Mathematical Morphology

Mathematical morphology is a way to study and modify shapes in an image.

We will start with binary images, where every pixel is either foreground or background.

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1. Binary Images

A binary image contains only two pixel values:

1 = foreground
0 = background

The blue pixels form the foreground object.

Instead of listing every 0 and 1, morphology often describes an image by storing only the locations of its foreground pixels. We call this collection of foreground locations \(X\).

\[ X \subseteq \mathbb{Z}^2 \]

Read this as: \(X\) is a collection of pixel locations on a 2D grid.

2. Matrix View and Location View

The same binary image can be described in two ways.

Matrix View

Store every pixel as a 0 or 1.

\[ \begin{bmatrix} 0 & 1 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{bmatrix} \]

Location View

Store only the locations where the image contains a 1.

\[ X = \{ (1,0), (2,0), (1,1) \} \]

Key idea: Both representations describe the same binary image. Morphology usually uses the location view because it makes shape operations easier to describe.

3. The Structuring Element

Morphology uses a small pattern called a structuring element.

We call the structuring element \(B\).

The structuring element acts like a small probe that moves across the image.

The locations containing a 1 are the active parts of the structuring element.

Purple cell: the origin, also called the anchor point.

\[ B \subseteq \mathbb{Z}^2 \]

Read this as: \(B\) is a collection of active locations in the structuring element.

4. Moving the Structuring Element

To perform morphology, we move the structuring element across the image one location at a time.

We use \(z\) to represent the current location of the anchor.

\[ B_z \]

\(B_z\) means: the structuring element \(B\) moved to location \(z\).

In the interactive visualizer, the purple outline shows the current location \(z\).

5. Reflection

Dilation uses a reflected version of the structuring element.

\[ B^r \]

\(B^r\) means: the structuring element reflected around its anchor.

On a 2D grid, this can be viewed as rotating the structuring element by \(180^\circ\).

6. Dilation

Dilation usually makes foreground objects larger.

At each location \(z\), ask:

Does at least one active point of the structuring element overlap a foreground pixel?

If yes, the output at that location becomes 1.

\[ X \oplus B = \left\{ z \mid (B^r)_z \cap X \neq \varnothing \right\} \]

Plain English:
Keep \(z\) if the reflected structuring element overlaps the foreground at least once.

Dilation: one overlap is enough → output = 1

7. Erosion

Erosion usually makes foreground objects smaller.

At each location \(z\), ask:

Do all active points of the structuring element fit inside the foreground?

If yes, the output at that location becomes 1.

\[ X \ominus B = \left\{ z \mid B_z \subseteq X \right\} \]

Plain English:
Keep \(z\) only if the entire active structuring element fits inside the foreground.

Erosion: every active point must fit → output = 1

8. Dilation vs. Erosion

Operation	Question	Output = 1 When	Typical Effect
Dilation	Does anything overlap?	At least one active structuring-element point overlaps \(X\).	Expands foreground objects.
Erosion	Does everything fit?	Every active structuring-element point lies inside \(X\).	Shrinks foreground objects.

Easy way to remember:

Dilation: one overlap is enough.
Erosion: everything must fit.

9. Interactive Example

Move the structuring element across the image and watch how the condition changes at each location.

Green = condition satisfied
Red = condition not satisfied
Purple = anchor position

Interactive Morphology Visualizer

Structuring Element

Purple = Origin (0,0)

Reflected Kernel

180° reflection for Dilation / Convolution

Image Grid (Set X)

Editable. Click to toggle and jump window.

Output Array

Computation Result

Ready. Click "Auto-Play Sweep", "Step Forward", or click directly on the Image Grid.

10. What Do These Operations Do?

Dilation

Dilation can:

make objects larger,
fill small gaps,
connect nearby regions,
make thin foreground regions thicker.

Erosion

Erosion can:

make objects smaller,
remove small foreground regions,
break narrow connections,
separate nearby objects.

11. Summary

A binary image contains foreground pixels and background pixels.

We call the collection of foreground locations \(X\).

A structuring element, \(B\), is a small pattern used to examine the shape of \(X\).

The anchor tells us where the structuring element is currently positioned.

Dilation produces an output of 1 when at least one active point overlaps the foreground.

Erosion produces an output of 1 only when every active point fits inside the foreground.

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