Optical Flow Estimation: An Error Analysis of Gradient-Based Methods with Local Optimization
IEEE Transactions on Pattern Analysis and Machine Intelligence
Scale-Space and Edge Detection Using Anisotropic Diffusion
IEEE Transactions on Pattern Analysis and Machine Intelligence
Stochastic finite elements: a spectral approach
Stochastic finite elements: a spectral approach
Image selective smoothing and edge detection by nonlinear diffusion
SIAM Journal on Numerical Analysis
Robust computation of optical flow in a multi-scale differential framework
International Journal of Computer Vision
Stochastic analysis
SIAM Journal on Applied Mathematics
The statistics of optical flow
Computer Vision and Image Understanding
Spatio-Temporal Image Processing: Theory and Scientific Applications
Spatio-Temporal Image Processing: Theory and Scientific Applications
Variational Optic Flow Computation with a Spatio-Temporal Smoothness Constraint
Journal of Mathematical Imaging and Vision
Computer Vision: A Modern Approach
Computer Vision: A Modern Approach
The Wiener--Askey Polynomial Chaos for Stochastic Differential Equations
SIAM Journal on Scientific Computing
A stochastic projection method for fluid flow II.: random process
Journal of Computational Physics
Combining the Advantages of Local and Global Optic Flow Methods
Proceedings of the 24th DAGM Symposium on Pattern Recognition
Modeling uncertainty in flow simulations via generalized polynomial chaos
Journal of Computational Physics
Image Statistics and Anisotropic Diffusion
ICCV '03 Proceedings of the Ninth IEEE International Conference on Computer Vision - Volume 2
Smart Nonlinear Diffusion: A Probabilistic Approach
IEEE Transactions on Pattern Analysis and Machine Intelligence
Morphological image sequence processing
Computing and Visualization in Science
Lucas/Kanade meets Horn/Schunck: combining local and global optic flow methods
International Journal of Computer Vision
Highly Accurate Optic Flow Computation with Theoretically Justified Warping
International Journal of Computer Vision
Piecewise-Smooth Dense Optical Flow via Level Sets
International Journal of Computer Vision
Galerkin Finite Element Methods for Parabolic Problems (Springer Series in Computational Mathematics)
Diffusion-Like reconstruction schemes from linear data models
DAGM'06 Proceedings of the 28th conference on Pattern Recognition
Ambrosio-tortorelli segmentation of stochastic images
ECCV'10 Proceedings of the 11th European conference on Computer vision: Part V
Fast parameter sensitivity analysis of PDE-based image processing methods
ECCV'12 Proceedings of the 12th European conference on Computer Vision - Volume Part VII
International Journal of Computer Vision
Segmentation of Stochastic Images using Level Set Propagation with Uncertain Speed
Journal of Mathematical Imaging and Vision
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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.