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

Space adaptive and hierarchical Bayesian variational models for image restoration

The main contribution of this thesis is the proposal of novel space-variant regularization or penalty terms motivated by a strong statistical rational. In light of the connection between the classical variational framework and the Bayesian formulation, we will focus on the design of highly flexible priors characterized by a large number of unknown parameters. The latter will be automatically estimated by setting up a hierarchical modeling framework, i.e. introducing informative or non-informative hyperpriors depending on the information at hand on the parameters. More specifically, in the first part of the thesis we will focus on the restoration of natural images, by introducing highly parametrized distribution to model the local behavior of the gradients in the image. The resulting regularizers hold the potential to adapt to the local smoothness, directionality and sparsity in the data. The estimation of the unknown parameters will be addressed by means of non-informative hyperpriors, namely uniform distributions over the parameter domain, thus leading to the classical Maximum Likelihood approach. In the second part of the thesis, we will address the problem of designing suitable penalty terms for the recovery of sparse signals. The space-variance in the proposed penalties, corresponding to a family of informative hyperpriors, namely generalized gamma hyperpriors, will follow directly from the assumption of the independence of the components in the signal. The study of the properties of the resulting energy functionals will thus lead to the introduction of two hybrid algorithms, aimed at combining the strong sparsity promotion characterizing non-convex penalty terms with the desirable guarantees of convex optimization

Calculating Sparse and Dense Correspondences for Near-Isometric Shapes

Comparing and analysing digital models are basic techniques of geometric shape processing. These techniques have a variety of applications, such as extracting the domain knowledge contained in the growing number of digital models to simplify shape modelling. Another example application is the analysis of real-world objects, which itself has a variety of applications, such as medical examinations, medical and agricultural research, and infrastructure maintenance. As methods to digitalize physical objects mature, any advances in the analysis of digital shapes lead to progress in the analysis of real-world objects. Global shape properties, like volume and surface area, are simple to compare but contain only very limited information. Much more information is contained in local shape differences, such as where and how a plant grew. Sadly the computation of local shape differences is hard as it requires knowledge of corresponding point pairs, i.e. points on both shapes that correspond to each other. The following article thesis (cumulative dissertation) discusses several recent publications for the computation of corresponding points: - Geodesic distances between points, i.e. distances along the surface, are fundamental for several shape processing tasks as well as several shape matching techniques. Chapter 3 introduces and analyses fast and accurate bounds on geodesic distances. - When building a shape space on a set of shapes, misaligned correspondences lead to points moving along the surfaces and finally to a larger shape space. Chapter 4 shows that this also works the other way around, that is good correspondences are obtain by optimizing them to generate a compact shape space. - Representing correspondences with a “functional map” has a variety of advantages. Chapter 5 shows that representing the correspondence map as an alignment of Green’s functions of the Laplace operator has similar advantages, but is much less dependent on the number of eigenvectors used for the computations. - Quadratic assignment problems were recently shown to reliably yield sparse correspondences. Chapter 6 compares state-of-the-art convex relaxations of graphics and vision with methods from discrete optimization on typical quadratic assignment problems emerging in shape matching

Geodesic Active Fields:A Geometric Framework for Image Registration

Image registration is the concept of mapping homologous points in a pair of images. In other words, one is looking for an underlying deformation field that matches one image to a target image. The spectrum of applications of image registration is extremely large: It ranges from bio-medical imaging and computer vision, to remote sensing or geographic information systems, and even involves consumer electronics. Mathematically, image registration is an inverse problem that is ill-posed, which means that the exact solution might not exist or not be unique. In order to render the problem tractable, it is usual to write the problem as an energy minimization, and to introduce additional regularity constraints on the unknown data. In the case of image registration, one often minimizes an image mismatch energy, and adds an additive penalty on the deformation field regularity as smoothness prior. Here, we focus on the registration of the human cerebral cortex. Precise cortical registration is required, for example, in statistical group studies in functional MR imaging, or in the analysis of brain connectivity. In particular, we work with spherical inflations of the extracted hemispherical surface and associated features, such as cortical mean curvature. Spatial mapping between cortical surfaces can then be achieved by registering the respective spherical feature maps. Despite the simplified spherical geometry, inter-subject registration remains a challenging task, mainly due to the complexity and inter-subject variability of the involved brain structures. In this thesis, we therefore present a registration scheme, which takes the peculiarities of the spherical feature maps into particular consideration. First, we realize that we need an appropriate hierarchical representation, so as to coarsely align based on the important structures with greater inter-subject stability, before taking smaller and more variable details into account. Based on arguments from brain morphogenesis, we propose an anisotropic scale-space of mean-curvature maps, built around the Beltrami framework. Second, inspired by concepts from vision-related elements of psycho-physical Gestalt theory, we hypothesize that anisotropic Beltrami regularization better suits the requirements of image registration regularization, compared to traditional Gaussian filtering. Different objects in an image should be allowed to move separately, and regularization should be limited to within the individual Gestalts. We render the regularization feature-preserving by limiting diffusion across edges in the deformation field, which is in clear contrast to the indifferent linear smoothing. We do so by embedding the deformation field as a manifold in higher-dimensional space, and minimize the associated Beltrami energy which represents the hyperarea of this embedded manifold as measure of deformation field regularity. Further, instead of simply adding this regularity penalty to the image mismatch in lieu of the standard penalty, we propose to incorporate the local image mismatch as weighting function into the Beltrami energy. The image registration problem is thus reformulated as a weighted minimal surface problem. This approach has several appealing aspects, including (1) invariance to re-parametrization and ability to work with images defined on non-flat, Riemannian domains (e.g., curved surfaces, scalespaces), and (2) intrinsic modulation of the local regularization strength as a function of the local image mismatch and/or noise level. On a side note, we show that the proposed scheme can easily keep up with recent trends in image registration towards using diffeomorphic and inverse consistent deformation models. The proposed registration scheme, called Geodesic Active Fields (GAF), is non-linear and non-convex. Therefore we propose an efficient optimization scheme, based on splitting. Data-mismatch and deformation field regularity are optimized over two different deformation fields, which are constrained to be equal. The constraint is addressed using an augmented Lagrangian scheme, and the resulting optimization problem is solved efficiently using alternate minimization of simpler sub-problems. In particular, we show that the proposed method can easily compete with state-of-the-art registration methods, such as Demons. Finally, we provide an implementation of the fast GAF method on the sphere, so as to register the triangulated cortical feature maps. We build an automatic parcellation algorithm for the human cerebral cortex, which combines the delineations available on a set of atlas brains in a Bayesian approach, so as to automatically delineate the corresponding regions on a subject brain given its feature map. In a leave-one-out cross-validation study on 39 brain surfaces with 35 manually delineated gyral regions, we show that the pairwise subject-atlas registration with the proposed spherical registration scheme significantly improves the individual alignment of cortical labels between subject and atlas brains, and, consequently, that the estimated automatic parcellations after label fusion are of better quality

PDE-based image compression based on edges and optimal data

This thesis investigates image compression with partial differential equations (PDEs) based on edges and optimal data. It first presents a lossy compression method for cartoon-like images. Edges together with some adjacent pixel values are extracted and encoded. During decoding, information not covered by this data is reconstructed by PDE-based inpainting with homogeneous diffusion. The result is a compression codec based on perceptual meaningful image features which is able to outperform JPEG and JPEG2000. In contrast, the second part of the thesis focuses on the optimal selection of inpainting data. The proposed methods allow to recover a general image from only 4% of all pixels almost perfectly, even with homogeneous diffusion inpainting. A simple conceptual encoding shows the potential of an optimal data selection for image compression: The results beat the quality of JPEG2000 when anisotropic diffusion is used for inpainting. Finally, the thesis shows that the combination of the concepts allows for further improvements.Die vorliegende Arbeit untersucht die Bildkompression mit partiellen Differentialgleichungen (PDEs), basierend auf Kanten und optimalen Daten. Sie stellt zunächst ein verlustbehaftetes Kompressionsverfahren für cartoonartige Bilder vor. Dazu werden Kanten zusammen mit einigen benachbarten Pixelwerten extrahiert und anschließend kodiert. Während der Dekodierung, werden Informationen, die durch die gespeicherten Daten nicht abgedeckt sind, mittels PDE-basiertem Inpainting mit homogenener Diffusion rekonstruiert. Das Ergebnis ist ein Kompressionscodec, der auf visuell bedeutsamen Bildmerkmalen basiert und in der Lage ist, die Qualität von JPEG und JPEG2000 zu übertreffen. Im Gegensatz dazu konzentriert sich der zweite Teil der Arbeit auf die optimale Auswahl von Inpaintingdaten. Die vorgeschlagenen Methoden ermöglichen es, ein gewöhnliches Bild aus nur 4% aller Pixel nahezu perfekt wiederherzustellen, selbst mit homogenem Diffusionsinpainting. Eine einfache konzeptuelle Kodierung zeigt das Potential einer optimierten Datenauswahl auf: Die Ergebnisse übersteigen die Qualität von JPEG2000, sofern das Inpainting mit einem anisotropen Diffusionsprozess erfolgt. Schließlich zeigt die Arbeit, dass weitere Verbesserungen durch die Kombination der Konzepte erreicht werden können

Structured Learning with Manifold Representations of Natural Data Variations

According to the manifold hypothesis, natural variations in high-dimensional data lie on or near a low-dimensional, nonlinear manifold. Additionally, many identity-preserving transformations are shared among classes of data which can allow for an efficient representation of data variations: a limited set of transformations can describe a majority of variations in many classes. This work demonstrates the learning of generative models of identity-preserving transformations on data manifolds in order to analyze, generate, and exploit the natural variations in data for machine learning tasks. The introduced transformation representations are incorporated into several novel models to highlight the ability to generate realistic samples of semantically meaningful transformations, to generalize transformations beyond their source domain, and to estimate transformations between data samples. We first develop a model for learning 3D manifold-based transformations from 2D projected inputs which can be used to perform depth inference from 2D moving inputs. We then confirm that our generative model of transformations can be generalized across classes by defining two transfer learning tasks that map transformations learned from a rich dataset to previously unseen data. Next, we develop the manifold autoencoder, which learns low-dimensional manifold structure from complex data in the latent space of an autoencoder and adapts the latent space to accommodate this structure. Finally, we introduce the Variational Autoencoder with Learned Latent Structure (VAELLS) which incorporates a learnable manifold model into the fully probabilistic generative framework of a variational autoencoder.Ph.D

A regularization approach for reconstruction and visualization of 3-D data

Esta tesis trata sobre reconstrucción de superficies a partir de imágenes de rango utilizando algunas extensiones de la Regularización de Tikhonov, que produce Splines aplicables a datos en n dimensiones. La idea central es que estos splines se pueden obtener mediante la teoría de regularización, utilizando un equilibrio entre la suavidad y la fidelidad a los datos, por tanto, serán aplicables tanto en la interpolación como en la aproximación de datos exactos o ruidosos. En esta tesis proponemos un enfoque variacional que incluye los datos e información a priori acerca de la solución, dada en forma de funcionales. Solucionamos problemas de optimización que resultan ser una extensión de la teoría de Tikhonov, con el propósito de incluir funcionales con propiedades locales y globales que pueden ser ajustadas mediante parámetros de regularización. El a priori es analizado en términos de las propiedades físicas y geométricas de los funcionales para luego ser agregados a la formulación variacional. Los resultados obtenidos se prueban con datos para reconstrucción de superficies, mostrando notables propiedades de reproducción y aproximación. En particular, utilizamos la reconstrucción de superficies para ilustrar las aplicaciones prácticas, pero nuestro enfoque tiene muchas más aplicaciones. En el centro de nuestra propuesta esta la teoría general de problemas inversos y las aplicaciones de algunas ideas provenientes del análisis funcional. Los splines que obtenemos son combinaciones lineales de las soluciones fundamentales de ciertos operadores en derivadas parciales, frecuentes en la teoría de la elasticidad y no se hace ninguna suposición previa sobre el modelo estadístico de los datos de entrada, de manera que se pueden tomar en términos de una inferencia estadística no paramétrica. Estos splines son implementables en una forma muy estable y se pueden aplicar en problemas de interpolación y suavizado. / Abstract: This thesis is about surface reconstruction from range images using some extensions of Tikhonov regularization that produces splines applicable on n-dimensional data. The central idea is that these splines can be obtained by regularization theory, using a trade-off between fidelity to data and smoothness properties; as a consequence, they are applicable both in interpolation and approximation of exact or noisy data. We propose a variational framework that includes data and a priori information about the solution, given in the form of functionals. We solve optimization problems which are extensions of Tikhonov theory, in order to include functionals with local and global features that can be tuned by regularization parameters. The a priori is thought in terms of geometric and physical properties of functionals and then added to the variational formulation. The results obtained are tested on data for surface reconstruction, showing remarkable reproducing and approximating properties. In this case we use surface reconstruction to illustrate practical applications; nevertheless, our approach has many other applications. In the core of our approach is the general theory of inverse problems and the application of some abstract ideas from functional analysis. The splines obtained are linear combinations of certain fundamental solutions of partial differential operators from elasticity theory and no prior assumption is made on a statistical model for the input data, so it can be thought in terms of nonparametric statistical inference. They are implementable in a very stable form and can be applied for both interpolation and smoothing problems.Doctorad