A Computational Approach to Edge Detection
IEEE Transactions on Pattern Analysis and Machine Intelligence
Scale-Space and Edge Detection Using Anisotropic Diffusion
IEEE Transactions on Pattern Analysis and Machine Intelligence
Ten lectures on wavelets
Image selective smoothing and edge detection by nonlinear diffusion
SIAM Journal on Numerical Analysis
Nonlinear Image Filtering with Edge and Corner Enhancement
IEEE Transactions on Pattern Analysis and Machine Intelligence
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
Geometric shock-capturing eno schemes for subpixel interpolation, computation and curve evolution
Graphical Models and Image Processing
High-Order Total Variation-Based Image Restoration
SIAM Journal on Scientific Computing
ADI schemes for higher-order nonlinear diffusion equations
Applied Numerical Mathematics
Geometric surface processing via normal maps
ACM Transactions on Graphics (TOG)
SIAM Journal on Scientific Computing
A note on the numerical solution of high-order differential equations
Journal of Computational and Applied Mathematics
Image Sharpening by Flows Based on Triple Well Potentials
Journal of Mathematical Imaging and Vision
Polynomial Fitting for Edge Detection in Irregularly Sampled Signals and Images
SIAM Journal on Numerical Analysis
A windowed Fourier pseudospectral method for hyperbolic conservation laws
Journal of Computational Physics
Image Processing And Analysis: Variational, Pde, Wavelet, And Stochastic Methods
Image Processing And Analysis: Variational, Pde, Wavelet, And Stochastic Methods
Discontinuity detection in multivariate space for stochastic simulations
Journal of Computational Physics
A Wavelet Tour of Signal Processing, Third Edition: The Sparse Way
A Wavelet Tour of Signal Processing, Third Edition: The Sparse Way
Markov Random Field Modeling in Image Analysis
Markov Random Field Modeling in Image Analysis
A novel PDE based image restoration: Convection-diffusion equation for image denoising
Journal of Computational and Applied Mathematics
Development of EMD-based denoising methods inspired by wavelet thresholding
IEEE Transactions on Signal Processing
Differential geometry based solvation model I: Eulerian formulation
Journal of Computational Physics
Iterative Filtering Decomposition Based on Local Spectral Evolution Kernel
Journal of Scientific Computing
IEEE Transactions on Image Processing
Fourth-order partial differential equations for noise removal
IEEE Transactions on Image Processing
Forward-and-backward diffusion processes for adaptive image enhancement and denoising
IEEE Transactions on Image Processing
IEEE Transactions on Image Processing
Image change detection algorithms: a systematic survey
IEEE Transactions on Image Processing
Strong-continuation, contrast-invariant inpainting with a third-order optimal PDE
IEEE Transactions on Image Processing
Multiscale geometric modeling of macromolecules I: Cartesian representation
Journal of Computational Physics
Hi-index | 0.01 |
Partial differential equation (PDE) based methods have become some of the most powerful tools for exploring the fundamental problems in signal processing, image processing, computer vision, machine vision and artificial intelligence in the past two decades. The advantages of PDE based approaches are that they can be made fully automatic, robust for the analysis of images, videos and high dimensional data. A fundamental question is whether one can use PDEs to perform all the basic tasks in the image processing. If one can devise PDEs to perform full-scale mode decomposition for signals and images, the modes thus generated would be very useful for secondary processing to meet the needs in various types of signal and image processing. Despite of great progress in PDE based image analysis in the past two decades, the basic roles of PDEs in image/signal analysis are only limited to PDE based low-pass filters, and their applications to noise removal, edge detection, segmentation, etc. At present, it is not clear how to construct PDE based methods for full-scale mode decomposition. The above-mentioned limitation of most current PDE based image/signal processing methods is addressed in the proposed work, in which we introduce a family of mode decomposition evolution equations (MoDEEs) for a vast variety of applications. The MoDEEs are constructed as an extension of a PDE based high-pass filter (Wei and Jia in Europhys. Lett. 59(6):814---819, 2002) by using arbitrarily high order PDE based low-pass filters introduced by Wei (IEEE Signal Process. Lett. 6(7):165---167, 1999). The use of arbitrarily high order PDEs is essential to the frequency localization in the mode decomposition. Similar to the wavelet transform, the present MoDEEs have a controllable time-frequency localization and allow a perfect reconstruction of the original function. Therefore, the MoDEE operation is also called a PDE transform. However, modes generated from the present approach are in the spatial or time domain and can be easily used for secondary processing. Various simplifications of the proposed MoDEEs, including a linearized version, and an algebraic version, are discussed for computational convenience. The Fourier pseudospectral method, which is unconditionally stable for linearized high order MoDEEs, is utilized in our computation. Validation is carried out to mode separation of high frequency adjacent modes. Applications are considered to signal and image denoising, image edge detection, feature extraction, enhancement etc. It is hoped that this work enhances the understanding of high order PDEs and yields robust and useful tools for image and signal analysis.