EP04_03 — 🌱 Flat Binary Dilation (mm.dil0)

4.9.3 EP04_03 🌱 Flat Binary Dilation (mm.dil0)

In particle microscopy and in OCR of worn license plates, thin or discontinuous traces need to be “thickened” for recognition to work. Morphological dilation does exactly that: it expands bright regions using a structuring element \(B\) — the same operation implemented in morph.py as mm::dil0(f, B), used when \(B\) is flat (without weights, only \(0\)/\(1\)). See Figure 4.32 for a simulation of this EP.

4.9.3.1 📋 Implementation Guidelines

Image dimensions: Read the integers \(L\) (rows) and \(C\) (columns) from \(f\).
Dimensions of \(B\): Read the integers \(L_B\) (rows) and \(C_B\) (columns) of the structuring element.
Structuring element: Read the matrix \(B\) with values \(0\) or \(1\), row by row.
Data: Read the matrix \(f\) (the original image), row by row.
Reflection: Construct \(B_{ref}\), the version of \(B\) reflected by \(180°\) (rows and columns reversed) — exactly as mm::dil0 does internally.
Neighborhood without padding: For each pixel \((y,x)\), traverse the positions \((by,bx)\) of \(B_{ref}\) centered at \((y,x)\), using the offset \[ v_y = y + by + o_y,\quad v_x = x + bx + o_x,\quad o_y=-\tfrac{L_B}{2}+0{,}5,\quad o_x=-\tfrac{C_B}{2}+0{,}5 \] Discard every \((v_y,v_x)\) outside \([0,L)\times[0,C)\) — do not pad with zeros.
Mapping: Compute each output pixel as the maximum between \(f(y,x)\) and all valid \(f(v_y,v_x)\) whose corresponding position in \(B_{ref}\) is \(1\): \[ g(y,x) = \max\Big(f(y,x),\ \max_{\substack{(v_y,v_x)\ \text{valid}\\ B_{ref}(by,bx)=1}} f(v_y,v_x)\Big) \]
Output: Display the matrix \(g\) with dimensions \(L \times C\).

4.9.3.2 📌 Computational Constraints

No padding: Never invent neighbors outside the image; use only those that actually exist.
Mandatory reflection: \(B\) must be reflected before being applied (this is what distinguishes mm::dil0 from a simple maximum search).
Edge robustness: If no valid position of \(B_{ref}=1\) falls within the domain for a given pixel, it maintains its original value.

4.9.3.3 🧠 Theoretical Foundation

Concept	Meaning	Visual Impact
Dilation	\(g \geq f\) always (extensive)	Bright regions grow, dark holes shrink
Larger \(B\)	Wider neighborhood	More aggressive growth
Reflection of \(B\)	\(B_{ref}(y,x) = B(-y,-x)\)	Ensures the formal Minkowski definition of dilation

4.9.3.4 📦 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 the matrix \(B\).
Next \(L\) lines: integer elements of the matrix \(f\).

Output:

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

4.9.3.5 📌 Examples

Input	Output	Remark
3 3 3 3 0 1 0 1 1 1 0 1 0 0 0 0 0 9 0 0 0 0	0 9 0 9 9 9 0 9 0	Symmetric cross \(B\): isolated point expands into a cross
1 4 1 3 1 1 1 10 200 5 80	200 200 200 80	Horizontal \(B\): each pixel “pulls” the maximum of row neighbors

🌱 Simulator EP04_03: Planar Dilation (mm.dil0) g = f ⊕ B

Toggle structuring element B (or select presets) and click cells of the original image f to light up or erase pixels.

Structuring Element B (Click to Toggle 0/1)

Original Image f (5×5)

Dilated g (f ⊕ B)

      g(y,x) = max over valid neighbors of reflected B
    

Figure 4.32: Simulator EP04_03: Binary Dilation (g = f ⊕ B)

%%writefile EP04_03.cpp
// your solution

Overwriting EP04_03.cpp

TestSuite("EP04_03.cpp").run()

✔️ EP04_03.cases already exists in casos/
📋 5 case(s) loaded from casos/EP04_03.cases

🔍 Testing C++: EP04_03.cpp
⚠️ EP04_03.cpp: Empty file (fewer than 3 lines). Tests skipped.