Properties of Higher Order Nonlinear Diffusion Filtering

Authors:
S. Didas;J. Weickert;B. Burgeth
Affiliations:
Fraunhofer-Institut für Techno- und Wirtschaftsmathematik, Abteilung Bildverarbeitung, Kaiserslautern, Germany 67663;Mathematical Image Analysis Group, Faculty of Mathematics and Computer Science, Saarland University, Saarbrücken, Germany 66041;Mathematical Image Analysis Group, Faculty of Mathematics and Computer Science, Saarland University, Saarbrücken, Germany 66041
Venue:
Journal of Mathematical Imaging and Vision
Year:
2009

Citing 26
Cited 10

Matrix analysis

Matrix analysis
Scale-Space and Edge Detection Using Anisotropic Diffusion

IEEE Transactions on Pattern Analysis and Machine Intelligence
Biased anisotropic diffusion: a unified regularization and diffusion approach to edge detection

Image and Vision Computing - Special issue on the first ECCV 1990
Image selective smoothing and edge detection by nonlinear diffusion

SIAM Journal on Numerical Analysis
Nonlinear total variation based noise removal algorithms

Proceedings of the eleventh annual international conference of the Center for Nonlinear Studies on Experimental mathematics : computational issues in nonlinear science: computational issues in nonlinear science
Numerical solution of partial differential equations

Numerical solution of partial differential equations
Denoising with higher order derivatives of bounded variation and an application to parameter estimation

Computing
LCIS: a boundary hierarchy for detail-preserving contrast reduction

Proceedings of the 26th annual conference on Computer graphics and interactive techniques
Relations Between Regularization and Diffusion Filtering

Journal of Mathematical Imaging and Vision
Spectral methods in MatLab

Spectral methods in MatLab
A Theoretical Framework for Convex Regularizers in PDE-Based Computation of Image Motion

International Journal of Computer Vision
High-Order Total Variation-Based Image Restoration

SIAM Journal on Scientific Computing
Minimizers of Cost-Functions Involving Nonsmooth Data-Fidelity Terms. Application to the Processing of Outliers

SIAM Journal on Numerical Analysis
Tube Methods for BV Regularization

Journal of Mathematical Imaging and Vision
Vector-Valued Image Regularization with PDEs: A Common Framework for Different Applications

IEEE Transactions on Pattern Analysis and Machine Intelligence
A Note on the Dual Treatment of Higher-Order Regularization Functionals

Computing
Iterative Image Restoration Combining Total Variation Minimization and a Second-Order Functional

International Journal of Computer Vision
Splines in Higher Order TV Regularization

International Journal of Computer Vision
Direct shape-from-shading with adaptive higher order regularisation

SSVM'07 Proceedings of the 1st international conference on Scale space and variational methods in computer vision
Stability and local feature enhancement of higher order nonlinear diffusion filtering

PR'05 Proceedings of the 27th DAGM conference on Pattern Recognition
Regularity and scale-space properties of fractional high order linear filtering

Scale-Space'05 Proceedings of the 5th international conference on Scale Space and PDE Methods in Computer Vision
Relations between higher order TV regularization and support vector regression

Scale-Space'05 Proceedings of the 5th international conference on Scale Space and PDE Methods in Computer Vision
Deterministic edge-preserving regularization in computed imaging

IEEE Transactions on Image Processing
Adaptive smoothing respecting feature directions

IEEE Transactions on Image Processing
Fourth-order partial differential equations for noise removal

IEEE Transactions on Image Processing
Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time

IEEE Transactions on Image Processing

Ramp preserving Perona-Malik model

Signal Processing
Two Enhanced Fourth Order Diffusion Models for Image Denoising

Journal of Mathematical Imaging and Vision
Image structure preserving denoising using generalized fractional time integrals

Signal Processing
Correspondence between frame shrinkage and high-order nonlinear diffusion

Applied Numerical Mathematics
Fully fractional anisotropic diffusion for image denoising

Mathematical and Computer Modelling: An International Journal
Fast communication: Image denoising using modified Perona-Malik model based on directional Laplacian

Signal Processing
On a System of Adaptive Coupled PDEs for Image Restoration

Journal of Mathematical Imaging and Vision
A fast adaptive reweighted residual-feedback iterative algorithm for fractional-order total variation regularized multiplicative noise removal of partly-textured images

Signal Processing
A coupled variational model for image denoising using a duality strategy and split Bregman

Multidimensional Systems and Signal Processing
A Combined First and Second Order Variational Approach for Image Reconstruction

Journal of Mathematical Imaging and Vision

Quantified Score

Hi-index	0.00

Visualization

Abstract

This paper provides a mathematical analysis of higher order variational methods and nonlinear diffusion filtering for image denoising. Besides the average grey value, it is shown that higher order diffusion filters preserve higher moments of the initial data. While a maximum-minimum principle in general does not hold for higher order filters, we derive stability in the 2-norm in the continuous and discrete setting. Considering the filters in terms of forward and backward diffusion, one can explain how not only the preservation, but also the enhancement of certain features in the given data is possible. Numerical results show the improved denoising capabilities of higher order filtering compared to the classical methods.