Bibliography

Adelson, E. and Bergen, J. (1991). The plenoptic function and the elements of early vision. In Computational Models of Vi.sual Processing, 3-20. MIT Press.

Akaike, H. (1974). A n~w look at the statistical model identification. IEEE' 'Iransactions on Automatic Control, 19, 6, 716-723.

Amir, A. and Lindenbaum, M. (1996). Grouping based non-additive verifica­tion. Technical Report 9518, Center for Intelligent Systems, Technion.

Astrom, K. and Wittenmark, B. (1984). Computer Controlled Systems. Addi­son Wesley.

Baker, S., Szeliski, R., and Anandan, P. (1998). A layered approach to stereo reconstruction. In Proc. Conf. Computer Vision and Pattern Recognition, 434-441.

Bar-Shalom, Y. and Fortmann, T. (1988). Tracking and Data Association. Academic Press.

Bartels, R., Beatty, J., and Barsky, B. (1987). An Introduction to Splines for use in Computer Gmphics and Geometric Modeling. Morgan Kaufmann.

Baumberg, A. and Hogg, D. (1994). Learning flexible models from image se­quences. In Proc. 3rd European Conf. Computer Vision, 299-308. Springer­Verlag.

Black, M. and Fleet, D. (1999). Probabilistic detection and tracking of motion discontinuities. In Proc. 7th Int. Conf. on Computer Vision, 551-558.

Black, M. and Jepson, A. (1998). A probabilistic framework for matching temporal trajectories: Condensation-based recognition of gestures and ex­pressions. In Proc. 5th European Conf. Computer Vision, 1, 909-924, Freiburg, Germany. Springer Verlag.

Black, M. and Yacoob, Y. (1995). Tracking and recognizing rigid and non-rigid facial motions using local parametric models of image motion. In Proc. 5th Int. Conf. on Computer Vision, 374-381.

156 Bibliography

Blake, A., Bascle, B., Isard, M., and MacCormick, J. (1998). Statistical models of visual shape and motion. Phil. Trans. R. Soc. A., 356, 1283-1302.

Blake, A., Curwen, R., and Zisserman, A. (1993). A framework for spatio­temporal control in the tracking of visual contours. Int. J. Computer Vision, 11,2, 127-145.

Blake, A. and Isard, M. (1998). Active Contours. Springer-Verlag.

Blake, A., !sard, M., and Reynard, D. (1995). Learning to track the visual motion of contours. J. Artificial Intelligence, 78, 101-134.

Blake, A. and Yuille, A., editors (1992). Active Vision. MIT.

Bobick, A. and Wilson, A. (1995). A state-based technique for the summari­sation and recognition of gesture. In Proc. 5th Int. Con! on Computer. Vision, 382-388.

Boden, M. (1990). The Philosophy of Artificial Intelligence. Oxford University Press.

Bozic, S. (1979). Digital and Kalman Filtering. Arnold.

Bregler, C. and Malik, J. (1998). Tracking people with twists and exponential maps. In Proc. Conf. Computer Vision and Pattern Recognition, 8-15.

Cameron, A. and Durrant-Whyte, H. (1988). Optimal sensor placement. Tech­nical Report OURRG-99-1O, Oxford University Robotics Research Group.

Carpenter, J., Clifford, P., and Fearnhead, P. (1997). An improved particle filter for non-linear problems. Technical report, Dept. of Statistics, University of Oxford. Available from www.stats.ox.ac. uk/-clifford/index.html.

Carpenter, J., Clifford, P., and Fearnhead, P. (1999). bust simulation-based filters for evolving data sets. port, Dept. of Statistics, University of Oxford. www.stats.ox.ac.uk/-clifford/index.html.

Building ro­Technical re­

Available from

Chandler, D. (1987). Introduction to Statistical Mecha.nics. Oxford University Press.

Chellappa, R. and Chatterjee, S. (1985). Classification of textures using Gaus­sian Markov random fields. IEEE Trans. Acoust., Speech, Signal Process­ing, ASSP-33, 959-963.

Cootes, T., Page, G., Jackson, C., and Taylor, C. (1996). Statistical grey­level models for object location and identification. J. Image and Vision Computing, 14,533-540.

Cootes, T. and Taylor, C. (1992). Active shape models. In Proc. British Machine Vision Can!, 265-275.

Bibliography 157

Cootes, T. and Taylor, C. (1996). Locating objects of varying shape using statistical feature detectors. In Proc. 4th European Conf. Computer Vision, 465-474.

Cootes, T., Taylor, C., Cooper, D., and Graham, J. (1995). Active shape models - their training and application. Computer Vision and Image Understanding, 61, 1, 38-59.

Cox, 1. and Hingorani, S. (1996). An efficient implementation of Reid's multiple hypothesis tracking algorithm and its evaluation for the purpose of visual tracking. IEEE Trans. on Pattern Analysis and Machine Intelligence, 18, 2,138-150.

Crisan, D., Gaines, J., and Lyons, T. (1998). Convergence of a branching particle method to the solution of the Zakai equation. SIAM Journal on Applied Mathematics, 58, 5, 1568-1590.

Crisan, D. and Lyons, T. (1997). Nonlinear filtering and measure-valued pro­cesses. Probability Theory and Related Fields, 109, 2, 217-244.

Crisan, D. and Lyons, T. (1999). A particle approximation of the solution of the Kushner-Stratonovitch equation. Probability Theory and Related Fields, 115, 4, 549-578.

Curwen, R. and Blake, A. (1992). Dynamic contours: real-time active splines. In Blake, A. and Yuille, A., editors, Active Vision, 39-58. MIT.

De Souza, K. M. A., Kent, J. T., and Mardia, K. V. (1997). Bayesian analysis of highly variable images. In The Art and Science of Bayes-ian Image Analysis. 17th Leeds Annual Statistical Research Workshop, Leeds.

Del Moral, P. (1998). Measure-valued processes and interacting particle sys­tems: application to nonlinear filtering problems. The Annals of Applied Probability, 8, 2, 438-495.

Deutscher, J., Blake, A., North, B., and Bascle, B. (1999). Tracking through singularities and discontinuities by random sampling. In Proc. 7th Int. Conf. on Computer Vision, 1144-1150.

Doucet, A. (1998). On sequential simulation-based methods for Bayesian filtering. Technical Report CUED/F-INFENG/TR310, Dept. of Engineering, University of Cambridge. Available from ~.stats.bris.ac.uk:81/MCMC/pages/list.html.

Duda, R. and Hart, P. (1973). Pattern Classification and Scene Analysis. John Wiley and Sons.

Eliot, T. S. (1942). Little Gidding. Faber.

Fleck, M., Forsyth, D., and Bregler, C. (1996). Finding naked people. In Proc. 4th European Conf. Computer Vision, 593-602, Cambridge, England.

158 Bibliograpby

Fu, K. (1982). Syntactic Pattern Recognition. Prentice-HalL

Gelb, A., editor (1974). Applied Optimal Estimation. MIT Press, Cambridge, MA.

Geman, D., Geman, S., Graffigne, C., and Dong, P. (1990). Boundary detec­tion by constrained optimisation. IEEE Trans. on Pattern Analysis and Machine Intelligence, 12, 7, 609-628.

Geman, S. and Geman, D. (1984). Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Trans. on Pattern Analysis and Machine Intelligence, 6, 6, 721-74l.

Geweke, J. (1989). Bayesian inference in econometric models using Monte Carlo integration. Econometrica, 57, 1317-1339.

Gidas, B. (1989). A renormalisation group approach to jmage processing prob- . lerns. IEEE Trans. on Pattern Analysis and Machine Intelligence, 11, 2, 164-180.

Gidas, B. and Mumford, D. (1999). Stochastic models for generic images. Available from www. dam. brown. edu/people/mwnf ord/.

Gilks, W., Richardson, S., and Spiegelhalter, D. (1996). Markov Chain Monte Carlo in Practice. Chapman and Hall.

Gordon, N. (1997). A hybrid bootstrap filter for target tracking in clutter. IEEE Trans. Aero. Elec. Systems, 33, 353-358.

Gordon, N., Salmond, D., and Smith, A. (1993). Novel approach to nonlinear/non-Gaussian Bayesian state estimation. lEE Proc. F, Radar and Signal Processing, 140, 2, 107-113.

Green, P. (1995). Reversible jump ]\,'{arkov chain Monte Carlo computation and Bayesian model determination. Biometrika, 82, 4, 711-32.

Grenander, U. (1976-1981). Lectures in Pattern Theory I, II and III. Springer.

Grenander, D., Chow, Y., and Keenan, D. (1991a). HANDS. A Pattern Theo­retic Study of Biological Shapes. Springer-Verlag. New York.

Grenander, U., Chow, Y., and Keenan, D. (1991b). HANDS. A Pattern Theo­retical Study of Biological Shapes. Springer-Verlag. New York.

Grirnson, W. (1990). Object Recognition by Computer. MIT Press.

Grirnson, W. and Huttenlocher, D. (1990). On the verification of hypothe­sised matches in model-based recognition. In Proc. 1st European Conf. Computer Vision, 489-498.

Grimson, W., Huttenlocher, D., and Jacobs, D. (1992). A study of affine match­ing with bounded sensor error. In Proc. 2nd European Conf. Computer Vision, 291-306.

Bibliography 159

Hallinan, P., Gordon, G., Yuille, A., Giblin, P., and Mumford, D. (1999). Two­and Three-Dimensional Patterns of the Face. A. K. Peters.

Handschin; J. (1970). Monte Carlo techniques for prediction and filtering of non-linear stochastic processes. Automatica, 6, 555-563.

Handschin, J. and Mayne, D. (1969). Monte Carlo techniques to estimate the conditional expectation in multi-stage non-linear filtering. International Journal of Control, 9, 5, 547-559.

Hardy, G. H. (1940). A Mathematician's Apology. Cambridge University Press.

Haritaoglu, L, Harwood, D., and Davis, L. (1998). w4 s: a real-time system for detecting and tracking people in 2.5D. In Proc. 5th European Conf. Computer Vision, 1,877-892, Freiburg, Germany. Springer-Verlag.

Heap, T. and Hogg, D. (1998). Wormholes in shape--space: Tracking through· discontinuous changes in shape. In Proc. 6th Int. Conf. on Computer Vision.

Hinton, G., Williams, C., and Revow, M. (1992). Adaptive elastic models for hand-printed character recognition. Advances in Neural Information Processing Systems, 4, 512-519.

Intille, S., Davis, J., and Bobick, A. (1997). Real-time closed-world tracking. In Proc. Conf. Computer Vision and Pattern Recognition, 697-703.

Irani, M. and Anandan, P. (1998). A unified approach to moving object detec­tion in 2D and 3D scenes. IEEE Trans. on Pattern Analysis and Machine Intelligence, 20, 6, 577-589.

Isard, M. (1998). Visual motion analysis by probabilistic propagation of condi­tional density. PhD thesis, University of Oxford.

Isard, M. and Blake, A. (1996). Visual tracking by stochastic propagation of conditional density. In Proc. 4th European Conf. Computer Vision, 343-356, Cambridge, England.

Isard, M. and Blake, A. (1998a). Condensation - conditional density propa­gation for visual tracking. Int. J. Computer Vision, 28, I, 5-28.

Isard, M. and Blake, A. (1998b). Icondensation: unifying low-level and high­level tracking in a stochastic framework. In Proc. 5th European Conf. Computer Vision, 893-908.

Isard, M. and Blake, A. (1998c). A mixed-state Condensation tracker with automatic model switching. In Proc. 6th Int. Conf. on Computer Vi.sion, 107-112.

Isard, M. and MacCormick, J. (2000). Hand tracking for vision-based drawing. Technical report, Visual Dynamics Group, Dept. Eng. Science, University of Oxford. Available from www.robots.ox.ac . ukrvdg.

160 Bibliography

Jollike, 1. (1986.). Principal Component Analysis. Springer-Verlag.

Jones, M. and Rehg, J. (1998). Statistical color models with applications to skin detection. In Proc. Con/. Computer Vision and Pattern Recognition. Also available from www.crl.research.digital.com.

Karatzas, r and Shreve, S. (1988). Brownian Motion and Stochastic Calculus. Springer-Verlag.

Kass, M., Witkin, A., and Terzopoulos, D. (1987). Snakes: active contour models. In Proc. 1st Int. Conf. on Computer Vision, 259-268.

Kaucic, R. (1997). Lip tmcking for audio-visual speech recognition. PhD thesis, University of Oxford.

Kent, J., Mardia, K., and Walder, A. (1996). Conditio!;!al cyclic Markov random . fields. Adv. Appl. Prob. (SGSA), 28, 1-12. -

Kitagawa, G. (1996). Monte Carlo filter and smoother for non-Gaussian nonlin­ear state space models. Journal of Computational and Gmphical Statistics, 5, 1, 1-25.

Kjeldsen, R. and Kender, J. (1996). Toward the use of gesture in traditional user interfaces. In Proc. 2nd Int. Conf. on A utomatic Face and Gesture Recognition, 151-156.

Koller, D., Weber, J., and Malik, J. (1994). Robust multiple car tracking with occlusion reasoning. In Proc. 3m European Con/. Computer Vision, 189-196. Springer-Verlag.

Kong, A., Liu, S., and Wong, W. (1994). Sequential imputations and Bayesian missing data problems. J .. Am. Stat. Assoc., 89,425, 278-288.

Langevin, P. (1908). Sur la theorie du mouvement brownien. C. R. Acad. Sci. Paris, 146, 530-533.

Leung, T., Burl, M., and Perona, P. (1995). Finding faces in cluttered scenes using random graph matching. In Proc. IEEE PAMI Con/. , 637-644, Cambridge.

Liggett, T. (1985). Intemcting Particle Systems. Springer-Verlag.

Liu, J, and Chen, R. (1995). Blind deconvolution via sequential imputations. J. Am. Stat. Assoc., 90, 430, 567-576.

Liu, J. and Chen, R. (1998). Sequential Monte Carlo methods for dynamic systems. J. Amer. Statist. Assoc., 93, 1032-1043.

Lowe, D. (1987a). Three-dimensional object recognition from single two­dimensional images. J. Artificial Intelligence, 31, 355-395.

Bibliography 161

Lowe, D. (1987b). The viewpoint consistency constraint. Int. J. Computer Vision, 1, 57-72.

Lowe, D. (1991). Fitting parameterised 3D models to images. IEEE TIuns. on Pattern Analysis and Machine Intelligence, 13, 5, 441-450.

Lowe, D. (1992). Robust model-based motion tracking through the integrat.ion of search and estimation. Int. J. Computer Vision, 8, 2, 113-122.

Lutkepohl, H. (1993). Introduction to Multiple Time Series Analysis. Springer­Verlag, 2nd edition.

MacCormick, J. and Blake, A. (1998a). A probabilistic contour discriminant for object localisation. In Proc. 6th Int. ConI. on Computer Vision, 390-395.

MacCormick, J. and Blake, A. (1998b). Spatial depend~nce in the observation. of visual contours. In'Proc. 5th European Conf. Computer Vision, 765-781.

MacCormick, J. and Blake, A. (1999). A probabilistic exclusion principle for tracking multiple objects. In Proc. 7th Int. Conf. on Computer Vision, 1, 572-578.

MacCormick, J. and Isard, M. (2000). Partitioned sampling, articulated ob­jects, and interface-quality hand tracking. In European Conj. Computer Vision.

Mackay, D. (1999). Introduction to Monte Carlo methods. In Jordan, M., editor, Learning in Graphical Models. MIT Press.

Malik, J., Belongie, S., Shi, J., and Leung, T. (1999). Textons, contours and regions: cue integration in image segmentation. In Proc. 7th Int. Conf. on Computer Vision, 918-925.

McKean, H. (1967). Propagation of chaos for a class of nonlinear parabolic equations. In Aziz, A., editor, Lecture Series in Differential Equations (7), 177-194. Catholic Univ., Washington D.C.

Metropolis, N., Rosenbluth, A., Rosenbluth, M., Teller, A., and Teller, E. (1953). Equation of state calculations by fast computing machines. Journal oj Chemical Physics, 6, 1087-1092.

Miller, M., Srivastava, A., and Grenander, U. (1995). Conditional-mean estima­tion via jump-diffusion processes in multiple target tracking/recognition. IEEE Transactions on Signal Processing, 43, 11,2678-2690.

Mumford, D. (1993). Elastica and computer vision. In Algebraic Geometry and its Applications, 507-518. Springer-Verlag.

Mumford, D. and Gidas, B. (1999). Stochastic models for generic images. Available from www. dam. brown. edu/people/mumf ord/.

162 Bibliography

Mumford, D. and Nitzberg, M. (1990). The 2.1D sketch. In Proc. 3rd Interna­tional Conf. Computer Vision, 138-144.

Mumford, D. and Shah, J. (1985). Boundary detection by minimising func­tionals. In Proc. Conf. Computer Vision and Pattern Recognition, 22-26.

Mumfordr D. and Shah, J. (1989). Optimal approximation by piecewise smooth functions. Comm. Pure and Appl. Math., 42, 577--685.

Neal, R. (1998). Annealed importance sampling. Technical Report 9805, Dept. of Statistics, University of Toronto.

North, B. (1998). Learning dynamical models for visual tracking. PhD thesis, Department of Engineering Science, Oxford University.

Papoulis, A. (1990). Probability and Statistics. Prentic0-Hall.

Paragios, N. and Deriche, R. (1998). A PDE-based level-set approach for de­tection and tracking of moving objects. In Proc. 6th International Conf. Computer Vision, 1139-45.

Pavlovic, V., Rehg, J., Cham, T., and Murphy, K. (1999). A dymanic Bayesian network approach to figure tracking using learned dynamic models. Proc. 7th Int. Conf. on Computer Vision.

Perez, P. and Heitz, F. (1996). Restriction of a Markov random field on a graph and multiresolution statistical image modelling. IEEE Trans. Information Theory, 42, 1, 180-190.

Peterfreund, N. {1998}. Robust tracking with spatio-velocity snakes: Kalman filtering approach. In Proc. 6th International Conf. Computer Vision, 433-39.

Pitt, M. and Shepherd, N. (1997). Filtering via simulation and auxiliary particle filters. Technical report, Nuffield College, University of Oxford.

Propp, J. and Wilson, D. (1996). Exact sampling with coupled !\·larkov chains and applications in statistical mechanics. Random Structures and Algo­rithms, 9, 223-252.

Rabiner, L. and Bing-Hwang, J. (1993). Fundamentals of Speech Recognition. Prentice-Hall.

Rasmussen, C. and Hager, G. (1998). Joint probabilistic techniques for tracking multi-part objects. In Proc. Conf. Computer Vision and Pattern Recogni­tion, 16-21.

Reid, D. (1979). An algorithm for tracking multiple targets. IEEE Transactions on Automatic Control, 24, 6, 843-854.

Bibliography 163

Reynard, D., Wildenberg, A., Blake, A., and Marchant, J. (1996). Learning dynamics of complex motions from image sequences. In Proc. 4th European Conf. Computer Vision, 357-368, Cambridge, England.

Ripley, B. (1987). Stochastic Simulation. New York: Wiley.

Ripley, B, and Sutherland, A. (1990). Finding spiral structures in images of galaxies. Phil. Trans. R. Soc. Lond. A., 332, 1627,477-485.

Robertson, R. and Schwertassek, R. (1988). Dynamics of Multibody Systems. Springer-Verlag.

Rothwell, C. (1996). Reasoning about occlusions during hypothesis verification. In Proc. 4th European Conf. Computer Vision, 599-609.

Rothwell, C., Zisserman, A., Mundy, J., and Forsyth, D. (1992). Efficient model library access by prQjectively invariant indexing Iunctions. In Pmc. Conf.· Computer Vision and Pattern Recognition, 109-114.

Rubin, D. (1988). Using the SIR algorithm to simulate posterior distributions. In Bayesian Statistics, 3, 395-402. Oxford University Press.

Rue, H. and Husby, O. (1997). Identification of partly destroyed objects using deformable templates. Technical Report 1/97, NTNU Statistics. available from www.math.ntnu.no/preprint/statistics.

Shao, Y. and Mayhew, J. (1996). A stochastic framework for object localisation. In Proc. British Machine Vision Conf., 203-212.

Shimshoni, 1. and Ponce, J. (1995). Probabilistic 3D object recognition. In Proc. 5th Int. Conf. on Computer Vision, 488-493.

Smith, A. and Gelfand, A. (1992). Bayesian statistics without tears: a sampling-resampling perspective. The American Statistician, 46, 84-88.

Soh, J., Chun, B., and Wang, M. (1994). An edge-based approach to moving object location. In Image and Video Processing II, 2182, 132-41. SPIE.

Stewart, L. (1983). Bayesian analysis using Monte Carlo integration - a pow­erful methodology for handling some difficult problems. The Stat., 32, 195-200.

Stockman, G., Kopstein, S., and Benett, S. (1982). Matching images to mod­els for registration and object detection via clustering. IEEE Trans. on Pattern Analysis and Machine Intelligence, 3, 3, 229-241.

Sullivan, J., Blake, A., Isard, M., and MacCormick, J. (1999). Object localisa­tion by Bayesian correlation. In Proc. 7th Int. Conf. on Computer Vision, 1068-1075.

Terzopoulos, D. and Szeliski, R. (1992). '!racking with Kalman snakes. In Blake, A. and Yuille, A., editors, Active Vision, 3-20. l\lIT.

164 Bibliography

Uhlenbeck, G. and Ornstein, L. (1930). On the theory of Brownian motion. Physical Review, 36, 823-84l.

Wang, J.and Adelson, E. (1993). Layered representation for motion analysis. In Proc. Conf. Computer Vision ~nd Pattern Recognition, 361-366.

Wildenberg, A. (1997). Learning and initialisation for visual tracking. PhD thesis, University of Oxford.

Williams, L. and Thornber, K. (1999). A comparison of measures for detecting natural shapes in cluttered backgrounds. In Proc. 5th European Con/. Computer Vision, 432-448.

Winkler, G. (1995). Image An~lysis, Random Fields and Dynamic Monte Carlo Methods. Springer-Verlag.

Wu, Y. N., Zhu, S. C., and Liu, X. (1999). Equivalence of Julesz and Gibbs texture ensembles. In Proc. 7th Int. Con/. on Computer Vision, 1025-32.

Zhu, S., Wu, Y., and Mumford, D. (1997). :Minimax entropy principle and its application to texture modelling. Neural Computation, 9, 1627-60.

Zhu, S., Wu, Y., and Mumford, D. (1998). Filters, random fields and maximum entropy (FRAME). Int. J. Computer Vision, 27, 2, 107-126.

Index

"', 153 x,153 *, 153 1-, 153 active contours, 2-4,36 active shape model, 4 Adelson, E., 3 afline, 5, 39, 71, 77, 81, 122 AI, see artificial intelligence AIe,51 Akaike, H., 51 algorithm(s)

computer vision (general dis­cussion), 1-2

Condensation, see Condensation algorithm

learning, see learning object localisation, see object

localisation of Propp and Wilson, 108 partitioned sampling, see par­

titioned sampling stochastic, 16, 28, 69, 144 tracking, see tracking

Amir, A., 93 Anandan, P., 3, 119 annealed importance sampling, 145 application of dynamics, see dynam-

ics, application of ARP, see auto-regressive process articulated objects, 5, 134-141 artificial intelligence, 1, 3, 144 Astrom, K., 62 asymptotic correctness of Conden­

sation, see Condensation algorithm

attached measurement line regime,

7,50,52,57,60,73,95 auto-regressive process, 5, 6, 60, 62-

64 autocorrelation, 56

b(n),bL(n); 1M B-spline, see spline background hypothesis, 60, 61 background models, 38-45, 55, 67,

113, 144, see also clutter and occlusion, 92 clutter feature probabilities, 40,

41,47,50,52,55,73,101, 123

for colour, 87 for tracking, 8, 37, 68 used in likelihood ratio, 46, 50,

53, 71, 89 background subtraction, 67, 68,113,

137, 138 Baker, S., 3 Bar-Shalom, Y., 113 Barsky, B., 4 Bartels, R., 4 Bascle, B., 3, 24, 37 Baumberg, A., 121 Bayes' theorem, 9, 57, 58, 69 Bayesian correlation, 66, 136 Bayesian filtering, 36 Bayesian inference, 57, 66, 95, 110 Beatty, J., 4 Belongie, S., 4 Benett, S., 66--68 Bergen, J., 3 bimodal distribution, 21, 22 Bing-Hwang, J., 5 Black, M., 5, 36

166 Stochastic A.1gorithms Eor Visual Tracking

Blake, Ao, 3-5, 8-11, 24, 25, 36, 37, 46, 66, 71, 78, 79, 84, 89, 136, 137, 144-147

blob, 2, 112-114 Bobick, Ao, 5, 112, 113 Boden, Mo, 1 bootstrap filter, 36 Borel measure, 146 boundary density function, 39, 45,

52, 59, 73, 114, 123 boundary feature, 39, 43, 49, 75,

119 boundary feature distribution, 123 boundary of tracked object, 4, 39,

40, 47, 48, 50, 52, 53, 55, 60,61,63,93,95,105,112, U5, 116, 118, 121

Bozic, So, 37 branched partitioned sampling, see

partitioned sampling, branched Bregler, Co, 5, 86 Brownian motion, 121 Burl, Mo, 38, 67, 68, 92

ex(i), c(i), c, 154 camera, 8, 68, 90, 91, 119, 140 Cameron, Ao, 66 Carpenter, Jo, 27, 32, 36, 133 Cartesian product, 65, 126, 134 Cham, To, 6 Chandler, Do, 98, 103, 109 change-point, 74 Chatterjee, So, 4 Chellappa, R., 4 Chen, R., 30, 35, 133 Chow, Yo, 38, 66, 67 Chun, Bo, 68 Clifford, Po, 27, 32, 36, 133 CLR, see contour likelihood ratio clutter, see also background models

feature probabilities, see back­ground models

in static scenes, 68, 74, 85 tracking in, 36, 37, 55, 66, 86,

123, 138 colour

inference using, 2, 114 skin-coloured object, 91, 138,

139 tracking using, see robustified

colour contour likelihood compact, 11, 17, 146 Condensation algorithm, 2, 8-37, 84,

118 alternatives to, 37 asymptotic correctness of, 8, 11,

21, 24-26 conventional, 131-135 history of, 35-36 ICondensation, see ICondensa-

tion limitations of, 124 need for random numbers in,

24 partitioned sampling and, 124-

141 performance of, 33, 133-134 tracking using, 85-90 with mixed states, 119, 131

Condensation diagram, 8, 10, 17, 19, 20, 24, 118, 135

well-formed, 24, 130 Condensation theorem, 8,24-26,130 conditional independence, 9 configuration of the target object,

5,7, 9, 38, 42, 60, 61, 65, 68, 84, 95, 114, 119

configuration space, 5, 9, 11,36,65, 69, 121, 134, 146, see also shape space

for multiple objects, 76, 114, 126

consciousness, 1, 144 continuous contour likelihood ratio,

60--64 contour, 4

configuration space, see config­uration space

template, see template contour likelihood, 2, 6, 38-59, 144

localisation using, 65-80, 92-111

Index

tracking using, see tracking, con­tour

contour likelihood ratio, 50-51, 85, 92

contour saliency, 61 contour tra.cking, see tracking, con­

tour contour, factored sampling of, 66-

80 contour, orientation of, 7 contours, active, 2-4, 36 control point, 4-6, 147 convergence

of estimates, 26, 69, 70, 96 using importance sampling,

103-107 weak, see weak convergence

convergence of distributions, 14-15 convolution with dynamics, see dy-

namics, application of Cooper, D., 4, 5, 67 Cootes, T., 4, 5, 66, 67 correlation, 136, 144, see also auto­

correlation Bayesian, see Bayesian corre­

lation of nearby measurements, 57, 60 of templates, 66, 67

correspondence space, 65, 67, 68 covariance, 37, 84, 85, 87 Cox, 1.,114 Crisan, D., 36 curse of dimensionality, 124 Curwen, R., 4, 79, 84, 146, 147 cyclic Markov random field, 93, 97,

101, 106

Davis, J., 112, 113 Davis, L., 112, 113 de Souza, K. M. A., 66, 73, 74 decomposition of dynamics, 119, 130,

134, 136 deformation, see shape space Del Moral, P., 14, 21, 24, 26, 36,

149, 150 dependence between particles, 33

167

Deriche, R., 113 deterministic particle set, 16, 17, 29 deterministic resampling, see resam-

pling, deterministic Deutscher, J., 37 diffusion, see dynamics dimension of configuration space, 3-

5, 37, 65-67, 81, 89, 114, 121, 124, 130, 147

dimensionality, curse of, 124 distance

between measurement lines, 55-57,59,60

between splines, 4, 77 distributio~-

convergence of, see convergence Gaussian, see Gaussian posterior, see posterior represented by particle set, see

represent target, see target distribution

distribution-valued distribution, 15 Dong, P., 38 Doucet, A., 27,36, 133 Duda, R., 67, 68 Durrant-Whyte, H., 66 dynamics

application of, 9-11, 18-20, 151 random numbers for, 24

decomposition of, see decom­position

learnt, 89, 122 modelling of, 5-6, 9, 55, 118,

119, 121, 145

econometrics, 36 edge convolution operator, 55, 56 edge feature, 38, 95

detector, 7 edge, strength of, 66 edgel, 9, 112 effective sample size, 20, 22, 26-35,

126, 134 Eliot, T. S., 144 encouraged site, 105-110 energy function, 96, 97

168 Stochastic Algorithms for Visual Tracking

estimate MAP, see MAP minimum mean square error, see

MMSE estimation theory, 69 estimator, 25, 26

maXimum likelihood, see max­imum likelihood

Monte Carlo, see Monte Carlo number of targets, 75-79

Euclidean metric, 146 Euclidean similarities, 5 Euler-Lagrange equation, 82 exclusion principle, 78, 112-122 extended Kalman filter, 37

face recognition, 38 factored sampling, 25, 125, 126

for object localisation, 66-75, 80--85, 94, 102

relation to particle sets, 25-26 Fearnhead, P., 27, 32, 36, 133 feature, see boundary feature; edge

feature; background mod­els

feature detector, 7, 56 feature innovation, see innovation features, 2, 9, 67

autocorrelation of, 56 generative model for, 38-54, 92,

95 nearest, 64

filter, 8, 55, see also Kalman filter; particle filter; Condensa­tion algorithm; Bayesian fil­tering; bootstrap filter; Monte Carlo filter; PDAF

fixed measurement line regime, 7, 52, 57-59, 62

Fleck, M., 86 Fleet, D., 36 foreground model, 47, 88, 144 Forsyth, D., 67, 86 Fortmann, T., 113 Fu, K., 38

g(e), 154

Gaines, J., 36 Gaussian

assumptions in Kalman filter, 37

boundary density function, 114 Gaussian noise, 5

in colour values, 87 in dynamics, 9 in edge features, 39, 73

Gaussian, mixture of, 37 Gelb, A., 9, 37, 62 Gelfand, A., 36 Geman, D., 38 Geman, S., 38 generative model, 2,38-54,92,147':'"

149 for RGB values, 87-89 Markov, 95--96 with exclusion principle, 114-

115 generic contour likelihood, 42-45 gesture, 102, 140 gesture recognition, 36, 141 Geweke, J., 27, 33,125 Gibbs distribution, 105, 106 Gibbs sampling, 103 Giblin, P., 38 Gidas, B., 38, 55, 98, 144 Gilks, W., 108 Gordon, G., 38 Gordon, N., 36, 114 gradient threshold, 73-75 Graffigne, C., 38 Graham, J., 4, 5, 67 graph matching, 68 Green, P., 67, 74 Grenander, U., 25, 26,38,66, 67 Grimson, W., 65, 67, 68, 92

Haar measure, 146 Hager, G., 112, 113 Hallinan, P., 38 Handschin, J., 36 hardware, 66, 67, 137, 140 Hardy, G. H., 52 Haritaoglu, I., 112, 113

Index

Hart, Po, 67, 68 Harwood, Do, 112, 113 head tracking, 86-90 head-and-shoulders model, 71, 74,

112, 115 Heap, To, 96 Heitz, Fo, 98 hierarchical search, 124 Hingorani, So, 114 Hinton, Go, 38 hit, 81 hit threshold, 81 hit time, 81 Hogg, Do, 36, 121 Hough transform, 67, 68 . human vision, 1, 3, 65, 144, 147 Husby, 0o, 93 Huttenlocher, Do, 68, 92 hypothesis, see background hypoth-

esis; object hypothesis; mul­tiple hypothesis tracking

ICondensation, 137 image feature, see feature image model, see generative model image processing, 38 image segmentation, see segmenta-

tion image texture, see texture importance function, 105, 128, 137 importance resampling, 128 importance reweighting function, 131 importance sampling, 103-107 importance sampling, annealed, 145 independence

of colour values, 87 of distinct targets' dynamics,

130 of measurement lines, 41, 55-

57, 76, 92, 94 independence generative models, 94 initialisation of tracking algorithms,

84-86, 89, 137 inner product, 4, 147 innovation, feature, 7, 42, 60 interior feature, 39,41

169

interior-exterior likelihood, 46-48,71, 73,147-149

intersection innovation, 42, 95, 114 intersection number, 42, 43, 95,114,

115,121 Intille, So, 112, 113 Irani, Mo, 119 Isard, Mo, 3-5, 8-11, 24, 25, 36,

37,46,66,71,78,89,135-138, 144, 145

Ising model, 93, 97, 109

Jackson, Co, 66 Jacobs, D., 68, 92 Jepson, A:; 36 jitter, 138 Jollike, I., 5 Jones, Mo, 86

Kalman filter, 5, 9,37, 55, 79, 85, 140, 143

Karatzas, I., 62, 64 Kass, M., 4 Kaucic, Ro, 4 Keenan, Do, 38, 66, 67 Kender, Jo, 86 Kent, Jo, 66, 73, 74, 97 kernel, 37 Kitagawa, Go, 36 Kjeldsen, R., 86 Koller, Do, 67, 68, 112 Kong, Ao, 30, 32 Kopstein, So, 66-68

Lagrangian, 82 Langevin, Po, 61 law of large numbers, 26, 33 layered sampling, 78 layers, 3, 119 learning, 5, 6, 36, 55, 79, 144 length of measurement lines, 41, 42,

73,123 Leung, To, 4, 38, 67, 68, 92 Liggett, To, 36 likelihood, see contour likelihood;

observation density likelihood function, 25, 36, 37

170 Stochastic Algorithms for Visual Tracking

for hand tracking, 136 incorporating exclusion princi­

ple, 115-118 one-target and two-target, 117 peaks in, 37, 70, 80, 94, 99,

100, 117, 127 likelihood- ratio, see contour likeli-

hood ratio Lindenbaum, Mo, 93 Liu, Jo, 30, 35, 133 Liu, So, 30, 32 Liu, Xo, 144 localisation, see object localisation lock, on target, 84-86, 88, 89 log-likelihood, 46, 53, 62. Lowe, Do, 55, 67, 68, 92 Lutkepohl, Ho, 5 Lyons, To, 36

MacCormick, Jo, 3, 24, 66, 78, 135, 136, 138, 144

Mackay, Do, 25, 96 Mahalanobis distance, 146 Malik, Jo, 4, 5, 67, 68, 112 manifold, 5, 79, 146 MAP estimate, 69, 78, 79 Marchant, Jo, 4 Mardia, K., 66, 73, 74, 97 Markov chain, 108, 109 Markov chain Monte Carlo, see MCMC Markov likelihood, 55, 92-111 Markov process, 62 Markov random field, 3, 93-96, 98,

101, 106, 110 partition function, 109

maximum a posteriori, see MAP maximum likelihood estimate, 6, 39,

69-71, 73, 77, 123 MayIiew, Jo, 66,67 Mayne, Do, 36 McKean, Ho, 36 MCMC, 66, 93, 96, 108 measure theory, 14, 18, 146, 150 measurement density, 118, see also

observation density

measurement line, 6, 38, 40, 60, 92, 123

distance between, see distance encouraged, see encouraged independence of, see indepen-

dence length of, see length selection of, 54-55

measurement line regime, see fixed; attached

measurement methodology, 6-7 measurement point, 6 medical images, 93 metric, see also spline metric

on co1illguration space, 146-147 Metropolis, No, 108 Miller, Mo, 66, 67 minimum mean squared error, 69,

73 mixture of Gaussians, 37 MLE, see maximum likelihood esti­

mate MMSE, see minimum mean squared

error Monte Carlo estimator, 27, 29 Monte Carlo filter, 36 Monte Carlo integration, 16, 25, 31,

33,96, 106, 109 Monte Carlo particle set, see parti­

cle set Monte Carlo sampling, 103, 106, 107,

109 mosaicking, 119 motion, see dynamics motion classification, 36 motion detection, 67, 68 multi-modal, 37, 86 multiple hypothesis tracking, 114 multiplication, operation on parti-

cle set or density, 10, 17-18, 118, 129

Mumford, Do, 38, 55, 66, 119, 144 Mundy, Jo, 67 Murphy, K., 6

vx(i), v(i), v, 154

Index

Neal, R., 145 necklace, 97, 106 Nitzberg, Mo, 119 noise, see Gaussian noise non-Boltzmann sampling, 103 non-detection probability, 39-42, 44,

50~59, 73,75,99,115,123 normal distribution, see Gaussian normal lines, see measurement lines normalisation

of weights in particle set, 33 North, Bo, 4, 6, 37, 39

object hypothesis, 61 object localisation, 1, 67

for tracker initialisation, 84-86 using contour likelihoods, 65-

80 using Markov likelihood, 92-111

object recognition, 65 observation density, 9, 10, 86, 119,

131, see also likelihood func­tion

occlusion and exclusion principle, see ex­

clusion principle and partitioned sampling, 124,

130-132 modelling of, 41, 92-111 prior for, 96-98 reasoning about, 112, 119, 121,

122 tracking despite, 122

one-body density, 118, 130, 131 opaque objects

and exclusion principle, 119-122

likelihood for, 46-48 operations on particle sets, see par­

. ticle sets operator on distributions, 18, 24,

150 order statistic, 49 order statistic likelihood, 49-50, 52,

89 orientation of contour, see contour

Ornstein, Lo, 61

p,153 pc(n; z), 154 P(X),153 P(P(X)), 153 Page, Go, 66 Papoulis, Ao, 69 Paragios, No, 113 particle, 11 particle set, 2, 9-11, 69, 70

171

and factored sampling, see fac­tored resampling

application of dynamics to, see dy-uamics

dependence between particles, see dependence

deterministic, see determinis-tic

effect of dynamics on, 151 effect of resampling on, 150 effectiveness of, see effective sam-

ple size multiplication by function, see

multiplication normalisation of, see normali­

sation operations on, 10, 16-24 representation of distribution by,

see represent resampling of, see resampling standard Monte Carlo, 15, 16,

18,25,27,31,33,34,150 weighted resampling of, see weighted

resampling partition function, 105, 106, 109-

110 partitioned sampling, 2, 36, 78, 114,

122, 124-141, 145, 152 branched, 131-133, 145 for articulated objects, 134-141 performance of, 133-134

pattern theory, 25 Pavlovic, Vo, 6 PCA, see principal components anal­

ysis

172 Stochastic Algorithms for Visual Tracking

PDAF, see probabilistic data asso­ciation filter

peak in likelihood, see likelihood func­

tion in posterior, see posterior

Perez, P.,1)8 performance, see Condensation al­

gorithm; partitioned sam­pling

Perona, P., 38, 67, 68, 92 Peterfreund, N., 55 Pitt, M., 36 plenoptic function, 3 Poisson distribution, 77 Poisson likelihood, 45-46,52,57,71 Poisson process, density of, 73, 74 Ponce, J., 68, 92 pose space, 65-67 posterior, 9, 11, 31, 58, 69, 77, 78,

94, 118, 125, 126 peak in, 108, 110, 135

principal components analysis, 5, 89, 121

prior, 11, 25, 55, 58, 66, 69, 81, 96, 125

learning, 79 on occlusion, see occlusion uniform on rotations, 146

probabilistic data association filter, 113,114

probabilistic exclusion principle, see exclusion principle

Probabilistic Membership Function, 66

probabilities, background, see back­ground

probability distribution represented by particle set, see

represent probability measure, 14 probability, non-detection, see non­

detection Propp, J., 108 pruning, 37, 67, 68, 114

qij, 154 quatemions, 146

Rabiner, L., 5 random field, see Markov random

field random process, 56: 57, 60, 61 random resampling: see resampling,

random random sample, 10, 14, 36, 80-84,

see also MCMC random variable, 26

in particle set, 11, 13 random walk, 60 Rasmussen,C, 112, 113 reHection principle, 64 Rehg, J., 6, 86 Reid, D., 114 reinitialisation, 86, 139 relaxation labelling, 67 renormalisation group theory, 98 replication of particles, 11, 12, 22 representation

of distribution by particle set, 10, 15-16, 118, 127, 128

resampling and effective sample size, 33-

35 deterministic, 22-24 lemma of Del Moral, 149-152 operation on particle sets, 10,

20-24, 118, 131, 137 previous algorithms using, 36 random, 20-22, 150 symbol in Condensation diagram,

23 transition kernel, 149, 150 weighted, 127-130, 133

symbol for, 130 resampling operator, 150, 151 Revow: M., 38 Reynard, D, 4 Richardson, S., 108 Ripley, B., 50, 66, 67, 103 Robertson, R.: 5

Index

robustified colour contour likelihood, 86-90, 136

robustified colour likelihood, 88 robustness parameter, 88 Rosenbluth, Ao, 108 Rosenbluth, Mo, 108 rotations, ·prior on, 71, 100, 146 Rothwell, C., 67, 93 Rubin, Do, 36 Rue, Ro, 93

Salmond, Do, 36 sampling, see Gibbs sampling; MCMC;

Monte Carlo sampling; al­gorithms, stochastic; fac­tored sampling; importance sampling; layered sampling; partitioned sampling; ran­dom sample; resampling

sampling importance resampling, 36 Schwertassek, R., 5 SDE, see stochastic differential equa­

tion search, see hierarchical search; like­

lihood function, peaks in; posterior, peaks in

of correspondence space, see cor­respondence space

of pose space, see pose space segmentation, 3, 112 separation

between distinct targets, 112, 114

of measurement lines, see mea­surement line

sequence of random variables in particle

set, 11, 13 video, see video sequence

sequeritial Bayesian ffitering, 36 Shah, Jo, 38, 66 Shao, Yo, 66, 67 shape, see also active shape model;

active contours topological changes in, 36

shape space, 4-5, 36, 39, 136

affine, 122 Shepherd, No, 36 Shi, Jo, 4 Shimshoni, I., 68, 92 Shreve, So, 62, 64 simulation, 63, 96

of a particle set, 24 of Bayes' rule, 9

173

SIR, see sampling importance re-sampling

site,encouraged, see encouraged site site, in MRF, 95, 98 skin colour, 86-89, see also colour Smith, Ao, 36 smooth, 82 -smoothing, 8 snake, 4 SNR,67 Soh, Jo, 68 span, 6, 60 spatial process, 41, 45 speech, 5 Spiegelhalter, Do, 108 spline, 4-7, 58-60, 136, 137, 146,

147 spline measure, 146 spline metric, 77, 146 Srivastava, Ao, 66, 67 state vector, 95 stationary distribution of Markov chain,

108, 109 stationary process, 62 statistical mechanics, 36 statistics, 1 Stewart, Lo, 33 stochastic algorithms, see algorithms stochastic differential equation, 60-

62,64 stochastic dynamics, see dynamics Stockman, Go, 6(H)8 Sullivan, Jo, 66, 78, 136, 144 survival diagnostic, 28-33, 124, 126,

133, 135 survival rate, 66, 124, 125 Sutherland, Ao, 66, 67 symmetric group, 44

174

Szeliski, R., 3, 4, 79, 84

target, see target object target distribution, 15 target object, 4, 9, 38, 65, 92, 124

estimating number of, 75-79 separation between, see sepa-

ration Taylor, C., 4, 5, 66, 67 Teller, A., 108 Teller, E., 108 template, 5, 38, 51, 57, 66, 70, 100,

121 colour, 136

Terzopoulos, D., 4, 79, 84 thermodynamics, 96 Thornber, K, 61 topology, of contour, 36 tracking

contour, 8, 37, 84-90, 124 head, see head tracking in clutter, see clutter initialisation, see initialisation multiple hypothesis, 114 multiple objects, 112-122 using colour, see colour

transition kernel, 19, 149, 150 transition matrix, 109, 119 transition probability, 123 travelling salesman, 66 tree, partitioned sampling on, 145 TUring machine, 1

Uhlenbeck, G., 61 umbrella sampling, 109 unimodal, 37

variance of hitting times, 84 of particle set estimator, 26, 28 of random process, 64 reduction by importance sam­

pling, 103 video sequence, 1,8, 11,38, 55, 65,

79, 86, 90, 140 virtual pen, 140 virtual workspace, 140, 143

Stochastic Algorithms for Visua.l Tracking

vision, see computer vision; human vision

visual tracking, see tracking

Walder, A., 97 Wang, J., 3 Wang, ~ ... 1., 68 weak convergence, 14, 15 weak law of large numbers, 26 web site for this book, 3, 90, 91,

113, 122, 140 Weber, J., 67, 68, 112 weight, 11 weighted resampling, see resampling,

_weighted well-formed, 24, 130 'Wiener process, 61 Wildenberg, A., 4 Williams, C., 38 Williams, 1., 61 Wilson, A., 5 Wilson, D., 108 Winkler, G., 56, 96 wire-frame model, 112

tracking using, 118-119 Witkin, A., 4 \Vittenmark, B., 62 Wong, W., 30, 32 Wu, Y., 38, 55, 144

X,153 x, 153

Yacoob, Y., 5 Yuille, A., 4, 38

Zhu, S., 38, 55, 144 Zisserman, A., 4, 67, 79, 84, 146,

147