Write short notes on the Preliminaries of Mathematical Morphology.
Mathematical morphology is based on the language of set theory, making it a unified and powerful tool for solving various image processing problems.
1. Representation of Images as Sets
In morphological image processing, objects in images are represented using sets:
For binary images, the set elements belong to the 2D integer space Z², where each element is a 2D vector denoting the coordinates of a foreground pixel.
For grayscale digital images, sets are represented in Z³. Here, each element has two components for pixel location and one for intensity value.
Sets in higher-dimensional spaces can represent other attributes like color or time-varying features.
This figure shows how sets are converted into digital images:
Left: Graphical set
Center: Embedded in rectangular background
Right: Digital grid image
2. Objects and Structuring Elements (SEs)
Morphological operations use two types of sets:
Objects → defined as sets of foreground pixels in the image
Structuring Elements (SEs) → small sets that probe the image
SEs may contain:
Foreground pixels (1)
Background pixels (0)
“Don’t care” elements (×) – whose value doesn’t matter during operation
These × elements can be ignored or adjusted based on value-matching needs.
This shows the SE format, where no border is present around SE, unlike the image. This is important because SEs are used like convolution kernels.
3. Embedding Sets into Arrays
Since images are stored as rectangular arrays, arbitrary-shaped sets must be embedded in arrays for processing.
All non-object pixels are filled with a background value
This embedding is essential to allow sliding SE operations (like in convolution)
4. Set Reflection
Reflection of a structuring element B about its origin is denoted as B̂ (read as B-hat):
This means each point (x,y) in B becomes (−x,−y) in B̂. This operation is equivalent to rotating the SE by 180°.
This figure illustrates several digital SEs and their reflections. The dot shows the origin.
5. Set Translation
Translation of a set B by a point z = (z₁, z₂) is denoted by:
