Time Adaptive Critical Variable Exponent Vectorial Diffusion Flows Applications in Image Processing


Variable exponent spaces have found interesting applications in real world problems. Recently, there have been considerable interest in utilizing variational and evolution problems based on variable exponents for imaging applications. One of the main class of partial differential equations (PDEs) is p(x)-Laplacian. In imaging applications the variable exponent can approach the critical value 1 and this poses unique challenges in proving existence of solutions. In this work, we develop some additional functional framework to study time-dependent parabolic variable exponent flows. Specifically, we consider bounded vectorial partial variation (BVPV) space and its variable exponent counterpart. We prove the existence of weak solutions of critical vectorial p(t,x)-Laplacian flow in variable exponent BVPV space via an abstract nonlinear Cauchy problem. For non-time dependent variable exponent based critical vectorial p(x)-Laplacian flow we obtain semigroup solution. We provide detailed experimental results on color image restoration using various example for the variable exponents and compare them traditional PDE based image processing procedures. Our results indicate the applicability of variable exponent Laplacian flows in image processing in general and image restoration in particular.

Image Restoration Results I. Color Images

USC-SIPI Miscellaneous color dataset - 3 channels

Noisy USC-SIPI Miscellaneous color test images

pc computed using the channel-wise multiscale eigenevalues exponent map

pc-PDE result with channel-wise multiscale eigenvalues exponent map at iteration T = 100

pm computed using the multichannel multiscale eigenevalues exponent map

pm-PDE result with multichannel multiscale eigenvalues exponent map at iteration T = 100

Image Restoration Results II. Multispectral Images

CAVE multispectral dataset 32 images - 31 channels

To Be Added.

Comparison with other vectorial diffusion schemes

To Be Added.

Other Applications: I. Edge detection

Edge detection with the time dependent exponent maps - 3 channels

To Be Added.

Other Applications: III. Image fusion

EPFL RGB-NIR scene dataset of 477 images in 9 categories

RGB + Near-infrared (NIR) - 4 channels

To Be Added.


  1. V. B. S. Prasath, D. Vorotnikov. On time adaptive critical variable exponent vectorial diffusion flows and their applications in image processing I. Analysis. Nonlinear Analysis, 168, pp. 176-197, Mar 2018. doi:10.1016/j.na.2017.11.013. Preliminary version at arXiv, March 2016. doi:10.48550/arXiv.1603.06337. This part explains the theoretical analysis of the diffusion models.

  2. V. B. S. Prasath, D. Vorotnikov. On time adaptive critical variable exponent vectorial diffusion flows and their applications in image processing II. Experiments. In Preparation, 2022. doi:10.48550/arXiv.22xx.12345 This part presents the image processing applications of the diffusion models.

See also an earlier work where grayscale image processing was undertaken with structure tensor eigenvalues based p(x) map:

V. B. S. Prasath, D. Vorotnikov, R. Pelapur, Shani Jose, G. Seetharaman, K. Palaniappan. Multiscale Tikhonov-total variation image restoration using spatially varying edge coherence exponent. IEEE Transactions on Image Processing, 24(12), 5220-5235, Dec 2015. doi:10.1109/TIP.2015.2479471 (Project)

For anisotropic diffusion of mono channel (gray-scale images), please visit this project page.