Understanding Affine and Projective Transformations in Photogrammetry

In photogrammetry, transformations are essential tools for mapping points from one coordinate system to another. Two of the most widely used transformations are Affine and Projective Transformations. Both play a crucial role in tasks like image rectification, 3D modeling, and georeferencing. Here’s a breakdown of these transformations:

1. Affine Transformation

An Affine Transformation is a linear mapping method that preserves points, straight lines, and planes. It allows for translation, rotation, scaling, and shearing, making it versatile for many photogrammetric applications.

Mathematically, a 2D affine transformation can be expressed as:$$\begin{bmatrix} x’ \\ y’ \end{bmatrix} = \begin{bmatrix} a_1 & a_2 \\ a_4 & a_5 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} + \begin{bmatrix} a_3 \\ a_6 \end{bmatrix}$$[x′y′​]=[a1​a4​​a2​a5​​][xy​]+[a3​a6​​]

Here:

• $x, y$x,y are the original coordinates
• $x’, y’$x′,y′ are the transformed coordinates
• $a_1$a1​ to $a_6$a6​ are transformation parameters representing rotation, scaling, shearing, and translation.

Affine transformations require at least 3 non-collinear points to solve for the parameters using least-squares estimation.

Key Features:

• Preserves parallelism of lines
• Can handle non-uniform scaling and skewing
• Suitable for most 2D-to-2D georeferencing tasks

2. Projective Transformation

A Projective Transformation, also known as a homography, goes a step further by accounting for perspective distortions. This makes it ideal for mapping between images taken from different viewpoints, such as in aerial or satellite imagery.

The 2D projective transformation is given by:$$x’ = \frac{a_1 x + a_2 y + a_3}{a_7 x + a_8 y + 1}, \quad y’ = \frac{a_4 x + a_5 y + a_6}{a_7 x + a_8 y + 1}$$x′=a7​x+a8​y+1a1​x+a2​y+a3​​,y′=a7​x+a8​y+1a4​x+a5​y+a6​​

Here:

• The denominator introduces perspective effects
• At least 4 points are needed to compute the 8 parameters

Applications:

• Correcting image distortions due to camera perspective
• Transforming images to a common plane for mosaicking
• Key step in 3D reconstruction and space resection techniques

Choosing Between Affine and Projective

• Use Affine when images or coordinate sets differ only by rotation, scaling, and translation. It’s simpler and computationally faster.
• Use Projective when perspective distortion is significant, such as aerial images captured at different angles or heights.

In essence, affine transformations handle linear adjustments, while projective transformations account for the complexities of perspective. Understanding both allows photogrammetrists to align images accurately and build reliable 3D models.