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

Optimising Different Feature Types for Inpainting-based Image Representations

Inpainting-based image compression is a promising alternative to classical transform-based lossy codecs. Typically it stores a carefully selected subset of all pixel locations and their colour values. In the decoding phase the missing information is reconstructed by an inpainting process such as homogeneous diffusion inpainting. Optimising the stored data is the key for achieving good performance. A few heuristic approaches also advocate alternative feature types such as derivative data and construct dedicated inpainting concepts. However, one still lacks a general approach that allows to optimise and inpaint the data simultaneously w.r.t. a collection of different feature types, their locations, and their values. Our paper closes this gap. We introduce a generalised inpainting process that can handle arbitrary features which can be expressed as linear equality constraints. This includes e.g. colour values and derivatives of any order. We propose a fully automatic algorithm that aims at finding the optimal features from a given collection as well as their locations and their function values within a specified total feature density. Its performance is demonstrated with a novel set of features that also includes local averages. Our experiments show that it clearly outperforms the popular inpainting with optimised colour data with the same density

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

Diffusion-based inpainting for coding remote-sensing data

Inpainting techniques based on partial differential equations (PDEs) such as diffusion processes are gaining growing importance as a novel family of image compression methods. Nevertheless, the application of inpainting in the field of hyperspectral imagery has been mainly focused on filling in missing information or dead pixels due to sensor failures. In this paper we propose a novel PDE-based inpainting algorithm to compress hyperspectral images. The method inpaints separately the known data in the spatial and in the spectral dimensions. Then it applies a prediction model to the final inpainting solution to obtain a representation much closer to the original image. Experimental results over a set of hyperspectral images indicate that the proposed algorithm can perform better than a recent proposed extension to prediction-based standard CCSDS-123.0 at low bitrate, better than JPEG 2000 Part 2 with the DWT 9/7 as a spectral transform at all bit-rates, and competitive to JPEG 2000 with principal component analysis (PCA), the optimal spectral decorrelation transform for Gaussian sources

Anisotropic osmosis filtering for shadow removal in images

We present an anisotropic extension of the isotropic osmosis model that has been introduced by Weickert et al.~(Weickert, 2013) for visual computing applications, and we adapt it specifically to shadow removal applications. We show that in the integrable setting, linear anisotropic osmosis minimises an energy that involves a suitable quadratic form which models local directional structures. In our shadow removal applications we estimate the local structure via a modified tensor voting approach (Moreno, 2012) and use this information within an anisotropic diffusion inpainting that resembles edge-enhancing anisotropic diffusion inpainting (Weickert, 2006, Gali\'c, 2008). Our numerical scheme combines the nonnegativity preserving stencil of Fehrenbach and Mirebeau (Fehrenbach, 2014) with an exact time stepping based on highly accurate polynomial approximations of the matrix exponential. The resulting anisotropic model is tested on several synthetic and natural images corrupted by constant shadows. We show that it outperforms isotropic osmosis, since it does not suffer from blurring artefacts at the shadow boundaries

Connecting mathematical models for image processing and neural networks

This thesis deals with the connections between mathematical models for image processing and deep learning. While data-driven deep learning models such as neural networks are flexible and well performing, they are often used as a black box. This makes it hard to provide theoretical model guarantees and scientific insights. On the other hand, more traditional, model-driven approaches such as diffusion, wavelet shrinkage, and variational models offer a rich set of mathematical foundations. Our goal is to transfer these foundations to neural networks. To this end, we pursue three strategies. First, we design trainable variants of traditional models and reduce their parameter set after training to obtain transparent and adaptive models. Moreover, we investigate the architectural design of numerical solvers for partial differential equations and translate them into building blocks of popular neural network architectures. This yields criteria for stable networks and inspires novel design concepts. Lastly, we present novel hybrid models for inpainting that rely on our theoretical findings. These strategies provide three ways for combining the best of the two worlds of model- and data-driven approaches. Our work contributes to the overarching goal of closing the gap between these worlds that still exists in performance and understanding.Gegenstand dieser Arbeit sind die Zusammenhänge zwischen mathematischen Modellen zur Bildverarbeitung und Deep Learning. Während datengetriebene Modelle des Deep Learning wie z.B. neuronale Netze flexibel sind und gute Ergebnisse liefern, werden sie oft als Black Box eingesetzt. Das macht es schwierig, theoretische Modellgarantien zu liefern und wissenschaftliche Erkenntnisse zu gewinnen. Im Gegensatz dazu bieten traditionellere, modellgetriebene Ansätze wie Diffusion, Wavelet Shrinkage und Variationsansätze eine Fülle von mathematischen Grundlagen. Unser Ziel ist es, diese auf neuronale Netze zu übertragen. Zu diesem Zweck verfolgen wir drei Strategien. Zunächst entwerfen wir trainierbare Varianten von traditionellen Modellen und reduzieren ihren Parametersatz, um transparente und adaptive Modelle zu erhalten. Außerdem untersuchen wir die Architekturen von numerischen Lösern für partielle Differentialgleichungen und übersetzen sie in Bausteine von populären neuronalen Netzwerken. Daraus ergeben sich Kriterien für stabile Netzwerke und neue Designkonzepte. Schließlich präsentieren wir neuartige hybride Modelle für Inpainting, die auf unseren theoretischen Erkenntnissen beruhen. Diese Strategien bieten drei Möglichkeiten, das Beste aus den beiden Welten der modell- und datengetriebenen Ansätzen zu vereinen. Diese Arbeit liefert einen Beitrag zum übergeordneten Ziel, die Lücke zwischen den zwei Welten zu schließen, die noch in Bezug auf Leistung und Modellverständnis besteht.ERC Advanced Grant INCOVI