Properties of the Fundamental Matrix in the Rectification Of Stereo Pairs of Images

Abstract

Introduction. This paper addresses the problem of stereo pair rectification based on a given fundamental matrix, which encodes the epipolar geometry between two images of a static scene acquired from different viewpoints. Rectification is a key preprocessing step, as it transforms the images in such a way that corresponding points lie on the same horizontal scanlines, which significantly simplifies the correspondence search and subsequent depth estimation in many computer vision tasks. Unlike classical ap proaches that construct rectifying transformations through general linear transformations, we exploit the intrinsic structure of the fundamental matrix. The conditions are explored under which the fundamental matrix can be transformed into a canonical form corresponding to a perfectly rectified stereo configuration with horizontal epipolar lines.

The Purpose of the paper is to investigate the problem of rectification of a stereo pair of images based on a given fundamental matrix describing the epipolar geometry between two projections of the scene.

Methods. An analytical approach is proposed to construct transformations that bring the fundamental matrix to a canonical form corresponding to a perfectly aligned stereo pair with horizontal epipolar lines.

Results. The main contribution of the paper is the reduction of the rectification problem to the problem of finding two rotation matrices satisfying a set of constraints derived from the properties of the fundamental matrix. As a result, the original problem of determining arbitrary projective transformations is replaced by a more tractable problem with significantly fewer degrees of freedom.

Conclusions. An important feature of the proposed approach is that it does not require knowledge of the intrinsic calibration parameters of the cameras. This makes the method applicable in uncalibrated settings, where only the fundamental matrix is available. Such scenarios frequently arise in practical applications, including structure-from-motion pipelines, stereo vision systems, and 3D scene reconstruction from image pairs. The proposed framework can serve as a basis for developing efficient numerical algorithms for stereo rectification. Future work may include the study of numerical stability, robustness to noise in the estimation of the fundamental matrix, and extensions to multi-view settings.