Geometric Transformations

CMSC 178IP - Module 06

Noel Jeffrey Pinton
Department of Computer Science
University of the Philippines Cebu

Geometric Transformations

CMSC 178IP - Module 06

Noel Jeffrey Pinton
Department of Computer Science
University of the Philippines Cebu

Learning Objectives

By the end of this module, you will be able to:

Apply basic geometric transformations (translation, rotation, scaling)
Understand homogeneous coordinates and transformation matrices
Implement affine and projective transformations
Choose appropriate interpolation methods
Perform image registration and alignment

Basic Transformations

Translation, rotation, and scaling

Basic Geometric Transformations

Translation, rotation, and scaling are the fundamental building blocks

Rotation Visualization

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

Rotation by angle θ around the origin

Homogeneous Coordinates

Unified representation for all transformations

Homogeneous Coordinates

Homogeneous Coordinates: Add an extra dimension to represent 2D points as 3D vectors.

(x, y) \rightarrow (x, y, 1)

Enables all transformations (including translation) as matrix multiplication.

Knowledge Check

Think About It

Why do we use homogeneous coordinates in computer graphics?

Click the blurred area to reveal the answer

Affine Transformations

Preserving parallel lines

Affine Transformation Matrix

\begin{bmatrix} x' \\ y' \\ 1 \end{bmatrix} = \begin{bmatrix} a & b & t_x \\ c & d & t_y \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ 1 \end{bmatrix}

Combines rotation, scaling, shearing, and translation

Transformation Hierarchy

From rigid to projective: each level adds more degrees of freedom

Transformation Composition

Matrix multiplication allows combining transformations:

T_{combined} = T_3 \cdot T_2 \cdot T_1

Order matters! Transformations are applied right-to-left.

Knowledge Check

Think About It

If you want to rotate an image around its center, what sequence of transformations is needed?

Click the blurred area to reveal the answer

Projective Transformations

Homography and perspective

Projective Transformation

Homography (8 DOF):

\begin{bmatrix} x' \\ y' \\ w \end{bmatrix} = \begin{bmatrix} h_{11} & h_{12} & h_{13} \\ h_{21} & h_{22} & h_{23} \\ h_{31} & h_{32} & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ 1 \end{bmatrix}

Final coordinates: $(x'/w, y'/w)$

Perspective Correction

Correcting perspective distortion in document scanning and photography

Interpolation

Resampling pixel values

Interpolation Methods

Nearest Neighbor
Fast, blocky artifacts

Bilinear
Smooth, slight blur

Bicubic
High quality, slower

Lanczos
Best quality, slowest

Interpolation Artifacts

Trade-off: Speed vs. quality. Nearest neighbor for speed, bicubic/Lanczos for quality.

Knowledge Check

Think About It

When would you prefer nearest-neighbor over bicubic interpolation?

Click the blurred area to reveal the answer

Image Warping

Non-linear transformations for artistic effects and correction

Polar Transforms

Polar Transform: Convert from Cartesian to polar coordinates.

r = \sqrt{x^2 + y^2}, \quad \theta = \arctan(y/x)

Useful for analyzing circular patterns and rotational symmetry.

Image Registration

Aligning images from different sources

Registration Pipeline

Registration Steps:

Feature detection
Feature matching
Transformation estimation
Image warping

Feature Matching for Registration

Corresponding points are used to estimate the transformation

Image Registration Applications

Medical imaging, satellite imagery, panorama stitching

Implementation

import cv2
import numpy as np

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

# Rotation matrix (center, angle, scale)
M = cv2.getRotationMatrix2D((cols/2, rows/2), 45, 1.0)
rotated = cv2.warpAffine(img, M, (cols, rows))

# Perspective transform
pts1 = np.float32([[0,0], [300,0], [0,300], [300,300]])
pts2 = np.float32([[0,0], [300,0], [50,300], [250,300]])
H = cv2.getPerspectiveTransform(pts1, pts2)
warped = cv2.warpPerspective(img, H, (cols, rows))

Summary

Key Takeaways

Basic transforms: Translation, rotation, scaling
Homogeneous coordinates: Unified matrix representation
Affine: Preserves parallel lines (6 DOF)
Projective: Homography for perspective (8 DOF)
Interpolation: Nearest, bilinear, bicubic - trade speed vs quality
Registration: Feature detection → matching → transform → warp

Questions?

Thank you for your attention!

Next: Module 07 - Feature Extraction

End of Module 06

Geometric Transformations

Questions?