U. Grenander, Y. Chos, and D. M. Keenan. A Pattern Theoretical Study of Biological Shapes. SpringerVerlag, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....for new critical areas. Finally, the use of new observations to evaluate the fitness of the new hypotheses provides a close tracking of the evolution of the state space. A mathematical justification of the approximation given by the particle filter can be stated in terms of factored sampling [28] and the method of composition [72] Factored sampling provides a justification to approximate the posterior by a weighted set of particles, as expressed in Equation (2.7) The method of composition justifies that the particles obtained by sampling from the dynamic prior are independent and ....

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

.... distribution p 0 (X) for the configuration X , and an observation likelihood L(X) p(ZjX) where Z j Z(I) is some finitedimensional representation of the image I , then the posterior density for X is given by p(X jZ) p 0 (X)p(ZjX) 2) This can be done very effectively by factored sampling [16] which produces a weighted particle set f(s (1) 1 ) s (N) N )g, of size N that approximates the posterior [7] From this approximation of the distribution fusion of inference about X from different sensors, over time and across scales. It also allows a structured way of ....

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

....treatment of background modelling for use in visual curve trackers. The new methods are tested using a real time tracker based on snakes deforming over time [18, 10, 3] represented by B spline curves [22, 8] The tracker runs at video field rate (50Hz) and is stabilised using a shape template [14, 16, 5, 30] incorporated into the dynamical model used as a predictor. It runs at video field rate (50Hz) in a cycle of prediction and measurement. The background modelling technique described here is not restricted to curves; it could also be applied to real time trackers based on polygons or other ....

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

....p 0 (X)p(ZjX) 1) In more straightforward, Gaussian cases, 1) can be computed in closed form. In the non Gaussian cases commonly for a version of this paper with colour figures and a movie of figure 15 arising, for example in image clutter or with multiple models, sampling methods are needed [8], and random sampling underlies the development of Bayesian correlation here. Relation to previous work Key elements of the work presented here are: IB Intensity Based observations, not just edges. FL Foreground Learning in terms of probability distributions estimated from one or more training ....

....are to be valid. For instance, assuming independence across adjacent pixels is unjustified, and leads to exaggerated variations in the likelihood p(ZjX) for even minor perturbations of X . There are three outstanding precursors to Bayesian correlation; one concerns random diffeomorphisms [8]; the second is an algorithm [17] for registration by maximisation of mutual information; third is localisation by foreground background learning [7] Attributes of these and other important prior work are summarised in table 1, in terms of elements of Bayesian correlation as listed above. 2 ....

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

....use the image sequence itself as features, the probability density for our trajectory family is nonlinear. Also any possible initialization heuristic for the trajectory of the unknown point track i might be far of. We therefore adopt a stochastic estimation technique based on factored sampling [7] to find the most likely values for m i . Factored sampling is a Monte Carlo technique for estimation of conditional probability densities. Let us assume we are trying to estimate the function p(X Z) where X and Z are continuous random variables statistically related by some unknown dependency. ....

....recipe. Let x 1 , xN be N samples drawn from the prior p(X) and let us generate a third random variable Y by choosing samples y = x s at random with probabilities p(Z X = x s ) k=1 p(Z X = x k ) 8) It has been shown that the probability distribution of Y tends to p(X Z) as ## [7]. An approximation of the mean of the posterior can be computed as E[X Z] s=1 x s p s (9) In our case p(X Z) p(m i Z) where Z are measurements derived from the image sequence. We evaluate each hypothesis (sample) m i for m i by computing for each frame f the sum of squared di#erences ....

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

....on modest workstations with some success. Several researchers have described Kalman filter formalisms for tracking of curves and surfaces [34, 35] This paper is based on a particular linear filter for curves which incorporates a mean shape a template , as used by a number of researchers [19, 21, 9, 38]. The tracker used here [11, 12] also has an affine invariance mechanism to accommodate 3D rigid transformations of planar shapes. We refer to this as the un trained tracker, not yet tuned for motion. Tuning for specific motions, it transpires, is a key ingredient in achieving robust tracking. ....

....The static effect is that learning characterises the likely configurations both shapes and positions of the visible contour; in the tracker this helps maintain a closer match between the tracked and the actual contour. Algorithms to learn static models have been demonstrated previously [21, 15] but such algorithms are unable to exploit training sets that are gathered as time sequences. For static algorithms, permuting the order of elements in a training set has no effect on the model learned. Our algorithm exploits the time sequence structure by simultaneously learning static and ....

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

.... R NX (which may, for example, represent the outline of a curve using a low dimensional parameterisation) and the observed image at time t is denoted Z t , with measurement history Z t = Z 1 ; Z t ) The representation used for probability distributions is derived from factored sampling [3, 9], where it was applied to static images. Factored sampling is a Bayesian technique to approximate a distribution p(XjZ) which applies when p(XjZ) is too complicated to sample directly, but when the prior p(X) can be sampled, and the measurement density p(ZjX) can be evaluated. The algorithm ....

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

....techniques can be used. The first use of such an iterative solution was proposed by Geman and Geman [11] for restoration of an image represented by mixed variables, both continuous (pixels) and discrete (the line process ) Sampling methods for recovery of a parametric curve x by sampling [24, 14, 25] have generally used spatial Markov processes as the underlying probabilistic model p(x) The basic method is factored sampling [14] It is useful when the conditional observation probability p(zjx) can be evaluated pointwise and sampling it is not feasible and when, conversely, the prior p(x) can ....

.... represented by mixed variables, both continuous (pixels) and discrete (the line process ) Sampling methods for recovery of a parametric curve x by sampling [24, 14, 25] have generally used spatial Markov processes as the underlying probabilistic model p(x) The basic method is factored sampling [14]. It is useful when the conditional observation probability p(zjx) can be evaluated pointwise and sampling it is not feasible and when, conversely, the prior p(x) can be sampled but not evaluated. The algorithm estimates means of properties f(x) e.g. moments) of the posterior p(xjz) by first ....

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

....any idea of an underlying true image which is being quantised by the imaging hardware. In particular, the same image, viewed at a different pixel resolution, would have an entirely different model. Similar iterative simulation techniques, under the name of factored sampling have been applied by Grenander et al. 1991) to find hands in noisy images. The factored sampling approach is to approximate the posterior density p(XjZ) by a discrete set of realisations from the prior p(X) weighted by the observation density p(ZjX) and this is discussed more fully in section 3.3 on page 39. Rather than using a model prior ....

....to obtain p(XjZ) kp(ZjX)p(X) 3.6) where k is a normalisation constant that is independent of X. In cases where p(ZjX) is sufficiently complex that p(XjZ) 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 0 from a distribution p(X) that approximates the posterior p(XjZ) First a sample set fs (1) s (N) g is generated from the prior density p(X) and then each index i 2 f1; ....

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

.... velocity are encoded in a state vector X 2 R NX (which may, for example, represent the outline of a curve using a low dimensional parameterisation) and images observed at time t are denoted Z t , with measurement history Z t = Z 1 ; Z t ) The Bayesian technique of factored sampling [20, 11] is a random sampling method to approximate a distribution p(XjZ) which applies when p(XjZ) is too complicated to sample directly, but when the prior p(X) can be sampled, and the measurement density p(ZjX) can be evaluated. Factored sampling proceeds by generating a set of N samples fs (n) g ....

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

....The results of a sample of 1; 000 configurations are shown ranked by value of their contour discriminant. The table displays the cases in which D 1, indicating a configuration that is more target like than clutter like. 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(XjZ) First a sample set fs (1) s (N) g is generated from the prior density p(x) and then a sample X = X i ; i 2 f1; ....

....of their contour discriminant. The table displays the cases in which D 1, indicating a configuration that is more target like than clutter like. 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(XjZ) First a sample set fs (1) s (N) g is generated from the prior density p(x) and then a sample X = X i ; i 2 f1; Ng is chosen with probability i = p(ZjX = s (i) P N j=1 ....

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

....which for which an informative prior distribution can be constructed (e.g. they are often planar with regular outline) Although architectural scenes are chosen to illustrate the basic principles the method proposed is representative of a general approach to segmentation. Taking inspiration from [6], rather than using an implicit model for the prior probability of a segmentation an explicit model is de ned and used. A solution to the nal problem (c) that of nding the texture map, is also found by considering the texture as a set of hidden variables. The layout of the paper is as follows. ....

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

....and 3d. See for example [2] for hand xrays, 3] 4] and [5] for MRI images of the brain. A comparison of the different approaches can be found in [6] Currently 3d versions of these algorithms are being used for brain mapping. One dimensional elastic models have also been extensively studied see [7], and [8] 9] There are several limitations to these elastic models. First the matching criterion seeks to minimize the mean square difference between the pixel intensities of the deformed template and that of the data image. On one hand this is a very well defined criterion however it does not ....

U. Grenander, Y. Chow, and D.M. Keenan, A Pattern Theoretical Study of Biological Shape, Springer Verlag, New York, 1991.

....motions. The second algorithm is a system identification algorithm based on ideas from adaptive control theory [2] and maximum likelihood estimation in statistics [22] Previous approaches to learning shape variability have used statistical models to represent a family of possible shapes [14, 9] but statically. In contrast the learning method reported here is dynamic, using and modelling temporal image sequences. Example motions are tracked by a general purpose tracker based on the assumption of default object dynamics. The tracked motion is used as a training set for the new algorithm ....

U. Grenander, Y. Chow, andD. M. Keenan. HANDS. A Pattern Theoretical Study of Biological Shapes. Springer-Verlag. New York, 1991.

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U. Grenander, Y. Chos, and D. M. Keenan. A Pattern Theoretical Study of Biological Shapes. SpringerVerlag, 1991.

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Y. Chow, U. Grenander, and D. M. Keenan. HANDS, A pattern Theoretical Study of Biological Shapes. Springer-Verlag, 1991.

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