Kruppa Equations Research Articles

A case against Kruppa's equations for camera self-calibration

We consider the self-calibration problem for perspective cameras and especially the classical Kruppa equation approach. It is known that for several common types of camera motion, self-calibration is degenerate, which manifests itself through the existence of ambiguous solutions. The author previously (1997, 1999) studied these critical motion sequences and showed their importance for practical applications. Here, we reveal a type of camera motion that is not critical for the generic self-calibration problem, but for which the Kruppa equation approach fails. This is the case if the optical centers of all cameras lie on a sphere and if the optical axes pass through the sphere's center, a very natural situation for 3D object modeling from images. Results of simulated experiments demonstrate the instability of numerical self-calibration algorithms in near-degenerate configurations.

Algebraic derivation of the Kruppa equations and a new algorithm for self-calibration of cameras

An algebraic derivation of the Kruppa equations is presented. This derivation is much simpler and thus easier to understand than the conventional derivation based on projective geometry. Using the newly derived Kruppa equations, we further propose a new algorithm for determining a camera’s intrinsic parameters. Experimental results and discussions are given for both synthetic and real images.

Kruppa's equations derived from the fundamental matrix

The purpose of this paper is to give a specific form for Kruppa's equations in terms of the fundamental matrix. Kruppa's equations can be written explicitly in terms of the singular value decomposition (SVD) of the fundamental matrix.

Motion of an uncalibrated stereo rig: self-calibration and metric reconstruction

We address in this paper the problem of self-calibration and metric reconstruction (up to a scale factor) from one unknown motion of an uncalibrated stereo rig. The epipolar constraint is first formulated for two uncalibrated images. The problem then becomes one of estimating unknowns such that the discrepancy from the epipolar constraint, in terms of sum of squared distances between points and their corresponding epipolar lines, is minimized. Although the full self-calibration is theoretically possible, we assume in this paper that the coordinates of the principal point of each camera are known. Then, the initialization of the unknowns can be done based on our previous work on self-calibration of a single moving camera, which requires to solve a set of so-called Kruppa equations. Redundancy of the information contained in a sequence of stereo images makes this method more robust than using a sequence of monocular images. Real data has been used to test the proposed method, and the results obtained are quite good. We also show experimentally that it is very difficult to estimate precisely the coordinates of the principal points of cameras. A variation of as high as several dozen pixels in the principal point coordinates does not affect significantly the 3-D reconstruction.