Question 5 of 9: Bundle-Adjustment Aerotriangulation — Counting Unknowns, Measurements, Equations, DoF

Question 5: Bundle-Adjustment Aerotriangulation — Counting Unknowns, Measurements, Equations, DoF (20 marks — 5.1 5, 5.2 6, 5.3 6, 5.4 3)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. A block of 4 overlapping images with 10 object points: four 3-D control points (▲ 1, 2, 9, 10) held fixed, and six pass/tie points (+ 3, 4, 5, 6, 7, 8) to be determined. Each point is measured (imaged) on every photo in which it appears.

Find. The type and count of unknowns (5.1), measurements (5.2), equations and their mathematical basis (5.3), and the statistical degrees of freedom (5.4).

[Figure not reproduced: The block as printed: two strips of two heavily overlapping photos (1&2 above, 3&4 below) whose ends overlap in a narrow band. Points 1, 2, 3, 4 lie in the overlap of photos 1&2 (2 rays each); 5&6 lie in the band common to all four photos (4 rays each); 7, 8, 9, 10 lie in the overlap. See the official exam paper.]

5.1 Unknowns (type and number). Two kinds of unknowns enter a bundle block, because both the cameras and the new ground points are solved simultaneously:

• Exterior-orientation (EO) parameters — 6 per image ($X_L,Y_L,Z_L,\omega,\phi,\kappa$): $6\times4=24$.
• Object-space coordinates of the tie points — 3 per point ($X,Y,Z$) for the 6 pass/tie points: $3\times6=18$. The four control points are held fixed, so they contribute no unknowns.

$$\text{Unknowns}=24+18=\boxed{42}.$$

5.2 Measurements (type and number). The measurements are the photo (image) coordinates $(x,y)$ of every point on every photo in which it appears — i.e. two measured coordinates per image ray. Counting rays from the figure:

Each ray supplies 2 measured coordinates, so

$$\text{Measurements}=2\times24=\boxed{48\ \text{photo coordinates}}.$$

5.3 Mathematical basis and number of equations. The mathematical model is the pair of collinearity equations (one for $x$, one for $y$) written for every image ray — the condition that each exposure station, image point and object point are collinear. That is 2 equations per ray:

$$\text{Equations}=2\times24=\boxed{48\ \text{collinearity equations}}.$$

5.4 Statistical degrees of freedom. The redundancy (degrees of freedom) is the number of observation equations minus the number of unknowns:

$$\text{DoF}=48-42=\boxed{6}.$$