Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations
Journal of Computational Physics
Stochastic finite elements: a spectral approach
Stochastic finite elements: a spectral approach
Evolutionary fronts for topology-independent shape modeling and recovery
ECCV '94 Proceedings of the third European conference on Computer vision (vol. 1)
Robust computation of optical flow in a multi-scale differential framework
International Journal of Computer Vision
International Journal of Computer Vision
Computer Vision and Image Understanding - Special issue on empirical evaluation of computer vision algorithms
Finite Element Method for Elliptic Problems
Finite Element Method for Elliptic Problems
The Wiener--Askey Polynomial Chaos for Stochastic Differential Equations
SIAM Journal on Scientific Computing
On Advances in Statistical Modeling of Natural Images
Journal of Mathematical Imaging and Vision
Adaptive Projection Operators in Multiresolution Scientific Visualization
IEEE Transactions on Visualization and Computer Graphics
Lucas/Kanade meets Horn/Schunck: combining local and global optic flow methods
International Journal of Computer Vision
Numerical Challenges in the Use of Polynomial Chaos Representations for Stochastic Processes
SIAM Journal on Scientific Computing
Predicting shock dynamics in the presence of uncertainties
Journal of Computational Physics - Special issue: Uncertainty quantification in simulation science
Computational aspects of the stochastic finite element method
Computing and Visualization in Science
Building Blocks for Computer Vision with Stochastic Partial Differential Equations
International Journal of Computer Vision
Mumford-Shah regularizer with spatial coherence
SSVM'07 Proceedings of the 1st international conference on Scale space and variational methods in computer vision
Ambrosio-tortorelli segmentation of stochastic images
ECCV'10 Proceedings of the 11th European conference on Computer vision: Part V
On the low-rank approximation by the pivoted Cholesky decomposition
Applied Numerical Mathematics
IEEE Transactions on Image Processing
From a Non-Local Ambrosio-Tortorelli Phase Field to a Randomized Part Hierarchy Tree
Journal of Mathematical Imaging and Vision
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We discuss an extension of the Ambrosio-Tortorelli approximation of the Mumford-Shah functional for the segmentation of images with uncertain gray values resulting from measurement errors and noise. Our approach yields a reliable precision estimate for the segmentation result, and it allows us to quantify the robustness of edges in noisy images and under gray value uncertainty. We develop an ansatz space for such images by identifying gray values with random variables. The use of these stochastic images in the minimization of energies of Ambrosio-Tortorelli type leads to stochastic partial differential equations for a stochastic smoothed version of the original image and a stochastic phase field for the edge set. For the discretization of these equations we utilize the generalized polynomial chaos expansion and the generalized spectral decomposition (GSD) method. In contrast to the simple classical sampling technique, this approach allows for an efficient determination of the stochastic properties of the output image and edge set by computations on an optimally small set of random variables. Also, we use an adaptive grid approach for the spatial dimensions to further improve the performance, and we extend an edge linking method for the classical Ambrosio-Tortorelli model for use with our stochastic model. The performance of the method is demonstrated on artificial data and a data set from a digital camera as well as real medical ultrasound data. A comparison of the intrusive GSD discretization with a stochastic collocation and a Monte Carlo sampling is shown.