Aerial Photogrammetry

Aerial photograph geometry

The geometry of an aerial photograph is set by the orientation of the camera axis at exposure. The camera axis is the line from the optical centre of the objective lens, perpendicular to the photographic plane. It meets the photograph at the photographic centre, which is the principal point of a vertical photograph.

A vertical photograph is one whose camera axis is perpendicular to the datum, so the photo plane is parallel to the datum plane. An oblique photograph is taken with the axis intentionally inclined from the vertical, and a tilted photograph has its axis departing from the nadir (plumb) line. A photo plane normal to the datum does not by itself make a photograph vertical.

On a tilted photograph four reference points are used. The principal point is where the optical axis meets the photo. The nadir point is where the plumb line from the exposure station meets the photo. The isocentre is the point where the bisector of the angle of tilt meets the photo, lying between the principal point and the nadir. The axis of tilt is the line through the isocentre perpendicular to the principal line.

The scale of a tilted photograph is not uniform. It is larger on one side of the isocentre and smaller on the other, so it is constant only on a truly vertical photograph. Relief displacement on a tilted photograph is radial from the nadir point.

For a vertical photograph over terrain at elevation h, with flying height H above datum and focal length f, the local scale is S = f/(H-h). The scale increases with terrain elevation, so it is not uniform over undulating ground. For small h/H, S \approx \frac{f}{H}\left(1+\frac{h}{H}\right), so the change in scale is nearly linear in h. The average scale is S_{avg} = f/(H-h_{avg}). For a constant flying height it depends on the focal length and on the topography through the mean ground elevation. It does not depend on the physical size of the print or on the size of objects on the ground.

The height above ground follows from H' = f/S. The flying height above datum is then H = H' + h_{avg}. Example: with f = 0.15 m and scale 1/6000, H' = 900 m. If the average terrain is 250 m, then H = 1150 m.

A scale along a line is the ratio of the photo length to the corresponding ground length, expressed in the same units. Example: a 40 m ground line measuring 8 mm on the photo has scale 8/40000 = 1/5000.

In a low oblique photograph, parallel ground lines that are not parallel to the picture plane converge toward a vanishing point. The scale changes continuously along the flight direction, the horizon does not lie within a low oblique frame, and square ground cells are distorted into trapezoids of differing size. Converging parallel lines and trapezoidal cells therefore indicate that the camera axis is tilted away from the nadir and that the scale is not constant.

Aerial photo coverage and overlap

The ground coverage of one photograph is found from the format side s and the photo scale 1/m, where m = 1/S. The ground side is L = s\,m, and the ground area is (sm)^2. Example: a 23 cm format at scale 1:15000 gives a ground side of 3450 m, so the area is about 11.9 km^2.

For a required ground area A, the ground side is \sqrt{A}, the scale factor is m = \sqrt{A}/s, and the flying height above ground is H' = m f. Example: for A = 4 km^2, a 0.20 m format, and f = 0.10 m, the ground side is 2000 m, m = 10000, and H' = 1000 m.

Let p be the end lap (longitudinal overlap) and q the side lap. The net ground advance along the flight line is (1-p)L, and the net advance between adjacent strips is (1-q)L. The number of strips is N_s = \lceil W/((1-q)L)\rceil + 1, where W is the width across the flight lines. The number of photographs per strip is N_p = \lceil D/((1-p)L)\rceil + 1, where D is the length along the flight line. The total is N_s \times N_p. Example: for a 10 km by 6 km area flown along the 10 km side, with L = 2 km, p = 60\% and q = 30\%, N_p = \lceil 12.5\rceil + 1 = 14, N_s = \lceil 4.29\rceil + 1 = 6, and the total is 84.

When the end lap p is at least 50%, the common area of three successive photographs is (2p-1)L. Example: with 60% end lap, the three-photo common region is 20% of the ground side.