Previous work of the authors developed a theoretically well-founded scheme (FNS) for finding the minimiser of a class of cost functions. Various problems in video analysis, stereo vision, ellipse-fitting, etc, may be expressed in terms of finding such a minimiser. However, in common with many other approaches, it is necessary to correct the minimiser as a post-process if an ancillary constraint is also to be satisfied. In this paper we develop the first integrated scheme (CFNS) for simultaneously minimising the cost function and satisfying the constraint. Preliminary experiments in the domain of fundamental-matrix estimation show that CFNS generates rank-2 estimates with smaller cost function values than rank-2 corrected FNS estimates. Furthermore, when compared with the HartleyZisserman Gold Standard method, CFNS is seen to generate results of comparable quality in a fraction of the time.

Previous work of the authors developed a theoretically well-founded scheme (FNS) for finding the minimiser of a class of cost functions. Various problems in video analysis, stereo vision, ellipse-fitting, etc, may be expressed in terms of finding such a minimiser. However, in common with many other approaches, it is necessary to correct the minimiser as a post-process if an ancillary constraint is also to be satisfied. In this paper we develop the first integrated scheme (CFNS) for simultaneously minimising the cost function and satisfying the constraint. Preliminary experiments in the domain of fundamental-matrix estimation show that CFNS generates rank-2 estimates with smaller cost function values than rank-2 corrected FNS estimates. Furthermore, when compared with the HartleyZisserman Gold Standard method, CFNS is seen to generate results of comparable quality in a fraction of the time.

In recent work the authors proposed a wide-ranging method for estimating parameters that constrain image feature locations and satisfy a constraint not involving image 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.

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 non-degenerate 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 eightpoint method of Hartley. 1.

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 eightpoint 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. 1.

A recently proposed argument to explain the improved performance of the eight-point algorithm that results from using normalized data [IEEE Trans. Pattern Anal. Mach. Intell., 25(9):1172–1177, 2003] relies upon adoption of a certain model for statistical data distribution. Under this model, the cost function that underlies the algorithm operating on the normalized data is statistically more advantageous than the cost function that underpins the algorithm using unnormalized data. Here we extend this explanation by introducing a more refined, structured model for data distribution. Under the extended model, the normalized eight-point algorithm turns out to be approximately consistent in a statistical sense. The proposed extension provides a link between the existing statistical rationalization of the normalized eight-point algorithm and the approach of Mühlich and Mester for enhancing total least squares estimation methods via equilibration. Our contribution forms part of a wider effort to rationalize and interrelate foundational methods in vision parameter estimation. 1.

The eight-point scheme is the simplest and fastest scheme for estimating the fundamental matrix (FM) from a number of noisy correspondences. As it ignores the fact that the FM must be singular, the resulting FM estimate is often inaccurate. Existing schemes that take the singularity constraint into consideration are several times slower and significantly more difficult to implement and understand. This paper describes extended versions of the eight-point (8P) and the weighted eight-point (W8P) schemes that effectively take the singularity constraint into consideration without sacrificing the efficiency and the simplicity of both schemes. The proposed schemes are respectively called the extended eight-point scheme (E8P) and the extended weighted eight-point scheme (EW8P). The E8P scheme was experimentally found to give exactly the same results as Hartley's algebraic distance minimization scheme while being almost as fast as the simplest scheme (i.e., the 8P scheme). At the expense of extra calculations per iteration, the EW8P scheme permits the use of geometric cost functions and, more importantly, robust weighting functions. It was experimentally found to give near-optimal results while being 8-16 times faster than the more complicated schemes such as Levenberg-Marquardt schemes. The FM estimates obtained by the E8P and the EW8P schemes perfectly satisfy the singularity constraint, eliminating the need to enforce the rank-2 constraint in a post-processing step.