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## CONDENSATION-Conditional Density Propagation for Visual Tracking (1998)

### Citations

5104 | Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images
- Geman, Geman
- 1984
(Show Context)
Citation Context ...al with temporal image sequences. A standard problem in statistical pattern recognition is to find an object parameterised as x with prior p(x), using data z from a single image. The posterior density p(x |z) represents all the knowledge about x that is deducible from the data. It can be evaluated, in principle, by applying Bayes’ rule (Papoulis, 1990) to obtain p(x |z) = kp(z |x)p(x) (6) where k is a normalisation constant that is independent of x. In cases where p(z |x) is sufficiently complex that p(x |z) cannot be evaluated simply in closed form, iterative sampling techniques can be used (Geman and Geman, 1984; Ripley and Sutherland, 1990; Grenander et al., 1991; Storvik, 1994). The factored sampling algorithm (Grenander et al., 1991) generates a random variate x from a distribution p(x) that approximates the posterior p(x |z). First, a sample-set {s(1), . . . , s(N )} is generated from the prior density p(x) and then an index Condensation—Conditional Density Propagation for Visual Tracking 9 Figure 3. Factored sampling: a set of points s(n), the centres of the blobs in the figure, is sampled randomly from a prior density p(x). Each sample is assigned a weight πi (depicted by blob area) in proport... |

3944 | Snakes - active contour models
- Kass, Witkin, et al.
- 1998
(Show Context)
Citation Context ...able geometric models of a moving object are available, they can be matched effectively to image data, though usually at considerable computational cost (Hogg, 1983; Lowe, 1991; Sullivan, 1992; Huttenlocher et al., 1993). Once an object has been located approximately, tracking it in subsequent images becomes more efficient computationally (Lowe, 1992), especially if motion is modelled as well as shape (Gennery, 1992; Harris, 1992). One important facility is the modelling of curve segments which interact with images (Fischler and Elschlager, 1973; Yuille and Hallinan, 1992) or image sequences (Kass et al., 1987; Dickmanns, and Graefe, 1988). This is more general than modelling entire objects but more clutter-resistant than applying signal-processing to low-level corners or edges. The methods to be discussed here have been applied at this level, to segments of parametric B-spline curves (Bartels et al., 1987) tracking over image sequences (Menet et al., 1990; Cipolla and Blake, 1990). The B-spline curves could, in theory, be parameterised by their control points. In practice, this allows too many degrees of freedom for stable tracking and it is necessary to restrict the curve to a low-dimensional par... |

3922 |
Random sample consensus, a paradigm for model fitting with applications to image analysis and automated cartography
- Fischler, Bolles
- 1981
(Show Context)
Citation Context ...ne of a finite set of measurements zt = {zt,1, . . . , zt,k} at time n is to be associated with Condensation—Conditional Density Propagation for Visual Tracking 25 the state xt at time t . Heuristic mechanisms such as the validation gate and the probabilistic data-association filter (PDAF) (Bar-Shalom and Fortmann, 1988) attempt to deal with the ambiguity of association. Alternatively it can, in principle, be dealt with exactly by “multiple hypothesis filtering” but with computational cost that grows exponentially over time and which is therefore ruled out in practice. The “RANSAC” algorithm (Fischler and Bolles, 1981) deals probabilistically with multiple observations but the observations have to be discrete, and there is no mechanism for temporal propagation. More complex methods including the Joint PDAF (JPDAF) (Bar-Shalom and Fortmann, 1988; Rao, 1992) address the more difficult problem of associating not simply single features but subsequences of Zt with the state. However, these methods rely on the existence of discrete features. In contour tracking the features are continuous curves and so are not naturally amenable to discrete association. A.4. Direct Integration Finally, one very general approach t... |

2013 |
Fundamentals of speech recognition
- Rabiner, Juang
- 1993
(Show Context)
Citation Context ...des of p(z |x) become narrow. One approach is “importance sampling” (Ripley, 1987) in which a heuristically chosen distribution, approximating p(z |x), is used to concentrate random sampling around modes. However, this has the drawback that the prior p(x) must be repeatedly evaluated whereas, in temporal propagation, the prior (prediction) p(xt |zt−1) cannot be evaluated pointwise, only sampled. Finally, it is striking that the density propagation equation (4) in the Condensation algorithm is a continuous form of the propagation rule of the “forward algorithm” for Hidden Markov Models (HMMs) (Rabiner and Bing-Hwang, 1993). The integral over continuous states in (5) becomes a summation over discrete states in the HMM, with p(xt |xt−1) represented by a transition matrix. This suggests a natural opportunity to combine the two so that mixed discrete/continuous states could be propagated over time. This would allow switching between multiple models, for instance, walk-trot-canter-gallop, each model represented by a stochastic differential equation, with transitions governed by a discrete conditional probability matrix. It seems likely that such a system could be executed as a Condensation tracker. A further challen... |

1732 |
Novel approach to nonlinear/nonGaussian Bayesian state estimation
- Gordon, Salmond, et al.
- 1993
(Show Context)
Citation Context .... Note that, in the case that the density p(z |x) is normal, the mean obtained by factored sampling is consistent with an estimate obtained more conventionally, and efficiently, from linear least squares estimation. For multi-modal distributions which cannot be approximated as normal, so that linear estimators are unusable, estimates of mean x by factored sampling continue to apply. 4. The CONDENSATION Algorithm The Condensation algorithm is based on factored sampling but extended to apply iteratively to successive images in a sequence. The same sampling strategy has been developed elsewhere (Gordon, et al., 1993; Kitagawa, 1996), presented as developments of MonteCarlo methods. Jump-diffusion tracking (Miller et al., 1995) may also be related to the approach described here. Given that the process at each time-step is a selfcontained iteration of factored sampling, the output of an iteration will be a weighted, time-stamped sample-set, denoted {s(n)t , n = 1, . . . , N } with weights π (n) t , representing approximately the conditional statedensity p(xt |Zt ) at time t . How is this sample-set obtained? Clearly, the process must begin with a prior density and the effective prior for time-step t should... |

1366 |
Veterling Numerical Recipes
- Press, Flannery, et al.
- 1988
(Show Context)
Citation Context ...reases (Fig. 3). Note that posterior mean properties E[g(x) |z] can be generated directly from the samples {s(n)} by weighting with pz(x) to give: E[g(x) |z] ≈ ∑N n=1 g ( s(n) ) pz ( s(n) ) ∑N n=1 pz ( s(n) ) . (7) For example, the mean can be estimated using g(x) = x (illustrated in Fig. 4) and the variance using g(x) = xxT . In the case that p(x) is a spatial Gauss-Markov process, Gibbs sampling from p(x) has been used to generate the random variates {s(1), . . . , s(N )}. Otherwise, for low-dimensional parameterisations as in this paper, standard, direct methods can be used for Gaussians2 (Press et al., 1988). Note that, in the case that the density p(z |x) is normal, the mean obtained by factored sampling is consistent with an estimate obtained more conventionally, and efficiently, from linear least squares estimation. For multi-modal distributions which cannot be approximated as normal, so that linear estimators are unusable, estimates of mean x by factored sampling continue to apply. 4. The CONDENSATION Algorithm The Condensation algorithm is based on factored sampling but extended to apply iteratively to successive images in a sequence. The same sampling strategy has been developed elsewhere (... |

1223 | Applied optimal estimation - Gelb - 1974 |

949 |
Tracking and Data Association
- Bar-Shalom, Fortmann
- 1988
(Show Context)
Citation Context ...observations that is also used later in computational experiments. 6.1. One-Dimensional Observations in Clutter In one dimension, observations reduce to a set of scalar positions {z = (z1, z2, . . . , zM)} and the observation density has the form p(z |x) where x is one-dimensional position. The multiplicity of measurements reflects the presence of clutter so either one of the events φm = {true measurement is zm}, m = 1, . . . ,M occurs, or else the target object is not visible with probability q = 1 − ∑m P(φm). Such reasoning about clutter and false alarms is commonly used in target tracking (Bar-Shalom and Fortmann, 1988). Now the observation density can be expressed as p(z |x) = qp(z |clutter)+ M∑ m=1 p(z |x, φm)P(φm). A reasonable functional form for this can be obtained by making some specific assumptions: that3 P(φm) = p,∀m, that the clutter is a Poisson process along the line with spatial density λ and that any true target measurement is unbiased and normally distributed with standard deviation σ . This leads to p(z |x) ∝ 1 + 1√ 2πσα ∑ m exp ( − ν 2 m 2σ 2 ) (13) where α = qλ and νm = zm − x , and is illustrated in Fig. 7. Peaks in the density function correspond to measured features and the state density... |

927 |
Linear Statistical Inference and its Applications
- Rao
- 1973
(Show Context)
Citation Context ... (using Lemma 2) = ∫ xt−1 p(xt |xt−1)p(xt−1 |Zt−1) which is the required result. 26 Isard and Blake Appendix C: Asymptotic Correctness of the CONDENSATION Algorithm The Condensation algorithm is validated here by a probabilistic argument showing that the sample-set representation of conditional density is correct, asymptotically, as the size N of the sample set at each time-step gets large. The argument is based on the one by Grenander et al. (1991) to justify their factored sampling algorithm for interpretation of static images. They use the standard probabilistic tool of “weak convergence” (Rao, 1973) and the “weak law of large numbers” to show that a posterior distribution inferred by factored sampling can be made arbitrarily accurate by choosing N sufficiently large. No formal indication is given as to how large N should be for a given level of accuracy, something which is determined, in practice, by experimentation. In the proof that follows, the correctness proof for factored sampling of a static image is made inductive so that it can be applied to successive images in a sequence. This would be sufficient to apply several independent images to the estimation of a static underlying obje... |

659 | Contour tracking by stochastic propagation of conditional density - Isard, Blake - 1996 |

590 |
Monte Carlo filter and smoother for non-Gaussian nonlinear state space models
- Kitagawa
- 1996
(Show Context)
Citation Context ...se that the density p(z |x) is normal, the mean obtained by factored sampling is consistent with an estimate obtained more conventionally, and efficiently, from linear least squares estimation. For multi-modal distributions which cannot be approximated as normal, so that linear estimators are unusable, estimates of mean x by factored sampling continue to apply. 4. The CONDENSATION Algorithm The Condensation algorithm is based on factored sampling but extended to apply iteratively to successive images in a sequence. The same sampling strategy has been developed elsewhere (Gordon, et al., 1993; Kitagawa, 1996), presented as developments of MonteCarlo methods. Jump-diffusion tracking (Miller et al., 1995) may also be related to the approach described here. Given that the process at each time-step is a selfcontained iteration of factored sampling, the output of an iteration will be a weighted, time-stamped sample-set, denoted {s(n)t , n = 1, . . . , N } with weights π (n) t , representing approximately the conditional statedensity p(xt |Zt ) at time t . How is this sample-set obtained? Clearly, the process must begin with a prior density and the effective prior for time-step t should be p(xt |Zt−1). ... |

567 |
Stochastic Simulation
- Ripley
- 1987
(Show Context)
Citation Context ... combines the configurations of the three people. An alternative is to attempt to develop a mode finder capable of pin-pointing several modes when present. More generally, there is a need for “operators” to interrogate densities: for instance, an operator to find a person moving to the right, or to find the tallest person. Perhaps such operators could be formulated as hypothesis tests applied to sample sets. A third question concerns the random sampling scheme and its efficiency. Factored sampling can be inefficient as the modes of p(z |x) become narrow. One approach is “importance sampling” (Ripley, 1987) in which a heuristically chosen distribution, approximating p(z |x), is used to concentrate random sampling around modes. However, this has the drawback that the prior p(x) must be repeatedly evaluated whereas, in temporal propagation, the prior (prediction) p(xt |zt−1) cannot be evaluated pointwise, only sampled. Finally, it is striking that the density propagation equation (4) in the Condensation algorithm is a continuous form of the propagation rule of the “forward algorithm” for Hidden Markov Models (HMMs) (Rabiner and Bing-Hwang, 1993). The integral over continuous states in (5) becomes ... |

467 |
Recognition by linear combinations of models
- Ullman, Basri
- 1991
(Show Context)
Citation Context ...jects but more clutter-resistant than applying signal-processing to low-level corners or edges. The methods to be discussed here have been applied at this level, to segments of parametric B-spline curves (Bartels et al., 1987) tracking over image sequences (Menet et al., 1990; Cipolla and Blake, 1990). The B-spline curves could, in theory, be parameterised by their control points. In practice, this allows too many degrees of freedom for stable tracking and it is necessary to restrict the curve to a low-dimensional parameter x, for example, over an affine space (Koenderink and Van Doorn, 1991; Ullman and Basri, 1991; Blake et al., 1993), or more generally allowing a “shape-space” of non-rigid motion (Cootes et al., 1993). 6 Isard and Blake Finally, prior probability densities can be defined over the curves (Cootes et al., 1993) represented by appropriate parameter vectors x, and also over their motions (Terzopoulos and Metaxas, 1991; Blake et al., 1993), and this constitutes a powerful facility for tracking. Reasonable defaults can be chosen for those densities. However, it is obviously more satisfactory to measure or estimate them from data-sequences (x1, x2, . . .). Algorithms to do this, assuming Gaus... |

416 |
The Representation and Matching of Pictorial Structures
- Fischler, Elschlager
- 1973
(Show Context)
Citation Context ...thods have arisen in computer vision for modelling shape and motion. When suitable geometric models of a moving object are available, they can be matched effectively to image data, though usually at considerable computational cost (Hogg, 1983; Lowe, 1991; Sullivan, 1992; Huttenlocher et al., 1993). Once an object has been located approximately, tracking it in subsequent images becomes more efficient computationally (Lowe, 1992), especially if motion is modelled as well as shape (Gennery, 1992; Harris, 1992). One important facility is the modelling of curve segments which interact with images (Fischler and Elschlager, 1973; Yuille and Hallinan, 1992) or image sequences (Kass et al., 1987; Dickmanns, and Graefe, 1988). This is more general than modelling entire objects but more clutter-resistant than applying signal-processing to low-level corners or edges. The methods to be discussed here have been applied at this level, to segments of parametric B-spline curves (Bartels et al., 1987) tracking over image sequences (Menet et al., 1990; Cipolla and Blake, 1990). The B-spline curves could, in theory, be parameterised by their control points. In practice, this allows too many degrees of freedom for stable tracking ... |

398 |
Dynamic 3D models with local and global deformation: Deformable superquadrics
- Terzopoulos, Metaxas
- 1991
(Show Context)
Citation Context ... curves could, in theory, be parameterised by their control points. In practice, this allows too many degrees of freedom for stable tracking and it is necessary to restrict the curve to a low-dimensional parameter x, for example, over an affine space (Koenderink and Van Doorn, 1991; Ullman and Basri, 1991; Blake et al., 1993), or more generally allowing a “shape-space” of non-rigid motion (Cootes et al., 1993). 6 Isard and Blake Finally, prior probability densities can be defined over the curves (Cootes et al., 1993) represented by appropriate parameter vectors x, and also over their motions (Terzopoulos and Metaxas, 1991; Blake et al., 1993), and this constitutes a powerful facility for tracking. Reasonable defaults can be chosen for those densities. However, it is obviously more satisfactory to measure or estimate them from data-sequences (x1, x2, . . .). Algorithms to do this, assuming Gaussian densities, are known in the control-theory literature (Goodwin and Sin, 1984) and have been applied in computer vision (Blake and Isard, 1994; Baumberg and Hogg, 1995). Given the learned prior, and an observation density that characterises the statistical variability of image data z given a curve state x, a posterior... |

340 |
Adaptive filtering prediction and control
- Goodwin, Sin
- 1984
(Show Context)
Citation Context ...pace” of non-rigid motion (Cootes et al., 1993). 6 Isard and Blake Finally, prior probability densities can be defined over the curves (Cootes et al., 1993) represented by appropriate parameter vectors x, and also over their motions (Terzopoulos and Metaxas, 1991; Blake et al., 1993), and this constitutes a powerful facility for tracking. Reasonable defaults can be chosen for those densities. However, it is obviously more satisfactory to measure or estimate them from data-sequences (x1, x2, . . .). Algorithms to do this, assuming Gaussian densities, are known in the control-theory literature (Goodwin and Sin, 1984) and have been applied in computer vision (Blake and Isard, 1994; Baumberg and Hogg, 1995). Given the learned prior, and an observation density that characterises the statistical variability of image data z given a curve state x, a posterior distribution can, in principle, be estimated for xt given zt at successive times t . 1.2. Kalman Filters and Data-Association Spatio-temporal estimation, the tracking of shape and position over time, has been dealt with thoroughly by Kalman filtering, in the relatively clutter-free case in which p(xt ) can satisfactorily be modelled as Gaussian (Dickmanns ... |

327 | Affine structure from motion. - Koenderink, Doorn - 1990 |

318 |
An Introduction to Splines for Use in Computer Graphics and Geometric Modeling
- Bartels, Beatty, et al.
- 1987
(Show Context)
Citation Context ...es becomes more efficient computationally (Lowe, 1992), especially if motion is modelled as well as shape (Gennery, 1992; Harris, 1992). One important facility is the modelling of curve segments which interact with images (Fischler and Elschlager, 1973; Yuille and Hallinan, 1992) or image sequences (Kass et al., 1987; Dickmanns, and Graefe, 1988). This is more general than modelling entire objects but more clutter-resistant than applying signal-processing to low-level corners or edges. The methods to be discussed here have been applied at this level, to segments of parametric B-spline curves (Bartels et al., 1987) tracking over image sequences (Menet et al., 1990; Cipolla and Blake, 1990). The B-spline curves could, in theory, be parameterised by their control points. In practice, this allows too many degrees of freedom for stable tracking and it is necessary to restrict the curve to a low-dimensional parameter x, for example, over an affine space (Koenderink and Van Doorn, 1991; Ullman and Basri, 1991; Blake et al., 1993), or more generally allowing a “shape-space” of non-rigid motion (Cootes et al., 1993). 6 Isard and Blake Finally, prior probability densities can be defined over the curves (Cootes e... |

261 |
Model-based vision: a program to see a walking person
- Hogg
- 1983
(Show Context)
Citation Context ...lve the resulting ambiguity by applying probabilistic models of object shape and motion to analyse the video-stream. The degree of generality of these models is pitched carefully: sufficiently specific for effective disambiguation but sufficiently general to be broadly applicable over entire classes of foreground objects. 1.1. Modelling Shape and Motion Effective methods have arisen in computer vision for modelling shape and motion. When suitable geometric models of a moving object are available, they can be matched effectively to image data, though usually at considerable computational cost (Hogg, 1983; Lowe, 1991; Sullivan, 1992; Huttenlocher et al., 1993). Once an object has been located approximately, tracking it in subsequent images becomes more efficient computationally (Lowe, 1992), especially if motion is modelled as well as shape (Gennery, 1992; Harris, 1992). One important facility is the modelling of curve segments which interact with images (Fischler and Elschlager, 1973; Yuille and Hallinan, 1992) or image sequences (Kass et al., 1987; Dickmanns, and Graefe, 1988). This is more general than modelling entire objects but more clutter-resistant than applying signal-processing to lo... |

259 | Kalman Filter-Based Algorithm for Estimating Depth from Image Sequence
- Matthies, Kanade, et al.
- 1989
(Show Context)
Citation Context ...berg and Hogg, 1995). Given the learned prior, and an observation density that characterises the statistical variability of image data z given a curve state x, a posterior distribution can, in principle, be estimated for xt given zt at successive times t . 1.2. Kalman Filters and Data-Association Spatio-temporal estimation, the tracking of shape and position over time, has been dealt with thoroughly by Kalman filtering, in the relatively clutter-free case in which p(xt ) can satisfactorily be modelled as Gaussian (Dickmanns and Graefe, 1988; Harris, 1992; Gennery, 1992; Rehg and Kanade, 1994; Matthies et al., 1989) Figure 1. Kalman filter as density propagation: in the case of Gaussian prior, process and observation densities, and assuming linear dynamics, the propagation process of Fig. 2 reduces to a diffusing Gaussian state density, represented completely by its evolving (multivariate) mean and variance—precisely what a Kalman filter computes. and can be applied to curves (Terzopoulos and Szeliski, 1992; Blake et al., 1993). These solutions work relatively poorly in clutter which causes the density for xt to be multi-modal and therefore non-Gaussian. With simple, discrete features such as points or c... |

227 |
Introduction to stochastic control theory
- Åström
- 1970
(Show Context)
Citation Context ...atures such as points or corners combinatorial data-association methods can be effective with clutter but combinatorial methods to do not apply naturally to curves. There remains a need for an appropriately general probabilistic mechanism to handle multi-modal density functions. 1.3. Temporal Propagation of Conditional Densities The Kalman filter as a recursive linear estimator is a special case, applying only to Gaussian densities, of a more general probability density propagation process. In continuous time this can be described in terms of diffusion, governed by a “Fokker-Planck” equation (Astrom, 1970), in which the density for xt drifts and spreads under the action of a stochastic model of its dynamics. In the simple Gaussian case, the diffusion is purely linear and the density function evolves as a Gaussian pulse that translates, spreads and is reinforced, remaining Gaussian throughout, as in Fig. 1, a process that is described analytically and exactly by the Kalman filter. The random component of the dynamical Condensation—Conditional Density Propagation for Visual Tracking 7 Figure 2. Probability density propagation: propagation is depicted here as it occurs over a discrete time-step. T... |

210 |
A framework for spatio-temporal control in the tracking of visual contours.
- Blake, Curwen, et al.
- 1993
(Show Context)
Citation Context ...esistant than applying signal-processing to low-level corners or edges. The methods to be discussed here have been applied at this level, to segments of parametric B-spline curves (Bartels et al., 1987) tracking over image sequences (Menet et al., 1990; Cipolla and Blake, 1990). The B-spline curves could, in theory, be parameterised by their control points. In practice, this allows too many degrees of freedom for stable tracking and it is necessary to restrict the curve to a low-dimensional parameter x, for example, over an affine space (Koenderink and Van Doorn, 1991; Ullman and Basri, 1991; Blake et al., 1993), or more generally allowing a “shape-space” of non-rigid motion (Cootes et al., 1993). 6 Isard and Blake Finally, prior probability densities can be defined over the curves (Cootes et al., 1993) represented by appropriate parameter vectors x, and also over their motions (Terzopoulos and Metaxas, 1991; Blake et al., 1993), and this constitutes a powerful facility for tracking. Reasonable defaults can be chosen for those densities. However, it is obviously more satisfactory to measure or estimate them from data-sequences (x1, x2, . . .). Algorithms to do this, assuming Gaussian densities, are k... |

197 | Visual tracking of high DOF articulated structures: an application to human hand tracking.
- REHG, KANADE
- 1994
(Show Context)
Citation Context ...e and Isard, 1994; Baumberg and Hogg, 1995). Given the learned prior, and an observation density that characterises the statistical variability of image data z given a curve state x, a posterior distribution can, in principle, be estimated for xt given zt at successive times t . 1.2. Kalman Filters and Data-Association Spatio-temporal estimation, the tracking of shape and position over time, has been dealt with thoroughly by Kalman filtering, in the relatively clutter-free case in which p(xt ) can satisfactorily be modelled as Gaussian (Dickmanns and Graefe, 1988; Harris, 1992; Gennery, 1992; Rehg and Kanade, 1994; Matthies et al., 1989) Figure 1. Kalman filter as density propagation: in the case of Gaussian prior, process and observation densities, and assuming linear dynamics, the propagation process of Fig. 2 reduces to a diffusing Gaussian state density, represented completely by its evolving (multivariate) mean and variance—precisely what a Kalman filter computes. and can be applied to curves (Terzopoulos and Szeliski, 1992; Blake et al., 1993). These solutions work relatively poorly in clutter which causes the density for xt to be multi-modal and therefore non-Gaussian. With simple, discrete feat... |

184 | Robust model–based motion tracking through the integration of search and estimation
- Lowe
- 1992
(Show Context)
Citation Context ...ntly specific for effective disambiguation but sufficiently general to be broadly applicable over entire classes of foreground objects. 1.1. Modelling Shape and Motion Effective methods have arisen in computer vision for modelling shape and motion. When suitable geometric models of a moving object are available, they can be matched effectively to image data, though usually at considerable computational cost (Hogg, 1983; Lowe, 1991; Sullivan, 1992; Huttenlocher et al., 1993). Once an object has been located approximately, tracking it in subsequent images becomes more efficient computationally (Lowe, 1992), especially if motion is modelled as well as shape (Gennery, 1992; Harris, 1992). One important facility is the modelling of curve segments which interact with images (Fischler and Elschlager, 1973; Yuille and Hallinan, 1992) or image sequences (Kass et al., 1987; Dickmanns, and Graefe, 1988). This is more general than modelling entire objects but more clutter-resistant than applying signal-processing to low-level corners or edges. The methods to be discussed here have been applied at this level, to segments of parametric B-spline curves (Bartels et al., 1987) tracking over image sequences (M... |

163 | Learning flexible models from image sequences
- Baumberg, Hogg
- 1994
(Show Context)
Citation Context ...ucted by applying an appropriate combination of three methods to build a W -matrix: 1. determining analytically combinations of contours derived from one or more views (Ullman and Basri, 1991; Koenderink and Van Doorn, 1991; Blake et al., 1993), a method that is usable both for affine spaces and for certain classes of articulated object; 2. capturing sequences of key frames of the object in different poses (Blake et al., 1995); 3. performing principal components analysis on a set of outlines of the deforming object (Cootes et al., 12 Isard and Blake Figure 6. The Condensation algorithm. 1993; Baumberg and Hogg, 1994) to derive a small set of representative contours. 5.2. Dynamical Model Exploiting earlier work on dynamical modelling (Blake et al., 1993, 1995), object dynamics are modelled as a second order process, conveniently represented in discrete time t as a second order linear difference equation: xt − x = A(xt−1 − x)+ Bwt (10) where wt are independent vectors of independent standard normal variables, the state-vector xt = ( Xt−1 Xt ) , (11) and where x is the mean value of the state and A, B are matrices representing the deterministic and stochastic components of the dynamical model, respectivel... |

155 |
Recursive Bayesian estimation using Gaussian sums. Automatica
- Sorenson, Alspach
- 1971
(Show Context)
Citation Context ...approach to filtering is then to approximate the dynamics by Taylor expansion as a linear process with timevarying coefficients and proceed as for linear Kalman filters. This generates a Gaussian representation of the evolving state-density which may be a good approximation depending on the nature of the non-linearity. This is the basis of the “Extended Kalman Filter” (EKF) (Gelb, 1974; Bar-Shalom and Fortmann, 1988). Alternatively, one can attempt a mixture representation, as earlier, but now allowing the weights w(m) also to vary over time. Unfortunately, even allowing dynamic re-weighting (Sorenson and Alspach, 1971) does not produce exact solutions for pt (x), because the individual Gaussian components do not remain Gaussian over time. For example, consider the case in which the process density p(xt |xt−1) is itself an additive mixture of k > 1 Gaussian components. According to the Bayesian propagation equation (5) each component of pt splits into k separate components in the transition from time n to time n + 1; the total number of components in pt grows exponentially as kt . Clearly, pt must be approximated at each time-step to prune back the number of components (Anderson and Moore, 1979) within some ... |

149 | Visual tracking of known three-dimensional objects.
- Gennery
- 1992
(Show Context)
Citation Context ...ral to be broadly applicable over entire classes of foreground objects. 1.1. Modelling Shape and Motion Effective methods have arisen in computer vision for modelling shape and motion. When suitable geometric models of a moving object are available, they can be matched effectively to image data, though usually at considerable computational cost (Hogg, 1983; Lowe, 1991; Sullivan, 1992; Huttenlocher et al., 1993). Once an object has been located approximately, tracking it in subsequent images becomes more efficient computationally (Lowe, 1992), especially if motion is modelled as well as shape (Gennery, 1992; Harris, 1992). One important facility is the modelling of curve segments which interact with images (Fischler and Elschlager, 1973; Yuille and Hallinan, 1992) or image sequences (Kass et al., 1987; Dickmanns, and Graefe, 1988). This is more general than modelling entire objects but more clutter-resistant than applying signal-processing to low-level corners or edges. The methods to be discussed here have been applied at this level, to segments of parametric B-spline curves (Bartels et al., 1987) tracking over image sequences (Menet et al., 1990; Cipolla and Blake, 1990). The B-spline curves c... |

118 |
Tracking with kalman snakes.
- Terzopoulos, Szeliski
- 1992
(Show Context)
Citation Context ... thoroughly by Kalman filtering, in the relatively clutter-free case in which p(xt ) can satisfactorily be modelled as Gaussian (Dickmanns and Graefe, 1988; Harris, 1992; Gennery, 1992; Rehg and Kanade, 1994; Matthies et al., 1989) Figure 1. Kalman filter as density propagation: in the case of Gaussian prior, process and observation densities, and assuming linear dynamics, the propagation process of Fig. 2 reduces to a diffusing Gaussian state density, represented completely by its evolving (multivariate) mean and variance—precisely what a Kalman filter computes. and can be applied to curves (Terzopoulos and Szeliski, 1992; Blake et al., 1993). These solutions work relatively poorly in clutter which causes the density for xt to be multi-modal and therefore non-Gaussian. With simple, discrete features such as points or corners combinatorial data-association methods can be effective with clutter but combinatorial methods to do not apply naturally to curves. There remains a need for an appropriately general probabilistic mechanism to handle multi-modal density functions. 1.3. Temporal Propagation of Conditional Densities The Kalman filter as a recursive linear estimator is a special case, applying only to Gaussian... |

111 | Learning to track the visual motion of contours
- Blake, Isard, et al.
- 1995
(Show Context)
Citation Context ...urve are represented by linear transformations of x . The Condensation algorithm itself does not demand necessarily a linear parameterisation though linearity is an attraction for another reason—the availability of algorithms to learn object dynamics. The algorithm could also be used, in principle, with nonlinear parameterised kinematics—for instance, representing an articulated hand in terms of joint angles (Rehg and Kanade, 1994). 5.1. Linear Parameterisations of Splines for Tracking We represent the state of a tracked object following methods established for tracking using a Kalman filter (Blake et al., 1995). Objects are modelled as a curve (or set of curves), typically though not necessarily the occluding contour, and represented at time t by a parameterised image curve r(s, t). The parameterisation is in terms of B-splines, so r(s, t) = (B(s) · Qx (t), B(s) · Qy(t)), for 0 ≤ s ≤ L , (8) where B(s) is a vector (B1(s), . . . , BNB (s)) T of B-spline basis functions, Qx and Qy are vectors of Bspline control point coordinates and L is the number of spans. It is usually desirable (Blake et al., 1993) to restrict the configuration of the spline to a shape-space of vectors X defined by( Qx Qy ) = W X ... |

106 |
Tracking non-rigid objects in complex scenes, tn ICCV93,
- Nob, Huttenlocher, et al.
- 1993
(Show Context)
Citation Context ...probabilistic models of object shape and motion to analyse the video-stream. The degree of generality of these models is pitched carefully: sufficiently specific for effective disambiguation but sufficiently general to be broadly applicable over entire classes of foreground objects. 1.1. Modelling Shape and Motion Effective methods have arisen in computer vision for modelling shape and motion. When suitable geometric models of a moving object are available, they can be matched effectively to image data, though usually at considerable computational cost (Hogg, 1983; Lowe, 1991; Sullivan, 1992; Huttenlocher et al., 1993). Once an object has been located approximately, tracking it in subsequent images becomes more efficient computationally (Lowe, 1992), especially if motion is modelled as well as shape (Gennery, 1992; Harris, 1992). One important facility is the modelling of curve segments which interact with images (Fischler and Elschlager, 1973; Yuille and Hallinan, 1992) or image sequences (Kass et al., 1987; Dickmanns, and Graefe, 1988). This is more general than modelling entire objects but more clutter-resistant than applying signal-processing to low-level corners or edges. The methods to be discussed he... |

102 |
Tracking with rigid models.
- Harris
- 1992
(Show Context)
Citation Context ...ly applicable over entire classes of foreground objects. 1.1. Modelling Shape and Motion Effective methods have arisen in computer vision for modelling shape and motion. When suitable geometric models of a moving object are available, they can be matched effectively to image data, though usually at considerable computational cost (Hogg, 1983; Lowe, 1991; Sullivan, 1992; Huttenlocher et al., 1993). Once an object has been located approximately, tracking it in subsequent images becomes more efficient computationally (Lowe, 1992), especially if motion is modelled as well as shape (Gennery, 1992; Harris, 1992). One important facility is the modelling of curve segments which interact with images (Fischler and Elschlager, 1973; Yuille and Hallinan, 1992) or image sequences (Kass et al., 1987; Dickmanns, and Graefe, 1988). This is more general than modelling entire objects but more clutter-resistant than applying signal-processing to low-level corners or edges. The methods to be discussed here have been applied at this level, to segments of parametric B-spline curves (Bartels et al., 1987) tracking over image sequences (Menet et al., 1990; Cipolla and Blake, 1990). The B-spline curves could, in theory... |

94 |
Probability and Statistics.
- Papoulis
- 1990
(Show Context)
Citation Context ...proach which is described in the following two sections. 3. Factored Sampling This section describes first the factored sampling algorithm dealing with non-Gaussian observations in single images. Then factored sampling is extended in the following section to deal with temporal image sequences. A standard problem in statistical pattern recognition is to find an object parameterised as x with prior p(x), using data z from a single image. The posterior density p(x |z) represents all the knowledge about x that is deducible from the data. It can be evaluated, in principle, by applying Bayes’ rule (Papoulis, 1990) to obtain p(x |z) = kp(z |x)p(x) (6) where k is a normalisation constant that is independent of x. In cases where p(z |x) is sufficiently complex that p(x |z) cannot be evaluated simply in closed form, iterative sampling techniques can be used (Geman and Geman, 1984; Ripley and Sutherland, 1990; Grenander et al., 1991; Storvik, 1994). The factored sampling algorithm (Grenander et al., 1991) generates a random variate x from a distribution p(x) that approximates the posterior p(x |z). First, a sample-set {s(1), . . . , s(N )} is generated from the prior density p(x) and then an index Condensa... |

70 | 3D position, attitude and shape input using video tracking of hands and lips.
- Blake, Isard
- 1994
(Show Context)
Citation Context ...e Finally, prior probability densities can be defined over the curves (Cootes et al., 1993) represented by appropriate parameter vectors x, and also over their motions (Terzopoulos and Metaxas, 1991; Blake et al., 1993), and this constitutes a powerful facility for tracking. Reasonable defaults can be chosen for those densities. However, it is obviously more satisfactory to measure or estimate them from data-sequences (x1, x2, . . .). Algorithms to do this, assuming Gaussian densities, are known in the control-theory literature (Goodwin and Sin, 1984) and have been applied in computer vision (Blake and Isard, 1994; Baumberg and Hogg, 1995). Given the learned prior, and an observation density that characterises the statistical variability of image data z given a curve state x, a posterior distribution can, in principle, be estimated for xt given zt at successive times t . 1.2. Kalman Filters and Data-Association Spatio-temporal estimation, the tracking of shape and position over time, has been dealt with thoroughly by Kalman filtering, in the relatively clutter-free case in which p(xt ) can satisfactorily be modelled as Gaussian (Dickmanns and Graefe, 1988; Harris, 1992; Gennery, 1992; Rehg and Kanade, ... |

64 |
B-snakes: Implementation and application to stereo.
- Menet, Saint-Marc, et al.
- 1990
(Show Context)
Citation Context ...), especially if motion is modelled as well as shape (Gennery, 1992; Harris, 1992). One important facility is the modelling of curve segments which interact with images (Fischler and Elschlager, 1973; Yuille and Hallinan, 1992) or image sequences (Kass et al., 1987; Dickmanns, and Graefe, 1988). This is more general than modelling entire objects but more clutter-resistant than applying signal-processing to low-level corners or edges. The methods to be discussed here have been applied at this level, to segments of parametric B-spline curves (Bartels et al., 1987) tracking over image sequences (Menet et al., 1990; Cipolla and Blake, 1990). The B-spline curves could, in theory, be parameterised by their control points. In practice, this allows too many degrees of freedom for stable tracking and it is necessary to restrict the curve to a low-dimensional parameter x, for example, over an affine space (Koenderink and Van Doorn, 1991; Ullman and Basri, 1991; Blake et al., 1993), or more generally allowing a “shape-space” of non-rigid motion (Cootes et al., 1993). 6 Isard and Blake Finally, prior probability densities can be defined over the curves (Cootes et al., 1993) represented by appropriate parameter ... |

63 | A Bayesian approach to dynamic contours through stochastic sampling and simulated annealing.
- Storvik
- 1994
(Show Context)
Citation Context ...recognition is to find an object parameterised as x with prior p(x), using data z from a single image. The posterior density p(x |z) represents all the knowledge about x that is deducible from the data. It can be evaluated, in principle, by applying Bayes’ rule (Papoulis, 1990) to obtain p(x |z) = kp(z |x)p(x) (6) where k is a normalisation constant that is independent of x. In cases where p(z |x) is sufficiently complex that p(x |z) cannot be evaluated simply in closed form, iterative sampling techniques can be used (Geman and Geman, 1984; Ripley and Sutherland, 1990; Grenander et al., 1991; Storvik, 1994). The factored sampling algorithm (Grenander et al., 1991) generates a random variate x from a distribution p(x) that approximates the posterior p(x |z). First, a sample-set {s(1), . . . , s(N )} is generated from the prior density p(x) and then an index Condensation—Conditional Density Propagation for Visual Tracking 9 Figure 3. Factored sampling: a set of points s(n), the centres of the blobs in the figure, is sampled randomly from a prior density p(x). Each sample is assigned a weight πi (depicted by blob area) in proportion to the value of the observation density p(z |x = s(n)). The weigh... |

60 |
The dynamic analysis of apparent contours.
- Cipolla, Blake
- 1990
(Show Context)
Citation Context ...ion is modelled as well as shape (Gennery, 1992; Harris, 1992). One important facility is the modelling of curve segments which interact with images (Fischler and Elschlager, 1973; Yuille and Hallinan, 1992) or image sequences (Kass et al., 1987; Dickmanns, and Graefe, 1988). This is more general than modelling entire objects but more clutter-resistant than applying signal-processing to low-level corners or edges. The methods to be discussed here have been applied at this level, to segments of parametric B-spline curves (Bartels et al., 1987) tracking over image sequences (Menet et al., 1990; Cipolla and Blake, 1990). The B-spline curves could, in theory, be parameterised by their control points. In practice, this allows too many degrees of freedom for stable tracking and it is necessary to restrict the curve to a low-dimensional parameter x, for example, over an affine space (Koenderink and Van Doorn, 1991; Ullman and Basri, 1991; Blake et al., 1993), or more generally allowing a “shape-space” of non-rigid motion (Cootes et al., 1993). 6 Isard and Blake Finally, prior probability densities can be defined over the curves (Cootes et al., 1993) represented by appropriate parameter vectors x, and also over t... |

60 |
Building and using flexible models incorporating grey-level information.
- Cootes, Taylor, et al.
- 1993
(Show Context)
Citation Context ...o be discussed here have been applied at this level, to segments of parametric B-spline curves (Bartels et al., 1987) tracking over image sequences (Menet et al., 1990; Cipolla and Blake, 1990). The B-spline curves could, in theory, be parameterised by their control points. In practice, this allows too many degrees of freedom for stable tracking and it is necessary to restrict the curve to a low-dimensional parameter x, for example, over an affine space (Koenderink and Van Doorn, 1991; Ullman and Basri, 1991; Blake et al., 1993), or more generally allowing a “shape-space” of non-rigid motion (Cootes et al., 1993). 6 Isard and Blake Finally, prior probability densities can be defined over the curves (Cootes et al., 1993) represented by appropriate parameter vectors x, and also over their motions (Terzopoulos and Metaxas, 1991; Blake et al., 1993), and this constitutes a powerful facility for tracking. Reasonable defaults can be chosen for those densities. However, it is obviously more satisfactory to measure or estimate them from data-sequences (x1, x2, . . .). Algorithms to do this, assuming Gaussian densities, are known in the control-theory literature (Goodwin and Sin, 1984) and have been applied in... |

59 | Visual interpretation of known objects in constrained scenes
- Sullivan
- 1992
(Show Context)
Citation Context ...ity by applying probabilistic models of object shape and motion to analyse the video-stream. The degree of generality of these models is pitched carefully: sufficiently specific for effective disambiguation but sufficiently general to be broadly applicable over entire classes of foreground objects. 1.1. Modelling Shape and Motion Effective methods have arisen in computer vision for modelling shape and motion. When suitable geometric models of a moving object are available, they can be matched effectively to image data, though usually at considerable computational cost (Hogg, 1983; Lowe, 1991; Sullivan, 1992; Huttenlocher et al., 1993). Once an object has been located approximately, tracking it in subsequent images becomes more efficient computationally (Lowe, 1992), especially if motion is modelled as well as shape (Gennery, 1992; Harris, 1992). One important facility is the modelling of curve segments which interact with images (Fischler and Elschlager, 1973; Yuille and Hallinan, 1992) or image sequences (Kass et al., 1987; Dickmanns, and Graefe, 1988). This is more general than modelling entire objects but more clutter-resistant than applying signal-processing to low-level corners or edges. Th... |

50 | Learning dynamics of complex motions from image sequences.
- Reynard, Wildenberg, et al.
- 1996
(Show Context)
Citation Context ... is a set of damped oscillators, whose modes, natural frequencies and damping constants are determined by A, driven by random accelerations coupled Condensation—Conditional Density Propagation for Visual Tracking 13 into the dynamics via B from the noise term Bw. While it is possible to set sensible defaults for A, x and B, it is more satisfactory and effective to estimate them from input data taken while the object performs typical motions. Methods for doing this via Maximum Likelihood Estimation are essential to the work described here and are described fully elsewhere (Blake et al., 1995; Reynard et al., 1996). The dynamical model can be re-expressed in such a way as to make quite clear that it is a temporal Markov chain: p(xt |xt−1) ∝ exp ( −1 2 ‖B−1((xt − x)− A(xt−1 − x))‖2 ) (12) where ‖ · · · ‖ is the Euclidean norm. It is therefore clear that the learned dynamical models are appropriate for use in the Condensation algorithm. 5.3. Initial Conditions Initial conditions for tracking can be determined by specifying the prior density p(x0), and if this is Gaussian, direct sampling can be used to initialise the Condensation algorithm. Alternatively, it is possible simply to allow the density p(xt ... |

43 |
Task-Directed Sensor Fusion and Planning - A Computational Approach
- Hager
- 1990
(Show Context)
Citation Context ...1992) address the more difficult problem of associating not simply single features but subsequences of Zt with the state. However, these methods rely on the existence of discrete features. In contour tracking the features are continuous curves and so are not naturally amenable to discrete association. A.4. Direct Integration Finally, one very general approach to non-linear filtering must be mentioned. This is simply to integrate (5) directly, using a suitable numerical representation of the state density such as finite elements. This in essence is what (Bucy, 1969) proposed and more recently (Hager, 1990) investigated with respect to robotics applications. It is usable in one or two dimensions but, complexity being exponential in the dimension, is altogether infeasible for problems of dimension around 6–20, typical of the tracking problems dealt with here. The Condensation algorithm is designed to offer a viable alternative. Appendix B: Derivation of the Sampling Rule The correctness of the sampling rule (4) in Section 2.3 is proved by first deriving two lemmas from the independence assumption (2). (This is similar to the derivation found in (Bar-Shalom and Fortmann, 1988), except that our ind... |

43 |
Deformable templates,”
- Yuille, Hallianan
- 1992
(Show Context)
Citation Context ...ision for modelling shape and motion. When suitable geometric models of a moving object are available, they can be matched effectively to image data, though usually at considerable computational cost (Hogg, 1983; Lowe, 1991; Sullivan, 1992; Huttenlocher et al., 1993). Once an object has been located approximately, tracking it in subsequent images becomes more efficient computationally (Lowe, 1992), especially if motion is modelled as well as shape (Gennery, 1992; Harris, 1992). One important facility is the modelling of curve segments which interact with images (Fischler and Elschlager, 1973; Yuille and Hallinan, 1992) or image sequences (Kass et al., 1987; Dickmanns, and Graefe, 1988). This is more general than modelling entire objects but more clutter-resistant than applying signal-processing to low-level corners or edges. The methods to be discussed here have been applied at this level, to segments of parametric B-spline curves (Bartels et al., 1987) tracking over image sequences (Menet et al., 1990; Cipolla and Blake, 1990). The B-spline curves could, in theory, be parameterised by their control points. In practice, this allows too many degrees of freedom for stable tracking and it is necessary to restr... |

40 | Generating spatiotemporal models from examples.
- Baumberg, Hogg
- 1995
(Show Context)
Citation Context ...ility densities can be defined over the curves (Cootes et al., 1993) represented by appropriate parameter vectors x, and also over their motions (Terzopoulos and Metaxas, 1991; Blake et al., 1993), and this constitutes a powerful facility for tracking. Reasonable defaults can be chosen for those densities. However, it is obviously more satisfactory to measure or estimate them from data-sequences (x1, x2, . . .). Algorithms to do this, assuming Gaussian densities, are known in the control-theory literature (Goodwin and Sin, 1984) and have been applied in computer vision (Blake and Isard, 1994; Baumberg and Hogg, 1995). Given the learned prior, and an observation density that characterises the statistical variability of image data z given a curve state x, a posterior distribution can, in principle, be estimated for xt given zt at successive times t . 1.2. Kalman Filters and Data-Association Spatio-temporal estimation, the tracking of shape and position over time, has been dealt with thoroughly by Kalman filtering, in the relatively clutter-free case in which p(xt ) can satisfactorily be modelled as Gaussian (Dickmanns and Graefe, 1988; Harris, 1992; Gennery, 1992; Rehg and Kanade, 1994; Matthies et al., 198... |

34 | Conditional-mean estimation via jump-diffusion processes in multiple target tracking/recognition.
- Miller, Srivasta, et al.
- 1995
(Show Context)
Citation Context ... with an estimate obtained more conventionally, and efficiently, from linear least squares estimation. For multi-modal distributions which cannot be approximated as normal, so that linear estimators are unusable, estimates of mean x by factored sampling continue to apply. 4. The CONDENSATION Algorithm The Condensation algorithm is based on factored sampling but extended to apply iteratively to successive images in a sequence. The same sampling strategy has been developed elsewhere (Gordon, et al., 1993; Kitagawa, 1996), presented as developments of MonteCarlo methods. Jump-diffusion tracking (Miller et al., 1995) may also be related to the approach described here. Given that the process at each time-step is a selfcontained iteration of factored sampling, the output of an iteration will be a weighted, time-stamped sample-set, denoted {s(n)t , n = 1, . . . , N } with weights π (n) t , representing approximately the conditional statedensity p(xt |Zt ) at time t . How is this sample-set obtained? Clearly, the process must begin with a prior density and the effective prior for time-step t should be p(xt |Zt−1). This prior is of course multi-modal in general and no functional representation of it is availab... |

19 |
Applications of dynamic monocular machine vision
- Dickmanns, Graefe
- 1988
(Show Context)
Citation Context ...Sin, 1984) and have been applied in computer vision (Blake and Isard, 1994; Baumberg and Hogg, 1995). Given the learned prior, and an observation density that characterises the statistical variability of image data z given a curve state x, a posterior distribution can, in principle, be estimated for xt given zt at successive times t . 1.2. Kalman Filters and Data-Association Spatio-temporal estimation, the tracking of shape and position over time, has been dealt with thoroughly by Kalman filtering, in the relatively clutter-free case in which p(xt ) can satisfactorily be modelled as Gaussian (Dickmanns and Graefe, 1988; Harris, 1992; Gennery, 1992; Rehg and Kanade, 1994; Matthies et al., 1989) Figure 1. Kalman filter as density propagation: in the case of Gaussian prior, process and observation densities, and assuming linear dynamics, the propagation process of Fig. 2 reduces to a diffusing Gaussian state density, represented completely by its evolving (multivariate) mean and variance—precisely what a Kalman filter computes. and can be applied to curves (Terzopoulos and Szeliski, 1992; Blake et al., 1993). These solutions work relatively poorly in clutter which causes the density for xt to be multi-modal an... |

18 |
Finding spiral structures in images of galaxies.
- Ripley, Sutherland
- 1990
(Show Context)
Citation Context ...sequences. A standard problem in statistical pattern recognition is to find an object parameterised as x with prior p(x), using data z from a single image. The posterior density p(x |z) represents all the knowledge about x that is deducible from the data. It can be evaluated, in principle, by applying Bayes’ rule (Papoulis, 1990) to obtain p(x |z) = kp(z |x)p(x) (6) where k is a normalisation constant that is independent of x. In cases where p(z |x) is sufficiently complex that p(x |z) cannot be evaluated simply in closed form, iterative sampling techniques can be used (Geman and Geman, 1984; Ripley and Sutherland, 1990; Grenander et al., 1991; Storvik, 1994). The factored sampling algorithm (Grenander et al., 1991) generates a random variate x from a distribution p(x) that approximates the posterior p(x |z). First, a sample-set {s(1), . . . , s(N )} is generated from the prior density p(x) and then an index Condensation—Conditional Density Propagation for Visual Tracking 9 Figure 3. Factored sampling: a set of points s(n), the centres of the blobs in the figure, is sampled randomly from a prior density p(x). Each sample is assigned a weight πi (depicted by blob area) in proportion to the value of the obser... |

15 |
Data association methods for tracking systems.
- Rao
- 1992
(Show Context)
Citation Context ...c data-association filter (PDAF) (Bar-Shalom and Fortmann, 1988) attempt to deal with the ambiguity of association. Alternatively it can, in principle, be dealt with exactly by “multiple hypothesis filtering” but with computational cost that grows exponentially over time and which is therefore ruled out in practice. The “RANSAC” algorithm (Fischler and Bolles, 1981) deals probabilistically with multiple observations but the observations have to be discrete, and there is no mechanism for temporal propagation. More complex methods including the Joint PDAF (JPDAF) (Bar-Shalom and Fortmann, 1988; Rao, 1992) address the more difficult problem of associating not simply single features but subsequences of Zt with the state. However, these methods rely on the existence of discrete features. In contour tracking the features are continuous curves and so are not naturally amenable to discrete association. A.4. Direct Integration Finally, one very general approach to non-linear filtering must be mentioned. This is simply to integrate (5) directly, using a suitable numerical representation of the state density such as finite elements. This in essence is what (Bucy, 1969) proposed and more recently (Hager... |

12 |
Bayes theorem and digital realizations for non-linear filters.
- Bucy
- 1969
(Show Context)
Citation Context ...F) (Bar-Shalom and Fortmann, 1988; Rao, 1992) address the more difficult problem of associating not simply single features but subsequences of Zt with the state. However, these methods rely on the existence of discrete features. In contour tracking the features are continuous curves and so are not naturally amenable to discrete association. A.4. Direct Integration Finally, one very general approach to non-linear filtering must be mentioned. This is simply to integrate (5) directly, using a suitable numerical representation of the state density such as finite elements. This in essence is what (Bucy, 1969) proposed and more recently (Hager, 1990) investigated with respect to robotics applications. It is usable in one or two dimensions but, complexity being exponential in the dimension, is altogether infeasible for problems of dimension around 6–20, typical of the tracking problems dealt with here. The Condensation algorithm is designed to offer a viable alternative. Appendix B: Derivation of the Sampling Rule The correctness of the sampling rule (4) in Section 2.3 is proved by first deriving two lemmas from the independence assumption (2). (This is similar to the derivation found in (Bar-Shalom... |

10 |
Statistical feature modelling for active contours.
- Rowe, Blake
- 1996
(Show Context)
Citation Context ...cing the camera (Fig. 9). One of the people moves from right to left, in front of the other two. The shape-space for tracking is built from a handdrawn template of head and shoulders (Fig. 8) which is then allowed to deform via planar affine transformations. A Kalman filter contour-tracker (Blake et al., 1993) with default motion parameters is able to track a single moving person just well enough to obtain a sequence of outline curves that is usable as training data. Given the high level of clutter, adequate performance with the Kalman filter is obtained here by means of background modelling (Rowe and Blake, 1996), a statistical form of background subtraction, which effectively removes clutter from the image data before it is tracked. It transpires, for this particular training set, that the learned motions comprise primarily horizontal translation, with vertical translation and horizontal and vertical shear present to a lesser degree. The learned shape and motion model can now be installed as p(xt |xt−1) in the Condensation algorithm which is run on a test sequence but without the benefit of background modelling, so that the background clutter is now visible to the tracker. Figure 10 shows how the sta... |

6 |
Fitting parameterised 3D models to images.
- Lowe
- 1991
(Show Context)
Citation Context ...lting ambiguity by applying probabilistic models of object shape and motion to analyse the video-stream. The degree of generality of these models is pitched carefully: sufficiently specific for effective disambiguation but sufficiently general to be broadly applicable over entire classes of foreground objects. 1.1. Modelling Shape and Motion Effective methods have arisen in computer vision for modelling shape and motion. When suitable geometric models of a moving object are available, they can be matched effectively to image data, though usually at considerable computational cost (Hogg, 1983; Lowe, 1991; Sullivan, 1992; Huttenlocher et al., 1993). Once an object has been located approximately, tracking it in subsequent images becomes more efficient computationally (Lowe, 1992), especially if motion is modelled as well as shape (Gennery, 1992; Harris, 1992). One important facility is the modelling of curve segments which interact with images (Fischler and Elschlager, 1973; Yuille and Hallinan, 1992) or image sequences (Kass et al., 1987; Dickmanns, and Graefe, 1988). This is more general than modelling entire objects but more clutter-resistant than applying signal-processing to low-level corn... |

1 | This paper has appeared in short form (Isard and Blake, - Notes - 1996 |

1 | The presence of clutter causes p(z |x) to be non-Gaussian, - Note - 1979 |