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)
Zippered polygon meshes from range images
SIGGRAPH '94 Proceedings of the 21st annual conference on Computer graphics and interactive techniques
International Journal of Computer Vision
Nonlinear preconditioning for diffuse interfaces
Journal of Computational Physics
The Wiener--Askey Polynomial Chaos for Stochastic Differential Equations
SIAM Journal on Scientific Computing
Numerical Challenges in the Use of Polynomial Chaos Representations for Stochastic Processes
SIAM Journal on Scientific Computing
Level Set Evolution without Re-Initialization: A New Variational Formulation
CVPR '05 Proceedings of the 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'05) - Volume 1 - Volume 01
Predicting shock dynamics in the presence of uncertainties
Journal of Computational Physics - Special issue: Uncertainty quantification in simulation science
Sharp interface tracking using the phase-field equation
Journal of Computational Physics
Journal of Computational Physics
Building Blocks for Computer Vision with Stochastic Partial Differential Equations
International Journal of Computer Vision
Intrusive Galerkin methods with upwinding for uncertain nonlinear hyperbolic systems
Journal of Computational Physics
Ambrosio-tortorelli segmentation of stochastic images
ECCV'10 Proceedings of the 11th European conference on Computer vision: Part V
Tensor-Structured Galerkin Approximation of Parametric and Stochastic Elliptic PDEs
SIAM Journal on Scientific Computing
A probabilistic multi-phase model for variational image segmentation
DAGM'06 Proceedings of the 28th conference on Pattern Recognition
On the low-rank approximation by the pivoted Cholesky decomposition
Applied Numerical Mathematics
IEEE Transactions on Image Processing
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We present an approach for the evolution of level sets under an uncertain velocity leading to stochastic level sets. The uncertain velocity can either be a random variable or a random field, i.e. a spatially varying random quantity, and it may result from measurement errors, noise, unknown material parameters or other sources of uncertainty. The use of stochastic level sets for the segmentation of images with uncertain gray values leads to stochastic domains, because the zero level set is not a single closed curve anymore. Instead, we have a band of possibly infinite thickness which contains all possible locations of the zero level set under the uncertainty. Thus, the approach allows for a probabilistic description of the segmented volume and the shape of the object. Due to numerical reasons, we use a parabolic approximation of the stochastic level set equation, which is a stochastic partial differential equation, and discretized the equation using the polynomial chaos and a stochastic finite difference scheme. For the verification of the intrusive discretization in the polynomial chaos we performed Monte Carlo and Stochastic Collocation simulations. We demonstrate the power of the stochastic level set approach by showing examples ranging from artificial tests to demonstrate individual aspects to a segmentation of objects in medical images.