Trends in Mathematical Imaging and Surface Processing

Motivated both by industrial applications and the challenge of new problems, one observes an increasing interest in the field of image and surface processing over the last years. It has become clear that even though the applications areas differ significantly the methodological overlap is enormous. Even if contributions to the field come from almost any discipline in mathematics, a major role is played by partial differential equations and in particular by geometric and variational modeling and by their numerical counterparts. The aim of the workshop was to gather a group of leading experts coming from mathematics, engineering and computer graphics to cover the main developments

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

Building Scene Models by Completing and Hallucinating Depth and Semantics

Building 3D scene models has been a longstanding goal of computer vision. The great progress in depth sensors brings us one step closer to achieving this in a single shot. However, depth sensors still produce imperfect measurements that are sparse and contain holes. While depth completion aims at tackling this issue, it ignores the fact that some regions of the scene are occluded by the foreground objects. Building a scene model would therefore require to hallucinate the depth behind these objects. In contrast with existing methods that either rely on manual input, or focus on the indoor scenario, we introduce a fully-automatic method to jointly complete and hallucinate depth and semantics in challenging outdoor scenes. To this end, we develop a two-layer model representing both the visible information and the hidden one. At the heart of our approach lies a formulation based on the Mumford-Shah functional, for which we derive an effective optimization strategy. Our experiments evidence that our approach can accurately fill the large holes in the input depth maps, segment the different kinds of objects in the scene, and hallucinate the depth and semantics behind the foreground objects

Extended patch prioritization for depth filling within constrained exemplar-based RGB-D image completion.

We address the problem of hole filling in depth images, obtained from either active or stereo sensing, for the purposes of depth image completion in an exemplar-based framework. Most existing exemplar-based inpainting techniques, designed for color image completion, do not perform well on depth information with object boundaries obstructed or surrounded by missing regions. In the proposed method, using both color (RGB) and depth (D) information available from a common-place RGB-D image, we explicitly modify the patch prioritization term utilized for target patch ordering to facilitate improved propagation of complex texture and linear structures within depth completion. Furthermore, the query space in the source region is constrained to increase the efficiency of the approach compared to other exemplar-driven methods. Evaluations demonstrate the efficacy of the proposed method compared to other contemporary completion techniques

Patch-based methods for variational image processing problems

Image Processing problems are notoriously difficult. To name a few of these difficulties, they are usually ill-posed, involve a huge number of unknowns (from one to several per pixel!), and images cannot be considered as the linear superposition of a few physical sources as they contain many different scales and non-linearities. However, if one considers instead of images as a whole small blocks (or patches) inside the pictures, many of these hurdles vanish and problems become much easier to solve, at the cost of increasing again the dimensionality of the data to process. Following the seminal NL-means algorithm in 2005-2006, methods that consider only the visual correlation between patches and ignore their spatial relationship are called non-local methods. While powerful, it is an arduous task to define non-local methods without using heuristic formulations or complex mathematical frameworks. On the other hand, another powerful property has brought global image processing algorithms one step further: it is the sparsity of images in well chosen representation basis. However, this property is difficult to embed naturally in non-local methods, yielding algorithms that are usually inefficient or circonvoluted. In this thesis, we explore alternative approaches to non-locality, with the goals of i) developing universal approaches that can handle local and non-local constraints and ii) leveraging the qualities of both non-locality and sparsity. For the first point, we will see that embedding the patches of an image into a graph-based framework can yield a simple algorithm that can switch from local to non-local diffusion, which we will apply to the problem of large area image inpainting. For the second point, we will first study a fast patch preselection process that is able to group patches according to their visual content. This preselection operator will then serve as input to a social sparsity enforcing operator that will create sparse groups of jointly sparse patches, thus exploiting all the redundancies present in the data, in a simple mathematical framework. Finally, we will study the problem of reconstructing plausible patches from a few binarized measurements. We will show that this task can be achieved in the case of popular binarized image keypoints descriptors, thus demonstrating a potential privacy issue in mobile visual recognition applications, but also opening a promising way to the design and the construction of a new generation of smart cameras

Deep spatial and tonal data optimisation for homogeneous diffusion inpainting

Difusion-based inpainting can reconstruct missing image areas with high quality from sparse data, provided that their location and their values are well optimised. This is particularly useful for applications such as image compression, where the original image is known. Selecting the known data constitutes a challenging optimisation problem, that has so far been only investigated with model-based approaches. So far, these methods require a choice between either high quality or high speed since qualitatively convincing algorithms rely on many time-consuming inpaintings. We propose the frst neural network architecture that allows fast optimisation of pixel positions and pixel values for homogeneous difusion inpainting. During training, we combine two optimisation networks with a neural network-based surrogate solver for difusion inpainting. This novel concept allows us to perform backpropagation based on inpainting results that approximate the solution of the inpainting equation. Without the need for a single inpainting during test time, our deep optimisation accelerates data selection by more than four orders of magnitude compared to common model-based approaches. This provides real-time performance with high quality results