Darren Gawley

On the fitting of surfaces to data with covariances

We consider the problem of estimating parameters of a model described by an equation of special form. Specific models arise in the analysis of a wide class of computer vision problems, including conic fitting and estimation of the fundamental matrix. We assume that noisy data are accompanied by (known) covariance matrices characterizing the uncertainty of the measurements. A cost function is first obtained by considering a maximum-likelihood formulation and applying certain necessary approximations that render the problem tractable. A Newton-like iterative scheme is then generated for determining a minimizer of the cost function. Unlike alternative approaches such as Sampsons method or the renormalization technique, the new scheme has as its theoretical limit the minimizer of the cost function. Furthermore, the scheme is simply expressed, efficient, and unsurpassed as a general technique in our testing. An important feature of the method is that it can serve as a basis for conducting theoretical comparison of various estimation approaches.

What value covariance information in estimating vision parameters

Many parameter estimation methods used in computer vision are able to utilise covariance information describing the uncertainty of data measurements. This paper considers the value of this information to the estimation process when applied to measured image point locations. Covariance matrices are first described and a procedure is then outlined whereby covariances may be associated with image features located via a measurement process. An empirical study is made of the conditions under which covariance information enables generation of improved parameter estimates. Also explored is the extent to which the noise should be anisotropic and inhomogeneous if improvements are to be obtained over covariance-free methods. Critical in this is the devising of synthetic experiments under which noise conditions can be precisely controlled. Given that covariance information is, in itself, subject to estimation error tests are also undertaken to determine the impact of imprecise covariance information upon the quality of parameter estimates. Finally, an experiment is carried out to assess the value of covariances in estimating the fundamental matrix from real images.

A new constrained parameter estimator for computer vision applications

A method of constrained parameter estimation is proposed for a class of computer vision problems. In a typical application, the parameters will describe a relationship between image feature locations, expressed as an equation linking the parameters and the image data, and will satisfy an ancillary constraint not involving the image data. A salient feature of the method is that it handles the ancillary constraint in an integrated fashion, not by means of a correction process operating upon results of unconstrained minimisation. The method is evaluated through experiments in fundamental matrix computation. Results are given for both synthetic and real images. It is demonstrated that the method produces results commensurate with, or superior to, previous approaches, with the advantage of being faster than comparable techniques.

From FNS to HEIV: a link between two vision parameter estimation methods

Problems requiring accurate determination of parameters from image-based quantities arise often in computer vision. Two recent, independently developed frameworks for estimating such parameters are the FNS and HEIV schemes. Here, it is shown that FNS and a core version of HEIV are essentially equivalent, solving a common underlying equation via different means. The analysis is driven by the search for a nondegenerate form of a certain generalized eigenvalue problem and effectively leads to a new derivation of the relevant case of the HEIV algorithm. This work may be seen as an extension of previous efforts to rationalize and interrelate a spectrum of estimators, including the renormalization method of Kanatani and the normalized eight-point method of Hartley.

FNS, CFNS and HEIV: A Unifying Approach

Estimation of parameters from image tokens is a central problem in computer vision. FNS, CFNS and HEIV are three recently developed methods for solving special but important cases of this problem. The schemes are means for finding unconstrained (FNS, HEIV) and constrained (CFNS) minimisers of cost functions. In earlier work of the authors, FNS, CFNS and a core version of HEIV were applied to a specific cost function. Here we extend the approach to more general cost functions. This allows the FNS, CFNS and HEIV methods to be placed within a common framework.

A new constrained parameter estimator: experiments in fundamental matrix computation

In recent work the authors proposed a wide-ranging method for estimating parameters that constrain image feature locations and satisfy a constraint not involvingimage data. The present work illustrates the use of the method with experiments concerning estimation of the fundamental matrix. Results are given for both synthetic and real images. It is demonstrated that the method gives results commensurate with, or superior to, previous approaches, with the advantage of being faster than comparable methods.

FNS and HEIV: relating two vision parameter estimation frameworks

Problems requiring accurate determination of parameters from image-based quantities arise often in computer vision. Two recent, independently developed frameworks for estimating such parameters are the FNS and HEIV schemes. Here it is shown that FNS (fundamental numerical scheme) and a core version of HEIV (heteroscedastic errors-in-variables) are essentially equivalent, solving a common underlying equation via different means. The analysis is driven by the search for a nondegenerate form of a certain generalised eigenvalue problem, and effectively leads to a new derivation of the relevant case of the HEIV algorithm. This work may be seen as an extension of previous efforts to rationalise and inter-relate a spectrum of estimators, including the renormalisation method of Kanatani and the normalised eight-point method of Hartley.

A statistical rationalisation of Hartley's normalised eight-point algorithm

The eight-point algorithm of Hartley occupies an important place in computer vision, notably as a means of providing an initial value of the fundamental matrix for use in iterative estimation methods. In this paper, a novel explanation is given for the improvement in performance of the eight-point algorithm that results from using normalised data. A first step is singling out a cost function that the normalised algorithm acts to minimise. The cost function is then shown to be statistically better founded than the cost function associated with the non-normalised algorithm. This augments the original argument that improved performance is due to the better conditioning of a pivotal matrix. Experimental results are given that support the adopted approach. This work continues a wider effort to place a variety of estimation techniques within a coherent framework.

Is covariance information useful in estimating vision parameters

This paper assesses some of the practical ramifications of recent developments in estimating vision parameters given information characterizing the uncertainty of the data. This uncertainty information may sometimes be estimated in association with the observation process, and is usually represented in the form of covariance matrices. An empirical study is made of the conditions under which improved parameter estimates can be obtained from data when covariance information is available. We explore, in the case of fundamental matrix estimation and conic fitting, the extent to which the noise should be anisotropic and inhomogeneous if improvements over traditional methods are to be obtained. Critical in this is the devising of synthetic experiments under which noise conditions can be precisely controlled. Given that covariance information is, in itself, subject to estimation error, testes are also undertaken to determine the impact of imprecise covariance information upon the quality of parameter estimates. We thus investigate the consequences for parameter estimation of inaccuracies in the characteristiziaton of noise that inevitably arise in practical computation.

A fast MLE-based method for estimating the fundamental matrix

We present a novel method for estimating the fundamental 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 minimiser 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.