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

Abstract: 

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 using different choices for the variable exponents (VarEx) that estimates edge maps, and compare them to traditional variational-PDE based image processing procedures in color, multispectral smoothing, restoration, and edge detection. These experimental results indicate the potential for applicability of variable exponent vectorial diffusion flows in multiple image processing tasks compared to traditional variational-PDE models.

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.

Edge Detection Results I. Color Images

Edge detection with the time dependent exponent maps - 3 channels

To Be Added.

Edge Detection Results II. Multispectral Images

CAVE multispectral dataset 32 images - 31 channels

To Be Added.

References:

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, 2025. doi:10.48550/arXiv.25xx.abcde This part presents the image processing applications (smoothing, restoration, edge detection) of the VarEx diffusion models.

See our earlier works where grayscale image processing - restoration, feature detection - were undertaken with structure tensor eigenvalues based variable exponent p(x) maps:

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)

V. B. S. Prasath, R. Pelapur, G. Seetharaman, K. Palaniappan. Multiscale structure tensor for improved feature extraction and image regularization. IEEE Transactions on Image Processing, 28(12), 6198-6210, Dec 2019. doi:10.1109/TIP.2019.2924799 (Project)

For gray-scale image processing projects, please visit this page.