DIP+CV · Programming Exercise

EP04_04 — 🪨 Binary Erosion by Flat Structuring Element (mm.ero0)

4.9.4 EP04_04 🪨 Binary Erosion by Flat Structuring Element (mm.ero0)

If dilation thickens, erosion thins. In cell counting systems, it is used to separate touching cells: by “eating away” the borders of each region, thin connections between objects disappear even before any counting is performed. In morph.py, this is the operation mm.ero0(f, B) — the exact dual of dilation, and the only one of the two that does not reflect the structuring element. See Figure 4.33 for a simulation of this exercise.

4.9.4.1 📋 Implementation Guidelines

  1. Image dimensions: Read the integers \(L\) (rows) and \(C\) (columns) from \(f\).
  2. Dimensions of \(B\): Read the integers \(L_B\) (rows) and \(C_B\) (columns) of the structuring element.
  3. Structuring element: Read the matrix \(B\) with values \(0\) or \(1\), row by row.
  4. Data: Read the matrix \(f\) (the original image), row by row.
  5. Neighborhood without padding (no reflection!): For each pixel \((y,x)\), traverse the positions \((by,bx)\) of \(B\) in the original order (without reflecting), using the same offset as in EP04_03: \[ v_y = y + by + o_y,\quad v_x = x + bx + o_x,\quad o_y=-\tfrac{L_B}{2}+0{,}5,\quad o_x=-\tfrac{C_B}{2}+0{,}5 \]

Discard every \((v_y,v_x)\) outside \([0,L)\times[0,C)\). 6. Mapping: Compute each output pixel as the minimum between \(f(y,x)\) and all valid \(f(v_y,v_x)\) whose corresponding position in \(B\) equals \(1\): \[ g(y,x) = \min\Big(f(y,x),\ \min_{\substack{(v_y,v_x)\ \text{valid}\\ B(by,bx)=1}} f(v_y,v_x)\Big) \] 7. Output: Display the matrix \(g\) with dimensions \(L \times C\).

4.9.4.2 📌 Computational Constraints

  • No reflection: Unlike dilation, \(B\) is used exactly as read — reflecting it here would be a serious conceptual error.
  • No padding: Neighbors outside the image are simply ignored, never treated as \(0\).
  • Edge robustness: If no valid position of \(B=1\) falls within the domain, the pixel retains its original value.

4.9.4.3 🧠 Theoretical Background

Concept Meaning Visual Impact
Erosion \(g \leq f\) always (anti-extensive) Bright regions shrink, point noise disappears
Duality \(\text{ero}(f,B) = -\text{dil}(-f, B_{ref})\) Erosion and dilation are mathematical “mirrors”
Larger \(B\) More aggressive erosion Thin objects disappear completely

4.9.4.4 📦 Input and Output Specification (VPL)

Input:

  • Line 1: Integer \(L\).
  • Line 2: Integer \(C\).
  • Line 3: Integer \(L_B\).
  • Line 4: Integer \(C_B\).
  • Next \(L_B\) lines: integer elements (\(0\) or \(1\)) of matrix \(B\).
  • Next \(L\) lines: integer elements of matrix \(f\).

Output:

  • Matrix \(g\) in \(L\) rows and \(C\) columns, integer values separated by spaces.

4.9.4.5 📌 Examples

Input Output Observation
3
3
3
3
0 1 0
1 1 1
0 1 0
9 9 9
9 0 9
9 9 9		 9 0 9
0 0 0
9 0 9		 The central “hole” (0) propagates in a cross pattern
1
4
1
3
1 1 1
10 200 5 80		 10 5 5 80 Horizontal \(B\): each pixel “pulls” the minimum of its row neighbors
🪨 Simulator EP04_04: Planar Erosion (mm.ero0) g = f ⊖ B

Switch the structuring element B (or select the presets) and click the cells of the original image f to turn pixels on or off.

Structuring Element B (Click to Toggle 0/1)
Original Image f (5×5)
Eroded g (f ⊖ B)
 
g(y,x) = min over valid neighbors of B (without reflection)
Figure 4.33: EP04_04 Simulator: Planar Binary Erosion (g = f ⊖ B) 
%%writefile EP04_04.py
# Python code
Overwriting EP04_04.py
TestSuite("EP04_04.py").run()
✔️ EP04_04.cases already exists in casos/
📋 5 case(s) loaded from casos/EP04_04.cases

🔍 Testing Python: EP04_04.py
⚠️ EP04_04.py: Empty file (fewer than 3 lines). Tests skipped.