Hough Line Transform with example

🎯 Goal

This post shows a real Hough voting case where votes actually accumulate
and a line is detected — not just theoretical curves.

We will:

1. Choose points that lie exactly on one line
2. Use a specific θ
3. Show accumulator peak formation

---

📐 Target Line (Ground Truth)

Assume the true line is:

\[y = x\]

This line can be written in Hough normal form.

For a line with slope 1:

• Normal angle \(\theta = 135^\circ\)
• Because the normal is perpendicular to the line

Recall: \(\rho = x \cos\theta + y \sin\theta\)

---

🧱 Step 1: Edge Points on the Line

Choose 3 exact edge points:

Point	x	y
P1	1	1
P2	2	2
P3	3	3

All points lie on \(y=x\).

---

🧮 Step 2: Fix θ = 135°

Compute constants:

\[\cos 135^\circ = -0.707\] \[\sin 135^\circ = 0.707\]

---

🗳️ Step 3: Compute ρ for Each Point

▶ Point P1 = (1,1)

\[\rho_1 = 1(-0.707) + 1(0.707) = 0\]

---

▶ Point P2 = (2,2)

\[\rho_2 = 2(-0.707) + 2(0.707) = 0\]

---

▶ Point P3 = (3,3)

\[\rho_3 = 3(-0.707) + 3(0.707) = 0\]

---

✅ Observation

All points produce the same (ρ, θ):

\[(\rho, \theta) = (0, 135^\circ)\]

This is the accumulator peak.

---

📊 Step 4: Accumulator Table

θ (deg)	ρ	Votes
135	0	3 ✅
others	*	≤1

Define detection threshold:

votes ≥ 3

→ Line detected.

---

🔄 Step 5: Recover the Line

Detected parameters:

\[x \cos\theta + y \sin\theta = \rho\]

Substitute values:

\[-0.707x + 0.707y = 0\]

Divide both sides by 0.707:

\[y = x\]

✔ Original line recovered.

---

🧠 Why Voting Works (Key Insight)

• Each point votes for all possible lines through it
• Collinear points vote for the same parameter
• Noise-free case → perfect intersection
• Noisy case → cluster of votes

This is why Hough is robust to gaps and noise.

---

⚠️ What Happens with Discretization

If:

• resolution is coarse \(\theta\)
• bins are wide \(\rho\)

Then:

• Votes still accumulate nearby
• Peak becomes a cluster, not a single bin

---

⚖️ Compare with LSM

Aspect	Hough	Least Squares
Needs full line	❌	✅
Robust to gaps	✅	❌
Uses voting	✅	❌
Uses fitting	❌	✅

---

🎯 Final Takeaway

This example shows real voting:

• Same \(\theta\)
• Same \(\rho\)
• Accumulator peak appears

Once you see this numerically,
Hough Transform stops being abstract 🚀