07. Morphology

Structural processing of images

From ~1960s

• Erosion: shrinks objects
• Dilation: expands objects
• Open: erode then dilate
  • Smooths images, removing small spurs, lines and noise
• Close: dilate then erode
  • Fills gaps and holes while preserving thin lines

Extracting quantitative descriptions of image components:

• Boundaries
• Skeletons
• Convex hulls

Pixels are either object or non-object pixels.

Structuring element: smaller matrix applied to the image

Binary erode:

• $A \otimes B = \{x: B_x \subseteq A \}$
• Structuring element placed centered around every pixel: remove if any pixel of the structuring element overlaps with a non-object pixel

Binary dilation:

• $A \otimes B = \{x: \hat{B_x} \cap A \neq \empty \}$
• Keep any pixels covered by the structuring element when placed at at least one location

Greyscale erode:

• Replace set operation with $\text{min}$ operation
• $D_G(A, B) = \min_{[j, k] \in B}{\{a[m - j, n - k], b[j, k]\}}$

Greyscale dilate:

• $D_G(A, B) = \max_{[j, k] \in B}{\{a[m - j, n - k], b[j, k]\}}$

Distance transform:

• Minimum distance of each pixel to non-object pixel
• Simple but inefficient: repeat erosion operation until all pixels gone; distance is the number of erosion operations required before the pixel disappeared
• Structuring element:
  • Chessboard: pixels sharing corners or edges are both have a distance of 1. 3 by 3 square structuring element
  • Manhattan: 3 by 3 cross structuring element

Skeleton transform:

• Reduces regions to one-pixel line thick borders
• Methods:
  • Distance transform: create by finding pixels with a distance of 1
  • Thinning: repeatedly thin image, retaining end points/connections

Convex hull:

• Follow outlines of object, except concavities