Symbols & Definitions

This page is a reference for the mathematical symbols used in binary mathematical morphology.

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\(X\) \(B\) \(z\) \(\mathbb{Z}\) \(\mathbb{Z}^2\) \(\in\) \(\subseteq\) \(\cap\) \(\cup\) \(\varnothing\) \(X^c\) \(\setminus\) \(|A|\)

Images and Coordinates

\(X\)

Binary Image / Foreground Set

\(X\) represents all pixel locations that belong to the foreground of a binary image.

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

If a foreground pixel is located at \((2,3)\), then \((2,3)\) belongs to \(X\).

\(B\)

Structuring Element

\(B\) is a small pattern used to examine the shape of the foreground set \(X\).

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

Only locations represented by 1 are active members of the structuring element.

\(z\)

Current Coordinate

\(z\) represents a position in the image.

\[ z=(x,y) \]

When the structuring element moves across the image, \(z\) represents the current position of its anchor.

\(\mathbb{Z}\)

Integers

\[ \mathbb{Z} = \{ \ldots,-2,-1,0,1,2,\ldots \} \]

Pixel coordinates use integer positions rather than arbitrary decimal coordinates.

\(\mathbb{Z}^2\)

2D Integer Grid

\[ \mathbb{Z}^2 = \{ (x,y) \mid x,y\in\mathbb{Z} \} \]

This represents all possible integer coordinate locations on a two-dimensional grid.

Think of \(\mathbb{Z}^2\) as the coordinate system used by the pixels in an image.

Basic Set Symbols

\(\in\)

Belongs To

\[ z \in X \]

This means that location \(z\) belongs to the foreground set \(X\).

\(\subseteq\)

Subset

\[ A \subseteq X \]

Every point in \(A\) is also contained in \(X\).

In erosion, \(B_z\subseteq X\) means that every active point of the moved structuring element fits inside the foreground.

\(\cap\)

Intersection

\[ A \cap X \]

The intersection contains the locations shared by both sets.

If \(A\cap X\neq\varnothing\), then \(A\) and \(X\) overlap somewhere.

\(\cup\)

Union

\[ A \cup X \]

The union contains every location that belongs to \(A\), \(X\), or both.

\(\varnothing\)

Empty Set

The empty set contains no locations.

\[ A\cap X=\varnothing \]

This means \(A\) and \(X\) do not overlap.

\(X^c\)

Complement / Background

\(X^c\) represents locations that are not part of the foreground \(X\).

\[ X^c = \{ z \mid z\notin X \} \]

For an image, the complement is normally understood relative to the finite image area being processed.

\(\setminus\)

Set Difference

\[ A\setminus B \]

Keep the points that belong to \(A\), but remove the points that also belong to \(B\).

\(|A|\)

Cardinality

Cardinality means the number of elements in a set.

\[ |A|=3 \]

This means that \(A\) contains three locations.

Images and Coordinates

Basic Set Symbols

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