661e432fd7a2bWeek6,7.pdf digital image processing

Digital Image Processing
Spring 2024
1
Zulaikha Kiran, 2024

Material taken from:
Digital Image Processing by Gonzalez and Woods – 4th Edition
Week 6, 7
2

Morphological Image Processing
3

• In image processing, we use morphology with two types of sets of pixels: objects
and structuring elements (SE’s).
• Typically, objects are defined as sets of foreground pixels.
• Structuring elements can be specified in terms of both foreground and background pixels.
• In forming rectangular arrays for digital image processing we assign a background
value to all pixels that are not members of object sets
4

Structuring elements
• A structuring element is a small image – used as a moving window
Structuring elements and their reflections about the origin
5

• Morphological Image processing is like spatial filtering, in that the
structuring element is moved like a window across the whole image,
to find values of pixels in a new image.
• The value of this new pixel depends on the operation performed
6

Basic Morphological Operations
• Erosion
• Erosion of A by B is the set of all points z such that B, translated by z, is
contained in A
• In a binary image, if any of the pixel (in the neighbourhood defined by
structuring element) is 0, then output is 0
• Dilation
• Dilation of A by B is the set of all displacements, z, such that the foreground
elements of B̂ overlap at least one element of A
7

Erosion
• Erosion of image f by structuring element s is given by f ⊖ s
• The structuring element s is positioned with its origin at (x, y) and the new
pixel value is determined using the rule:
𝑔 𝑥, 𝑦 =
1 𝑖𝑓 𝑠 𝑓𝑖𝑡𝑠 𝑓
0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
• For each foreground pixel, superimpose the structuring element on top of
the input image so that the origin of the structuring element coincides with
the input pixel position.
• If for every pixel in the structuring element, the corresponding pixel in the
image underneath is a foreground pixel, then the input pixel is left as it is.
• If any of the corresponding pixels in the image are background, however,
the input pixel is also set to background value.
8

Erosion
9

Zulaikha Kiran, 2024 10
Input Image Output Image
Structuring Element
https://penny-xu.github.io/blog/mathematical-
morphology

Erosion
11

• Erosion can split apart joined objects
• Erosion can remove extrusions
• Erosion shrinks objects
12

Erosion
• A binary image of a wire-
bond mask in which
foreground pixels are shown
in white.
• Image eroded using square
structuring elements of sizes
11x11 15x15 and 45x45
elements, respectively, all
valued 1.
13

• Count the number of objects using MATLAB
14

Dilation
• Dilation of image f by structuring element s is given by f ⊕ s
• The structuring element s is positioned with its origin at (x, y) and the new pixel
value is determined using the rule:
• 𝑔 𝑥, 𝑦 =
1 𝑖𝑓 𝑠 ℎ𝑖𝑡𝑠 𝑓
0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
• For each background pixel superimpose the structuring element on top of the
input image so that the origin of the structuring element coincides with the input
pixel position
• If at least one pixel in the structuring element coincides with a foreground pixel in
the image underneath, then the input pixel is set to the foreground value
• If all the corresponding pixels in the image are background, however, the input
pixel is left at the background value
15

Dilation
16

Zulaikha Kiran, 2024 17
Input Image Output Image
Structuring Element
https://penny-xu.github.io/blog/mathematical-
morphology

Dilation
18

• Dilation can
• Repair breaks
• Repair intrusions
• Enlarge objects
19

Compound Operators
• Combinations of erosion and dilation
• Opening
• Closing
21

Opening
• Erosion followed by dilation
• Denoted by f ⃝ s
• f ⃝ s = (f ⊖ s) ⊕ s
22

Opening
23

Closing
• Dilation followed by erosion
• Denoted by f ● s
• f ● s = (f ⊕ s) ⊖ s
24

Closing
25

Hit or Miss Transform
• I ◉B = {z|(B )Z ⊆I }
27

Morphological Algorithms
• Boundary Extraction
• Region Filling
• Connected Components Extraction
• Skeleton Extraction
30

Boundary Extraction
• The boundary of set A denoted by β(A) is obtained by first eroding A
by a suitable structuring element B and then taking the difference
between A and its erosion.
• ꞵ(A) = A – (A ⊖ B)
31

• Boundary extraction using a 3 x 3 square structuring element
32

Region Filling
• A hole may be defined as a background region surrounded by a
connected border of foreground pixels.
• Steps:
• Start from a known point p and take X0= p,
• Then take the next values of Xk as: Xk = (Xk-1 ⊕ B ) ⋂ AC
• Terminate iterations if Xk = Xk-1
• The intersection of dilation and the complement of A limits the result to
inside the region of interest
• The set union of Xk and A contains the filled set and its boundaries
33

Extraction of connected components
• Steps:
• Start from a known point p and take X0= p,
• Then take the next values of Xk as: Xk = (Xk-1 ⊕ B ) ⋂ A
• Terminate iterations if Xk = Xk-1
• The component Y is given as Y = Xk
36

Skeleton
• If z is a point of S(A), and (D)Z is the largest disk centered at z and
contained in A, one cannot find a larger disk (not necessarily centered
at z) containing (D)Z and simultaneously included in A. A disk (D)Z
satisfying these conditions is called a maximum disk.
• If (D)Z is a maximum disk, it touches the boundary of A at two or more
different places.
38

Skeleton
39

661e432fd7a2bWeek6,7.pdf digital image processing

More Related Content

Similar to 661e432fd7a2bWeek6,7.pdf digital image processing (20)

Recently uploaded (20)

661e432fd7a2bWeek6,7.pdf digital image processing