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@ARTICLE{Preusser:59280,
      author       = {Preusser, T. and Scharr, H. and Krajsek, K. and Kirby,
                      R.M.},
      title        = {{B}uilding blocks for computer vision with stochastic
                      partial differential equations},
      journal      = {International journal of computer vision},
      volume       = {80},
      issn         = {0920-5691},
      address      = {Dordrecht [u.a.]},
      publisher    = {Springer Science + Business Media B.V},
      reportid     = {PreJuSER-59280},
      pages        = {375 - 405},
      year         = {2008},
      note         = {Record converted from VDB: 12.11.2012},
      abstract     = {We discuss the basic concepts of computer vision with
                      stochastic partial differential equations (SPDEs). In
                      typical approaches based on partial differential equations
                      (PDEs), the end result in the best case is usually one value
                      per pixel, the "expected" value. Error estimates or even
                      full probability density functions PDFs are usually not
                      available. This paper provides a framework allowing one to
                      derive such PDFs, rendering computer vision approaches into
                      measurements fulfilling scientific standards due to full
                      error propagation. We identify the image data with random
                      fields in order to model images and image sequences which
                      carry uncertainty in their gray values, e.g. due to noise in
                      the acquisition process.The noisy behaviors of gray values
                      is modeled as stochastic processes which are approximated
                      with the method of generalized polynomial chaos
                      (Wiener-Askey-Chaos). The Wiener-Askey polynomial chaos is
                      combined with a standard spatial approximation based upon
                      piecewise multi-linear finite elements. We present the basic
                      building blocks needed for computer vision and image
                      processing in this stochastic setting, i.e. we discuss the
                      computation of stochastic moments, projections, gradient
                      magnitudes, edge indicators, structure tensors, etc. Finally
                      we show applications of our framework to derive stochastic
                      analogs of well known PDEs for de-noising and optical flow
                      extraction. These models are discretized with the stochastic
                      Galerkin method. Our selection of SPDE models allows us to
                      draw connections to the classical deterministic models as
                      well as to stochastic image processing not based on PDEs.
                      Several examples guide the reader through the presentation
                      and show the usefulness of the framework.},
      keywords     = {J (WoSType)},
      cin          = {ICG-3},
      ddc          = {004},
      cid          = {I:(DE-Juel1)ICG-3-20090406},
      pnm          = {Terrestrische Umwelt},
      pid          = {G:(DE-Juel1)FUEK407},
      shelfmark    = {Computer Science, Artificial Intelligence},
      typ          = {PUB:(DE-HGF)16},
      UT           = {WOS:000259370500006},
      doi          = {10.1007/s11263-008-0145-5},
      url          = {https://juser.fz-juelich.de/record/59280},
}