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Agenda

- Binary morphological processing
- Erosion and dilation
- Opening and closing
- Gray-scale morphological processing
- Erosion and dilation
- Morphological gradients

Roger S. Gaborski

2

Introduction

- Morphology: a branch of biology dealing with the form and structure of creatures
- Mathematical morphology:
- Extract image components based on shape

e.g. boundaries, skeletons, convex hull, etc

- Image denoise

e.g. reduce noise after edge detection

Roger S. Gaborski

3

Binary Morphological Processing

Non-linear image processing technique

Order of sequence of operations is important

Linear: (3+2)*3 = (5)*3=15

3*3+2*3=9+6=15

Non-linear: (3+2)2 = (5)2 =25 [sum, then square]

(3)2 + (2)2 =9+4=13 [square, then sum]

Based on geometric structure

Used for edge detection, noise removal and feature extraction

Used to ‘understand’ the shape/form of a binary image

Roger S. Gaborski

4

Image – Set of Pixels

Basic idea is to treat an object within an image as a set of pixels (or coordinates of pixels)

In binary images

Background pixels are set to 0 and appear black

Foreground pixels (objects) are 1 and appear white

Roger S. Gaborski

5

Morphological Image Processing

A-B = A- (A∩B)

From: Digital Image Processing, Gonzalez,Woods

And Eddins

Roger S. Gaborski

6

Translation of Object A by vector b

Define Translate object A by vector b:

At = { t I2 : t = a+b, a A }

Where I2is the two dimensional image space that contains the image

Definition of DILATION is the UNION of all the translations:

A B = U {t I2 : t = a+b, a A} for all b’s in B

Roger S. Gaborski

7

DILATION

A

A1= A B

B

B

a

a

Object B is one point located at (a,0) A1: Object A is translated by object B

Since dilation is the union of all the translations, A B = U Atwhere the set union U is for all the b’s in B, the dilation of rectangle A in the positive x direction by a results in rectangle A1 (same size as A, just translated to the right)

Roger S. Gaborski

8

DILATION – B has 2 Elements

A

A2 A1

(part of A1 is under A2)

-a a -a a

Object B is 2 points, (a,0), (-a,0)

There are two translations of A as result of two elements in B

Dilation is defined as the UNION of the objects A1 and A2.

NOT THE INTERSECTION

Roger S. Gaborski

9

DILATION

Image (A)

SE (B)

Round Structuring Element (SE) can be interpreted

as rolling the SE around the contour of the object.

New object has rounded corners and is larger by

½ width of the SE

Dilation

Roger S. Gaborski

10

DILATION

Countless

translation

Vectors

Another approach

Rounded corners

Image (A)

SE (B)

Dilation

Roger S. Gaborski

11

DILATION

Square corners

Image (A)

SE (B)

Square Structuring Element (SE) can be interpreted

as moving the SE around the contour of the object.

New object has square corners and is larger by

½ width of the SE

Dilation

Roger S. Gaborski

12

DILATION

Countless

translation

vectors

Another approach

Square corners

Image (A)

SE (B)

Dilation

Roger S. Gaborski

13

DILATION

The shape of B determines the final shape of the dilated object. B acts as a geometric filter that changes the geometric structure of A

Roger S. Gaborski

14

Morphological Image Processing

From: Digital Image Processing, Gonzalez,Woods

And Eddins

Roger S. Gaborski

15

Morphological Image Processing

Image A

Image B

~ A

A U B

A ∩B

A-B = A ∩(~B)

From: Digital Image Processing, Gonzalez,Woods

And Eddins

Roger S. Gaborski

16

Original Image

Translation Process

Dilated

Image

From: Digital Image Processing, Gonzalez,Woods

And Eddins

Roger S. Gaborski

17

imdilate

- IM2 = IMDILATE(IM,NHOOD) dilates the image IM, where NHOOD is a
- matrix of 0s and 1s that specifies the structuring element
- neighborhood. This is equivalent to the syntax IIMDILATE(IM,
- STREL(NHOOD)). IMDILATE determines the center element of the
- neighborhood by FLOOR((SIZE(NHOOD) + 1)/2).
- >> se = imrotate(eye(3),90)
- se =
- 0 0 1
- 0 1 0
- 1 0 0
- >> ctr=floor(size(se)+1)/2
- ctr =
- 2 2

1

2

3

1 2 3

Roger S. Gaborski

18

Im (original image)

Im2 (dialated image)

>> Im = zeros([13 19]);

>> Im(6,6:8)=1;

>> Im2 = imdilate(Im,se);

1

2

3

Roger S. Gaborski

19

1 2 3

INPUT IMAGE

DILATED IMAGE

>> I = zeros([13 19]);

>> I(6, 6:12)=1;

>> SE = imrotate(eye(5),90);

>> I2=imdilate(I,SE);

>> figure, imagesc(I)

>> figure, imagesc(SE)

>> figure, imagesc(I2)

1

2

3

4

5

SE

1 2 3 4 5

Roger S. Gaborski

20

INPUT IMAGE

DILATED IMAGE

I I2

1

2

3

4

5

>> I(6:9,6:13)=1;

>> figure, imagesc(I)

>> I2=imdilate(I,SE);

>> figure, imagesc(I2)

SE

1 2 3 4 5

Roger S. Gaborski

21

Morphological Image Processing

From: Digital Image Processing, Gonzalez,Woods

And Eddins

Roger S. Gaborski

23

Dilation and Erosion

DILATION: Adds pixels to the boundary of an object

EROSIN: Removes pixels from the boundary of an object

Number of pixels added or removed depends on size and shape of structuring element

Roger S. Gaborski

24

Original Image

Translation Process

Eroded

Image

From: Digital Image Processing, Gonzalez,Woods

And Eddins

Roger S. Gaborski

25

REODED IMAGE

2 pixel wide

INPUT IMAGE

>> I=zeros(13, 19); I(6:9,6:13)=1;

>> figure, imagesc(I)

>> I2=imerode(I,SE);

>> figure, imagesc(I2)

1

2

3

1 2 3

26

Roger S. Gaborski

SE = 3x1

Morphological Image Processing

Original Image

Erosion with a disk of radius 10

From: Digital Image Processing, Gonzalez,Woods

And Eddins

Erosion with a disk

of radius 5

Erosion with a disk

of radius 20

27

Roger S. Gaborski

Combinations

In most morphological applications dilation and erosion are used in combination

May use same or different structuring elements

Roger S. Gaborski

28

Morphological Opening and Closing

Opening of A by B

A o B = (AO B) B; imopen(A, B)

Erosion of A by B, followed by the dilation of the result by B

Closing of A by B

A B = (A B) O B; imclose(A, B)

Dilation of A by B, followed by the erosion of the result by B

Roger S. Gaborski

29

MATLAB Function strel

strel constructs structuring elements with various shapes and sizes

Syntax: se = strel(shape, parameters)

Example:

se = strel(‘octagon’, R);

R is the dimension – see help function

Roger S. Gaborski

30

- Erosion of A by B, followed by the dilation of the result by B

Erosion- if any element of structuring element overlaps with background output is 0

f (original image)

fe (eroded image)

FIRST - EROSION

>> se = strel('square', 20);

fe = imerode(f,se);

figure, imagesc(fe),title('fe')

Roger S. Gaborski

31

Outputs 1 at center of SE when at least one element of SE overlaps object

fe (eroded image)

fd (dilated image)

SECOND - DILATION

>> se = strel('square', 20);

fd = imdilate(fe,se);

figure, imagesc(fd),title('fd')

Roger S. Gaborski

32

figure, imagesc(FO),title('FO')

FO (opened image)

fd (dilated image in previous slide)

Roger S. Gaborski

33

What if we increased size of SE for DILATION operation??

se = 25 se = 30

se = strel('square', 30);

fd = imdilate(fe,se);

figure, imagesc(fd),title('fd')

se = strel('square', 25);

fd = imdilate(fe,se);

figure, imagesc(fd),title('fd')

Roger S. Gaborski

34

Dilation of A by B

Outputs 1 at center of SE when at least one element of SE overlaps object

se = strel('square', 20);

fd = imdilate(f,se);

figure, imagesc(fd),title('fd')

Roger S. Gaborski

35

Erosion- if any element of structuring element overlaps with background

output is 0

Roger S. Gaborski

36

Morphological Image Processing

original image

opening

opening + closing

From: Digital Image Processing, Gonzalez,Woods

And Eddins

Roger S. Gaborski

38

Hit or Miss Transformation

Usage: to identify specified configuration of pixels, e.g.

isolated foreground pixels

pixels at end of lines (end points)

Definition

A B = (A B1) ∩(AcB2)

A eroded by B1, intersects A complement eroded by B2 (two different structuring elements: B1 , B2)

Roger S. Gaborski

39

Hit or Miss Example

Find cross shape pixel configuration:

MATLAB Function: C = bwhitmiss(A, B1, B2)

Roger S. Gaborski

40

A eroded by B1

Complement of Original

Image and B2

Erosion of A complement

And B2

Intersection of eroded images

Roger S. Gaborski

41

From: Digital Image Processing, Gonzalez,Woods

And Eddins

A eroded by B1

Complement of Original

Image and B2

Erosion of A complement

And B2

Intersection of eroded images

Roger S. Gaborski

42

From: Digital Image Processing, Gonzalez,Woods

And Eddins

A eroded by B1

Complement of Original

Image and B2

Erosion of A complement

And B2

Intersection of eroded images

Roger S. Gaborski

43

From: Digital Image Processing, Gonzalez,Woods

And Eddins

A eroded by B1

Complement of Original

Image and B2

Erosion of A complement

And B2

Intersection of eroded images

Roger S. Gaborski

44

From: Digital Image Processing, Gonzalez,Woods

And Eddins

A eroded by B1

Complement of Original

Image and B2

Erosion of A complement

And B2

Intersection of eroded images

Roger S. Gaborski

45

From: Digital Image Processing, Gonzalez,Woods

And Eddins

A eroded by B1

Complement of Original

Image and B2

Erosion of A complement

And B2

Intersection of eroded images

Roger S. Gaborski

46

From: Digital Image Processing, Gonzalez,Woods

And Eddins

Hit or Miss

Have all the pixels in B1 (hits all pixels in B1), but none of the pixels in B2 (misses all pixels in B2)

More precisely, hit-and-miss operation

Roger S. Gaborski

47

Hit or Miss Example #2

Locate upper left hand corner pixels of objects in an image

Pixels that have east and south neighbors (Hits) and no NE, N, NW, W, SW Pixels (Misses)

B1 = B2 =

Don’t

Care about

SE

Roger S. Gaborski

48

Morphological Image Processing

G = bwhitmiss(f, B1, B2);

Figure, imshow(g)

From: Digital Image Processing, Gonzalez,Woods

And Eddins

Roger S. Gaborski

49

bwmorph(f, operation, n)

Implements various morphological operations based on combinations of dilations, erosions and look up table operations.

f: input binary image

operation: a string specifying the desired operation

n: a positive integer specifying iteration times (default: 1)

Roger S. Gaborski

50

Example 1: thinning

- To reduce binary objects or shapes in an image to strokes whose width is 1 pixel
- Matlab implementation

>> f = imread(‘fingerprint_cleaned.tif’);

>> g = bwmorph(f, ‘thin’, 1);

>> g2 = bwmorph(f, ‘thin’, 2);

>> g3 = bwmorph(f, ‘thin’, Inf);

Roger S. Gaborski

51

Thinned once

Thinned twice

Thinned until stability

Roger S. Gaborski

52

From: Digital Image Processing, Gonzalez,Woods

And Eddins

Morphological Image Processing

Example 2: skeletonization

From: Digital Image Processing, Gonzalez,Woods

And Eddins

Roger S. Gaborski

54

Grayscale Morphological Processing

Similar to spatial convolution

Dilation: local maximum operator

Erosion: local minimum operator

Roger S. Gaborski

55

Grayscale Morphological Processing

Morphological Gradient:

Dilated image – eroded image

Half-gradient by erosion:

Image – eroded image

Half-gradient by dilation:

Image – dilated image

Roger S. Gaborski

59

Morphological Gradient: Dilated image – Eroded image

Increase size of SE: more difference area in morph-gradient image

se = ones(7)

Roger S. Gaborski

61

Directional GradientsFunction of Structure Element

horizontal_gradient

= imdilate(I,seh) - imerode(I,seh);

vertical_gradient

= imdilate(I,sev) - imerode(I,sev);

Where

I : the input image

seh = strel([1 1 1]);

sev = strel([1;1;1]);

Roger S. Gaborski

63

Horizontal GradientGradient Perpendicular to Edge

Vertical edges detected using horizontal gradient

Roger S. Gaborski

64

Vertical GradientGradient Perpendicular to Edge

Horizontal edges detected using vertical gradient

Roger S. Gaborski

66

Label connected components

- How many objects

in the image?

- 6 squares
- 3 disks
- 7 letters of “a”
- 9 vertical bars
- Each object

= a connected region/component

Roger S. Gaborski

68

Removing background

resulted in loss of small stripes

Used gray scale gradient

Processing to find stripes

image by masks

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