Mathematical Approaches to Digital Image Inpainting

Image inpainting process is used to restore the damaged image or missing parts of an image. This technique is used in some applications, such as removal of text in images and photo restoration. There are different types of methods used in image inpainting, such as non-inear partial differential equations(PDEs), wavelet transformation and framelet transformation. We studied the usage of the current image inpainting methods and solved the Poisson equation using a five-point stencil method. We used a modified five-point stencil method to solve the same equation. It gave better results than the standard five-point stencil method. Using modified five-point stencil method results as the initial condition, we solved the iterative linear and non-linear diffusion PDE. We considered different types of diffusion conductivity and compared their results. When compared with PSNR values, the iterative linear diffusion PDE method gave the best results where as constant diffusion conductivity PDE gave the worst result. Furthermore, inverse diffusion conductivity PDE had given better results than that of the constant diffusion PDE. However, it was worse than the Gaussian and Lorentz diffusion conductivity PDE. Gaussian and Lorentz diffusion conductivity iterative linear PDE had given a better result for image inpainting. When we use any inpainting technique, we cannot restore the original image. We studied the relationship between the error of the image inpainting and the inpainted domain. Error is proportional to the value of the Greens function. There are two types of methods to find the Greens function. The first method is solving a Poisson equation for a different shape of domain, such as a circle, ellipse, triangle and rectangle. If the inpainting domain has a different shape, then it is difficult to find the error. We used the conformal mapping method to find the error. We also developed a formula for transformation from any polygon to the unit circle. Moreover, we applied the Schwarz Christoffel transformation to transform from the upper half plane to any polygon

Optimising Spatial and Tonal Data for PDE-based Inpainting

Some recent methods for lossy signal and image compression store only a few selected pixels and fill in the missing structures by inpainting with a partial differential equation (PDE). Suitable operators include the Laplacian, the biharmonic operator, and edge-enhancing anisotropic diffusion (EED). The quality of such approaches depends substantially on the selection of the data that is kept. Optimising this data in the domain and codomain gives rise to challenging mathematical problems that shall be addressed in our work. In the 1D case, we prove results that provide insights into the difficulty of this problem, and we give evidence that a splitting into spatial and tonal (i.e. function value) optimisation does hardly deteriorate the results. In the 2D setting, we present generic algorithms that achieve a high reconstruction quality even if the specified data is very sparse. To optimise the spatial data, we use a probabilistic sparsification, followed by a nonlocal pixel exchange that avoids getting trapped in bad local optima. After this spatial optimisation we perform a tonal optimisation that modifies the function values in order to reduce the global reconstruction error. For homogeneous diffusion inpainting, this comes down to a least squares problem for which we prove that it has a unique solution. We demonstrate that it can be found efficiently with a gradient descent approach that is accelerated with fast explicit diffusion (FED) cycles. Our framework allows to specify the desired density of the inpainting mask a priori. Moreover, is more generic than other data optimisation approaches for the sparse inpainting problem, since it can also be extended to nonlinear inpainting operators such as EED. This is exploited to achieve reconstructions with state-of-the-art quality. We also give an extensive literature survey on PDE-based image compression methods

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

Sobolev gradients and image interpolation

We present here a new image inpainting algorithm based on the Sobolev gradient method in conjunction with the Navier-Stokes model. The original model of Bertalmio et al is reformulated as a variational principle based on the minimization of a well chosen functional by a steepest descent method. This provides an alternative of the direct solving of a high-order partial differential equation and, consequently, allows to avoid complicated numerical schemes (min-mod limiters or anisotropic diffusion). We theoretically analyze our algorithm in an infinite dimensional setting using an evolution equation and obtain global existence and uniqueness results as well as the existence of an $\omega$-limit. Using a finite difference implementation, we demonstrate using various examples that the Sobolev gradient flow, due to its smoothing and preconditioning properties, is an effective tool for use in the image inpainting problem

The final publication is available at link.springer.com. An Optimal Control Approach to Find Sparse Data for Laplace Interpolation

Abstract. Finding optimal data for inpainting is a key problem in the context of partial differential equation-based image compression. We present a new model for optimising the data used for the reconstruction by the underlying homogeneous diffusion process. Our approach is based on an optimal control framework with a strictly convex cost functional containing an L1 term to enforce sparsity of the data and non-convex constraints. We propose a numerical approach that solves a series of convex optimisation problems with linear constraints. Our numerical examples show that it outperforms existing methods with respect to quality and computation time