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Electrical & Computer Engineering Dr. D. J. Jackson Lecture 14-
Computer Vision &
Digital Image Processing
Morphological Image Processing I
Introduction
• Morphology – a branch of biology concerned with
the form and structure of plants and animals
• Mathematical morphology – a tool for extracting
image components useful in the representation and
description of image shape including:
– Boundaries
– Skeletons
– Convex hull
• We will also look at morphological techniques for
– Filtering
– Thinning
– Pruning
Electrical & Computer Engineering Dr. D. J. Jackson Lecture 14-
Preview
• Language of mathematical morphology is set theory
• Sets in mathematical morphology represent objects in an
image
- For example, the set of all black pixels in a binary image is a complete morphological description of the image
• For binary images, sets are members of the 2-D integer
space Z^2
- Each element of the set is a tuple (2-D vector) whose coordinates are the (x,y) coordinates of a black (or white depending on convention) pixel in the image
• Gray-scale digital images are represented as sets in Z^3
- Coordinates and gray-scale value
• Higher dimensioned sets could represent attributes such as
color, time varying components, etc.
Basic Concepts from Set Theory
- Let A be a set in Z^2. If a = ( a 1 , a 2 ) is an element of A , then we write a ∈ A
- If a is not an element of A we write a ∉ A
- A set with no elements is called the null or empty set and is denoted by the symbol ∅
- A set is specified by the contents of two braces: {·}
- For binary images, the elements of the sets are the coordinates of pixels representing objects
- If we write an expression of the form C ={ w | w = - d , for d ∈ D } we mean that C is the set of elements, w , such that w is formed by multiplying each of the two coordinates of all the elements of set D by -
Electrical & Computer Engineering Dr. D. J. Jackson Lecture 14-
Basic Concepts from Set Theory
- The reflection of set B is defined as
B ={ w | w = - b , for b ∈ B }
- The translation of set B by point z = ( z 1 , z 2 ) is defined as
( B ) z = { c | c = b + z, for b ∈ B }
Structuring Elements
- Set reflection and translation are used extensively in morphological operations based on structuring elements (SE)
- An SE is a small set (or subimage) used to “probe” an area of interest for certain properties - May be of arbitrary shape and size - In practice an SE is generally a regular geometric shape (square, rectangle, diamond, etc.) - Generally padded to a rectangular array for image processing
Electrical & Computer Engineering Dr. D. J. Jackson Lecture 14-
Structuring Elements (continued)
• Two-dimensional, or flat, structuring elements
consist of a matrix of 0's and 1's, typically much
smaller than the image being processed.
• The center pixel of the structuring element, called
the origin, identifies the pixel of interest--the pixel
being processed.
• The pixels in the structuring element containing 1's
define the neighborhood of the structuring element.
Operation with a Structuring Element (example)
Electrical & Computer Engineering Dr. D. J. Jackson Lecture 14-
Structuring Elements: Matlab Functions
Dilation
• With A and B as sets in Z^2 , the dilation of A by B , denoted
A ⊕ B , is defined as
A ⊕ B = { z | ( B ) z ∩ A ≠ ∅}
• This formulation is based on the reflection of B about its
origin and shifting this reflection by z
• The dilation of A by B is the set of all displacements , z , such
that B and A overlap by at least one element
• Therefore, another expression for the dilation of A by B is
A ⊕ B = { z | [( B ) z ∩ A ] ⊆ A }
• Set B is the structuring element
Electrical & Computer Engineering Dr. D. J. Jackson Lecture 14-
Dilation Example
- B = B because B is symmetric with respect to its origin
- The dashed line shows the original set A and the solid boundary shows the limit beyond which any further displacements of the origin of B by z would cause the intersection of B and A to be empty
- All points inside this boundary constitute the dilation of A by B
- The second case shows more dilation vertically than horizontally
Dilation Application
• One simple application of dilation is for bridging gaps
• In the image below, the maximum break length is two pixels
• Although low pass filtering can be used to accomplish the
same task, this generates a gray-scale image that must then
be thresholded to produce a resulting binary image
Electrical & Computer Engineering Dr. D. J. Jackson Lecture 14-
Erosion
• With A and B as sets in Z^2 , the erosion of A by B , denoted
A B , is defined as
A B = { z | ( B ) z ⊆ A }
• In words, the erosion of A by B is the set of all points z such
that B , translated by z , is contained in A
• Dilation and erosion are duals of each other with respect to
set complementation and reflection
• Therefore
( A B ) c^ = Ac^ ⊕ B
and
( A ⊕ B ) c^ = Ac^ B
Erosion Example
Erosion and Dilation Application
• One simple application of erosion is for eliminating irrelevant
detail (in terms of size) from a binary image
• Note: In general, dilation does not restore fully the eroded
objects
Erosion of a binary image with a 13x13 size structuring element and subsequent dilation of the result with the same element.
