1 Introduction

The aim of the work in this paper is to be able to make measurements of world planes from their perspective images, and to accurately predict the uncertainty of these measurements. Thus a camera becomes a measurement device. Example applications of this device include measurement of interior scenes such as walls or floors for furniture placement and interior design purposes, and architectural measurements, where the size and position of windows, doors etc are determined [ 1 ].

The camera model for perspective images of planes is well known [ 13 ]: points on the world plane are mapped to points on the image plane by a plane to plane homography , also known as a plane projective transformation. A homography is described by a matrix H . Once this matrix is determined the back-projection of an image point to a point on the world plane is straightforward. The distance between two points on the world plane is simply computed from the Euclidean distance between their back-projected images.

However, a measurement is of little use unless its accuracy is known. Estimating the accuracy (or uncertainty) requires a proper treatment of the sources of error, not just the error in selecting the image points but also the errors in the homography matrix itself. The homography matrix error arises from the position errors of the point correspondences from which the matrix is computed.

In this paper we make three novel contributions: first, it is shown in section  4 that first order uncertainty analysis is sufficient for typical imaging arrangements. This is achieved by developing the analysis to second order and obtaining a bound on the truncation error. Second, it is shown that the first order analysis is exact for the affine part of the homography, and that an approximation is only involved for the non-linear part. Third, and most significant, in section  5 an expression is obtained for the covariance of the estimated H matrix by using first order matrix perturbation theory.

The uncertainty analysis developed here builds on and extends previous analysis of the uncertainty in relations estimated from homogeneous equations, for example homographies [ 10 , 11 ] and epipolar geometry [ 4 , 5 ]. It extends these results because it covers the cases both of where the matrix is exactly and is over determined by the world-image correspondences, and furthermore it is not adversely affected when the estimation matrix is near singular. This is explained in more detail in section  5 . The correctness of the uncertainty predictions has been extensively tested both by Monte Carlo simulation and by numerous experiments on real images.

Section  6 describes how the uncertainty analysis is applied to particular measurements taking account of the cumulative effects of different error sources, including the image point localisation and the homography matrix covariance.

Section  7 gives examples of predicting uncertainties and achieving a specified uncertainty by varying the number and distribution of correspondences. Both interior and architectural measurement examples are covered.