Digital image restoration by partial differential equations

Inpainting, art of improving the image quality, has achieved great development in recent years, reaching a high level of popularity after the restoration of images sent by the Hubble telescope, through mathematical methods and computational tools. However, the images are not always totally affected, suffering some loss of information in some areas, others simply are affected by the passage of time, so it is necessary to establish techniques to restore images in which the damage is not reversible using a filter applied to the entire image. This study established a method which allows to detect the damaged regions of the image and to perform a restoration process on the damaged regions, based on partial differential equations, thus achieving the construction of a recovered image. The method proposed in this paper could be applied to the restoration of grayscale and color images and even of artwork.La restauración digital de imágenes, definida como el arte de mejorar la calidad de las imágenes, ha logrado un amplio desarrollo en los últimos años, alcanzando un alto nivel de popularidad, desde que se utilizaron métodos matemáticos y computacionales para restaurar las imágenes distorsionadas, enviadas por el telescopio Hubble; sin embargo, las imágenes no siempre son afectadas de manera total, algunas sufren pérdida de información en algunas regiones, otras simplemente son afectadas por el paso del tiempo, por tanto, es necesario establecer técnicas que permitan restaurar imágenes, en las cuales, el daño no es reversible, mediante un filtro aplicado sobre toda la imagen. En este trabajo, se establece un método que permite detectar las regiones dañadas en la imagen, realizar un proceso de restauración, basado en ecuaciones diferenciales parciales, sobre las regiones dañadas, logrando, de este modo, construir la imagen recuperada. El método propuesto en este escrito podrá ser aplicado para la restauración de imágenes en blanco y negro, a color e, incluso, imágenes de obras de arte.Incluye referencias bibliográfica

Shock filters based on implicit cluster separation

One of the classic problems in low level vision is image restoration. An important contribution toward this effort has been the development of shock filters by Osher and Rudin (1990). It performs image deblurring using hyperbolic partial differential equations. In this paper we relate the notion of cluster separation from the field of pattern recognition to the shock filter formulation. A kind of shock filter is proposed based on the idea of gradient based separation of clusters. The proposed formulation is general enough as it can allow various models of density functions in the cluster separation process. The efficacy of the method is demonstrated through various examples

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 $\alpha$-order variation model, which can outperform the currently popular high order regularization models. There exist several previous works using total $\alpha$-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 $\alpha$-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

Segmentation and Restoration of Images on Surfaces by Parametric Active Contours with Topology Changes

In this article, a new method for segmentation and restoration of images on two-dimensional surfaces is given. Active contour models for image segmentation are extended to images on surfaces. The evolving curves on the surfaces are mathematically described using a parametric approach. For image restoration, a diffusion equation with Neumann boundary conditions is solved in a postprocessing step in the individual regions. Numerical schemes are presented which allow to efficiently compute segmentations and denoised versions of images on surfaces. Also topology changes of the evolving curves are detected and performed using a fast sub-routine. Finally, several experiments are presented where the developed methods are applied on different artificial and real images defined on different surfaces

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