Adaptive Coupled PDE System for Image Restoration


We consider a coupled system of partial differential equations (PDEs) based model for image restoration. Both the image and the edge variables are incorporated by coupling them into two different PDEs. It is shown that the initial-boundary value problem has global in time dissipative solutions (in a sense going back to P.-L. Lions), and several properties of these solutions are established. Some numerical examples are given to highlight the denoising nature of the proposed model along with some comparison results.

Comparison with Classical Schemes

Perona and Malik [2]

Catte et al [3]

ADAP Coupled PDE [1]

Comparison with Other PDE Schemes

Image u

Edge v

Method Noise |u0-u|

Example Medical Image Denoising Result

Input u0

ADAP Coupled PDE [1] u

Edge Variable v

Lets look at the Brain image denoising in a better way

Noisy input image

Smoothed output image

Image Decomposition

Note that the scheme with diffusion based PDEs naturally leads to a u0 = u + v + w model, where u - smooth part, v - edge part and w - noise/texture part, we provide the following example to illustrate it. See related publication (TBA) below and corresponding Project page (TBA) for more details.

Smooth (u)

Edges (v)

Residue (w = u0-(u+v))

Test Images:

(from USC-SIPI Miscellaneous set and other standard test images)

If you use the images presented here kindly cite Reference [1] below.


[1] V. B. S. prasath, D. Vorotnikov. On a system of adaptive coupled PDEs for image restoration. Journal of Mathematical Imaging and Vision, 48(1):35-52, 2014. doi:10.1007/s10851-012-0386-3. Preliminary version at arXiv, December 2011. doi:10.48550/arXiv.1112.2904.

Related Talk: Global dissipative solutions for generalized forward-backward diffusion equations, at The Third Ohio River Analysis Meeting, March 8-10, 2013, University of Cincinnati, OH, USA. Slides available at figshare: 10.6084/m9.figshare.646651.


$The first author (VBSP) thanks the Fields Institute, Toronto, Canada for their great hospitality and support during the Thematic Program on Inverse Problems and Imaging where part of this work was done. The second author (DV) was partially supported by CMUC and FCT (Portugal), through European Program COMPETE/FEDER.


[2] P. Perona and J. Malik. Scale-space and edge detection using anisotropic diffusion. IEEE Transactions on Pattern Analysis and Machine Intelligence, pp. 629-639, Vol. 12, No. 7, 1990.

[3] V. Catte, P. L. Lions, J.-M. Morel, T. Coll, Image selective smoothing and edge detection by nonlinear diffusion. SIAM Journal on Numerical Analysis, pp. 182-193 Vol. 29, No. 1, 1992.

[4] M. Nitzberg, T. Shiota, Nonlinear image filtering with edge and corner enhancement. IEEE Transactions on Pattern Analysis and Machine Intelligence, pp. 826-833, Vol. 14, No. 8, 1992.

[5] Y. Chen, S. E. Levine, Image recovery via diffusion tensor and time-delay regularization. Journal of Visual Communication and Image Representation, pp. 156-175, Vol. 13, No. 1-2, 2002.

[6] A. Belahmidi, A. Chambolle, Time-delay regularization of anisotropic diffusion and image processing. Mathematical Modelling and Numerical Analysis, pp. 231-251, Vol. 39, No. 2, 2005.

[7] H. Amann, Time-delayed Perona-Malik type problems. Acta Mathematica Universitatis Comenianae, pp. 15-38, Vol. LXXVI, No. 1, 2007.

For anisotropic diffusion of more than one channel (Vectorial = Color, Multispectral & Hyperspectral), please visit this page.