{"title": "Contour Organisation with the EM Algorithm", "book": "Advances in Neural Information Processing Systems", "page_first": 880, "page_last": 886, "abstract": null, "full_text": "Contour Organisation with the EM \n\nAlgorithm \n\nJ. A. F. Leite and E. R. Hancock \n\nDepartment of Computer Science \n\nUniversity of York, York, Y01 5DD, UK. \n\nAbstract \n\nThis paper describes how the early visual process of contour organ(cid:173)\nisation can be realised using the EM algorithm. The underlying \ncomputational representation is based on fine spline coverings. Ac(cid:173)\ncording to our EM approach the adjustment of spline parameters \ndraws on an iterative weighted least-squares fitting process. The \nexpectation step of our EM procedure computes the likelihood of \nthe data using a mixture model defined over the set of spline cover(cid:173)\nings. These splines are limited in their spatial extent using Gaus(cid:173)\nsian windowing functions. The maximisation of the likelihood leads \nto a set of linear equations in the spline parameters which solve the \nweighted least squares problem. We evaluate the technique on the \nlocalisation of road structures in aerial infra-red images. \n\n1 \n\nIntroduction \n\nDempster, Laird and Rubin's EM (expectation and maximisation) [1] algorithm was \noriginally introduced as a means of finding maximum likelihood solutions to prob(cid:173)\nlems posed in terms of incomplete data. The basic idea underlying the algorithm \nis to iterate between the expectation and maximisation modes until convergence is \nreached. Expectation involves computing a posteriori model probabilities using a \nmixture density specified in terms of a series of model parameters. In the max(cid:173)\nimisation phase, the model parameters are recomputed to maximise the expected \nvalue of the incomplete data likelihood. In fact, when viewed from this perspective, \nthe updating of a posteriori probabilities in the expectation phase would appear \nto have much in common with the probabilistic relaxation process extensively ex(cid:173)\nploited in low and intermediate level vision [9, 2] . Maximisation of the incomplete \n\n\fContour Organisation with the EM Algorithm \n\n881 \n\ndata likelihood is reminiscent of robust estimation where outlier reject is employed \nin the iterative re-computation of model parameters [7]. \n\nIt is these observations that motivate the study reported in this paper. We are \ninterested in the organisation of the output of local feature enhancement operators \ninto meaningful global contour structures [13, 2]. Despite providing one of the clas(cid:173)\nsical applications of relaxation labelling in low-level vision [9], successful solutions \nto the iterative curve reinforcement problem have proved to be surprisingly elusive \n[8, 12, 2]. Recently, two contrasting ideas have offered practical relaxation operat(cid:173)\nors. Zucker et al [13] have sought biologically plausible operators which draw on \nthe idea of computing a global curve organisation potential and locating consistent \nIn essence this biologically \nstructure using a form of local snake dynamics [11]. \ninspired model delivers a fine arrangement of local splines that minimise the curve \norganisation potential. Hancock and Kittler [2], on the other hand, appealed to a \nmore information theoretic motivation [4]. In an attempt to overcome some of the \nwell documented limitations of the original Rosenfeld, Hummel and Zucker relaxa(cid:173)\ntion operator [9] they have developed a Bayesian framework for relaxation labelling \n[4]. Of particular significance for the low-level curve enhancement problem is the un(cid:173)\nderlying statistical framework which makes a clear-cut distinction between the roles \nof uncertain image data and prior knowledge of contour structure. This framework \nhas allowed the output of local image operators to be represented in terms of Gaus(cid:173)\nsian measurement densities, while curve structure is represented by a dictionary of \nconsistent contour structures [2]. \n\nWhile both the fine-spline coverings of Zucker [13] and the dictionary-based re(cid:173)\nlaxation operator of Hancock and Kittler [2] have delivered practical solutions to \nthe curve reinforcement problem, they each suffer a number of shortcomings. For \ninstance, although the fine spline operator can achieve quasi-global curve organisa(cid:173)\ntion, it is based on an essentially ad hoc local compatibility model. While being \nmore information theoretic, the dictionary-based relaxation operator is limited by \nvirtue of the fact that in most practical applications the dictionary can only realist(cid:173)\nically be evaluated over at most a 3x3 pixel neighbourhood. Our aim in this paper \nis to bridge the methodological divide between the biologically inspired fine-spline \noperator and the statistical framework of dictionary-based relaxation. We develop \nan iterative spline fitting process using the EM algorithm of Dempster et al [1] . \nIn doing this we retain the statistical framework for representing filter responses \nthat has been used to great effect in the initialisation of dictionary-based relaxa(cid:173)\ntion. However, we overcome the limited contour representation of the dictionary by \ndrawing on local cubic splines. \n\n2 Prerequisites \n\nThe practical goal in this paper is the detection of line-features which manifest \nthemselves as intensity ridges of variable width in raw image data. Each pixel \nis characterised by a vector of measurements, ?* \n\n\", \n\n\" ~ + \n:--I-\n~ / ' \\ ~ \n(vi> + + ~ \n:f\\ ~ ~ \n\" :+ :+ :+ \n~ I ~'''' ~ \n:+ \n~ (Ie) \nI\u00bb ~ \n\nx ~ ,l) \n\n\"\"\"X (Ix) \n\n\" \n\nMI) \n\n, ) \n\nx \n\nIv) \n\n, \n\n(YIIII \n\n(j) \n\nso \n\nhy) \n\n(i!) \n\n(v) \n\nI,.) \n\n(ril) \n\n \n\n(i,) \n\n(a) \n\n(b) \n\n(c) \n\nFigure 2: Evolution of the spline in the fitting process. The image in (a) is the \njunction spline while the image in (b) is the branch spline. The first spline is shown \nin (i), and the subsequent ones from (ii) to (xi). The evolution of the corresponding \nspline probabilities is shown in (c). \n\n[7] Meer P., Mintz D., Rosenfeld A . and Kim D.Y., \"Robust Regression Methods for \nComputer Vision - A Review\", International Journal of Computer vision, 6, pp. \n59- 70, 1991. \n\n[8] Peleg S. and Rosenfeld A\" \n\n\"Determining Compatibility Coefficients for curve en(cid:173)\n\nhancement relaxation processes\", IEEE SMC, 8, pp. 548-555, 1978. \n\n[9] Rosenfeld A., Hummel R.A. and Zucker S.W., \"Scene labelling by relaxation opera(cid:173)\n\ntions\", IEEE Transactions SMC, SMC-6, pp400-433, 1976. \n\n[10] Sander P.T. and Zucker S.W ., \"Inferring surface structure and differential structure \n\nfrom 3D images\" , IEEE PAMI, 12, pp 833-854, 1990. \n\n[11] Terzopoulos D. , \"Regularisation of inverse problems involving discontinuities\" , IEEE \n\nPAMI, 8, pp 129-139, 1986. \n\n[12] Zucker, S.W., Hummel R.A., and Rosenfeld A., \"An application ofrelaxation labelling \n\nto line and curve enhancement\", IEEE TC, C-26, pp. 394-403, 1977. \n\n[13] Zucker S. , David C., Dobbins A. and Iverson L., \"The organisation of curve qe(cid:173)\n\ntection: coarse tangent fields and fine spline coverings\" , Proceedings of the Second \nInternational Conference on Computer Vision, pp. 577-586, 1988. \n\n\f", "award": [], "sourceid": 1291, "authors": [{"given_name": "Jos\u00e9", "family_name": "Leite", "institution": null}, {"given_name": "Edwin", "family_name": "Hancock", "institution": null}]}*