🔄 Homographies and Transformations

A homography is the most general planar (projective) transformation, mapping points from one plane to another. In computer vision, it aligns two images of the same flat surface taken from different viewpoints.

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🧠 What Is a Homography?

• It’s a 3x3 matrix H that warps one image to align with another.
• Used to model perspective changes (camera tilt, view shifts).
• Maps any 4 non-collinear points in one plane to 4 points in another.

Real-World Use Cases

Use Case	What Homography Does
Panorama Stitching	Warps one image to align with another
AR Projection	Projects virtual objects onto real surfaces
Image Rectification	Removes perspective distortion

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🔢 What Does a Homography Matrix Look Like?

It’s denoted H:

🐍 Full Code: Feature Matching + Homography (SIFT + FLANN)

import cv2
import numpy as np
import matplotlib.pyplot as plt

# 1. Load and extract features
img1 = cv2.imread("bird1.jpg", cv2.IMREAD_GRAYSCALE)
img2 = cv2.imread("bird2.jpg", cv2.IMREAD_GRAYSCALE)
sift = cv2.SIFT_create()
kp1, des1 = sift.detectAndCompute(img1, None)
kp2, des2 = sift.detectAndCompute(img2, None)

# 2. Match with FLANN + Lowe's ratio test
index_params = dict(algorithm=cv2.FLANN_INDEX_KDTREE, trees=5)
flann = cv2.FlannBasedMatcher(index_params, dict(checks=50))
matches = flann.knnMatch(des1, des2, k=2)
good = [m for m,n in matches if m.distance < 0.7*n.distance]

# 3. Compute homography
src_pts = np.float32([ kp1[m.queryIdx ].pt for m in good ]).reshape(-1,1,2)
dst_pts = np.float32([ kp2[m.trainIdx].pt for m in good ]).reshape(-1,1,2)
H, mask = cv2.findHomography(src_pts, dst_pts, cv2.RANSAC, 5.0)

# 4. Warp image1 to align with image2
h, w = img2.shape
warp = cv2.warpPerspective(cv2.imread("bird1.jpg"), H, (w, h))

# 5. Visualise matches and warped result
plt.figure(figsize=(16,6))
plt.subplot(1,2,1)
plt.imshow(cv2.cvtColor(cv2.drawMatches(
    cv2.imread("bird1.jpg"), kp1,
    cv2.imread("bird2.jpg"), kp2,
    good, None, flags=cv2.DrawMatchesFlags_NOT_DRAW_SINGLE_POINTS
), cv2.COLOR_BGR2RGB))
plt.title("Matched Keypoints (FLANN + SIFT)")
plt.axis("off")

plt.subplot(1,2,2)
plt.imshow(cv2.cvtColor(warp, cv2.COLOR_BGR2RGB))
plt.title("Image 1 Warped to Align with Image 2")
plt.axis("off")
plt.tight_layout()
plt.show()

📋 Recap: What This Code Does

Step	Description
1–2	Load images, detect SIFT features & compute descriptors
3–4	Match descriptors with FLANN & apply Lowe’s ratio test
5–6	Extract matched keypoint coordinates & compute the 3×3 homography
7	Warp the first image using warpPerspective()
8–9	Visualise the matched keypoints and the warped image

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🌐 Geometric Transformations — The Big Picture

Any operation that changes position, shape, or orientation of pixels.

Transformation	What It Does	Matrix Size
Translation	Move image left/right/up/down	2×3
Rotation	Rotate around a point	2×3
Scaling	Zoom in/out	2×3
Affine Transform	Preserve straight lines & parallelism	2×3
Homography	General 2D→2D transform (projective)	3×3

🐍 Code: Example of Affine Transformation

import cv2
import numpy as np
import matplotlib.pyplot as plt

# Load the image
image = cv2.imread('bird1.jpg')
image = cv2.cvtColor(image, cv2.COLOR_BGR2RGB)  # Convert to RGB for correct display in matplotlib

# Image dimensions
h, w = image.shape[:2]

# Define a 3×3 affine transformation matrix in homogeneous coordinates
# This example applies: scale, shear, rotate a bit, and translate
affine_matrix_3x3 = np.array([
    [1.2, 0.2, 50],   # a little scale and shear + translation in x
    [-0.2, 1.1, 30],  # a little rotation and scale + translation in y
    [0, 0, 1]         # homogeneous row
], dtype=np.float32)

# Extract the 2×3 part for cv2.warpAffine (since it expects 2x3)
affine_matrix_2x3 = affine_matrix_3x3[:2, :]

# Apply the affine transformation
transformed = cv2.warpAffine(image, affine_matrix_2x3, (w, h))

# Show the original and transformed image side-by-side
plt.figure(figsize=(10, 5))
plt.subplot(1, 2, 1)
plt.title("Original Image")
plt.imshow(image)
plt.axis('off')

plt.subplot(1, 2, 2)
plt.title("Affine Transformed Image")
plt.imshow(transformed)
plt.axis('off')

plt.tight_layout()
plt.show()

🔖 Homography — A Special Geometric Transformation

• Most general planar transform
• Models perspective (e.g. camera tilt)
• Maps any 4 points in one plane to 4 points in another

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Q&A

Question	Answer
Is homography a geometric transformation?	✅ Yes — it’s a type of geometric transform
Are all geometric transforms homographies?	❌ No — simpler transforms (scaling, rotation) are not

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Visual Example

• Affine transforms cannot correct perspective distortion
• Homographies straighten a tilted photo of a page, making it appear head-on

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✅ Summary

Term	Definition
Geometric Transformation	Any operation that repositions image pixels
Homography	Most general 2D transform (represented by a 3×3 matrix)
Use Cases	Align images, correct perspective, stitch panoramas