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Morphological Image Processing. Digital Image Processing. Bundit Thipakorn, Ph.D. Computer Engineering Department. Mathematical Morphology. Mathematical morphology is a tool that:. is based on set theory;

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slide1

Morphological Image Processing

Digital Image Processing

Bundit Thipakorn, Ph.D.

Computer Engineering Department

slide2

Mathematical Morphology

Mathematical morphology is a tool that:

  • is based on set theory;
  • concerns the study of form and structure (shape and properties of objects/regions of an image);
  • can be used for pre/post-processing within image analysis such as filtering, thinning, and pruning.
slide3

Morphology

= empty set

Ø

= is a member or element of

= ‘implies that’

Cont’d.

Basic Definitions:

1. Standard set notation:

The expression {p | <condition(s) > } means ‘the set of points, p, such that the listed <condition(s)> is (are) true’.

slide4

Morphology

Cont’d.

2. ‘Included’

The binary object can be identified as a subset of pixels A contained or ‘included’ within the total set of image pixels N. That is,

Therefore if p is an element of A then p is also an element of N. That is,

slide5

Morphology

Cont’d.

3. ‘Complement’

The ‘complement’ of the set A is the set which are not element of A, that is, all the white background pixels in the image. The complement of A can be denoted as Ac and defined as:

Therefore, if p is an element of A it cannot be an element of Ac, i.e.

slide6

Morphology

Cont’d.

4. ‘Transposition (Reflection)’

The ‘transposition’ of the set of pixels, A, is the set such that each pixel position is reflected about a defined origin. The ‘transposition’ of A, denoted Â, is defined as:

5. ‘Union’

The ‘union’ of two sets A and B is denoted by A  B and defined as:

slide7

Morphology

Cont’d.

6. ‘Intersection’

The ‘intersection’ of two sets A and B is denoted by A  B and defined as:

7. ‘Difference’

The ‘difference’ between two sets A and B is denoted by A\B and defined as:

slide8

Morphology

Cont’d.

This is analogous to the logical XOR (exclusive OR) operation. The resulting set is the set such that each pixel is belonged to either set A or set B.

8. ‘Translation’

The ‘translation’ of A by x = (x1, x2), denoted (A)x, is defined as:

A vector addition

slide9

Morphology

Cont’d.

Original Image

Reflection of set A

to produce set B.

slide10

Morphology

Cont’d.

slide11

Morphology

Cont’d.

A\B (A XOR B)

Translation, (A)x, of set A by vector x (2,2)

slide12

Morphology

Cont’d.

  • Set in mathematical morphology represent the shapes of objects in an image.
  • A binary image can be treated as a 2D point set of integer pairs (Z2). Each element of a set is a tuple (2-D vector) whose coordinates are the (x,y) coordinates of a black (by convention) pixel in the image.

X

= the origin (0,0)

Y

A discrete image ‘P’ = {(1,1), (1,2), (1,3), (2,1),

(2,2), (2,3), (3,1)}.

An example of point set

slide13

Morphology

Cont’d.

  • Gray-scale digital images can be represented as sets of integer triples (Z3); (x, y, intensity value).
  • Morphological operators usually take a binary image (point set A) and another small point set B called a “structuring element”.
  • B is expressed with respect to a local origin called the “representative point” and the pixel in the image A corresponding to the representative point is called the “current” pixel.
  • Usually, the structuring element is sized 3x3 and has its origin at the center pixel.
slide14

Morphology

Cont’d.

Generic morphological operator:

  • Determine the shape of structuring element.
  • The structuring element is shifted over the image.
  • At each pixel of the image elements of the structuring element are compared with the set of the underlying pixels.
  • If the two sets of elements match the condition defined by the set operator, the pixel underneath the origin of the structuring element is set to a pre-defined value (0 or 1 for binary image).
slide15

Dilation and Erosion

  • Two operations which are fundamental to morphological analysis of images are ‘dilation’ and ‘erosion’.
  • Almost all morphological operations can be defined in terms of these two basic operations.

Dilation (Grow, Expand)

Enlarge the boundaries of regions of foreground pixels and fill in holes in the object (a binary image).

slide16

Dilation

Cont’d.

Let A and B as sets in Z2. The dilation of A by B, denoted A  B, is defined as:

Dilation

i.e. An output pixel will be set at all points where the translated B (structuring element) and the image A have a ‘non-empty’ intersection.

Generally, the dilation process is performed by laying the structuring element on the image and shifting it across the image in a manner similar to convolution. At each pixel,

slide17

Dilation

Cont’d.

  • If the origin of the structuring element coincides with a ‘0’ in the image, there is no change; move to the next pixel.
  • If the origin of the structuring element coincides with a ‘1’ in the image, perform the OR logic operation on all pixels within the structuring element.
slide18

Dilation

Cont’d.

3x3 Structuring Element

(a) An Original Image

(b)The Dilation of (a)

slide19

Erosion

Erosion (Shrink, Reduce)

Shrink the boundaries of regions of foreground pixels and enlarge holes in the object (a binary image).

Let A and B as sets in Z2. The dilation of A by B, denoted A B, is defined as:

Erosion

slide20

Erosion

Cont’d.

i.e. The erosion of A by B is simply the set of all point of A such that B is a subset of A.

Generally, the erosion process is similar to dilation, but we turn pixels to ‘0’, not ‘1’. At each pixel,

  • If the origin of the structuring element coincides with a ‘0’ in the image, there is no change; move to the next pixel.
  • If the origin of the structuring element coincides with a ‘1’ in the image, and any of the ‘1’ pixels in the structuring element extend beyond the object
slide21

Erosion

Cont’d.

(‘1’ pixels) in the image, then change the ‘1’ pixel in the image to a ‘0’.

Erosions can be made directional by using less symmetrical structuring elements. For example, a structuring element that is 10 pixels wide and 1 pixel high will erode in a horizontal direction only.

slide22

Erosion

Cont’d.

3x3 Structuring Element

(a) An Original Image

(b)The Erosion of (a)

slide23

Opening and Closing

The opening of an image A by the structuring element B is denoted by and is defined as

Opening

Erosion followed by a dilation using the same structuring element.

Opening can be used to eliminate all pixels in regions that are too small to contain the structuring element.

slide24

Opening and Closing

The closing of an image A by the structuring element B is denoted by and is defined as

Cont’d.

Closing

Dilation followed by an erosion using the same structuring element.

Closing connects objects that are close to each other, fills up small holes, and smoothes the object outline by filling up narrow gulfs.

slide25

Opening and Closing

Cont’d.

3x3 Structuring Element

(a) An Original Image

(b)The Opening of (a)

slide26

Opening and Closing

Cont’d.

3x3 Structuring Element

(a) An Original Image

(b)The Closing of (a)

slide27

Hit-or-Miss Transformation

= size of the structuring element

The hit-or-miss transformation of an image A by the structuring element B = (B1, B2) is denoted by and is defined as

*

*

Hit-or-Miss

Shape detection or Find local patterns of pixels.

where B1 = X = pattern of pixels that we want to find,

B2 = W - X,

W = a small window enclosing “X”.

slide28

Hit-or-Miss

Cont’d.

slide29

Boundary Extraction

b(A) = A - (A B)

The boundary of set A, b(A), can be obtained by the following operations:

slide30

Thinning and Thickening

The thinning of set A by a structuring element B, denoted A B, can be defined as:

A B = A - (A B)

*

If a sequence of structuring elements {B} = {B1, B2, B3, …, Bn} is used where Bi is a rotated version of Bi-1. The process is to thin A by one pass with B1, then thin the result with one pass of B2, and so on, unit A is thinned with one pass of Bn.

slide32

Thickening

The thickening of set A by a structuring element B, denoted A B, can be defined as:

*

A B = A U (A B)