Morphological Operations

Morphological operations transform a region using a second region called a structuring element (SE). The SE acts as a probe that is slid across every pixel; its shape and size set the spatial scale at which features are grown, shrunk, or detected.

Structuring elements must be centred on the origin. Two common choices:

Mathematical foundation

All operations on this page are built from two primitives — the Minkowski sum and the Minkowski difference of two sets $A$ and $B$:

\[A \oplus B = \{ a + b \mid a \in A,\, b \in B \},\qquad A \ominus B = \{ c \mid B + c \subseteq A \}.\]

dilation(A, B) is A ⊕ B and erosion(A, B) is A ⊖ B; opening, closing, gradients, and boundaries are all defined as compositions of these two. The internal helpers Regions._minkowski_addition and Regions._minkowski_subtraction implement the primitives on Regions. Because the formulas treat A and B symmetrically, the structuring element can be any region — disks, boxes, polygons, line segments, even point clouds.

Erosion

erosion(a, se) shrinks a. A pixel is kept only when the SE, placed at that pixel, fits entirely inside a. Protrusions narrower than the SE are trimmed away.

julia> region5 = region_from_box(-3, -3, 3, 3);  # 7×7 = 49 pixels

julia> se_sq   = region_from_box(-1, -1, 1, 1);  # 3×3 square SE

julia> e = erosion(region5, se_sq);

julia> area(e)                                    # shrinks to 5×5
25

julia> contains(e, -2, 0)                        # 1 pixel inside the original boundary
true

julia> contains(e, -3, 0)                        # original boundary pixel — removed
false

A disk SE erodes isotropically. erosion(region_from_circle(0,0,r), region_from_circle(0,0,s)) gives region_from_circle(0,0,r-s) when r > s.

Dilation

dilation(a, se) grows a. A pixel is added for every pixel of the SE placed at every pixel of a — equivalently, every point reachable by sliding the SE over the region.

julia> small = region_from_box(-1, -1, 1, 1);   # 3×3

julia> d = dilation(small, se_sq);              # 3×3 dilated by 3×3

julia> area(d)                                  # grows to 5×5
25

julia> contains(d, 2, 0)                        # pixel added beyond original boundary
true

Dilation is the dual of erosion: dilation(a, se) == invert(erosion(invert(a), se)).

Opening

opening(a, se) = erosion followed by dilation with the same SE. It removes foreground features (isolated pixels, thin protrusions) smaller than the SE while leaving larger structures nearly unchanged. The result is always a subset of a.

julia> opening(region5, se_sq) == region5       # box survives — no features smaller than SE
true

julia> isempty(opening(Region([Run(0, 0:0)]), se_sq))  # isolated pixel is removed
true

Opening is idempotent: applying it twice gives the same result as applying it once.

Closing

closing(a, se) = dilation followed by erosion. It fills background features (narrow gaps, small holes) smaller than the SE while leaving the overall shape nearly unchanged. The result always contains a as a subset.

julia> gapped = union(region_from_box(-3, -3, -1, 3),   # two bars with a 1-column gap
                      region_from_box( 1, -3,  3, 3));

julia> area(gapped)                                      # 6 columns × 7 rows, gap missing
42

julia> c3 = closing(gapped, se_sq);

julia> contains(c3, 0, 0)                               # gap at column 0 is filled
true

julia> area(c3)                                         # gap plus rounded corners added
49

Morphological Gradient

morphological_gradient(a, se) = difference(dilation(a,se), erosion(a,se)). The result is a ring that straddles the boundary of a, extending one SE radius both inside and outside.

julia> grad = morphological_gradient(region5, se_sq);

julia> area(grad)                                       # 9×9 minus 5×5
56

julia> contains(grad, -3, 0)                           # boundary pixel — in the ring
true

julia> contains(grad, 0, 0)                            # interior pixel — not in the ring
false

Inner and Outer Boundary

inner_boundary and outer_boundary use a fixed 3×3 square SE and split the gradient into the layer that lies inside versus outside the region.

julia> ib = inner_boundary(region5);

julia> area(ib)                                        # one pixel ring inside
24

julia> contains(ib, -3, 0)                            # outermost layer of region5
true

julia> contains(ib, -2, 0)                            # second layer — not inner boundary
false

julia> ob = outer_boundary(region5);

julia> area(ob)                                        # one pixel ring outside
32

julia> contains(ob, -4, 0)                            # first pixel beyond boundary
true

julia> contains(ob, -3, 0)                            # the boundary itself — not outer boundary
false

Holes and fill_holes

holes(a) returns all enclosed background regions — connected components of the background inside the bounding box that do not touch any edge. fill_holes(a) fills them all.

julia> box_frame = difference(region_from_box(-3, -3, 3, 3),
                              region_from_box(-1, -1, 1, 1));  # 7×7 box with 3×3 hole

julia> hs = holes(box_frame);

julia> length(hs)                                         # one enclosed hole
1

julia> contains(hs[1], 0, 0)                             # the hole contains the origin
true

julia> filled_box = fill_holes(box_frame);

julia> area(filled_box)                                   # hole filled — back to 7×7
49

julia> contains(filled_box, 0, 0) && contains(filled_box, -3, 0)
true

Gear Example

Applying morphological operations to the segmented gear illustrates each operation's effect at the scale of the gear teeth (radius-5 disk SE for erosion/dilation, radius-8 for opening/closing):

se5 = region_from_circle(0, 0, 5)    # disk SE, radius 5
se8 = region_from_circle(0, 0, 8)    # disk SE, radius 8
se2 = region_from_circle(0, 0, 2)    # disk SE, radius 2

eroded  = erosion(gear, se5)          # teeth shaved inward by 5 px
dilated = dilation(gear, se5)         # teeth grown outward by 5 px
opened  = opening(gear, se8)          # small protrusions removed
closed  = closing(gear, se8)          # narrow inter-tooth gaps bridged
grad    = morphological_gradient(gear, se2)   # boundary ring
filled  = fill_holes(gear)            # spoke voids and shaft hole filled

Original	Eroded (r = 5)	Dilated (r = 5)

Opened (r = 8)	Closed (r = 8)	Gradient (r = 2)

Holes filled

The gradient image traces every edge in the gear — outer teeth profile, spoke edges, and the shaft hole — as a thin white ring on the darkened original. The filled image shows the gear outline with all interior voids removed: only the teeth profile remains.