Hovey, C. and Toglia, A., "Four-Point Planar Homography Algorithm for Rectification Photogrammetry: Development and Applications," SAE Technical Paper 2013-01-0780, 2013, doi:10.4271/2013-01-0780.
Photogrammetric techniques of analyzing vehicles and scenes for accident reconstruction are well documented and have appeared in various forms and levels of complexity over the years. Plane-to-plane rectification algorithms, frequently used for accident reconstruction, are subsets of a growing field of computer vision algorithms, which are rigorously developed in [1,2,3,4]. While these algorithms are well formulated, they are not well illustrated. It is often not clear how to leverage advancements in computer vision algorithms for the purposes of rectification photogrammetry in the context of accident reconstruction.Perhaps as expected, a second strategy exists in the literature, which describes the use, as opposed to the development, of commercial computer programs for rectification photogrammetry [5,6,7,8]. Commercial software applications provide a robust and wide array of photogrammetric analysis. However, their use does not promote learning or insight into the mathematics of rectification algorithms, as the commercial program is compiled and guarded by the company as its intellectual property.There is a gap between these two bodies of work. That gap is the elucidation of the mathematics and the algorithm used to connect planar homography mathematics with concrete examples and published results that can be used by analysts to validate their own algorithm implementation.This paper helps bridge that gap. We first develop, in a simple yet robust way, the mathematical underpinning of the four-point planar algorithm. Next, we state the algorithm, along with the underlying assumptions and requirements. Finally, we review and provide results from two case studies and a laboratory study, which can be used to validate subsequent implementations by other analyists. Novel to this work is the application of suitable pre-conditioned homography matrices to enable image rectification via direct solution using only four control points.