Homography and image alignment in Computer Vision - Practice Problems & Coding Challenges

Challenge - 5 Problems

🧠 Conceptual

intermediate

What does a homography matrix represent in image alignment?

In the context of aligning two images taken from different viewpoints, what does the homography matrix represent?

A A matrix that encodes the 3D depth information of the scene.

B A transformation that maps points from one image plane to another, assuming the scene is planar or the camera rotates around its optical center.

C A matrix that describes the color adjustment needed to match brightness between two images.

D A filter that removes noise from images before alignment.

💡 Hint

Think about how points in one flat image relate to points in another flat image when the camera moves.

The homography matrix is a 3x3 matrix that transforms points from one image plane to another under the assumption that the scene is flat or the camera rotates around its center. It does not handle color or depth information.

❓ Predict Output

intermediate

Output of homography application on a point

Given the homography matrix H = [[1, 0, 2], [0, 1, 3], [0, 0, 1]] and a point p = [4, 5], what is the transformed point p' after applying the homography?

Computer Vision

import numpy as np
H = np.array([[1, 0, 2], [0, 1, 3], [0, 0, 1]])
p = np.array([4, 5, 1])
p_prime = H @ p
p_prime_cartesian = p_prime[:2] / p_prime[2]
print(p_prime_cartesian.tolist())

A [6.0, 8.0]

B [4.0, 5.0]

C [2.0, 3.0]

D [8.0, 10.0]

💡 Hint

Remember to convert from homogeneous coordinates back to Cartesian coordinates by dividing by the last element.

Applying the homography adds 2 to x and 3 to y, so the point (4,5) becomes (6,8).

❓ Hyperparameter

advanced

Choosing the number of point correspondences for homography estimation

When estimating a homography matrix using point correspondences between two images, what is the minimum number of point pairs required, and why?

A 4 pairs, because a homography has 8 degrees of freedom and each point provides two equations.

B 3 pairs, because each point provides three equations for the homography.

C 8 pairs, because the homography matrix has 8 elements to solve for.

D 2 pairs, because each pair gives enough constraints for the transformation.

💡 Hint

Consider how many unknowns the homography matrix has and how many equations each point correspondence provides.

A homography matrix has 9 elements but is defined up to scale, so 8 degrees of freedom. Each point correspondence gives 2 equations, so 4 pairs are needed to solve.

❓ Metrics

advanced

Evaluating homography quality with reprojection error

Which metric best measures the quality of a computed homography matrix when aligning two images?

A Mean squared error of pixel intensities between the two images.

B Number of matched keypoints detected.

C Structural similarity index (SSIM) between the images.

D Average reprojection error between transformed points and their true correspondences.

💡 Hint

Think about how well the homography maps points from one image to the other.

The reprojection error measures the distance between points transformed by the homography and their actual matching points, directly assessing alignment accuracy.

🔧 Debug

expert

Why does this homography estimation fail with RANSAC?

Consider this Python snippet using OpenCV to estimate a homography with RANSAC:

import cv2
import numpy as np
pts_src = np.array([[10, 20], [30, 40], [50, 60], [70, 80]])
pts_dst = np.array([[12, 22], [32, 42], [52, 62], [72, 82]])
h, mask = cv2.findHomography(pts_src, pts_dst, cv2.RANSAC, 5.0)
print(h)

The output is None. What is the most likely reason?

A The points are perfectly aligned, so RANSAC returns None.

B The threshold value 5.0 is too small for RANSAC to accept any inliers.

C There are too few point correspondences; RANSAC requires more points to find a valid homography.

D The input points are not in homogeneous coordinates, causing failure.

💡 Hint

Recall the minimum number of points needed for homography estimation and how RANSAC works.

RANSAC requires more than the minimum number of points to robustly estimate a homography and identify inliers. With only 4 points, it often fails to find a consensus set.