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

Sparse variational regularization for visual motion estimation

The computation of visual motion is a key component in numerous computer vision tasks such as object detection, visual object tracking and activity recognition. Despite exten- sive research effort, efficient handling of motion discontinuities, occlusions and illumina- tion changes still remains elusive in visual motion estimation. The work presented in this thesis utilizes variational methods to handle the aforementioned problems because these methods allow the integration of various mathematical concepts into a single en- ergy minimization framework. This thesis applies the concepts from signal sparsity to the variational regularization for visual motion estimation. The regularization is designed in such a way that it handles motion discontinuities and can detect object occlusions

A METHOD FOR DENOISING IMAGE CONTOURS

The edge detection techniques have to compromise between sensitivity and noise. In order for the main contours to be uninterrupted, the level of sensitivity has to be raised, which however has the negative effect of producing a multitude of insignificant contours (noise). This article proposes a method of removing this noise, which acts directly on the binary representation of the image contours

From spline wavelet to sampling theory on circulant graphs and beyond– conceiving sparsity in graph signal processing

Graph Signal Processing (GSP), as the field concerned with the extension of classical signal processing concepts to the graph domain, is still at the beginning on the path toward providing a generalized theory of signal processing. As such, this thesis aspires to conceive the theory of sparse representations on graphs by traversing the cornerstones of wavelet and sampling theory on graphs. Beginning with the novel topic of graph spline wavelet theory, we introduce families of spline and e-spline wavelets, and associated filterbanks on circulant graphs, which lever- age an inherent vanishing moment property of circulant graph Laplacian matrices (and their parameterized generalizations), for the reproduction and annihilation of (exponen- tial) polynomial signals. Further, these families are shown to provide a stepping stone to generalized graph wavelet designs with adaptive (annihilation) properties. Circulant graphs, which serve as building blocks, facilitate intuitively equivalent signal processing concepts and operations, such that insights can be leveraged for and extended to more complex scenarios, including arbitrary undirected graphs, time-varying graphs, as well as associated signals with space- and time-variant properties, all the while retaining the focus on inducing sparse representations. Further, we shift from sparsity-inducing to sparsity-leveraging theory and present a novel sampling and graph coarsening framework for (wavelet-)sparse graph signals, inspired by Finite Rate of Innovation (FRI) theory and directly building upon (graph) spline wavelet theory. At its core, the introduced Graph-FRI-framework states that any K-sparse signal residing on the vertices of a circulant graph can be sampled and perfectly reconstructed from its dimensionality-reduced graph spectral representation of minimum size 2K, while the structure of an associated coarsened graph is simultaneously inferred. Extensions to arbitrary graphs can be enforced via suitable approximation schemes. Eventually, gained insights are unified in a graph-based image approximation framework which further leverages graph partitioning and re-labelling techniques for a maximally sparse graph wavelet representation.Open Acces

Geometric Surface Processing and Virtual Modeling

In this work we focus on two main topics "Geometric Surface Processing" and "Virtual Modeling". The inspiration and coordination for most of the research work contained in the thesis has been driven by the project New Interactive and Innovative Technologies for CAD (NIIT4CAD), funded by the European Eurostars Programme. NIIT4CAD has the ambitious aim of overcoming the limitations of the traditional approach to surface modeling of current 3D CAD systems by introducing new methodologies and technologies based on subdivision surfaces in a new virtual modeling framework. These innovations will allow designers and engineers to transform quickly and intuitively an idea of shape in a high-quality geometrical model suited for engineering and manufacturing purposes. One of the objective of the thesis is indeed the reconstruction and modeling of surfaces, representing arbitrary topology objects, starting from 3D irregular curve networks acquired through an ad-hoc smart-pen device. The thesis is organized in two main parts: "Geometric Surface Processing" and "Virtual Modeling". During the development of the geometric pipeline in our Virtual Modeling system, we faced many challenges that captured our interest and opened new areas of research and experimentation. In the first part, we present these theories and some applications to Geometric Surface Processing. This allowed us to better formalize and give a broader understanding on some of the techniques used in our latest advancements on virtual modeling and surface reconstruction. The research on both topics led to important results that have been published and presented in articles and conferences of international relevance

Analysis and Manipulation of Repetitive Structures of Varying Shape

Self-similarity and repetitions are ubiquitous in man-made and natural objects. Such structural regularities often relate to form, function, aesthetics, and design considerations. Discovering structural redundancies along with their dominant variations from 3D geometry not only allows us to better understand the underlying objects, but is also beneficial for several geometry processing tasks including compact representation, shape completion, and intuitive shape manipulation. To identify these repetitions, we present a novel detection algorithm based on analyzing a graph of surface features. We combine general feature detection schemes with a RANSAC-based randomized subgraph searching algorithm in order to reliably detect recurring patterns of locally unique structures. A subsequent segmentation step based on a simultaneous region growing is applied to verify that the actual data supports the patterns detected in the feature graphs. We introduce our graph based detection algorithm on the example of rigid repetitive structure detection. Then we extend the approach to allow more general deformations between the detected parts. We introduce subspace symmetries whereby we characterize similarity by requiring the set of repeating structures to form a low dimensional shape space. We discover these structures based on detecting linearly correlated correspondences among graphs of invariant features. The found symmetries along with the modeled variations are useful for a variety of applications including non-local and non-rigid denoising. Employing subspace symmetries for shape editing, we introduce a morphable part model for smart shape manipulation. The input geometry is converted to an assembly of deformable parts with appropriate boundary conditions. Our method uses self-similarities from a single model or corresponding parts of shape collections as training input and allows the user also to reassemble the identified parts in new configurations, thus exploiting both the discrete and continuous learned variations while ensuring appropriate boundary conditions across part boundaries. We obtain an interactive yet intuitive shape deformation framework producing realistic deformations on classes of objects that are difficult to edit using repetition-unaware deformation techniques

Image compression with anisotropic diffusion

Compression is an important field of digital image processing where well-engineered methods with high performance exist. Partial differential equations (PDEs), however, have not much been explored in this context so far. In our paper we introduce a novel framework for image compression that makes use of the interpolation qualities of edge-enhancing diffusion. Although this anisotropic diffusion equation with a diffusion tensor was originally proposed for image denoising, we show that it outperforms many other PDEs when sparse scattered data must be interpolated. To exploit this property for image compression, we consider an adaptive triangulation method for removing less significant pixels from the image. The remaining points serve as scattered interpolation data for the diffusion process. They can be coded in a compact way that reflects the B-tree structure of the triangulation. We supplement the coding step with a number of amendments such as error threshold adaptation, diffusion-based point selection, and specific quantisation strategies. Our experiments illustrate the usefulness of each of these modifications. They demonstrate that for high compression rates, our PDE-based approach does not only give far better results than the widely-used JPEG standard, but can even come close to the quality of the highly optimised JPEG2000 codec

Graph Signal Processing: Overview, Challenges and Applications

Research in Graph Signal Processing (GSP) aims to develop tools for processing data defined on irregular graph domains. In this paper we first provide an overview of core ideas in GSP and their connection to conventional digital signal processing. We then summarize recent developments in developing basic GSP tools, including methods for sampling, filtering or graph learning. Next, we review progress in several application areas using GSP, including processing and analysis of sensor network data, biological data, and applications to image processing and machine learning. We finish by providing a brief historical perspective to highlight how concepts recently developed in GSP build on top of prior research in other areas.Comment: To appear, Proceedings of the IEE

Machine Learning And Image Processing For Noise Removal And Robust Edge Detection In The Presence Of Mixed Noise

The central goal of this dissertation is to design and model a smoothing filter based on the random single and mixed noise distribution that would attenuate the effect of noise while preserving edge details. Only then could robust, integrated and resilient edge detection methods be deployed to overcome the ubiquitous presence of random noise in images. Random noise effects are modeled as those that could emanate from impulse noise, Gaussian noise and speckle noise. In the first step, evaluation of methods is performed based on an exhaustive review on the different types of denoising methods which focus on impulse noise, Gaussian noise and their related denoising filters. These include spatial filters (linear, non-linear and a combination of them), transform domain filters, neural network-based filters, numerical-based filters, fuzzy based filters, morphological filters, statistical filters, and supervised learning-based filters. In the second step, switching adaptive median and fixed weighted mean filter (SAMFWMF) which is a combination of linear and non-linear filters, is introduced in order to detect and remove impulse noise. Then, a robust edge detection method is applied which relies on an integrated process including non-maximum suppression, maximum sequence, thresholding and morphological operations. The results are obtained on MRI and natural images. In the third step, a combination of transform domain-based filter which is a combination of dual tree – complex wavelet transform (DT-CWT) and total variation, is introduced in order to detect and remove Gaussian noise as well as mixed Gaussian and Speckle noise. Then, a robust edge detection is applied in order to track the true edges. The results are obtained on medical ultrasound and natural images. In the fourth step, a smoothing filter, which is a feed-forward convolutional network (CNN) is introduced to assume a deep architecture, and supported through a specific learning algorithm, l2 loss function minimization, a regularization method, and batch normalization all integrated in order to detect and remove impulse noise as well as mixed impulse and Gaussian noise. Then, a robust edge detection is applied in order to track the true edges. The results are obtained on natural images for both specific and non-specific noise-level