# Fourth-order partial differential equations for noise removal

@article{You2000FourthorderPD, title={Fourth-order partial differential equations for noise removal}, author={Yu-Li You and Mostafa Kaveh}, journal={IEEE transactions on image processing : a publication of the IEEE Signal Processing Society}, year={2000}, volume={9 10}, pages={ 1723-30 } }

A class of fourth-order partial differential equations (PDEs) are proposed to optimize the trade-off between noise removal and edge preservation. The time evolution of these PDEs seeks to minimize a cost functional which is an increasing function of the absolute value of the Laplacian of the image intensity function. Since the Laplacian of an image at a pixel is zero if the image is planar in its neighborhood, these PDEs attempt to remove noise and preserve edges by approximating an observed…

## 809 Citations

### Fourth-order anisotropic diffusion equations for noise removal

- MathematicsProceedings 7th International Conference on Signal Processing, 2004. Proceedings. ICSP '04. 2004.
- 2004

A class of fourth-order PDEs are proposed to optimize the trade-of between noise removal and edge preservation. The time evolution of these PDEs seeks to minimize a cost functional which is an…

### Noise Removal Using Edge-Preserving Fourth-Order Partial Differential Equations

- Mathematics2009 2nd International Congress on Image and Signal Processing
- 2009

Over the last decade, partial differential equations (PDEs) have been justified as effective tools for image smoothing; they are able to achieve a good trade-off between noise removal and…

### An Anisotropic Fourth-Order Partial Differential Equation for Noise Removal

- EngineeringSSVM
- 2009

The proposed fourth-order nonlinear diffusion filter, which has an anisotropic behavior on the image features, produces a noticeable improvement in the quality of denoised images evaluated subjectively and quantitatively as well as a substantial increment of the convergence rate comparing to the classical filter.

### A fourth-order Partial Differential Equation model for multiplicative noise removal in images

- Mathematics2013 International Conference on Emerging Trends in Communication, Control, Signal Processing and Computing Applications (C2SPCA)
- 2013

In coherent imaging, the sensed images are usually corrupted with multiplicative data dependent noise. Unlike additive noise, the presence of multiplicative noise destroys the information content in…

### P(x) Harmonic Surface and Its Projection for Noise Removal

- Computer Science2009 International Conference on Information Engineering and Computer Science
- 2009

A novel PDE is proposed based on minimal surface and p(x) harmonic maps that behaves like TV model at edge regions while like heat transfer equation within homogeneous regions and combines the advantages of TV model and Gaussian process.

### An efficient feature-preserving PDE algorithm for image denoising based on a spatial-fractional anisotropic diffusion equation

- MathematicsArXiv
- 2021

An efficient feature-preserving fractional PDE algorithm is proposed for image denoising based on a nonlinear spatial-fractional anisotropic diffusion equation that keeps a remarkable balance between noise removal and feature preserving, and has an extremely high structural retention property.

### Fractional-Order Anisotropic Diffusion for Image Denoising

- MathematicsIEEE Transactions on Image Processing
- 2007

A new class of fractional-order anisotropic diffusion equations for noise removal are introduced which are Euler-Lagrange equations of a cost functional which is an increasing function of the absolute value of the fractional derivative of the image intensity function.

### Trainable fourth-order partial differential equations for image noise removal

- Computer Science
- 2021

This article aims to tackle the problem by introducing a fourth-order equation with flexible and trainable coefficients, and with the help of an optimal control problem, the coefficients are determined; therefore the proposed model adapts itself to each particular application.

### Detail-Preserving Fourth-Order Nonlinear PDE-Based Image Restoration Framework

- Mathematics
- 2020

This approach outperforms not only the classic 2D image filters that often generate the undesired blurring effect, but also some nonlinear second order partial differential equation based smoothing schemes that produce the blocky effect.

### A fourth-order partial differential equations method of noise removal

- Computer Science2011 4th International Congress on Image and Signal Processing
- 2011

A new method for image denoising based on a fourth-order partial differential equation (PDE) model and a mean curvature diffusion (MCD) model that can preserve fine structure and keep texture well while quickly removing image's noise.

## References

SHOWING 1-10 OF 15 REFERENCES

### Image enhancement using fourth order partial differential equations

- MathematicsConference Record of Thirty-Second Asilomar Conference on Signals, Systems and Computers (Cat. No.98CH36284)
- 1998

A class of fourth order partial differential equations (PDE) are proposed to optimize the trade-off between noise removal and edge preservation. The time evolution of these PDEs seeks to minimize a…

### Behavioral analysis of anisotropic diffusion in image processing

- MathematicsIEEE Trans. Image Process.
- 1996

It is demonstrated that an anisotrop diffusion is well posed when there exists a unique global minimum for the energy functional and that the ill posedness of a certain anisotropic diffusion is caused by the fact that its energy functional has an infinite number of global minima that are dense in the image space.

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

- Computer ScienceSIGGRAPH '99
- 1999

This work builds a similar hierarchy using multiple instances of a new low curvature image simplifier (LCIS), a partial differential equation inspired by anisotropic diffusion, and constructs a high detail, low contrast display image by compressing only the large features, then adding back all small details.

### Scale-Space and Edge Detection Using Anisotropic Diffusion

- Computer ScienceIEEE Trans. Pattern Anal. Mach. Intell.
- 1990

A new definition of scale-space is suggested, and a class of algorithms used to realize a diffusion process is introduced, chosen to vary spatially in such a way as to encourage intra Region smoothing rather than interregion smoothing.

### Iterative Methods for Total Variation Denoising

- Computer ScienceSIAM J. Sci. Comput.
- 1996

A fixed point algorithm for minimizing a TV penalized least squares functional is presented and compared with existing minimization schemes, and a variant of the cell-centered finite difference multigrid method of Ewing and Shen is implemented for solving the (large, sparse) linear subproblems.

### Image selective smoothing and edge detection by nonlinear diffusion. II

- Mathematics
- 1992

A new version of the Perona and Malik theory for edge detection and image restoration is proposed. This new version keeps all the improvements of the original model and avoids its drawbacks: it is…

### Curvature and the evolution of fronts

- Mathematics
- 1985

The evolution of a front propagating along its normal vector field with speedF dependent on curvatureK is considered. The change in total variation of the propagating front is shown to depend only…

### THE MATHEMATICS OF MOVING CONTACT LINES IN THIN LIQUID FILMS

- Physics
- 1998

Thin Films and Moving Contact Lines The motion of a liquid under the influence of surface tension is a phenomenon we experience every day when we take a shower, drink a cup of coffee, or turn on the…