Canny Edge Detector

🎯 What Canny Tries to Optimize

The Canny edge detector was designed to be an optimal edge detector under three criteria:

1. Good detection
   Detect real edges while minimizing false detections caused by noise.

2. Good localization
   The detected edge position should be as close as possible to the true edge.

3. Single response
   Each real edge should be detected only once (thin edges).

These goals directly define the algorithmic steps.

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✅ Inputs / Outputs

Input

• Grayscale image

Output

• Binary edge image

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🧠 High-Level Pipeline

Input Image
   ↓
Gaussian Smoothing
   ↓
Gradient (Magnitude & Direction)
   ↓
Non-Maximum Suppression
   ↓
Double Thresholding
   ↓
Edge Tracking by Hysteresis
   ↓
Final Edge Map

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1️⃣ Gaussian Smoothing (Noise Suppression)

Why?

Differentiation amplifies noise.
Edges must be detected after smoothing.

Gaussian Kernel

\[G(x,y) = \frac{1}{2\pi\sigma^2} \exp\left(-\frac{x^2 + y^2}{2\sigma^2}\right)\]

Smoothed image:

\[I_s = I * G\]

Practical Notes

• Kernel size: typically (5 \times 5) or (7 \times 7)
• (\sigma) controls noise vs detail trade-off

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2️⃣ Gradient Computation

Intensity Gradient

\[G_x = \frac{\partial I_s}{\partial x}, \quad G_y = \frac{\partial I_s}{\partial y}\]

Usually approximated using Sobel filters.

Gradient Magnitude

\[M(x,y) = \sqrt{G_x^2 + G_y^2}\]

Gradient Direction

\[\theta(x,y) = \operatorname{atan2}(G_y, G_x)\]

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3️⃣ Non-Maximum Suppression (Edge Thinning)

Purpose

Remove thick edges by keeping only local maxima of gradient magnitude.

Principle

For each pixel:

• Look along the gradient direction
• Compare with two neighboring pixels
• Keep the pixel only if it is the maximum

Direction Quantization

Gradient direction is quantized to:

• 0°
• 45°
• 90°
• 135°

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4️⃣ Double Thresholding

Why Two Thresholds?

A single threshold either:

• Breaks edges, or
• Keeps too much noise

Classification

• Strong edge:
  \(M(x,y) \ge T_H\)

• Weak edge:
  \(T_L \le M(x,y) < T_H\)

• Non-edge:
  \(M(x,y) < T_L\)

Typical choice:

• ( T_L = 0.4 \sim 0.5 \cdot T_H )

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5️⃣ Edge Tracking by Hysteresis

Core Idea

Weak edges are valid only if connected to strong edges.

Connectivity

• Use 8-connected neighborhood
• Ensures diagonal continuity

Algorithm

1. Start from all strong edges
2. Recursively include connected weak edges
3. Discard isolated weak edges

This step guarantees continuous edges.

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📐 Why Canny Works So Well

Problem	Solution
Noise	Gaussian smoothing
Thick edges	Non-maximum suppression
Broken edges	Hysteresis
False positives	Double thresholding

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⚠️ Common Implementation Pitfalls

• Too small (\sigma) → noisy edges
• Too large (\sigma) → lost fine details
• High (T_H) → broken edges
• Low (T_L) → noise leakage

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🧪 Minimal Pseudo-Code

smooth = gaussian_filter(I)
Gx, Gy = gradient(smooth)
M, theta = magnitude_direction(Gx, Gy)
N = non_maximum_suppression(M, theta)
E = hysteresis_threshold(N, TL, TH)

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🎯 Takeaway

Canny is not just an edge filter — it is a carefully engineered pipeline.

Each step directly corresponds to an optimality criterion:

• Smoothing → noise robustness
• Gradient → edge strength
• NMS → single response
• Hysteresis → edge continuity

Understanding this pipeline allows you to implement, tune, and debug Canny from scratch 🚀