A FAST MLE-BASED METHOD FOR ESTIMATING THE FUNDAMENTAL MATRIX W. Chojnacki, M. Brooks, A. van den Hengel, D. Gawley Department of Computer Science University of Adelaide Adelaide, SA 5005, Australia {wojtek|mjb|hengel|dg}@cs.adelaide.edu.au ABSTRACT We present a novel method for estimating the fundamen- tal matrix, a key problem arising in stereo vision. The method aims to minimise a cost function that is derived from maximum likelihood considerations. The respective min- imiser turns out to be significantly more accurate than the familiar algebraic least squares technique. Furthermore, the method is identical in accuracy to a Levenberg-Marquardt minimiser, while proving simpler and faster. 1. INTRODUCTION Many problems in computer vision may be couched in terms of parameter estimation. Accordingly, much effort has gone into the development of sophisticated techniques for gen- erating estimates of parameters. Some of these techniques utilise covariance information characterising uncertainty in the data [1, 2]. This paper is concerned with the applica- tion of a recently introduced covariance-based method [3] to the problem of estimating the fundamental matrix (see also [1,4–10]). However, we assume here that, as is often the case, covariance information is unavailable. A 3D point in a scene perspectively projected onto the image plane of a camera gives rise to an image point rep- resented by a pair (m 1 ,m 2 ) of coordinates, or equivalently, by the vector m =[m 1 ,m 2 , 1] T . A 3D point projected onto the image planes of two cameras endowed with sepa- rate coordinate systems gives rise to a pair of corresponding points. When represented by (m, m ′ ), this pair satisfies the epipolar equation m ′ T Fm =0, (1) where F =[f ij ] is a 3 × 3 fundamental matrix that incorpo- rates information about the relative orientation and internal geometry of the cameras [11]. The matrix F is subject to the rank-2 constraint det F =0. If we let θ =[f 11 ,f 12 ,f 13 ,f 21 ,f 22 ,f 23 ,f 31 ,f 32 ,f 33 ] T be the vector of parameters, x =[m 1 ,m 2 ,m ′ 1 ,m ′ 2 ] T be the vector of variables, and u(x)=[m 1 m ′ 1 ,m 2 m ′ 1 ,m ′ 1 ,m 1 m ′ 2 ,m 2 m ′ 2 ,m ′ 2 , m 1 ,m 2 , 1] T be the vector of transformed variables, then (1) can be rewritten as θ T u(x)=0. It is this latter form of the epipo- lar equation that we exploit to design a fast, accurate method for estimating the fundamental matrix given a set of corre- sponding points. In fashioning the technique, we shall not be concerned with the issues of robustness and parameter- isation. Accordingly, the performance of the new method will be gauged in isolation. 2. COST FUNCTIONS AND ESTIMATES Estimating the fundamental matrix will rest upon the use of cost functions measuring the extent to which the data and candidate estimates fail to satisfy the epipolar equa- tion. If—for simplicity—the rank-2 constraint is set aside, then, given a set of data {x 1 ,..., x n } and a cost function J = J (θ; x 1 ,..., x n ), a corresponding estimate  θ is de- fined as the parameter which minimises J : J (  θ) = min θ=0 J (θ; x 1 ,..., x n ). (2) Since (1) does not change if θ is multiplied by a non- zero scalar, we consider only cost functions satisfying J (tθ; x 1 ,..., x n )= J (θ; x 1 ,..., x n ) for any non-zero scalar t. For such functions,  θ satisfies (2) alongside t  θ for any non-zero scalar t, and so the corresponding estimate is defined only to within a scalar factor. 2.1. Algebraic least squares estimator A straightforward estimator is derived from the cost func- tion J ALS (θ; x 1 ,..., x n )= ‖θ‖ −2 n  i=1 θ T A i θ,