Analytic reduction of the Kruppa equations

Abstract

Given the fundamental matrix between a pair of images taken by a nonstationary projective camera with constant internal parameters, we show how to use the two independent Kruppa equations in order to explicitly cut down the number of parameters of the Kruppa matrix KKT by exactly two. Thus, we derive a procedure which results in a closed formula for the Kruppa matrix that depends on exactly three remaining parameters. This formula allows an easy incorporation of the positivity constraint and admits of an interpretation in terms of the image of the horopter. We focus on the general case where the camera motion is unknown and not restricted to some special type. Solutions of the Kruppa equations given three fundamental matrices have been attempted in the past by iterative numerical methods that are searching in multidimensional spaces. As an application of the reduced Kruppa matrix mentioned above we also outline how this problem can be analytically reduced to the determination of the real, positive roots of a polynomial of 14-th degree in one variable. © Springer-Verlag Berlin Heidelberg 2002.