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The spatial distortion possible in an aerial photo, in which elevated objects such as buildings appear to lean outward, is first illustrated; then, you are presented with a formula by which height of an object in a photo can be calculated. A schematic diagram helps to understand the parameters in another formula that applies to a stereo pair of successive photos along a flight line. The use of special instruments called stereo plotters that allow a systematic process of determining contour lines is described as one of the conventional methods for producing topographic maps.

Measuring Heights from Individual and Paired Images


Aerial photograph of downtown Long Beach, California.

11-11: Using this figure, determine the height h of the building to which are drawn white arrows to distances d (photo displacement from bottom to top) and r (to building top). On the actual photo (not your screen) d = 0.5 inch and r = 3.0 inches. Scale of the photo is 1:3600. Aircraft altitude is 1800 ft. ANSWER






ha = H - (Bf)/Pa

Since H and f control the photo scale, these values plus the appropriate parallaxes and the air base B are sufficient to establish the actual distance from the principal point of any location in one of the photos. For the point at a, the X and Y distances are given by Xa = B(xa/Pa and Ya = B(ya/pa. Similar equations hold for point b. The slope angle Θ between points a and b is just the angle whose tangent is ha/hb, where a is found to be higher than b. The real length of the slope between a and b is determined using the Pythagorean equation; thus: ab = [(Xa - Xb)2 - (Ya - Yb)2]1/2.

11-12: For the adventurous, try this question: For the inclined surface containing separated points M and N on a slope whose heights or relief we want to determine, we measure the distance (in the direction parallel to the flight line) on one photo between the base (here, point N which is lower than point M) of that slope and the principal point (the optical center of a photo = nadir point if camera is on a level aircraft and points straight down). Use the above equations to figure out the heights of points M and N in the diagram, their coordinate positions (in millimeters) on the left photo, the actual length of line MN, and the slope angle. Imput parameters: H = 4000 m; B = 400 m; f = 120 mm; Xm = - 20 m; Ym = - 40 m; Xm' = - 100 m; Xn = + 30 m; Yn = - 70 m; xn' = - 80 m. This problem is calculation intensive; do this question only if you are ambitious.ANSWER




11-13: A stereo pair was obtained from a photo mission flown at 2000 meters. We wish to determine the height of a water tower (storage tank on legs).The photo base length P between the two photos when properly placed for effective stereo viewing is 70 mm. The differential parallax dP for the tower, as measured by a stereometer, is 0.4 mm. What is the height? ANSWER

Using the above devices to specify a large enough number of elevation points in stereo photos to permit contouring is difficult and tedious. Instead, we apply sophisticated optical-mechanical stereoplotters. These devices use two or three movable projectors that pass light through stereo pairs made into glass-mounted film diapositives placed onto a tracing table (platen) which we can raise and lower. We produce a 3-D image or stereomodel, which an outside observer can view by some filter method. For black and white photos, we do this by the anaglyph method, in which we project one photo through a red filter and the other a cyan filter and then view them through eyeglasses with corresponding color filters. We project a small point of light or dot onto the stereomodel. When the dot appears to coincide with some part of the model surface (as seen through the viewer), that dot locates a particular elevation determined by the platen height and calibrated by control points. An experienced operator can then move the light along the model surface, such that it always remains coincident. This motion traces a contour line. They then raise (or lower) the platen table a fixed amount determined by the chosen contour interval and they contour the next elevation. This process continues until they trace the entire surface. Today, analytical stereoplotters are capable of semi-automating the contouring through computer processing of mathematically transformed data.

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Primary Author: Nicholas M. Short, Sr. email: