Wojciech Chojnacki

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.

Determining the egomotion of an uncalibrated camera from instantaneous optical flow

A procedure is described for self-calibration of a moving camera from instantaneous optical flow. Under certain assumptions this procedure allows the egomotion and some intrinsic parameters of the camera to be determined solely from the instantaneous positions and velocities of a set of image features. The proposed method relies on the use of a differential epipolar equation that relates optical flow to the egomotion and the internal geometry of the camera. A detailed derivation of this equation is presented. This aspect of the work may be seen as a recasting into an analytical framework of the pivotal research of Vieville Faugeras [Proceedings of the Fifth International Conference on Computer Vision (IEEE Computer Society Press, Los Alamitos, Calif., 1995), pp. 750–756]. The information about the cameras egomotion and internal geometry enters the differential epipolar equation via two matrices. It emerges that the optical flow determines the composite ratio of some of the entries of the two matrices. It is shown that a camera with unknown focal length undergoing arbitrary motion can be self-calibrated by means of closed-form expressions in the composite ratio. The corresponding formulas specify five egomotion parameters as well as the focal length and its derivative. A procedure is presented for reconstructing the viewed scene, up to a scale factor, from the derived self-calibration data and the optical flow data. Experimental results are given to demonstrate the correctness of the approach.

Rationalising the Renormalisation Method of Kanatani

The renormalisation technique of Kanatani is intended to iteratively minimise a cost function of a certain form while avoiding systematic bias inherent in the common method of minimisation due to Sampson. Within the computer vision community, the technique has generally proven difficult to absorb. This work presents an alternative derivation of the technique, and places it in the context of other approaches. We first show that the minimiser of the cost function must satisfy a special variational equation. A Newton-like, fundamental numerical scheme is presented with the property that its theoretical limit coincides with the minimiser. Standard statistical techniques are then employed to derive afresh several renormalisation schemes. The fundamental scheme proves pivotal in the rationalising of the renormalisation and other schemes, and enables us to show that the renormalisation schemes do not have as their theoretical limit the desired minimiser. The various minimisation schemes are finally subjected to a comparative performance analysis under controlled conditions.

Revisiting Hartley's normalized eight-point algorithm

Hartleys eight-point algorithm has maintained 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 normalized data. It is first established that the normalized algorithm acts to minimize a specific cost function. It is then shown that this cost function I!; statistically better founded than the cost function associated with the nonnormalized 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.

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.

Impossible and ambiguous shading patterns

A smooth object depicted in a monochrome image will often exhibit brightness variation, or shading. A problem much studied in computer vision has been that of how object shape may be recovered from image shading. When the imaging conditions are such that an overhead point-source illuminates a smooth Lambertian surface, the problem may be formulated as that of finding a solution to an eikonal equation. This article will focus on the existence and uniqueness of such solutions, reporting recent results obtained. With regard to existence, shading patterns are exhibited for which there is no corresponding object shape. Specifically, a necessary and sufficient condition is presented for a circularly symmetric eikonal equation to admit exclusively unbounded solutions; additionally, a sufficient condition is given for an eikonal equation to have no solution whatsoever. In connection with uniqueness, we consider eikonal equations, defined over a disc, such that the Euclidean norm of the gradient of any solution is circularly symmetric, vanishes exactly at the disc center, and diverges to infinity as the circumference of the disc is approached. Contrary to earlier influential work, a class of such eikonal equations is shown to possess simultaneously circularly symmetric and noncircularly symmetric bounded smooth solutions.

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.

REVISITING PENTLAND'S ESTIMATOR OF LIGHT SOURCE DIRECTION

We examine the pioneering method of Pentland [ J. Opt. Soc. Am.72, 448 ( 1982)] for automatically estimating direction of the “sun” (light source) from a single image. It is shown that, under the assumptions used in the derivation of the method, the estimate of source direction is erroneous. Specifically, it is shown that an image-based expression used in calculating source direction diverges to infinity as the density of image points is increased and that the formula involving this expression is therefore incorrect. When the method is implemented, the flaw manifests itself in the undesirable dependence of the estimator on image resolution. Supporting experimental evidence is given for this. An alternative source-direction estimator that is free of these drawbacks is proposed.

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.