210 research outputs found
A Total Fractional-Order Variation Model for Image Restoration with Non-homogeneous Boundary Conditions and its Numerical Solution
To overcome the weakness of a total variation based model for image
restoration, various high order (typically second order) regularization models
have been proposed and studied recently. In this paper we analyze and test a
fractional-order derivative based total -order variation model, which
can outperform the currently popular high order regularization models. There
exist several previous works using total -order variations for image
restoration; however first no analysis is done yet and second all tested
formulations, differing from each other, utilize the zero Dirichlet boundary
conditions which are not realistic (while non-zero boundary conditions violate
definitions of fractional-order derivatives). This paper first reviews some
results of fractional-order derivatives and then analyzes the theoretical
properties of the proposed total -order variational model rigorously.
It then develops four algorithms for solving the variational problem, one based
on the variational Split-Bregman idea and three based on direct solution of the
discretise-optimization problem. Numerical experiments show that, in terms of
restoration quality and solution efficiency, the proposed model can produce
highly competitive results, for smooth images, to two established high order
models: the mean curvature and the total generalized variation.Comment: 26 page
Image Inpainting and Enhancement using Fractional Order Variational Model
The intention of image inpainting is to complete or fill the corrupted or missing zones of an image by considering the knowledge from the source region. A novel fractional order variational image inpainting model in reference to Caputo definition is introduced in this article. First, the fractional differential, and its numerical methods are represented according to Caputo definition. Then, a fractional differential mask is represented in 8-directions. The complex diffusivity function is also defined to preserve the edges. Finally, the missing regions are filled by using variational model with fractional differentials of 8-directions. The simulation results and analysis display that the new model not only inpaints the missing regions, but also heightens the contrast of the image. The inpainted images have better visual quality than other fractional differential filters
Sobolev spaces with non-Muckenhoupt weights, fractional elliptic operators, and applications
We propose a new variational model in weighted Sobolev spaces with
non-standard weights and applications to image processing. We show that these
weights are, in general, not of Muckenhoupt type and therefore the classical
analysis tools may not apply. For special cases of the weights, the resulting
variational problem is known to be equivalent to the fractional Poisson
problem. The trace space for the weighted Sobolev space is identified to be
embedded in a weighted space. We propose a finite element scheme to solve
the Euler-Lagrange equations, and for the image denoising application we
propose an algorithm to identify the unknown weights. The approach is
illustrated on several test problems and it yields better results when compared
to the existing total variation techniques
ADI splitting schemes for a fourth-order nonlinear partial differential equation from image processing
We present directional operator splitting schemes for the numerical solution of a fourth-order, nonlinear partial differential evolution equation which arises in image processing. This equation constitutes the H−1-gradient flow of the total variation and represents a prototype of higher-order equations of similar type which are popular in imaging for denoising, deblurring and inpainting problems. The efficient numerical solution of this equation is very challenging due to the stiffness of most numerical schemes. We show that the combination of directional splitting schemes with implicit time-stepping provides a stable and computationally cheap numerical realisation of the equation
A Fractional Image Inpainting Model Using a Variant of Mumford-Shah Model
In this paper, we propose a fourth order PDE model for image inpainting based
on a variant of the famous Mumford-Shah (MS) image segmentation model.
Convexity splitting is used to discrtised the time and we replace the Laplacian
by its fractional counterpart in the time discretised scheme. Fourier spectral
method is used for space discretization. Consistency, stability and convergence
of the time discretised model has been carried out. The model is tested on some
standard test images and compared them with the result of some models existed
in the literature.Comment: 19 page
p-Laplace Variational Image Inpainting Model Using Riesz Fractional Differential Filter
In this paper, p-Laplace variational image inpainting model with symmetric Riesz fractional differential filter is proposed. Variational inpainting models are very useful to restore many smaller damaged regions of an image. Integer order variational image inpainting models (especially second and fourth order) work well to complete the unknown regions. However, in the process of inpainting with these models, any of the unindented visual effects such as staircasing, speckle noise, edge blurring, or loss in contrast are introduced. Recently, fractional derivative operators were applied by researchers to restore the damaged regions of the image. Experimentation with these operators for variational image inpainting led to the conclusion that second order symmetric Riesz fractional differential operator not only completes the damaged regions effectively, but also reducing unintended effects. In this article, The filling process of damaged regions is based on the fractional central curvature term. The proposed model is compared with integer order variational models and also GrunwaldLetnikov fractional derivative based variational inpainting in terms of peak signal to noise ratio, structural similarity and mutual information
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