Homography in Computer Vision (4-Point Solution and N-Point Least Squares)

📐 What Is Homography?

A homography is a projective transformation that maps points from one plane to another.

It is widely used for:

• Image stitching
• Perspective correction
• Planar object tracking
• Camera pose estimation (planar scenes)

A homography is represented by a 3×3 matrix:

\[\mathbf{H} = \begin{bmatrix} h_{11} & h_{12} & h_{13} \\ h_{21} & h_{22} & h_{23} \\ h_{31} & h_{32} & h_{33} \end{bmatrix}\]

Defined up to scale: \(\mathbf{H} \sim \lambda \mathbf{H}\)

So it has 8 degrees of freedom.

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🔗 Point Mapping Equation

Using homogeneous coordinates:

\[\begin{bmatrix} x' \\ y' \\ 1 \end{bmatrix} \sim \mathbf{H} \begin{bmatrix} x \\ y \\ 1 \end{bmatrix}\]

Which expands to:

\[x' = \frac{h_{11}x + h_{12}y + h_{13}}{h_{31}x + h_{32}y + h_{33}}\] \[y' = \frac{h_{21}x + h_{22}y + h_{23}}{h_{31}x + h_{32}y + h_{33}}\]

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🟦 Case 1: Homography from 4 Point Correspondences

Why 4 Points?

• Each point correspondence gives 2 independent equations
• Homography has 8 unknowns
• Minimum: 4 point pairs

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Linear Equation per Point

For correspondence ((x,y) \leftrightarrow (x’,y’)):

\[\begin{aligned} x'(h_{31}x + h_{32}y + h_{33}) &= h_{11}x + h_{12}y + h_{13} \\ y'(h_{31}x + h_{32}y + h_{33}) &= h_{21}x + h_{22}y + h_{23} \end{aligned}\]

Rearranged into linear form:

\[\mathbf{A} \mathbf{h} = \mathbf{0}\]

Where:

\[\mathbf{h} = [h_{11}, h_{12}, h_{13}, h_{21}, h_{22}, h_{23}, h_{31}, h_{32}, h_{33}]^T\]

Matrix (\mathbf{A}) has size 8×9.

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Solution

• Solve using SVD
• Solution = right singular vector corresponding to smallest singular value
• Reshape into 3×3 matrix

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🟩 Case 2: Homography from N Points (N ≥ 4)

When more than 4 correspondences are available, the system becomes overdetermined.

This improves:

• Noise robustness
• Numerical stability

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📊 Linear System (DLT Form)

For N point pairs:

\[\mathbf{A} \mathbf{h} = \mathbf{0}\]

Where:

• (\mathbf{A}) is (2N)×9
• (N > 4)

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🧮 Pseudo-Inverse / Least Squares View

We solve:

\[\min_{\mathbf{h}} \| \mathbf{A} \mathbf{h} \|^2\]

Subject to:

\[\| \mathbf{h} \| = 1\]

Solution

• Compute SVD of (\mathbf{A})
• Take eigenvector corresponding to smallest singular value
• Equivalent to solving with Moore–Penrose pseudo-inverse

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🔧 Normalized DLT (Important in Practice)

Before solving:

1. Normalize points (zero mean, unit variance)
2. Solve homography
3. Denormalize result

This greatly improves numerical stability.

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⚠️ Degenerate Configurations

Homography estimation fails if:

• Points are collinear
• All points lie on a line
• Insufficient spatial spread

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⚖️ 4-Point vs N-Point Comparison

Aspect	4 Points	N Points
Minimum	Yes	No
Noise robustness	❌	✅
Uses least squares	❌	✅
Typical use	Theory, exact	Real systems

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🧠 Practical Pipeline

Feature Match
   ↓
RANSAC (inliers)
   ↓
Normalized DLT
   ↓
Homography

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🎯 Takeaway

• Homography has 8 DOF
• 4 points give a minimal exact solution
• N points → least squares via SVD / pseudo-inverse
• Normalization is critical in real systems

Understanding this math is essential for stitching, tracking, and registration 🚀