Mathematical morphology

Mathematical morphology studies shape-transforming operations defined through a chosen structuring element. For foreground pixels represented as a set $A$ and a finite footprint $B$, morphological dilation iswhile morphological erosion is $A\ominus B=\{x:x+B\subseteq A\}$. Dilation adds footprint translates around the foreground; erosion retains placements whose entire footprint fits.

For a one-dimensional example, $A=\{0,1,2\}$ and $B=\{0,1\}$ give dilation $\{0,1,2,3\}$ and erosion $\{0,1\}$. Morphological opening, erosion followed by dilation with the same footprint, removes an isolated set $\{0,2\}$ for that footprint because neither placement contains two consecutive points. grayscale variants use local minima and maxima rather than binary membership. The footprint and boundary conventions determine the exact operation.^[ref]

Morphological opening applies morphological erosion and then morphological dilation with the same structuring element. For a foreground set $A$ and structuring element $B$,It keeps the union of all translated copies of $B$ contained in $A$. Consequently the result is contained in $A$, and repeating the operation changes nothing: morphological opening is idempotent.

For a one-dimensional example, take $A=\{0,1,2,5\}$ and $B=\{0,1\}$. Morphological erosion gives $\{0,1\}$ and the following morphological dilation restores $\{0,1,2\}$; the isolated point at five disappears. The surviving run is restored rather than shifted. In image processing, selecting the structuring element controls which bright features survive. Scikit-image demonstrates removal of bright regions too thin to contain the chosen footprint.^[ref]

A structuring element specifies the neighborhood shape and reference position used by an operation in mathematical morphology. In binary image processing, represent it as a finite set $B$ of offsets. Morphological erosion tests where a translated $B$ fits inside the foreground; morphological dilation adds its offsets to foreground positions.

A three-by-three square uses all nine offsets around its center, while a cross uses the center and its four horizontal or vertical neighbors. They treat diagonal features differently despite having the same width. Scikit-image's morphology demonstration uses a disk-shaped footprint whose radius controls the affected scale.^[ref] The reference position matters for asymmetric shapes, and finite-image border rules must specify how neighborhoods outside the image are treated.

For foreground set $A$ and structuring element $B$, morphological erosion isThus an output position is retained only when every offset in the translated structuring element lands inside the foreground. This can shrink regions and remove features narrower than the neighborhood.^[ref]

For $A=\{0,1,2,5\}$ and $B=\{0,1\}$, only starts zero and one have their successor in $A$, giving $A\ominus B=\{0,1\}$. The point five disappears. For grayscale data with a flat footprint, the analogous operation takes a local minimum. An image implementation additionally chooses border rules; the set formula assumes the ambient space outside $A$ is background.

For foreground set $A$ and structuring element $B$, morphological dilation is their Minkowski sum:Equivalently, place a translated copy of $B$ at each foreground position and take their union. For $A=\{0,2\}$ and $B=\{0,1\}$, the result is $\{0,1,2,3\}$, connecting the separated input points.

In image processing, morphological dilation can enlarge bright regions or bridge small gaps. The grayscale version for a flat footprint takes a local maximum, with the footprint reflected according to the stated offset convention.^[ref] Asymmetric footprints make reflection conventions visible; symmetric disks hide that distinction. Morphological dilation alone generally enlarges a region, whereas morphological opening combines it with morphological erosion.