Height from Stereo Parallax

Look at the same object from two positions and it appears to shift against its background. Photogrammetry measures that shift and turns it into height. Parallax here is the displacement of a point between the two photographs of a stereo pair, measured PARALLEL to the flight line: the absolute parallax $p_a$ at the object's base, and the parallax difference $\Delta p$ between its top and its base. The exact relation is $h = H\,\Delta p/(p_a + \Delta p)$.

The reason a point higher up has more parallax is worth seeing rather than taking on faith. A closer object shifts more between two viewpoints — hold a finger at arm's length, close one eye and then the other, and it jumps further than the wall behind it. The top of a building is closer to the camera than its base, so it shifts more, and the difference is exactly what $\Delta p$ measures. Everything else is similar triangles.

Most textbooks also give the approximation $h \approx H\,\Delta p/p_a$, dropping the $\Delta p$ from the denominator. It overstates the height by the factor $(p_a + \Delta p)/p_a$, which is well under a per cent for a tree on a high-altitude pair and tens of per cent for serious relief. This site uses the exact form, because there is no reason not to and because the approximation is a genuine source of quiet error in mountainous work.

The number that should shape how you think about this is the sensitivity. On a typical pair, one millimetre of parallax on the photograph is worth tens of metres of height on the ground. That is why parallax was measured with a floating mark and a parallax bar reading to hundredths of a millimetre rather than with a scale rule, and why the whole apparatus of stereoplotters existed. Run the equation backwards before planning a measurement: work out what $\Delta p$ the height you need to resolve would produce, and compare it against what your instrument or matcher can actually read. If the answer is a few hundredths of a millimetre, the height you get back will be noise.

Two conditions the geometry assumes, and reality breaks both. The pair must be truly vertical — tilt introduces errors this equation cannot express. And the parallax must be measured PARALLEL to the flight line. Any component measured across it is y-parallax, which is a symptom of tilt, of unequal flying heights, or of a badly oriented pair, and folding it into $\Delta p$ produces a confidently wrong answer. Clearing y-parallax is exactly what relative orientation does, and a stereo pair that will not clear it is telling you something about the flight rather than about the ground.

The last thing worth saying is that this equation has not been retired, it has been industrialised. A modern dense matcher is solving precisely this relation, for every pixel, across every overlapping pair in the block, tens of millions of times — and then a bundle adjustment reconciles all of it simultaneously with the camera positions and the interior orientation. Understanding the two-image case is what lets you look at a point cloud and know why the trees are noisy, why the north-facing slope is worse than the south-facing one, and why the edges of the block are weaker than the middle.