: thicken objects without joining disconnected 1s

Usual procedure: thin background of image and complement result

input image after many thickenings



Skeletons: S(A)

z is a point of S(A) if the largest disk (D)z

centered at z and contained in A, touches

the boundary of A at two or more different

places.



Skeletons: example

input image skeleton



Pruning: cleans up parasitic components that

remains after thinning or skeletonizing.

input image after pruning


Morphological Reconstruction

Geodesic dilation

Let F denote a marker image and G a mask image.

GBFFDG )1(

FDDFD n

GG

n

G

)1()1()(

limits the growth



Reconstruction by dilation

Is the geodesic dilation of marker F with respect to

mask G, iterated until stability is achieved, that is

until

FDFR k

G

D

G

)( FDFD k

G

k

G

)1()(



Reconstruction by dilation GBFFDG )1(

FDDFD n

GG

n

G

)1()1()(

FDFR k

G

D

G

)(

FDFD k

G

k

G

)1()(



Reconstruction by dilation GBFFDG )1(

FDDFD n

GG

n

G

)1()1()(

FDFR k

G

D

G

)(

FDFD k

G

k

G

)1()(

If the intersection between

the dilated marker and the

mask is non empty,

reconstruction by

dilation rebuilds the

mask from that!



Opening by reconstruction

Erosion first removes small objects and reconstruction by

dilation is executed from them. Formally

where indicates n erosions of F by B.

Note that:

F is the mask image

is the marker image

Restores exactly the shape of objects in the mask F that

remain after erosion

)][()( nBFRFO D

F

n

R

)( nBF

)( nBF



Opening by reconstruction (example): extract characters

that contain long vertical strokes

Text image (I) 918×2018 pixels

Character height aprox. 50 pixels Erosion with a SE (𝐵1) of 51×1 pixels

Opening of the eroded image Reconstruction of the eroded image

does not

restore the

original

𝐼 ⊖ 𝐵1

𝑂𝑅(1)

𝐼 𝐼 ∘ 𝐵1 SE (𝐵2)

of 3×3

pixels

restores

exactly the

original!

Summary



Structuring Elements

only the form matters used infrequently in

practice


Dilation and Erosion Dilation

The dilation of a gray scale image f by a flat SE b at any

location (x,y) is the maximum of the image in the region

coincident with b when the origin of b is (x,y).

Erosion

The erosion of a gray scale image f by a flat SE b at any

location (x,y) is the minimum of the image in the region

coincident with b when the origin of b is (x,y).

)},({max),)((),(

tysxfyxbfbts

)},({min)(),(

tysxfbfbts


Dilation and Erosion

Examples

X-ray image with

448×425 pixels

Erosion using a flat

disk SE with radius

of 2 pixels.

Dilation using the

same SE


Opening and Closing

Opening (same form)

Closing (same form)

Duality (same form)

bbfbf )(

bbfbf )(

bfbf cc ˆ bfbf cc ˆ


Opening and Closing

Opening and Closing

Original 1-D signal

Flat structuring element

Opening

Flat SE pushed down along the top of

the signal

Closing


Opening and Closing

Examples

X-ray image with

448×425 pixels

Opening using a flat

disk SE with radius

of 3 pixels.

Closing using an SE

with radius 5

Eliminates small/thin

bright regions.

Eliminates small/thin

dark regions.


Gray Scale Morphological Algorithms

Morphological smoothing

566×566 pixels

Cygnus Loop

Supernova, taken in

the X-ray banc by

Hubble telescope

Result of opening

and closing with a

disk SE of radius 1

pixel

Result of opening

and closing with a

disk SE of radius 5

pixels

Result of opening

and closing with a

disk SE of radius 3

pixels

bbf )(



Morphological gradient

512×512 image of a

head CT scan

Result of dilation

Gradient result Result of erosion

)()( bfbfg



Top-hat transformations

Bottom-hat transformations

One of the principal applications is in removing objects from an

image by using a SE that does not fit the objects to be removed.

Top-hat keeps light objects on a dark background.

Bottom-hat keeps dark object on a light background.

)( bffThat

fbfBhat )(



Top-hat application: correction of non-uniform illumination.

Input 600×600 image Thresholded image

Image opened

with a disk SE

of radius 40

Top-hat

transformation

Thresholded top-

hat image


Gray-scale Morphological Reconstruction

Geodesic dilation

Let f and g denote the marker and mask gray-scale

images, where f g

where denotes the point-wise minimum operator.

gbffDg )()()1(

fDDfD n

gg

n

g

)1()1()( )(



Geodesic erosion

Let f and g denote the marker and mask gray-scale images.

Its defined as

where denotes the point-wise maximum operator.

gbffEg )()()1(

fEEfE n

gg

n

g

)1()1()( )(



Reconstruction by dilation of a gray-scale mask image g

by a gray-scale marker image f, is the geodesic dilation of

f with respect go g iterated until stability is achieved

until

Reconstruction by erosion of g by f is similarly defined as

until

fDfR k

g

D

g

)( fDfD k

g

k

g

)1()(

fEfR k

g

E

g

)( fEfE k

g

k

g

)1()(



Opening by reconstruction of size n of an image f is

defined as the reconstruction by dilation of f from the

erosion of size n of f; that is

where indicates n erosions of f by b.

)][()( bnfRfO D

f

n

R

)( bnf



Top-hat by reconstruction consists of subtracting from an

image f its opening by reconstruction

fOffT n

Rhat

)(



Example

a) Input image 1134x1360 pixels

b) Opening by reconstruction of (a) using a horizontal line 71 pixels long in the erosion

c) Opening of (a) using the same line (just for comparison)

d) Top-hat by reconstruction

e) Top-hat (just for comparison)

f) Opening by reconstruction of (d) using a vertical line 11 pixels long

g) Dilation of (f) using a horizontal line 21 pixels long

h) Minimum of (d) and (g)

i) Using (h) as a marker and (g) as the mask and applying reconstruction by dilation with a vertical SE.


Next Topic

Segmentation

morphological image processing - ele – puc rioraul/imageanalysis/morphology.pdf · 9/3/2019...

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