Andrew D. J. Cross

Graph matching with a dual-step EM algorithm

This paper describes a new approach to matching geometric structure in 2D point-sets. The novel feature is to unify the tasks of estimating transformation geometry and identifying point-correspondence matches. Unification is realized by constructing a mixture model over the bipartite graph representing the correspondence match and by affecting optimization using the EM algorithm. According to our EM framework, the probabilities of structural correspondence gate contributions to the expected likelihood function used to estimate maximum likelihood transformation parameters. These gating probabilities measure the consistency of the matched neighborhoods in the graphs. The recovery of transformational geometry and hard correspondence matches are interleaved and are realized by applying coupled update operations to the expected log-likelihood function. We evaluate the technique on two real-world problems.

Inexact graph matching using genetic search

This paper describes a framework for performing relational graph matching using genetic search. There are three novel ingredients to the work. Firstly, we cast the optimisation process into a Bayesian framework by exploiting the recently reported global consistency measure of Wilson and Hancock as a fitness measure. The second novel idea is to realise the crossover process at the level of subgraphs, rather than employing string-based or random crossover. Finally, we accelerate convergence by employing a deterministic hill-climbing process prior to selection. Since we adopt the Bayesian consistency measure as a fitness function, the basic measure of relational distance underpinning the technique is Hamming distance. Our standpoint is that genetic search provides a more attractive means of performing stochastic discrete optimisation on the global consistency measure than alternatives such as simulated annealing. Moreover, the action of the optimisation process is easily understood in terms of its action in the Hamming distance domain. We demonstrate empirically not only that the method possesses polynomial convergence time but also that the convergence rate is more rapid than simulated annealing. We provide some experimental evaluation of the method in the matching of aerial stereograms and evaluate its sensitivity on synthetically generated graphs.

Corner detection via topographic analysis of vector-potential

Abstract This paper describes how corner detection can be realised using a new feature representation based on a magneto-static analogy. The idea is to compute a vector-potential by appealing to an analogy in which the Canny edge-map is regarded as an elementary current density residing on the image plane. In this paper, we demonstrate that corners are located at the saddle-points of the magnitude of the vector-potential. These points correspond to the intersections of saddle-ridge and saddle-valley structures, i.e. to junctions of the edge and symmetry lines. We describe a template-based method for locating the saddle-points. This involves performing a non-minimum suppression test in the direction of the vector-potential and a non-maximum suppression test in the orthogonal direction. Experimental results using both synthetic and real images are given. We investigate the angle and scale sensitivity of the new corner detector and compare it with a number of alternative corner detectors.

Genetic Search for Structural Matching

This paper describes a novel framework for performing relational graph matching using genetic search. The fitness measure is the recently reported global consistency measure of Wilson and Hancock. The basic measure of relational distance underpinning the technique is Hamming distance. Our standpoint is that genetic search provides a more attractive means of performing stochastic discrete optimisation on the global consistency measure than alternatives such as simulated annealing. Moreover, the action of the optimisation process is easily understood in terms of its action in the Hamming distance domain. We provide some experimental evaluation of the method in the matching of aerial stereograms.

Convergence of a hill-climbing genetic algorithm for graph matching

Abstract This paper presents a convergence analysis for the problem of consistent labelling using genetic search. The work builds on a recent empirical study of graph matching where we showed that a Bayesian consistency measure could be efficiently optimised using a hybrid genetic search procedure which incorporated a hill-climbing step. In the present study we return to the algorithm and provide some theoretical justification for its observed convergence behaviour. The novelty of the analysis is to demonstrate analytically that the hill-climbing step significantly accelerates convergence, and that the convergence rate is polynomial in the size of the node-set of the graphs being matched.

Structural Matching with Active Triangulations

This paper describes a novel approach to relational matching problems in machine vision. Rather than matching static scene descriptions, the approach adopts an active representation of the data to be matched. This representation is based on a Delaunay triangulation that is iteratively reconfigured to increase its degree of topological congruency with the model relational structure in a reconstructive matching process. The active reconfiguration of relational structures is controlled by a MAP update process. The final restored graph representation is optimal in the sense that it hasmaximum a posteriori probabilitywith respect to the available attributes for the objects under match. The benefits of the technique are demonstrated experimentally on the matching of cluttered synthetic aperture radar data to a model in the form of a digital map. The operational limits of the method are established in a simulation study.

Scale space vector fields for symmetry detection

This paper describes a vectorial representation that can be used to assess the symmetry of objects in 2D images. The method exploits the calculus of vector fields. Commencing from the gradient field extracted from filtered grey-scale images we construct a vector potential. Our image representation is based on the distribution of tangential gradient vectors residing on the image plane. By embedding the image plane in an augmented 3-dimensional space, we compute the vector potential by performing volume integration over the distribution of edge tangents. The associated vector field is computed by taking the curl of the vector potential. The auxiliary spatial dimension provides a natural scale-space sampling of the generating edge-tangent distribution; as the height above the image plane is increased, so the volume over which averaging is effected also increases. We extract edge and symmetry lines through a topographic analysis of the vector-field at various heights above the image plane. Symmetry axes are lines of where the curl of the vector potential vanishes; at edges the divergence of the vector potential vanishes.

Scale-space vector fields for feature analysis

This paper describes a vectorial representation that can be used to assess the symmetry of objects in 2D images. The method exploits a magneto-static analogy. Commencing from the gradient-field extracted from filtered grey-scale images we construct a vector-potential. Our magneto-static analogy is that tangential gradient vectors represent the elements of a current distribution on the image plane. By embedding the image plane in an augmented 3-dimensional space, we compute the vector potential by performing volume integration over the current distribution. The associated magnetic field is computed by taking the curl of the vector-potential. The auxiliary spatial dimension provides a natural scale-space sampling of the generating current distribution; as the height above the image plane is increased, so the volume over which averaging is effected also increases. We extract edge and symmetry lines through a topographic analysis of the vector-field at various heights above the image plane. Symmetry axes are lines where the curl of the vector-potential vanishes; at edges the divergence of the vector-potential vanishes.

Inexact Graph Matching with Genetic Search

This paper describes a novel framework for performing relational graph matching using genetic search. The fitness measure is Bayesian in origin. It gauges relational consistency at both the symbolic and attribute levels. The basic measure of symbolic consistency is Hamming distance, while attribute consistency is measured using Mahalanobis distance. We provide examples of the performance on synthetic graphs containing significant levels of clutter. We also demonstrate that the technique is capable of resolving multiple graphs with significant overlap. The performance advantages over deterministic hill climbing are also demonstrated.

Relational matching with stochastic optimisation

This paper describes a novel framework for performing relational graph matching by stochastic optimisation. The starting point for this study is a configurational probability measure which gauges the consistency of relational matches using a compound exponential function of Hamming distance. In order to overcome some of the well documented shortcomings of deterministic updating, we develop two contrasting stochastic optimisation strategies. The first of these exploits the apparatus of statistical physics to compute the Boltzmann distribution that models the configurational probability measure so that relational matching may be performed by simulated annealing. The second approach is a genetic hill climbing algorithm which is motivated by the way in which our configurational probability measure models matching errors. Because the genetic optimisation commences with a pool of random matches, it obviates the need for accurate initialisation. Moreover, it can recover consistent matches exploiting only structural information from the graphs under match.