Image Transformations¶

Image Filtering vs Image Warping¶

Two fundamental ways to transform an image:

Image filtering: changes the range (pixel intensity values). Output g(x,y) = h(f(x,y)).
Image warping: changes the domain (pixel coordinates/positions). Output g(x',y') = f(T(x,y)).

Filtering modifies what each pixel contains; warping modifies where each pixel is located.

Parametric Global Warping¶

Goal: find a single transformation function T with parameters that maps every point p = (x, y) to p' = (x', y') uniformly across the entire image.

Matrix representation:

\[p' = M \cdot p\]

where M encodes all transformation parameters. For 2D transforms, M is initially a 2x2 matrix applied to [x, y]^T to produce [x', y']^T.

2D Scaling¶

\[\begin{split}\begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} s_x & 0 \\ 0 & s_y \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}\end{split}\]

Uniform scaling: s_x = s_y
Non-uniform scaling changes aspect ratio

2D Rotation¶

\[\begin{split}\begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}\end{split}\]

Derived from trigonometric identities for rotating a point by angle theta.

2D Shear¶

\[\begin{split}M = \begin{bmatrix} 1 & sh_x \\ sh_y & 1 \end{bmatrix}\end{split}\]

Off-diagonal terms produce shear in x and y directions.

Mirroring¶

diag(-1, 1): mirror about y-axis (flip horizontal)
diag(1, -1): mirror about x-axis (flip vertical)
diag(-1, -1): mirror about origin (flip both)

Properties of 2D Linear Transformations¶

Origin maps to origin (no translation possible)
Lines map to lines
Parallel lines remain parallel
Ratios are preserved

Homogeneous Coordinates¶

The 2x2 matrix cannot represent translation (which requires addition, not multiplication). Solution: extend to homogeneous coordinates.

Convert 2D point (x, y) to 3-vector (x, y, w) where:

The 2D point is recovered as (x/w, y/w)
Typically w = 1, so (x, y, 1)
w = 0 represents a point at infinity
(0, 0, 0) is undefined

This allows all transformations (including translation) to be expressed as a single 3x3 matrix multiplication.

2D Transformations in Homogeneous Coordinates¶

Translation¶

\[\begin{split}\begin{bmatrix} x' \\ y' \\ 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 & t_x \\ 0 & 1 & t_y \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ 1 \end{bmatrix}\end{split}\]

Scale¶

\[\begin{split}\begin{bmatrix} x' \\ y' \\ 1 \end{bmatrix} = \begin{bmatrix} s_x & 0 & 0 \\ 0 & s_y & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ 1 \end{bmatrix}\end{split}\]

Rotation¶

\[\begin{split}\begin{bmatrix} x' \\ y' \\ 1 \end{bmatrix} = \begin{bmatrix} \cos\theta & -\sin\theta & 0 \\ \sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ 1 \end{bmatrix}\end{split}\]

Shear¶

\[\begin{split}\begin{bmatrix} x' \\ y' \\ 1 \end{bmatrix} = \begin{bmatrix} 1 & sh_x & 0 \\ sh_y & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ 1 \end{bmatrix}\end{split}\]

Composite transformations are achieved by multiplying matrices (translation + rotation + shear, etc.).

Affine Transformations¶

Combines linear transformations (rotation, scale, shear) with translation.

\[\begin{split}\begin{bmatrix} x' \\ y' \\ 1 \end{bmatrix} = \begin{bmatrix} a & b & c \\ d & e & f \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ 1 \end{bmatrix}\end{split}\]

DOF: 6 (parameters a, b, c, d, e, f)
Point correspondences needed: 3
Properties preserved: lines map to lines, parallel lines remain parallel
Not preserved: origin does not necessarily map to origin

Projective Transformations¶

Combines affine transformation with a projective warp.

\[\begin{split}\begin{bmatrix} x' \\ y' \\ w' \end{bmatrix} = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ 1 \end{bmatrix}\end{split}\]

DOF: 8 (bottom-right element is fixed at 1)
Point correspondences needed: 4
Properties preserved: straight lines remain straight
Not preserved: parallelism, ratios

Summary of Transformation Types¶

Transformation	DOF	Correspondences	Preserved
Translation	2	1	Orientation
Rigid (Euclidean)	3	2	Lengths, angles
Similarity	4	2	Angles
Affine	6	3	Parallelism, straight lines
Projective	8	4	Straight lines only

Code Demos¶

Translation¶

Use cv2.warpAffine with a 2x3 transformation matrix:

import cv2
import numpy as np

img = cv2.imread('image.png')
rows, cols = img.shape[:2]

# Translation matrix: shift by tx=100, ty=50
M = np.float32([[1, 0, 100],
                [0, 1, 50]])

translated = cv2.warpAffine(img, M, (cols, rows))

Rotation¶

Use cv2.getRotationMatrix2D to build the rotation matrix:

# Rotate 45 degrees around origin (0,0)
M = cv2.getRotationMatrix2D((0, 0), 45, 1)
rotated = cv2.warpAffine(img, M, (cols, rows))

# Rotate -45 degrees around image center
center = (cols // 2, rows // 2)
M = cv2.getRotationMatrix2D(center, -45, 1)
rotated_center = cv2.warpAffine(img, M, (cols, rows))

Scale and Shear¶

# Scale by 1.5x in both directions
scaled = cv2.resize(img, None, fx=1.5, fy=1.5)

# Horizontal shear (off-diagonal term)
M = np.float32([[1, 0.5, 0],
                [0, 1,   0]])
sheared = cv2.warpAffine(img, M, (cols, rows))

Affine Warp¶

Compute from 3 point correspondences:

pts1 = np.float32([[p1], [p2], [p3]])
pts2 = np.float32([[p1'], [p2'], [p3']])

M = cv2.getAffineTransform(pts1, pts2)
warped = cv2.warpAffine(img, M, (cols, rows))

Perspective Warp¶

Compute from 4 point correspondences:

pts1 = np.float32([[p1], [p2], [p3], [p4]])
pts2 = np.float32([[p1'], [p2'], [p3'], [p4']])

M = cv2.getPerspectiveTransform(pts1, pts2)
warped = cv2.warpPerspective(img, M, (cols, rows))

Forward vs Inverse Warping¶

Forward Warping¶

For each pixel in source f(x,y), compute destination T(x,y) in output.
Problem: destination may land between pixel locations.
Solution: distribute (splat) color among neighboring pixels.
Risk: holes — some output pixels may never be mapped to.

Inverse Warping¶

For each pixel in output, compute source location T^{-1}(x',y') in input.
Problem: source location may land between pixel locations.
Solution: interpolate color from neighboring source pixels.
Advantage: no holes in output (every output pixel is filled).
Requirement: need an invertible warp function T^{-1}.

Inverse warping is generally preferred because it eliminates holes. Invertibility is straightforward for rigid transforms (rotation, translation, scale) but becomes harder for non-global or complex warps.

Reference¶

Szeliski, Computer Vision: Algorithms and Applications, Chapter 2.