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  • Electronic Resource  (2)
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  • 2000-2004  (2)
  • 2001  (2)
  • AMS Classification. 65L05; 22E60.  (1)
  • Monte Carlo  (1)
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  • Electronic Resource  (2)
  • E-Resource
  • Loose Leaf
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  • 2000-2004  (2)
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  • 1
    Electronic Resource
    Electronic Resource
    Springer
    Foundations of computational mathematics 1 (2001), S. 129-160 
    ISSN: 1615-3383
    Keywords: AMS Classification. 65L05; 22E60.
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract. In this paper we develop in a systematic manner the theory of time-stepping methods based on the Cayley transform. Such methods can be applied to discretize differential equations that evolve in some Lie groups, in particular in the orthogonal group and the symplectic group. Unlike many other Lie-group solvers, they do not require the evaluation of matrix exponentials. Similarly to the theory of Magnus expansions in [13], we identify terms in a Cayley expansion with rooted trees, which can be constructed recursively. Each such term is an integral over a polytope but all such integrals can be evaluated to high order by using special quadrature formulas similar to the construction in [13]. Truncated Cayley expansions (with exact integrals) need not be time-symmetric, hence the method does not display the usual advantages associated with time symmetry, e.g., even order of approximation. However, time symmetry (with its attendant benefits) is attained when exact integrals are replaced by certain quadrature formulas.
    Type of Medium: Electronic Resource
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  • 2
    Electronic Resource
    Electronic Resource
    Springer
    International journal of computer vision 44 (2001), S. 111-135 
    ISSN: 1573-1405
    Keywords: vision ; object location ; Monte Carlo ; filter-bank ; statistical independence
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Notes: Abstract A Bayesian approach to intensity-based object localisation is presented that employs a learned probabilistic model of image filter-bank output, applied via Monte Carlo methods, to escape the inefficiency of exhaustive search. An adequate probabilistic account of image data requires intensities both in the foreground (i.e. over the object), and in the background, to be modelled. Some previous approaches to object localisation by Monte Carlo methods have used models which, we claim, do not fully address the issue of the statistical independence of image intensities. It is addressed here by applying to each image a bank of filters whose outputs are approximately statistically independent. Distributions of the responses of individual filters, over foreground and background, are learned from training data. These distributions are then used to define a joint distribution for the output of the filter bank, conditioned on object configuration, and this serves as an observation likelihood for use in probabilistic inference about localisation. The effectiveness of probabilistic object localisation in image clutter, using Bayesian Localisation, is illustrated. Because it is a Monte Carlo method, it produces not simply a single estimate of object configuration, but an entire sample from the posterior distribution for the configuration. This makes sequential inference of configuration possible. Two examples are illustrated here: coarse to fine scale inference, and propagation of configuration estimates over time, in image sequences.
    Type of Medium: Electronic Resource
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