Showing papers by "Stephen J. Maybank published in 1995"

Robust detection of degenerate configurations for the fundamental matrix

Abstract: New methods are reported for the detection of multiple solutions (degeneracy) when estimating the fundamental matrix, with specific emphasis on robustness in the presence of data contamination (outliers). The fundamental matrix can be used as a first step in the recovery of structure from motion. If the set of correspondences is degenerate then this structure cannot be accurately recovered and many solutions will explain the data equally well. It is essential that we are alerted to such eventualities. However, current feature matchers are very prone to mismatching, giving a high rate of contamination within the data. Such contamination can make a degenerate data set appear non degenerate, thus the need for robust methods becomes apparent. The paper presents such methods with a particular emphasis on providing a method that will work on real imagery and with an automated (non perfect) feature detector and matcher. It is demonstrated that proper modelling of degeneracy in the presence of outliers enables the detection of outliers which would otherwise be missed. Results using real image sequences are presented. All processing, point matching, degeneracy detection and outlier detection is automatic. >

Probabilistic analysis of the application of the cross ratio to model based vision: misclassification

Abstract: The cross ratio of four colinear points is of fundamental importance in model based vision, because it is the simplest numerical property of an object that is invariant under projection to an image. It provides a basis for algorithms to recognise objects from images without first estimating the position and orientation of the camera. A quantitative analysis of the effectiveness of the cross ratio in model based vision is made. A given imageI of four colinear points is classified by making comparisons between the measured cross ratio τ of the four image points and the cross ratios stored in the model database. The imageI is accepted as a projection of an objectO σ with cross ratio σ if |τ−σ|≤ntu, wheren is the standard deviation of the image noise,t is a threshold andu=∥∇τ∥. The performance of the cross ratio is described quantitatively by the probability of rejectionR, the probability of false alarmF and the probability of misclassificationp σ(ς), defined for two model cross ratios σ, ς. The trade off between these different probabilities is determined byt. It is assumed that in the absence of an object the image points have identical Gaussian distributions, and that in the presence of an object the image points have the appropriate conditional densities. The measurements of the image points are subject to small random Gaussian perturbations. Under these assumptions the trade offs betweenR,F andp σ(ς) are given to a good approximation byR=2(1−Ф(t)),F=r F ∈t, $$\sqrt {p_\sigma (\varsigma )} = e_\sigma \in t\left| {\sigma - \varsigma } \right|^{ - 1} $$ ∈t|σ−ς|−1, where e is the relative noise level, Ф is cumulative distribution function for the normal distribution,r F is constant, ande σ is a function of σ only. The trade off betweenR andF is obtained in Maybank (1994). In this paper the trade off betweenR andp σ(ς) is obtained. It is conjectured that the general form of the above trade offs betweenR,F andp σ(ς) is the same for a range of invariants useful in model based vision. The conjecture prompts the following definition: an invariant which has trade offs betweenR,F,p σ(ς) of the above form is said to benon-degenerate for model based vision. The consequences of the trade off betweenR andp σ(ς) are examined. In particular, it is shown that for a fixed overall probability of misclassification there is a maximum possible model cross ratio σ m , and there is a maximum possible numberN of models. Approximate expressions for σ m andN are obtained. They indicate that in practice a model database containing only cross ratio values can have a size of order at most ten, for a physically plausible level of image noise, and for a probability of misclassification of the order 0.1.

The critical line congruence for reconstruction from three images

Abstract: Let three projected images of a set of lines in space be given Then, in general, the relative positions of the lines can be reconstructed uniquely up to a collineation of space Reconstruction fails to be unique for certain critical sets of lines It is known that each critical set is parameterised by a Bordiga surface in ℙ4 A new proof of this result is given In addition, it is shown that every Bordiga surface parameterises a critical set of lines The proof involves an explicit construction of the second or spurious set of lines which projects down to the same three images as the veridical set of lines

Relation between 3D invariants and 2D invariants

Abstract: Polynomial relations are established between the invariants of certain mixed sets of points and lines and the invariants of their projected images. The relations are obtained using the properties of a rational curve, in fact a twisted cubic, which is a covariant of the given set of points and lines.