EP03_09 — 📡 3×3 Median Filter

3.14.9 EP03_09 📡 3×3 Median Filter

Synthetic aperture radar (SAR) images used in environmental and military monitoring suffer from a specific type of noise called speckle, which has characteristics similar to salt-and-pepper noise. The median filter is the standard method for removing this noise because it preserves the edges of structures while eliminating spurious points.

See Figure 3.34 for a simulation of this exercise.

3.14.9.1 📋 Implementation Guidelines

Dimensions: Read the integers \(L\) (rows) and \(C\) (columns).
Data: Read the pixel matrix \(f\).
3×3 Median Filter: For each internal pixel \((i,j)\) with \(1 \le i < L-1\), \(1 \le j < C-1\):
- Collect the 9 pixels from the \(3 \times 3\) neighborhood: \(\{f(i+s, j+t) : s,t \in \{-1,0,1\}\}\).
- Sort the 9 values in ascending order.
- Assign \(g(i,j)\) to the central value (index position 4, considering index 0).

\[g(i,j) = \text{median}\{f(i+s, j+t) : s,t \in \{-1,0,1\}\}\]

Border: Copy directly: \(g(i,j) = f(i,j)\).
Output: Display the filtered \(L \times C\) matrix.

3.14.9.2 📌 Computational Constraints

Window: Always \(3 \times 3 = 9\) elements.
Median: The central element of the sorted sequence (index 4 from 0 to 8).
No clipping: The median of values in \([0, 255]\) remains in \([0, 255]\).
Nonlinear: The median filter cannot be expressed as a linear convolution.

3.14.9.3 🧠 Theoretical Background

Noise	Mean Filter	Median Filter
Salt and pepper (0 or 255)	Spreads the noise	Removes without distorting edges
Gaussian	Reduces effectively	Reduces partially

3.14.9.4 📦 Input and Output Specification (VPL)

Input:

Line 1: Integer \(L\).
Line 2: Integer \(C\).
Following lines: Matrix elements.

Output:

Filtered \(L \times C\) matrix.

3.14.9.5 📌 Examples

Input	Output	Observation
3 3 100 100 100 100 0 100 100 100 100	100 100 100 100 100 100 100 100 100	Black point removed: median of 8×100+1×0 = 100
3 3 50 50 50 50 255 50 50 50 50	50 50 50 50 50 50 50 50 50	White point (salt) removed

📉 Simulator EP03_09: 3×3 Median Filter g = Median(Neighbors)

Inject impulsive noise (salt and pepper) and hover over internal pixels of the result to inspect the sorting of the neighborhood vector and the noise removal.

Salt (255) and pepper (0) noise — ~30% of internal pixels affected

Image f — With Noise Salt (255) and pepper (0) visible

Result g — Without Noise Hover to inspect

3×3 Neighborhood Vector — Sorted Hover over an internal pixel of the result to visualize

—

Legend:

3×3 Window

Central Pixel

Border (Copied)

Median

Noise (Removed)

      Hover over an internal pixel of the result to see the sorting process.
    

Figure 3.34: EP03_09 Simulator: 3×3 Median Filter

%%writefile EP03_09.py
# Python code

Overwriting EP03_09.py

TestSuite("EP03_09.py").run()

✔️ EP03_09.cases already exists in casos/
📋 5 case(s) loaded from casos/EP03_09.cases

🔍 Testing Python: EP03_09.py
⚠️ EP03_09.py: Empty file (fewer than 3 lines). Tests skipped.