A plane measuring device
Introduction
The aim of the work in this paper is to be able to make measurements of world planes from their perspective images, and 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.
The camera model for perspective images of planes is well known [1]: 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 3×3 matrix H. Once the 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, 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 [2] and epipolar geometry [3], [4]. It extends these results because it covers the cases where the matrix is exactly determined and the case where the matrix is over-determined, by the world-image correspondences. Furthermore, the analysis 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.
6 Uncertainty in point localization, 7 Uncertainty in distance measurement describe how the uncertainty analysis is applied to particular measurements taking account of the cumulative effects of different error sources, including the image point localization and the homography matrix covariance.
Section 8 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.
Section snippets
The camera model
We describe here the camera model which consists of central projection specialised to planes.
Fig. 1 shows the imaging process. The notation used is that points on the world plane are represented by upper case vectors, X, and their corresponding images are represented by lower case vectors x, where x and X are homogeneous 3-vectors, and Under perspective projection corresponding points are related by [1], [5]:where H is a 3×3 homogeneous matrix, and “=” is equality up
Computing the plane to plane homography
From Eq. (1) each image to world point correspondence provides two equations linear in the H matrix elements. They are
For n correspondences we obtain a system of 2n equation in 8 unknowns. If n=4 then an exact solution is obtained. Otherwise, if n>4, the matrix is over determined, and for non-perfect data H is estimated by a suitable minimisation scheme.
The covariance of the estimated H matrix depends both on the errors in the position
First and second order uncertainty analysis
To avoid unnecessarily complicated algebra the comparison between first and second order analysis is developed for a line to line homography. The one-dimensional case illustrates all the ideas involved, and the algebraic expressions are easily interpreted. The generalisation to 3×3 matrices is straight-forward and does not provide any new insights here.
In the one-dimensional case Eq. (1) reduces to:where H2×2 is a 2×2 homography matrix. For the geometry shown in Fig. 2 the matrix is
The covariance of the estimated homography
In this section we compute the covariance of the homography H estimated from n image-world point correspondences. We consider all the computation points to be measured with error modelled as an homogeneous, isotropic Gaussian noise process. For the image computation points we define σx=σy=σ, and for the world ones ΣX=ΣY=Σ. It is not strictly necessary to have such idealised distributions but this has not been found to be a restriction in practice.
From Section 3.4 we seek the eigenvector h with
Uncertainty in point localization
There may be errors in the world and image points used to compute the homography, and there may be errors in the image points back projected to make world measurements. All of these uncertainties must be taken into account in order to compute a cumulative uncertainty for the world measurement.
In this section we list the formulae used to compute the uncertainty for measurements under various error situations. The first order analysis is assumed sufficient. The uncertainty in the homography is
Uncertainty in distance measurement
The distance d between two world points X1 and X2 will also have an associated uncertainty.
Let the two end points of the image of the segment we want to measure be x1=(x1,y1)T and x2=(x2,y2)T
The two corresponding points on the world plane in homogeneous coordinates are: U1=(U1,V1,W1)T and U2=(U2,V2,W2)T where Ui=H(xi,yi,1)T. In inhomogeneous coordinates they are: Xi=(Xi,Yi)T with Xi=Ui/Wi and Yi=Vi/Wi;
The distance between the two points X1 and X2 on the world plane is:and its
Examples and applications
The correctness of the uncertainity analysis is demonstrated in this section on a number of examples. It is shown that the ground truth measurements always lie within the estimated error bounds. Furthermore, the utility of the analysis is illustrated. The covariance expression predicts uncertainty given the number of image-world computation points and their distributions. It is thus possible to decide where correspondences are required in order to achieve a particular desired measurement
Conclusion and extensions
We have developed a first order model of uncertainty prediction which takes account of all the errors involved in estimating the homography and employing it for measurements. The validity has been demonstrated in a number of applications. The approach can be applied to any world measurement, lengths and parallelism have been demonstrated, and area and angles can also be computed.
It is worth noting that although the theory has been developed here for plane to plane homographies, the same
Acknowledgements
We would like to thank Andrew Fitzgibbon for the development and maintenance of the Targetjr/IUE software. This software was supported financially by the UK EPSRC IUE Implementation Project GR/L05969.
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