A Computational Approach to Edge Detection
IEEE Transactions on Pattern Analysis and Machine Intelligence
Stochastic finite elements: a spectral approach
Stochastic finite elements: a spectral approach
The nature of statistical learning theory
The nature of statistical learning theory
The Wiener--Askey Polynomial Chaos for Stochastic Differential Equations
SIAM Journal on Scientific Computing
Uncertainty propagation using Wiener-Haar expansions
Journal of Computational Physics
Multi-resolution analysis of wiener-type uncertainty propagation schemes
Journal of Computational Physics
Polynomial Fitting for Edge Detection in Irregularly Sampled Signals and Images
SIAM Journal on Numerical Analysis
High-Order Collocation Methods for Differential Equations with Random Inputs
SIAM Journal on Scientific Computing
Multi-Element Generalized Polynomial Chaos for Arbitrary Probability Measures
SIAM Journal on Scientific Computing
Adaptive Edge Detectors for Piecewise Smooth Data Based on the minmod Limiter
Journal of Scientific Computing
Sparse grid collocation schemes for stochastic natural convection problems
Journal of Computational Physics
Determining the locations and discontinuities in the derivatives of functions
Applied Numerical Mathematics
Characterization of discontinuities in high-dimensional stochastic problems on adaptive sparse grids
Journal of Computational Physics
Iterative Filtering Decomposition Based on Local Spectral Evolution Kernel
Journal of Scientific Computing
Mode Decomposition Evolution Equations
Journal of Scientific Computing
Journal of Computational Physics
Edge Detection by Adaptive Splitting II. The Three-Dimensional Case
Journal of Scientific Computing
Uncertainty Quantification given Discontinuous Model Response and a Limited Number of Model Runs
SIAM Journal on Scientific Computing
Journal of Computational Physics
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Edge detection has traditionally been associated with detecting physical space jump discontinuities in one dimension, e.g. seismic signals, and two dimensions, e.g. digital images. Hence most of the research on edge detection algorithms is restricted to these contexts. High dimension edge detection can be of significant importance, however. For instance, stochastic variants of classical differential equations not only have variables in space/time dimensions, but additional dimensions are often introduced to the problem by the nature of the random inputs. The stochastic solutions to such problems sometimes contain discontinuities in the corresponding random space and a prior knowledge of jump locations can be very helpful in increasing the accuracy of the final solution. Traditional edge detection methods typically require uniform grid point distribution. They also often involve the computation of gradients and/or Laplacians, which can become very complicated to compute as the number of dimensions increases. The polynomial annihilation edge detection method, on the other hand, is more flexible in terms of its geometric specifications and is furthermore relatively easy to apply. This paper discusses the numerical implementation of the polynomial annihilation edge detection method to high dimensional functions that arise when solving stochastic partial differential equations.