Projective Transformation in Photogrammetry

Projective transformation is a fundamental mapping in photogrammetry that models the perspective geometry of imaging systems.
2D Projective Transformation
A point $(x, y)$(x,y) is mapped to $(x’, y’)$(x′,y′) as:
$$x’ = \frac{a_1 x + a_2 y + a_3}{a_7 x + a_8 y + 1}$$x′=a7​x+a8​y+1a1​x+a2​y+a3​​ $$y’ = \frac{a_4 x + a_5 y + a_6}{a_7 x + a_8 y + 1}$$y′=a7​x+a8​y+1a4​x+a5​y+a6​​ This rational form allows modeling of perspective distortion.

Linearized Form for Estimation
To estimate the parameters, we rearrange the equations:
$$x'(a_7 x + a_8 y + 1) – (a_1 x + a_2 y + a_3) = 0$$x′(a7​x+a8​y+1)−(a1​x+a2​y+a3​)=0 $$y'(a_7 x + a_8 y + 1) – (a_4 x + a_5 y + a_6) = 0$$y′(a7​x+a8​y+1)−(a4​x+a5​y+a6​)=0 Expanding:
$$x’ a_7 x + x’ a_8 y + x’ – a_1 x – a_2 y – a_3 = 0$$x′a7​x+x′a8​y+x′−a1​x−a2​y−a3​=0 $$y’ a_7 x + y’ a_8 y + y’ – a_4 x – a_5 y – a_6 = 0$$y′a7​x+y′a8​y+y′−a4​x−a5​y−a6​=0 These equations are linear with respect to the unknown parameters.

Matrix Form
For multiple points, the system is written as:
$$\mathbf{L} = \mathbf{A}\mathbf{X}$$L=AX where:
$$\mathbf{X} = \begin{bmatrix} a_1 \\ a_2 \\ a_3 \\ a_4 \\ a_5 \\ a_6 \\ a_7 \\ a_8 \end{bmatrix}$$X=​a1​a2​a3​a4​a5​a6​a7​a8​​​ The least squares solution is:
$$\hat{\mathbf{X}} = (\mathbf{A}^T \mathbf{A})^{-1} \mathbf{A}^T \mathbf{L}$$X^=(ATA)−1ATL
3D to 2D Projective Transformation
For mapping $(x, y, z)$(x,y,z) to image coordinates:
$$x’ = \frac{a_1 x + a_2 y + a_3 z + a_4}{a_9 x + a_{10} y + a_{11} z + 1}$$x′=a9​x+a10​y+a11​z+1a1​x+a2​y+a3​z+a4​​ $$y’ = \frac{a_5 x + a_6 y + a_7 z + a_8}{a_9 x + a_{10} y + a_{11} z + 1}$$y′=a9​x+a10​y+a11​z+1a5​x+a6​y+a7​z+a8​​ This model contains 11 parameters and is widely used in photogrammetry (e.g., DLT).

Key Notes

Minimum of 4 points required for 2D case
Parameters solved using least squares
Captures perspective effects
More general than affine transformation