Erosion And Dilation

Morphological erosion and dilation algorithms.

Erosion2d: Performs a two-dimensional erosion using a structuring element matching with a square or a cross.
Erosion3d: Performs a three-dimensional erosion using a structuring element matching with a full or partial cube.
Dilation2d: Performs a two-dimensional dilation using a structuring element matching with a square or a cross.
Dilation3d: Performs a three-dimensional dilation using a structuring element matching with a full or partial cube.
ErosionDisk2d: Performs a two-dimensional erosion using a structuring element matching with a disk.
ErosionDisk3d: Performs a three-dimensional erosion using a structuring element matching with an oriented disk.
ErosionBall3d: Performs a three-dimensional erosion using a structuring element matching with a sphere.
DilationDisk2d: Performs a two-dimensional dilation using a structuring element matching with a disk.
DilationDisk3d: Performs a three-dimensional dilation using a structuring element matching with an oriented disk.
DilationBall3d: Performs a three-dimensional erosion using a structuring element matching with a sphere.
ErosionLine2d: Performs a two-dimensional erosion using a structuring element matching with a line.
ErosionLine3d: Performs a three-dimensional erosion using a structuring element matching with a line.
DilationLine2d: Performs a two-dimensional dilation using a structuring element matching with a line.
DilationLine3d: Performs a three-dimensional dilation using a structuring element matching with a line.
ErosionColor2d: Performs an erosion on a true-color image using a structuring element matching with a square.
DilationColor2d: Performs a dilation on a true-color image using a structuring element matching with a square.
SelectiveErosion2d: Erodes objects of a two-dimensional binary image conditionally to a local constraint.
SelectiveErosion3d: Erodes objects of a three-dimensional binary image conditionally to a local constraint.
SelectiveDilation2d: Dilates objects of a two-dimensional binary image conditionally to a local constraint.
SelectiveDilation3d: Dilates objects of a three-dimensional binary image conditionally to a local constraint.

For an introduction: section Mathematical Morphology.

Introduction to Erosion

In an erosion, pixel values within the structuring element are set to the minimum value of the element. In a binary image, an erosion removes isolated points and small particles, shrinks other particles, discards peaks at the object boundaries, and disconnects some particles.

Erosion operations are reiterative: repeating an erosion or dilation of size 1 N times has the same effect as performing a single erosion with a structuring element of size N.

In Figure 1 the binary image is

$I$ , and

$X$ denotes the set of points with a value of 1. The erosion of

$I$ by the structuring element

$B$ results in the set of points

$x$ , where the disk representing

$B$ and centered on

$x$ is totally included in the set of points

$X$ . The erosion of

$I$ can be denoted as

$E_B(I)$ or

$E(I)$ .

<b> Figure 1.</b> Erosion applied to a binary image

Figure 1. Erosion applied to a binary image

The eroded set of

$X$ by the structuring element

$B$ is:

$E(X)=\left\{ x ~; ~ B_x \subset X \right\}$
It may also be written as:

$E(X)=X \ominus B$
The value of the structuring element

$B$ varies depending on the type of erosion.
On a gray level image, the erosion by the structuring element

$B$ is the search for the minimal value of intensities within

$B$ .

2D image : $E\left\{I(x,y)\right\}=Min\left\{I(x_i,y_j);(x_i,y_j) ~ \in B \right\}$ .
3D image : $E\left\{I(x,y,z)\right\}=Min\left\{I(x_i,y_j,z_k);(x_i,y_j,z_k) ~ \in B \right\}$ .

When the point hits the edge of the image, the structuring element is composed of the intersection of

$B$ with the points of the structuring element totally within the image, and not the points outside the image.

Introduction to Dilation

A dilation is the opposite of an erosion. In a dilation, pixel values within the structuring element are set to the maximum value of the pixel neighborhood. In a binary image, a dilation fills the small holes inside particles and gulfs at the object boundaries, enlarges the size of the particles, and may connect neighboring particles.

Dilation operations are reiterative: repeating a dilation of size 1 N times has the same effect as performing a single dilation with a structuring element of size N.

In Figure 2, the binary image is

$I$ , and

$X$ denotes the set of points with a value of 1. The dilation of

$I$ by the structuring element

$B$ results in the set of points

$x$ , where the disk representing

$B$ and centered on

$x$ has a non-empty intersection with the set of points

$X$ . The dilation of

$I$ can be denoted as

$D_B(I)$ or

$D(I)$ .

<b> Figure 2.</b> Dilation applied to a binary image

Figure 2. Dilation applied to a binary image

The dilated set of X by the structuring element B is:

$D(X)=\left\{ x ~; ~ B_x \cap X \neq \varnothing \right\}$
It may also be expressed as:

$D(X)=X \oplus B$
The value of the structuring element

$B$ varies depending on the type of dilation.
On a gray level image, the dilation by the structuring element B is the search for the maximum value of intensities within

$B$ :

2D image : $D\left\{I(x,y)\right\}=Max\left\{I(x_i,y_j);(x_i,y_j) ~ \in B \right\}$ .
3D image : $D\left\{I(x,y,z)\right\}=Max\left\{I(x_i,y_j,z_k);(x_i,y_j,z_k) ~ \in B \right\}$ .

When the point hits the edge of the image, the structuring element is composed of the intersection of B with the points of the structuring element totally within the image, and not the points outside the image.