Morphological Operations

Let I an image (grayscale or binary image) and let H a mask (structuring element) where H(i,j)∈R, for (i,j)∈Z^2

• Dilation: is defined as the maximum of the values in H added to the values of the current subimage of I,
  

(I⊕H)(u,v)=max{I(u+i,v+j)+H(i,j)} for ((i,j)∈H)⁡

Properties:

Brighter (dark features are reduced, bright features are thickened, background is brighter)

• Erosion:  is defined as the minimum of the values in H differences to the values of the current subimage of I,
  

(I⊖H)(u,v)=min⁡{I(u+i,v+j)-H(i,j)} for ((i,j)∈H)⁡

Properties:

Darker (bright features are reduced, dark features are thickened, background is darker)

• Opening: denotes an erosion followed by a dilation with the same structuring element H,
  

I∘H=(I⊖H)⊕H

Properties:

Remove point or noise.

• Closing: denotes a dilation followed by an erosion with the same structuring element H,
  

I⋅H=(I⊕H)⊖H

 Properties:

Remove gaps.

• Morphological Gradient: The dilation thickens regions in an image and the erosion shrinks them. Therefore, their difference emphasizes the boundaries between regions. If the SE is relatively small, homogeneous areas will not be affected by dilation and erosion, so the subtraction tends to eliminate them. The net result is an image with the gradient-like effect.
  

(I⊕H)-(I⊖H)

The follow operations, extract small elements and details from given images. These operations are used for various image processing tasks, such as feature extraction, background equalization, image enhancement, and others.

• Top-hat Transform (or White-hat transform): of a gray-scale image I is defines as I minus its opening:
  

I-(I∘H)

Properties:

This operation is used for light objects on a dark background.

• Bottom-hat Transform (or Black-hat transform): of a gray-scale image I is defines as its closing minus I:
  

(I⋅H)-I

Properties:

This operation is used for dark objects on a light background.

Source:

• Digital Image Processing - An Algorithmic Introduction Using Java. Wilhelm Burger and Mark J. Burge
  
• Lamar University - pdf: Morphological Image Processing: Gray‐scale morphology