127 research outputs found

    Connecting mathematical models for image processing and neural networks

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    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

    A PDE Approach to Data-driven Sub-Riemannian Geodesics in SE(2)

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    We present a new flexible wavefront propagation algorithm for the boundary value problem for sub-Riemannian (SR) geodesics in the roto-translation group SE(2)=R2⋊S1SE(2) = \mathbb{R}^2 \rtimes S^1 with a metric tensor depending on a smooth external cost C:SE(2)→[δ,1]\mathcal{C}:SE(2) \to [\delta,1], δ>0\delta>0, computed from image data. The method consists of a first step where a SR-distance map is computed as a viscosity solution of a Hamilton-Jacobi-Bellman (HJB) system derived via Pontryagin's Maximum Principle (PMP). Subsequent backward integration, again relying on PMP, gives the SR-geodesics. For C=1\mathcal{C}=1 we show that our method produces the global minimizers. Comparison with exact solutions shows a remarkable accuracy of the SR-spheres and the SR-geodesics. We present numerical computations of Maxwell points and cusp points, which we again verify for the uniform cost case C=1\mathcal{C}=1. Regarding image analysis applications, tracking of elongated structures in retinal and synthetic images show that our line tracking generically deals with crossings. We show the benefits of including the sub-Riemannian geometry.Comment: Extended version of SSVM 2015 conference article "Data-driven Sub-Riemannian Geodesics in SE(2)

    Anisotropy Across Fields and Scales

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    This open access book focuses on processing, modeling, and visualization of anisotropy information, which are often addressed by employing sophisticated mathematical constructs such as tensors and other higher-order descriptors. It also discusses adaptations of such constructs to problems encountered in seemingly dissimilar areas of medical imaging, physical sciences, and engineering. Featuring original research contributions as well as insightful reviews for scientists interested in handling anisotropy information, it covers topics such as pertinent geometric and algebraic properties of tensors and tensor fields, challenges faced in processing and visualizing different types of data, statistical techniques for data processing, and specific applications like mapping white-matter fiber tracts in the brain. The book helps readers grasp the current challenges in the field and provides information on the techniques devised to address them. Further, it facilitates the transfer of knowledge between different disciplines in order to advance the research frontiers in these areas. This multidisciplinary book presents, in part, the outcomes of the seventh in a series of Dagstuhl seminars devoted to visualization and processing of tensor fields and higher-order descriptors, which was held in Dagstuhl, Germany, on October 28–November 2, 2018

    Anisotropy Across Fields and Scales

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    This open access book focuses on processing, modeling, and visualization of anisotropy information, which are often addressed by employing sophisticated mathematical constructs such as tensors and other higher-order descriptors. It also discusses adaptations of such constructs to problems encountered in seemingly dissimilar areas of medical imaging, physical sciences, and engineering. Featuring original research contributions as well as insightful reviews for scientists interested in handling anisotropy information, it covers topics such as pertinent geometric and algebraic properties of tensors and tensor fields, challenges faced in processing and visualizing different types of data, statistical techniques for data processing, and specific applications like mapping white-matter fiber tracts in the brain. The book helps readers grasp the current challenges in the field and provides information on the techniques devised to address them. Further, it facilitates the transfer of knowledge between different disciplines in order to advance the research frontiers in these areas. This multidisciplinary book presents, in part, the outcomes of the seventh in a series of Dagstuhl seminars devoted to visualization and processing of tensor fields and higher-order descriptors, which was held in Dagstuhl, Germany, on October 28–November 2, 2018

    An evaluation of partial differential equations based digital inpainting algorithms

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    Partial Differential equations (PDEs) have been used to model various phenomena/tasks in different scientific and engineering endeavours. This thesis is devoted to modelling image inpainting by numerical implementations of certain PDEs. The main objectives of image inpainting include reconstructing damaged parts and filling-in regions in which data/colour information are missing. Different automatic and semi-automatic approaches to image inpainting have been developed including PDE-based, texture synthesis-based, exemplar-based, and hybrid approaches. Various challenges remain unresolved in reconstructing large size missing regions and/or missing areas with highly textured surroundings. Our main aim is to address such challenges by developing new advanced schemes with particular focus on using PDEs of different orders to preserve continuity of textural and geometric information in the surrounding of missing regions. We first investigated the problem of partial colour restoration in an image region whose greyscale channel is intact. A PDE-based solution is known that is modelled as minimising total variation of gradients in the different colour channels. We extend the applicability of this model to partial inpainting in other 3-channels colour spaces (such as RGB where information is missing in any of the two colours), simply by exploiting the known linear/affine relationships between different colouring models in the derivation of a modified PDE solution obtained by using the Euler-Lagrange minimisation of the corresponding gradient Total Variation (TV). We also developed two TV models on the relations between greyscale and colour channels using the Laplacian operator and the directional derivatives of gradients. The corresponding Euler-Lagrange minimisation yields two new PDEs of different orders for partial colourisation. We implemented these solutions in both spatial and frequency domains. We measure the success of these models by evaluating known image quality measures in inpainted regions for sufficiently large datasets and scenarios. The results reveal that our schemes compare well with existing algorithms, but inpainting large regions remains a challenge. Secondly, we investigate the Total Inpainting (TI) problem where all colour channels are missing in an image region. Reviewing and implementing existing PDE-based total inpainting methods reveal that high order PDEs, applied to each colour channel separately, perform well but are influenced by the size of the region and the quantity of texture surrounding it. Here we developed a TI scheme that benefits from our partial inpainting approach and apply two PDE methods to recover the missing regions in the image. First, we extract the (Y, Cb, Cr) of the image outside the missing region, apply the above PDE methods for reconstructing the missing regions in the luminance channel (Y), and then use the colourisation method to recover the missing (Cb, Cr) colours in the region. We shall demonstrate that compared to existing TI algorithms, our proposed method (using 2 PDE methods) performs well when tested on large datasets of natural and face images. Furthermore, this helps understanding of the impact of the texture in the surrounding areas on inpainting and opens new research directions. Thirdly, we investigate existing Exemplar-Based Inpainting (EBI) methods that do not use PDEs but simultaneously propagate the texture and structure into the missing region by finding similar patches within the rest of image and copying them into the boundary of the missing region. The order of patch propagation is determined by a priority function, and the similarity is determined by matching criteria. We shall exploit recently emerging Topological Data Analysis (TDA) tools to create innovative EBI schemes, referred to as TEBI. TDA studies shapes of data/objects to quantify image texture in terms of connectivity and closeness properties of certain data landmarks. Such quantifications help determine the appropriate size of patch propagation and will be used to modify the patch propagation priority function using the geometrical properties of curvature of isophotes, and to improve the matching criteria of patches by calculating the correlation coefficients from the spatial, gradient and Laplacian domains. The performance of this TEBI method will be tested by applying it to natural dataset images, resulting in improved inpainting when compared with other EBI methods. Fourthly, the recent hybrid-based inpainting techniques are reviewed and a number of highly performing innovative hybrid techniques that combine the use of high order PDE methods with the TEBI method for the simultaneous rebuilding of the missing texture and structure regions in an image are proposed. Such a hybrid scheme first decomposes the image into texture and structure components, and then the missing regions in these components are recovered by TEBI and PDE based methods respectively. The performance of our hybrid schemes will be compared with two existing hybrid algorithms. Fifthly, we turn our attention to inpainting large missing regions, and develop an innovative inpainting scheme that uses the concept of seam carving to reduce this problem to that of inpainting a smaller size missing region that can be dealt with efficiently using the inpainting schemes developed above. Seam carving resizes images based on content-awareness of the image for both reduction and expansion without affecting those image regions that have rich information. The missing region of the seam-carved version will be recovered by the TEBI method, original image size is restored by adding the removed seams and the missing parts of the added seams are then repaired using a high order PDE inpainting scheme. The benefits of this approach in dealing with large missing regions are demonstrated. The extensive performance testing of the developed inpainting methods shows that these methods significantly outperform existing inpainting methods for such a challenging task. However, the performance is still not acceptable in recovering large missing regions in high texture and structure images, and hence we shall identify remaining challenges to be investigated in the future. We shall also extend our work by investigating recently developed deep learning based image/video colourisation, with the aim of overcoming its limitations and shortcoming. Finally, we should also describe our on-going research into using TDA to detect recently growing serious “malicious” use of inpainting to create Fake images/videos

    Introducing anisotropic tensor to high order variational model for image restoration

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    Second order total variation (SOTV) models have advantages for image restoration over their first order counterparts including their ability to remove the staircase artefact in the restored image. However, such models tend to blur the reconstructed image when discretised for numerical solution [1–5]. To overcome this drawback, we introduce a new tensor weighted second order (TWSO) model for image restoration. Specifically, we develop a novel regulariser for the SOTV model that uses the Frobenius norm of the product of the isotropic SOTV Hessian matrix and an anisotropic tensor. We then adapt the alternating direction method of multipliers (ADMM) to solve the proposed model by breaking down the original problem into several subproblems. All the subproblems have closed-forms and can be solved efficiently. The proposed method is compared with state-of-the-art approaches such as tensor-based anisotropic diffusion, total generalised variation, and Euler's elastica. We validate the proposed TWSO model using extensive experimental results on a large number of images from the Berkeley BSDS500. We also demonstrate that our method effectively reduces both the staircase and blurring effects and outperforms existing approaches for image inpainting and denoising applications

    Inpainting large missing regions based on Seam Carving

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    Inpainting techniques are developed to recover missing image information. Existing inpainting approaches are: Partial Differential Equations Based Inpainting (PDE-BI) and Exemplar-Based Inpainting (EBI). PDE-BI methods used to fill in the missing information via information propagation from neighbouring areas. However, it can only reconstruct successfully small missing regions that are surrounded by limited texture. However, EBI methods are used to recover large regions with richly-textured/structured areas around them, moreover, artefacts are likely to occur. This paper proposes a technique to reduce the missing region size based on seam carving approach, which enables EBI and PDE-BI to recover the missing part. In our proposal, seam carving is used to reduce only the size of the missing region, to be subsequently recovered using EBI method. The added extra paths resulting from the added seams is repaired using PDE-BI. This method outperformed the state-of-art EBI methods

    Understanding and advancing PDE-based image compression

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    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

    Trends in Mathematical Imaging and Surface Processing

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    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
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