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

Theoretical Foundation of the Weighted Laplace Inpainting Problem

Laplace interpolation is a popular approach in image inpainting using partial differential equations. The classic approach considers the Laplace equation with mixed boundary conditions. Recently a more general formulation has been proposed where the differential operator consists of a point-wise convex combination of the Laplacian and the known image data. We provide the first detailed analysis on existence and uniqueness of solutions for the arising mixed boundary value problem. Our approach considers the corresponding weak formulation and aims at using the Theorem of Lax-Milgram to assert the existence of a solution. To this end we have to resort to weighted Sobolev spaces. Our analysis shows that solutions do not exist unconditionally. The weights need some regularity and fulfil certain growth conditions. The results from this work complement findings which were previously only available for a discrete setup.Comment: 16 pages, 2 Figure

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

Image Restoration using Automatic Damaged Regions Detection and Machine Learning-Based Inpainting Technique

In this dissertation we propose two novel image restoration schemes. The first pertains to automatic detection of damaged regions in old photographs and digital images of cracked paintings. In cases when inpainting mask generation cannot be completely automatic, our detection algorithm facilitates precise mask creation, particularly useful for images containing damage that is tedious to annotate or difficult to geometrically define. The main contribution of this dissertation is the development and utilization of a new inpainting technique, region hiding, to repair a single image by training a convolutional neural network on various transformations of that image. Region hiding is also effective in object removal tasks. Lastly, we present a segmentation system for distinguishing glands, stroma, and cells in slide images, in addition to current results, as one component of an ongoing project to aid in colon cancer prognostication

3D exemplar-based image inpainting in electron microscopy

In electron microscopy (EM) a common problem is the non-availability of data, which causes artefacts in reconstructions. In this thesis the goal is to generate artificial data where missing in EM by using exemplar-based inpainting (EBI). We implement an accelerated 3D version tailored to applications in EM, which reduces reconstruction times from days to minutes. We develop intelligent sampling strategies to find optimal data as input for reconstruction methods. Further, we investigate approaches to reduce electron dose and acquisition time. Sparse sampling followed by inpainting is the most promising approach. As common evaluation measures may lead to misinterpretation of results in EM and falsify a subsequent analysis, we propose to use application driven metrics and demonstrate this in a segmentation task. A further application of our technique is the artificial generation of projections in tiltbased EM. EBI is used to generate missing projections, such that the full angular range is covered. Subsequent reconstructions are significantly enhanced in terms of resolution, which facilitates further analysis of samples. In conclusion, EBI proves promising when used as an additional data generation step to tackle the non-availability of data in EM, which is evaluated in selected applications. Enhancing adaptive sampling methods and refining EBI, especially considering the mutual influence, promotes higher throughput in EM using less electron dose while not lessening quality.Ein häufig vorkommendes Problem in der Elektronenmikroskopie (EM) ist die Nichtverfügbarkeit von Daten, was zu Artefakten in Rekonstruktionen führt. In dieser Arbeit ist es das Ziel fehlende Daten in der EM künstlich zu erzeugen, was durch Exemplar-basiertes Inpainting (EBI) realisiert wird. Wir implementieren eine auf EM zugeschnittene beschleunigte 3D Version, welche es ermöglicht, Rekonstruktionszeiten von Tagen auf Minuten zu reduzieren. Wir entwickeln intelligente Abtaststrategien, um optimale Datenpunkte für die Rekonstruktion zu erhalten. Ansätze zur Reduzierung von Elektronendosis und Aufnahmezeit werden untersucht. Unterabtastung gefolgt von Inpainting führt zu den besten Resultaten. Evaluationsmaße zur Beurteilung der Rekonstruktionsqualität helfen in der EM oft nicht und können zu falschen Schlüssen führen, weswegen anwendungsbasierte Metriken die bessere Wahl darstellen. Dies demonstrieren wir anhand eines Beispiels. Die künstliche Erzeugung von Projektionen in der neigungsbasierten Elektronentomographie ist eine weitere Anwendung. EBI wird verwendet um fehlende Projektionen zu generieren. Daraus resultierende Rekonstruktionen weisen eine deutlich erhöhte Auflösung auf. EBI ist ein vielversprechender Ansatz, um nicht verfügbare Daten in der EM zu generieren. Dies wird auf Basis verschiedener Anwendungen gezeigt und evaluiert. Adaptive Aufnahmestrategien und EBI können also zu einem höheren Durchsatz in der EM führen, ohne die Bildqualität merklich zu verschlechtern

Understanding and advancing PDE-based image compression

This thesis is dedicated to image compression with partial differential equations (PDEs). PDE-based codecs store only a small amount of image points and propagate their information into the unknown image areas during the decompression step. For certain classes of images, PDE-based compression can already outperform the current quasi-standard, JPEG2000. However, the reasons for this success are not yet fully understood, and PDE-based compression is still in a proof-of-concept stage. With a probabilistic justification for anisotropic diffusion, we contribute to a deeper insight into design principles for PDE-based codecs. Moreover, by analysing the interaction between efficient storage methods and image reconstruction with diffusion, we can rank PDEs according to their practical value in compression. Based on these observations, we advance PDE-based compression towards practical viability: First, we present a new hybrid codec that combines PDE- and patch-based interpolation to deal with highly textured images. Furthermore, a new video player demonstrates the real-time capacities of PDE-based image interpolation and a new region of interest coding algorithm represents important image areas with high accuracy. Finally, we propose a new framework for diffusion-based image colourisation that we use to build an efficient codec for colour images. Experiments on real world image databases show that our new method is qualitatively competitive to current state-of-the-art codecs.Diese Dissertation ist der Bildkompression mit partiellen Differentialgleichungen (PDEs, partial differential equations) gewidmet. PDE-Codecs speichern nur einen geringen Anteil aller Bildpunkte und transportieren deren Information in fehlende Bildregionen. In einigen Fällen kann PDE-basierte Kompression den aktuellen Quasi-Standard, JPEG2000, bereits schlagen. Allerdings sind die Gründe für diesen Erfolg noch nicht vollständig erforscht, und PDE-basierte Kompression befindet sich derzeit noch im Anfangsstadium. Wir tragen durch eine probabilistische Rechtfertigung anisotroper Diffusion zu einem tieferen Verständnis PDE-basierten Codec-Designs bei. Eine Analyse der Interaktion zwischen effizienten Speicherverfahren und Bildrekonstruktion erlaubt es uns, PDEs nach ihrem Nutzen für die Kompression zu beurteilen. Anhand dieser Einsichten entwickeln wir PDE-basierte Kompression hinsichtlich ihrer praktischen Nutzbarkeit weiter: Wir stellen einen Hybrid-Codec für hochtexturierte Bilder vor, der umgebungsbasierte Interpolation mit PDEs kombiniert. Ein neuer Video-Dekodierer demonstriert die Echtzeitfähigkeit PDE-basierter Interpolation und eine Region-of-Interest-Methode erlaubt es, wichtige Bildbereiche mit hoher Genauigkeit zu speichern. Schlussendlich stellen wir ein neues diffusionsbasiertes Kolorierungsverfahren vor, welches uns effiziente Kompression von Farbbildern ermöglicht. Experimente auf Realwelt-Bilddatenbanken zeigen die Konkurrenzfähigkeit dieses Verfahrens auf

Understanding and advancing PDE-based image compression

This thesis is dedicated to image compression with partial differential equations (PDEs). PDE-based codecs store only a small amount of image points and propagate their information into the unknown image areas during the decompression step. For certain classes of images, PDE-based compression can already outperform the current quasi-standard, JPEG2000. However, the reasons for this success are not yet fully understood, and PDE-based compression is still in a proof-of-concept stage. With a probabilistic justification for anisotropic diffusion, we contribute to a deeper insight into design principles for PDE-based codecs. Moreover, by analysing the interaction between efficient storage methods and image reconstruction with diffusion, we can rank PDEs according to their practical value in compression. Based on these observations, we advance PDE-based compression towards practical viability: First, we present a new hybrid codec that combines PDE- and patch-based interpolation to deal with highly textured images. Furthermore, a new video player demonstrates the real-time capacities of PDE-based image interpolation and a new region of interest coding algorithm represents important image areas with high accuracy. Finally, we propose a new framework for diffusion-based image colourisation that we use to build an efficient codec for colour images. Experiments on real world image databases show that our new method is qualitatively competitive to current state-of-the-art codecs.Diese Dissertation ist der Bildkompression mit partiellen Differentialgleichungen (PDEs, partial differential equations) gewidmet. PDE-Codecs speichern nur einen geringen Anteil aller Bildpunkte und transportieren deren Information in fehlende Bildregionen. In einigen Fällen kann PDE-basierte Kompression den aktuellen Quasi-Standard, JPEG2000, bereits schlagen. Allerdings sind die Gründe für diesen Erfolg noch nicht vollständig erforscht, und PDE-basierte Kompression befindet sich derzeit noch im Anfangsstadium. Wir tragen durch eine probabilistische Rechtfertigung anisotroper Diffusion zu einem tieferen Verständnis PDE-basierten Codec-Designs bei. Eine Analyse der Interaktion zwischen effizienten Speicherverfahren und Bildrekonstruktion erlaubt es uns, PDEs nach ihrem Nutzen für die Kompression zu beurteilen. Anhand dieser Einsichten entwickeln wir PDE-basierte Kompression hinsichtlich ihrer praktischen Nutzbarkeit weiter: Wir stellen einen Hybrid-Codec für hochtexturierte Bilder vor, der umgebungsbasierte Interpolation mit PDEs kombiniert. Ein neuer Video-Dekodierer demonstriert die Echtzeitfähigkeit PDE-basierter Interpolation und eine Region-of-Interest-Methode erlaubt es, wichtige Bildbereiche mit hoher Genauigkeit zu speichern. Schlussendlich stellen wir ein neues diffusionsbasiertes Kolorierungsverfahren vor, welches uns effiziente Kompression von Farbbildern ermöglicht. Experimente auf Realwelt-Bilddatenbanken zeigen die Konkurrenzfähigkeit dieses Verfahrens auf