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Morphological Operator Properties

Published by Ka Ming Lui on August 28, 2026

Dilation, erosion, opening, and closing describe what transformation is applied. Operator properties describe how that transformation behaves mathematically.

This page assumes that dilation, erosion, opening, closing, structuring-element reflection, and the origin of \(B\) have already been introduced. Here, we focus on relationships and properties rather than repeating how those operations are calculated.
Quick Navigation
1. From Operations to Properties 2. Increasing 3. Extensive 4. Anti-Extensive 5. Idempotent 6. Translation Invariance 7. Linearity 8. Distributivity 9. De Morgan’s Laws 10. Morphological Duality 11. Why the Origin Matters 12. Operator Comparison 13. Properties vs. Topology 14. Interactive Explorer

1. From Operations to Properties

Let \(\Psi\) represent an image operator. Applying it to a binary image \(X\) produces another set:

\[ X \xrightarrow{\Psi} \Psi(X). \]
The Question Changes
Start with an Input \(X\)
↓
Apply a Known Morphological Operator \(\Psi\)
↓
Obtain \(\Psi(X)\)
↓
Ask Which Mathematical Relationships Must Hold
We are no longer asking how a particular output pixel is calculated. We are asking what relationships the complete operator preserves.

2. Increasing

An operator \(\Psi\) is increasing when

\[ X\subseteq Y \quad\Rightarrow\quad \Psi(X)\subseteq\Psi(Y). \]

Before the Operator

Y
X
\(X\subseteq Y\)

After the Same Operator

\(\Psi(Y)\)
\(\Psi(X)\)
\(\Psi(X)\subseteq\Psi(Y)\)
If \(X\) begins inside \(Y\), an increasing operator does not reverse that ordering.
Dilation, erosion, opening, and closing are increasing.

3. Extensive

An operator is extensive when the original set is always contained in the output:

\[ X\subseteq\Psi(X). \]
\(\Psi(X)\)
X
An extensive operator never loses an original foreground point. It may keep the same foreground or add more.

Closing is extensive:

\[ X \subseteq X\bullet B. \]

Dilation is extensive when the origin belongs to \(B\):

\[ 0\in B \quad\Rightarrow\quad X\subseteq X\oplus B. \]

4. Anti-Extensive

An operator is anti-extensive when its output remains inside the original set:

\[ \Psi(X)\subseteq X. \]
X
\(\Psi(X)\)
An anti-extensive operator may remove foreground points, but it cannot create new foreground outside the original set.

Opening is anti-extensive:

\[ X\circ B \subseteq X. \]

Erosion is anti-extensive when the origin belongs to \(B\):

\[ 0\in B \quad\Rightarrow\quad X\ominus B \subseteq X. \]

5. Idempotent

An operator is idempotent if applying it again does not change the result:

\[ \Psi(\Psi(X)) = \Psi(X). \]
\(X\)
↓
Apply \(\Psi\)
↓
\(\Psi(X)\)
↓
Apply the Same \(\Psi\) Again
↓
No Further Change

Opening and closing are idempotent:

\[ (X\circ B)\circ B = X\circ B \] \[ (X\bullet B)\bullet B = X\bullet B. \]
Dilation and erosion are generally not idempotent. Repeating them normally continues changing the image.

6. Translation Invariance

Let \(X_z\) represent \(X\) translated by displacement \(z\). A translation-invariant operator satisfies

\[ \Psi(X_z) = (\Psi(X))_z. \]
Translate \(X\) by \(z\), Then Apply \(\Psi\)
↓
Same Result
↓
Apply \(\Psi\) First, Then Translate the Output by \(z\)
Dilation, erosion, opening, and closing are translation invariant in their mathematical definitions.
Padding and cropping can create apparent differences at the edge of a finite stored image. That is a boundary-handling issue, not a failure of the underlying mathematical property.

7. Linearity and Why Morphology Is Nonlinear

For an ordinary numerical image operator \(T\), linearity means

\[ T(af+bg) = aT(f)+bT(g). \]
Σ

Ordinary Linear Filtering

A fixed convolution kernel combines samples using weighted sums.

\[ (af+bg)*h = a(f*h)+b(g*h). \]
max

Mathematical Morphology

Morphological operations are based on ordering, geometric fitting, set relations, maxima, or minima.

For example, consider a local maximum operator \(T\):

\[ f=(1,0,0), \qquad g=(0,1,0) \] \[ T(f)=1, \qquad T(g)=1 \] \[ T(f+g) = 1 \neq 2 = T(f)+T(g). \]
Mathematical morphology is generally nonlinear under ordinary arithmetic superposition.
For binary sets, ordinary vector addition is not the natural way to combine sets in the first place. The comparison becomes especially clear in gray-scale morphology, where dilation and erosion are based on maximum and minimum operations.

8. Distributivity Is Not the Same as Linearity

A nonlinear operator can still obey useful algebraic identities.

Dilation distributes over union:

\[ (X\cup Y)\oplus B = (X\oplus B) \cup (Y\oplus B). \]

Erosion distributes over intersection:

\[ (X\cap Y)\ominus B = (X\ominus B) \cap (Y\ominus B). \]

Dilation

Union ↔ Dilation

Dilating a union is equivalent to dilating the two sets separately and then taking their union.

Erosion

Intersection ↔ Erosion

Eroding an intersection is equivalent to eroding the two sets separately and then intersecting the results.

These identities are called distributive properties. They do not mean that dilation or erosion is linear in the ordinary vector-space sense.

9. De Morgan’s Laws

Let \(\Omega\) be the binary-image universe. The complement of a set \(A\) is

\[ A^c = \Omega\setminus A. \]

De Morgan’s laws state

\[ (A\cup C)^c = A^c\cap C^c \] \[ (A\cap C)^c = A^c\cup C^c. \]
Complement Exchanges Union and Intersection
Union, Then Complement
\((A\cup C)^c\)
=
Complement, Then Intersect
\(A^c\cap C^c\)
Intersection, Then Complement
\((A\cap C)^c\)
=
Complement, Then Union
\(A^c\cup C^c\)
Complement changes union into intersection and intersection into union.

10. Morphological Duality

De Morgan’s laws lead naturally to the dual relationships between morphological operators.

Dilation and erosion are dual:

\[ (X\oplus B)^c = X^c\ominus B^r \] \[ (X\ominus B)^c = X^c\oplus B^r. \]

Opening and closing form another dual pair:

\[ (X\circ B)^c = X^c\bullet B^r \] \[ (X\bullet B)^c = X^c\circ B^r. \]
Apply an Operator to \(X\)
↓
Complement the Result
↓
Equivalent Result
↓
Complement \(X\), Reflect \(B\), and Apply the Dual Operator
Reflection \(B^r\) is not being reintroduced here. Its role was established earlier; here it appears because it is required by the duality identity.

11. Why the Origin Matters for Some Properties

We do not need to redefine the structuring-element origin here, but one detail affects two important properties:

\[ 0\in B. \]

Origin Included

0
\(0\in B\)

The anchor location is an active member of the structuring element.

Origin Not Included

0
\(0\notin B\)

The anchor still defines the reference location, but that location is not part of \(B\).

When \(0\in B\), dilation is extensive and erosion is anti-extensive.
Opening remains anti-extensive and closing remains extensive without requiring this extra origin condition.

12. Comparing the Main Operators

Operator Increasing Extensive Anti-Extensive Idempotent Translation Invariant Ordinary Linear
Dilation Yes If \(0\in B\) Generally no Generally no Yes No
Erosion Yes Generally no If \(0\in B\) Generally no Yes No
Opening Yes No Yes Yes Yes No
Closing Yes Yes No Yes Yes No

13. Operator Properties vs. Connectivity and Topology

Properties such as increasing and idempotent describe relationships between sets. They do not automatically say that the topology of an object is preserved.

+

Components Can Merge

A transformation can connect two previously separate regions.

−

Components Can Split

A thin connection can disappear and separate one region into multiple components.

○

Holes Can Change

Small background regions can appear, disappear, or change depending on the transformation.

This is why connectivity and topology are studied separately from increasing, extensive, anti-extensive, and idempotent properties.

14. Interactive Property Explorer

The visualizer below does not repeat the moving structuring-element demonstration. Instead, it compares complete images in the relationships required by each property.

Edit the Example

Change the binary set \(X\) and the active positions of \(B\).

The purple center marks the fixed origin. You may include or exclude that location from \(B\) to see why the origin condition matters.

Test a Relationship

Test increasing, extensive, anti-extensive, idempotent, translation invariance, distributivity, De Morgan’s laws, or morphological duality.

The tool constructs the comparison images automatically.

15. Summary

Property Definition or Main Idea
Increasing \(X\subseteq Y \Rightarrow \Psi(X)\subseteq\Psi(Y)\)
Extensive \(X\subseteq\Psi(X)\)
Anti-Extensive \(\Psi(X)\subseteq X\)
Idempotent \(\Psi(\Psi(X))=\Psi(X)\)
Translation Invariant \(\Psi(X_z)=(\Psi(X))_z\)
Ordinary Linearity \(T(af+bg)=aT(f)+bT(g)\). Basic morphology is generally nonlinear under ordinary arithmetic superposition.
Distributivity Dilation distributes over union and erosion distributes over intersection.
De Morgan Complement exchanges union and intersection.
Morphological Duality Complement exchanges dilation with erosion and opening with closing, using \(B^r\).
Morphological Operator
Property to Test
Input Set X
Click any cell to toggle foreground and background.
Structuring Element B
Origin included
Relationship Being Tested
Current Example
✓ Holds for this example
Origin Condition
Translation z
Second Set
A second binary set is used when the selected identity requires two sets.
Comparison
Edit X or B and the selected property test will update.
Published inDigital Image Processing

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