Translation of "Zur Ermittlung eines Objektes aus zwei Perspektiven mit innerer Orientierung" by Erwin Kruppa (1913)

Erwin Kruppa’s 1913 paper, Erwin Kruppa, “Zur Ermittlung eines Objektes aus zwei Perspektiven mit innerer Orientierung”, Sitzungsberichte der Mathematisch-Naturwissenschaftlichen Kaiserlichen Akademie der Wissenschaften, Vol. 122 (1913), pp. 1939–1948, which may be translated as “To determine a 3D object from two perspective views with known inner orientation”, is a landmark paper in Computer Vision because it provides the first five-point algorithm for relative pose estimation. Kruppa showed that (a finite number of solutions for) the relative pose between two calibrated images of a rigid object can be computed from five point matches between the images. Kruppa’s work also gained attention in the topic of camera self-calibration, as presented in (Maybank and Faugeras , 1992). Since the paper is still relevant today (more than a hundred citations within the last ten years) and the paper is not available online, we ordered a copy from the German National Library in Frankfurt and provide an English translation along with the German original. We also adapt the terminology to a modern jargon and provide some clarifications (highlighted in sans-serif font). For a historical review of geometric computer vision, the reader is referred to the recent survey paper (Sturm , 2011). Summary of Kruppa’s Approach It is well known (and was already known by the time Kruppa’s paper was written) that the epipolar geometry between two uncalibrated perspective images can be determined from seven point correspondences, up to three possible solutions in general. Kruppa showed that, when the images are calibrated, five correspondences are sufficient. The epipolar geometry, combined with the calibration information, then allows to recover the relative pose of the two images (up to scale), in turn enabling to determine the shape of the 3D scene (also up to scale). He proved that, in general, there are up to 11 solutions (22 possible relative poses, coming in 11 pairs of solutions corresponding to relative poses related by rotating one camera by 180 degrees about the baseline joining the cameras’ optical centers, since such a rotation leaves the reconstructed 3D shape invariant). In 1988, it was shown by (Demazure , 1988) that there are actually at most 10 distinct solutions. Later, (Nistér , 2004) developed the first algorithm that guarantees to indeed provide at most 10 solutions. Kruppa’s approach can be briefly outlined as follows. As mentioned, it was already known how to compute the epipolar geometry from seven point correspondences. When using only five such correspondences, two additional constraints are required. Kruppa’s insight was that the knowledge of the cameras’ calibration delivers such constraints. In particular, knowing the calibration of a camera is equivalent to knowing the image of the absolute conic (IAC) in that camera. The absolute conic is invariant to similarity transformations and thus the IAC is invariant to camera displacements. Hence, the IAC is directly determined by the camera’s intrinsic parameters and indeed, the IAC fully encodes the intrinsic parameters of a perspective camera. Kruppa essentially formulated what could be called epipolar constraints for images of conics. For images of points, the epipolar constraint states that the two image points lie on corresponding epipolar lines. As for images of conics, we have: each of the two image conics has two epipolar lines tangent to it – these two pairs of epipolar lines must correspond to one another. This observation gives rise to two constraints which, together with five point correspondences, allow to compute the ∗We thank the publishing house of the Austrian Academy of Sciences (ÖAW, https://verlag.oeaw.ac.at/) for granting permission to reproduce the original paper by Erwin Kruppa. 1 ar X iv :1 80 1. 01 45 4v 1 [ cs .C V ] 2 5 D ec 2 01 7 epipolar geometry and, subsequently, the relative pose and the 3D shape of the scene (all these up to the stated number of possible solutions). The term “Kruppa’s equations” was probably coined by (Maybank and Faugeras , 1992), who showed how to use the epipolar constraint for images of conics, for self-calibration: whereas Kruppa used the known calibration to compute epipolar geometry, Maybank and Faugeras showed that known epipolar geometry allows to compute the camera calibration, even in the absence of a calibration grid in the scene (to be precise, depending on the number of intrinsic parameters to be computed, one needs to provide the epipolar geometry for more than one image pair).