During the 20th century various concepts for conservation-restoration and in particular for presentation and retouching were postulated, according to the respective phases of development of the discipline. An overview over a series of relevant aspects, methods and problems is given with the aim of facilitating definitions forming the basis of automated virtual completion of works of art.

• Thematic program: Optimal transportation structures, gradient flows and entropy methods for Applied PDEs (2007)
• Event: Mini-workshop on "PDE's and Variational Tools in Image Inpainting" (2007)

In the last 15 years the artists working process comes more and more in the influence by computer-controlled surfaces like screens, mouses, keyboards and special-designed software. Most of the software and hardware was developed from classical analog image generation tools like the use of pencils or photographic methods. This main focus often clouds new capacities in the interaction between artist and computer-aided methods in image editing and image generation. So the proposal is to research new aspects in interrelation between artist and tool using transdiscipline science methods.

• Thematic program: Optimal transportation structures, gradient flows and entropy methods for Applied PDEs (2007)
• Event: Mini-workshop on "PDE's and Variational Tools in Image Inpainting" (2007)

Second order variational inpainting methods, like total variation inpainting (cf. Rudin-Osher-Fatemi), have drawbacks as in the connection of edges over large distances or the continuous propagation of level lines into the damaged domain. In an attempt to solve both the connectivity principle and the so called staircasing effect resulting from second order image diffusions, a number of third and fourth order diffusions have been suggested for image inpainting. A new approach in the class of fourth order inpainting algorithms is inpainting of binary images using the Cahn- Hilliard equation proposed in Bertozzi-Esedoglu-Gillette. In this talk I will present some analytic and numerical results for Cahn-Hilliard inpainting. Besides this also other variations of this higher order approach for grayvalue images will be discussed.

• Thematic program: Optimal transportation structures, gradient flows and entropy methods for Applied PDEs (2007)
• Event: Mini-workshop on "PDE's and Variational Tools in Image Inpainting" (2007)

In this talk we discuss resent results on the regularity of the free boundaries in a certain type of parabolic free boundary problems. Mathematically the problem is formulated as follows. Let a function $u$ and an open set $\Omega \subset \mathbb{R}^{n+1}_+=\{(x,t): x \in \mathbb{R}^n, t \in \mathbb{R}, x_1>0\}, n \geqslant 2$ solve the following problem: $$ H(u)=\chi_{\Omega } \quad \text{in} \quad Q_1^+, \qquad u=|Du|=0 \quad \text{in} \quad Q_1^+ \setminus \Omega, \qquad u=0 \quad \text{on} \quad \Pi, $$ where $H=\Delta -\partial_t$ is the heat operator, $\chi_{\Omega }$ denotes the characteristic function of $\Omega $, $Q_1$ is the unit cylinder in $\mathbb{R}^{n+1}$, $Q_1^+=Q_1 \cap \mathbb{R}^{n+1}_+$, $\Pi =\{(x,t): x_1=0\}$, and the first equation is understood in the weak (distributional) sense.

• Thematic program: Optimal transportation structures, gradient flows and entropy methods for Applied PDEs (2007)
• Event: Mini-workshop on "PDE's and Variational Tools in Image Inpainting" (2007)

In many image acquisition processes only incomplete data is recorded for a variety of reasons, like for example defective sensors, superimposed texts or logos, or occlusions by other objects. Further, non-digitally stored image data like photographs or celluloid movies may suffer from fading, mechanical stress or simply from aging. Typical symptoms are blotches or torn out pieces, which both lead to image regions which contain no information. To restore the missing image data a multitude of inpainting algorithms has been devised in recent years. We present several inpainting algorithms, each suitable for completion of a different type of image data. Starting from the Ginzburg--Landau (GL) energy we derive an algorithm which can be used to inpaint levelsets. We show two possible applications: levelset-wise inpainting of images, and surface inpainting. Thereafter we show how to embed images into complex valued functions, such that GL inpainting can be directly applied to image functions, without the indirection using levelsets. The GL inpainting algorithm -- like most inpainting algorithms based on variational methods or PDEs -- does not handle textured images very well. Therefore we add a texture synthesis algorithm based on Markov Random Fields to supplement the missing texture information. Finally, we consider inpainting of movies. Since consecutive frames of a movie usually differ very little, one may find the image information missing in one frame in another frame nearby. To identify corresponding regions of different frames, we employ a kind of optical flow approach with a piecewise continuity constraint. If possible, the missing frame region is copied from undamaged frames, otherwise a still image inpainting is performed.

• Thematic program: Optimal transportation structures, gradient flows and entropy methods for Applied PDEs (2007)
• Event: Mini-workshop on "PDE's and Variational Tools in Image Inpainting" (2007)

A mean-field approach to modeling in economics and finance is presented. The model consists on a system of nonlinear diffusion equation related to a free-boundary value problem. It describes the idealized situation of two groups of people, buyers and vendors, trading one good; the resulting price of the good comes from a dynamical equilibrium.

• Thematic program: Optimal transportation structures, gradient flows and entropy methods for Applied PDEs (2007)
• Event: Mini-workshop on "PDE's and Variational Tools in Image Inpainting" (2007)

We present a new, fully automatic technique for wire and scratch removal that works well in both textured and non-textured areas of an image. Chan, Shen and Kang introduced a technique for inpainting using an Euler's elastica energy-based variational model that works well for repairing smooth areas of the image while maintaining edge detail. The technique is very slow, due to a stiff, 4th order PDE. Efros and Leung used texture synthesis techniques for inpainting and hole filling. This works well for areas of an image that contain repeating patterns. We have combined these two techniques to accelerate and constrain the solution of the 4th order PDE. Instead of a stiff minimization, we have a combinatorial optimization problem that is much quicker to solve.

• Thematic program: Optimal transportation structures, gradient flows and entropy methods for Applied PDEs (2007)
• Event: Mini-workshop on "PDE's and Variational Tools in Image Inpainting" (2007)

• Thematic program: Optimal transportation structures, gradient flows and entropy methods for Applied PDEs (2007)
• Event: Mini-workshop on "PDE's and Variational Tools in Image Inpainting" (2007)