The nonlinear diffusion model introduced by Perona and Malik (1990 IEEE
Trans. Pattern Anal. Mach. Intell. 12 629–39) is well suited to preserve salient
edges while restoring noisy images. This model overcomes well-known edge
smearing effects of the heat equation by using a gradient dependent diffusion
function. Despite providing better denoizing results, the analysis of the PM
scheme is difficult due to the forward-backward nature of the diffusion flow.
We study a related adaptive forward-backward diffusion equation which uses a
mollified inverse gradient term engrafted in the diffusion term of a general
nonlinear parabolic equation. We prove a series of existence, uniqueness and
regularity results for viscosity, weak and dissipative solutions for such forward-
backward diffusion flows. In particular, we introduce a novel functional
framework for wellposedness of flows of total variation type. A set of synthetic
and real image processing examples are used to illustrate the properties
and advantages of the proposed adaptive forward-backward diffusion flows.
@article{Surya:PDE-Restoration-IP-2015,
author = "V. B. S. Prasath and J. M. Urbano and D. Vorotnikov",
title = "Analysis of adaptive forward-backward diffusion flows with applications in image processing",
year = 2015,
journal = "Inverse Problems",
volume = 31,
number = 105008,
pages = "30pp",
keywords = "restoration, anisotropic diffusion",
doi = "10.1088/0266-5611/31/10/105008"
}
V. B. S. Prasath, J. M. Urbano, and D. Vorotnikov. Analysis of adaptive forward-backward diffusion flows with applications in image processing. Inverse Problems, volume 31, issue 105008, pages 30pp, 2015.
Edge preserving regularization using partial differential equation (PDE) based schemes are now widely used in image restoration. We propose an adaptive multiscale variable exponent-based anisotropic variational PDE scheme that avoids current limitations such as over smoothing and blockiness artifacts while still retaining and enhancing edge structures across scale. The innovative model automatically balances between Tikhonov and total variation (TV) regularization effects using scene content information by adopting a spatially varying edge coherence exponent term constructed from the eigenvalues of the smoothed structure tensor. The multiscale exponent model considered here leads to a novel denoising method which preserves edges and provides selective denoising without generating artifacts for both additive and multiplicative noise models. Mathematical analysis of the proposed method in variable exponent space demonstrates its robustness, and that the approach theoretically satisfies the maximum-minimum principle which guarantees that artificial edge regions are not created. Extensive experimental results on synthetic, natural and biomedical images indicate that the proposed Multiscale Tikhonov-Total Variation (MTTV) and Dynamical MTTV (D-MTTV) schemes perform better than sixteen other denoising algorithms in terms of several metrics including signal-to-noise ratio improvement and structure preservation. Promising extensions to handle multiplicative noise models and multichannel imagery are also provided.
@article{Surya:MTTV-TIP-2015,
author = "V. B. S. Prasath and D. Vorotnikov and R. Pelapur and S. Jose and G. Seetharaman and K. Palaniappan",
title = "Multiscale Tikhonov-total variation image restoration using spatially varying edge coherence exponent",
year = 2015,
journal = "IEEE Transactions on Image Processing",
volume = 24,
number = 12,
pages = "5220-5235",
keywords = "image restoration, regularization, anisotropic diffusion, total variation",
doi = "10.1109/TIP.2015.2479471",
url = "http://cell.missouri.edu/pages/MTTV/"
}
V. B. S. Prasath, D. Vorotnikov, R. Pelapur, S. Jose, G. Seetharaman, and K. Palaniappan. Multiscale Tikhonov-total variation image restoration using spatially varying edge coherence exponent. IEEE Transactions on Image Processing, volume 24, issue 12, pages 5220-5235, 2015.
Anisotropic diffusion is a key concept in digital image denoising and restoration. To improve
the anisotropic diffusion based schemes and to avoid the well-known drawbacks
such as edge blurring and ‘staircasing’ artifacts, in this paper, we consider a class of
weighted anisotropic diffusion partial differential equations (PDEs). By considering an
adaptive parameter within the usual divergence process, we retain the powerful denoising
capability of anisotropic diffusion PDE without any oscillating artifacts. A well-balanced
flow version of the proposed scheme is considered which adds an adaptive fidelity term to
the usual diffusion term. The scheme is general, in the sense that, different diffusion coefficient
functions can be utilized according to the need and imaging modality. To illustrate
the advantage of the proposed methodology, we provide some examples, which are applied
in restoring noisy synthetic and real digital images. A comparison study with other
anisotropic diffusion based schemes highlight the superiority of the proposed scheme.
@article{Surya:WeightedPDE-Restoration-NONRWA-2014,
author = "V. B. S. Prasath and D. Vorotnikov",
title = "Weighted and well-balanced anisotropic diffusion scheme for image denoising and restoration",
year = 2014,
journal = "Nonlinear Analysis: Real World Applications",
volume = 17,
pages = "33-46",
keywords = "restoration, anisotropic diffusion",
doi = "10.1016/j.nonrwa.2013.10.004"
}
V. B. S. Prasath and D. Vorotnikov. Weighted and well-balanced anisotropic diffusion scheme for image denoising and restoration. Nonlinear Analysis: Real World Applications, volume 17, pages 33-46, 2014.
Anisotropic diffusion is a key concept in digital image denoising and restoration. To improve
the anisotropic diffusion based schemes and to avoid the well-known drawbacks
such as edge blurring and ‘staircasing’ artifacts, in this paper, we consider a class of
weighted anisotropic diffusion partial differential equations (PDEs). By considering an
adaptive parameter within the usual divergence process, we retain the powerful denoising
capability of anisotropic diffusion PDE without any oscillating artifacts. A well-balanced
flow version of the proposed scheme is considered which adds an adaptive fidelity term to
the usual diffusion term. The scheme is general, in the sense that, different diffusion coefficient
functions can be utilized according to the need and imaging modality. To illustrate
the advantage of the proposed methodology, we provide some examples, which are applied
in restoring noisy synthetic and real digital images. A comparison study with other
anisotropic diffusion based schemes highlight the superiority of the proposed scheme.
@article{Surya:CoupledPDE-Restoration-JMIV-2014,
author = "V. B. S. Prasath and D. Vorotnikov",
title = "On a system of adaptive coupled PDEs for image restoration",
year = 2014,
journal = "Journal of Mathematical Imaging and Vision",
volume = 48,
number = 1,
pages = "35-52",
keywords = "restoration, anisotropic diffusion",
doi = "10.1007/s10851-012-0386-3",
url = "http://dx.doi.org/10.6084/m9.figshare.646651"
}
V. B. S. Prasath and D. Vorotnikov. On a system of adaptive coupled PDEs for image restoration. Journal of Mathematical Imaging and Vision, volume 48, issue 1, pages 35-52, 2014.