EP03_08 — 🧭 Sobel Gradient: Gx and Gy

3.14.8 EP03_08 🧭 Sobel Gradient: Gx and Gy

In Mars exploration rovers (such as Perseverance), obstacle detection is performed in real time by stereoscopic cameras. The Sobel operator computes the directional gradient of the scene and is used in the edge detection algorithm to identify rocks, cracks, and terrain unevenness that could compromise navigation.

See Figure 3.33 for a simulation of this EP.

3.14.8.1 📋 Implementation Guidelines

Dimensions: Read the integers \(L\) (rows) and \(C\) (columns).
Data: Read the matrix \(f\).
Gx and Gy: For each internal pixel \((i,j)\) with \(1 \le i < L-1\), \(1 \le j < C-1\):

\[G_x(i,j) = [f(i-1,j+1) + 2f(i,j+1) + f(i+1,j+1)] - [f(i-1,j-1) + 2f(i,j-1) + f(i+1,j-1)]\]

\[G_y(i,j) = [f(i+1,j-1) + 2f(i+1,j) + f(i+1,j+1)] - [f(i-1,j-1) + 2f(i-1,j) + f(i-1,j+1)]\]

Magnitude: \(|\nabla f(i,j)| = \text{clip}(\text{round}(\sqrt{G_x^2 + G_y^2}))\).
Edge: Edge pixels receive magnitude 0.
Output: Display the \(L \times C\) magnitude.

3.14.8.2 📌 Computational Constraints

Rounding: Apply round before converting to integer.
Saturation: \(\text{clip}(x) = \max(0, \min(255, x))\).
Square root: Use \(\sqrt{G_x^2 + G_y^2}\) (not the approximation \(|G_x| + |G_y|\)).

3.14.8.3 🧠 Theoretical Foundation

Operator	Detects	Diagonal coefficients
\(G_x\)	Vertical edges	\(\pm 1\)
\(G_y\)	Horizontal edges	\(\pm 1\)
\(\|\nabla f\|\)	All edges	Combined

3.14.8.4 📦 Input and Output Specification (VPL)

Input:

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

Output:

Gradient magnitude, \(L \times C\) matrix.

3.14.8.5 📌 Examples

Input	Output	Observation
3 3 0 0 0 0 0 0 0 0 0	0 0 0 0 0 0 0 0 0	Null image: zero gradient
3 3 0 0 255 0 0 255 0 0 255	0 0 0 0 255 0 0 0 0	Central vertical edge: high Gx

🧭 Simulator EP03_08: Sobel Gradient (Gx and Gy) |∇f| = √(Gx² + Gy²)

Analyze the horizontal (Gx) and vertical (Gy) decomposition of the Sobel operator and hover over the magnitude pixels to inspect the 3×3 neighborhood.

Sobel Kernels:

−1

−2

−1

−2

−1

Original Image f 5×5 pixel matrix

Magnitude |∇f| √(Gx² + Gy²)

Gx — Horizontal Gradient Blue = Negative · White = Zero · Bright Blue = Positive

Gy — Vertical Gradient Amber = Negative · White = Zero · Bright Amber = Positive

Legend:

Inspected 3×3 Neighborhood

Central Pixel

Edge (Forced to 0)

      Hover over an inner magnitude pixel to see the Gx and Gy decomposition.
    

Figure 3.33: EP03_08 Simulator: Sobel Gradient (Gx and Gy)

%%writefile EP03_08.py
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TestSuite("EP03_08.py").run()

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⚠️ EP03_08.py: Empty file (fewer than 3 lines). Tests skipped.