LEAST SQUARES

The technique of least-squares estimation is called upon when a statistically most probable solution is required for a mathematical problem that has multiple possible solutions. Consider the two-photo photogrammetric determination of an object point by spatial intersection of two rays. If the two rays perfectly intersect in space then there is a unique solution to the target point position. But what happens if there are 3 non-perfectly intersecting rays, or even 4,5 or more? With three rays there are three possible combinations of two-ray intersections, all of which produce solutions that will likely be slightly different. But which is correct? Or, which is most probable? Logically, we would assume that the most probable solution is represented by the average of all the possible unique solutions to this over determined problem (more measurements than necessary to obtain a single unique solution). This could be a lengthy computation, for in the case of 5 intersecting rays, for example, there are 10 possible unique solutions. The least-squares estimation allows us to determine the most probable solution via a single computation. This then improves the accuracy and reliability of the photogrammetric orientation process.