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000059280 0247_ $$2DOI$$a10.1007/s11263-008-0145-5
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000059280 041__ $$aeng
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000059280 084__ $$2WoS$$aComputer Science, Artificial Intelligence
000059280 1001_ $$0P:(DE-HGF)0$$aPreusser, T.$$b0
000059280 245__ $$aBuilding blocks for computer vision with stochastic partial differential equations
000059280 260__ $$aDordrecht [u.a.]$$bSpringer Science + Business Media B.V$$c2008
000059280 300__ $$a375 - 405
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000059280 440_0 $$019561$$aInternational Journal of Computer Vision$$v80$$x0920-5691$$y3
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000059280 520__ $$aWe discuss the basic concepts of computer vision with stochastic partial differential equations (SPDEs). In typical approaches based on partial differential equations (PDEs), the end result in the best case is usually one value per pixel, the "expected" value. Error estimates or even full probability density functions PDFs are usually not available. This paper provides a framework allowing one to derive such PDFs, rendering computer vision approaches into measurements fulfilling scientific standards due to full error propagation. We identify the image data with random fields in order to model images and image sequences which carry uncertainty in their gray values, e.g. due to noise in the acquisition process.The noisy behaviors of gray values is modeled as stochastic processes which are approximated with the method of generalized polynomial chaos (Wiener-Askey-Chaos). The Wiener-Askey polynomial chaos is combined with a standard spatial approximation based upon piecewise multi-linear finite elements. We present the basic building blocks needed for computer vision and image processing in this stochastic setting, i.e. we discuss the computation of stochastic moments, projections, gradient magnitudes, edge indicators, structure tensors, etc. Finally we show applications of our framework to derive stochastic analogs of well known PDEs for de-noising and optical flow extraction. These models are discretized with the stochastic Galerkin method. Our selection of SPDE models allows us to draw connections to the classical deterministic models as well as to stochastic image processing not based on PDEs. Several examples guide the reader through the presentation and show the usefulness of the framework.
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000059280 65320 $$2Author$$aimage processing
000059280 65320 $$2Author$$aerror propagation
000059280 65320 $$2Author$$arandom fields
000059280 65320 $$2Author$$apolynomial chaos
000059280 65320 $$2Author$$astochastic partial differential equations
000059280 65320 $$2Author$$astochastic galerkin method
000059280 65320 $$2Author$$astochastic finite element method
000059280 650_7 $$2WoSType$$aJ
000059280 7001_ $$0P:(DE-Juel1)129394$$aScharr, H.$$b1$$uFZJ
000059280 7001_ $$0P:(DE-Juel1)129347$$aKrajsek, K.$$b2$$uFZJ
000059280 7001_ $$0P:(DE-HGF)0$$aKirby, R.M.$$b3
000059280 773__ $$0PERI:(DE-600)1479903-0$$a10.1007/s11263-008-0145-5$$gVol. 80, p. 375 - 405$$p375 - 405$$q80<375 - 405$$tInternational journal of computer vision$$v80$$x0920-5691$$y2008
000059280 8567_ $$uhttp://dx.doi.org/10.1007/s11263-008-0145-5
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