Bayesian Dictionary Learning for Single and Coupled Feature Spaces

Over-complete bases offer the flexibility to represent much wider range of signals with more elementary basis atoms than signal dimension. The use of over-complete dictionaries for sparse representation has been a new trend recently and has increasingly become recognized as providing high performance for applications such as denoise, image super-resolution, inpaiting, compression, blind source separation and linear unmixing. This dissertation studies the dictionary learning for single or coupled feature spaces and its application in image restoration tasks. A Bayesian strategy using a beta process prior is applied to solve both problems. Firstly, we illustrate how to generalize the existing beta process dictionary learning method (BP) to learn dictionary for single feature space. The advantage of this approach is that the number of dictionary atoms and their relative importance may be inferred non-parametrically. Next, we propose a new beta process joint dictionary learning method (BP-JDL) for coupled feature spaces, where the learned dictionaries also reflect the relationship between the two spaces. Compared to previous couple feature spaces dictionary learning algorithms, our algorithm not only provides dictionaries that customized to each feature space, but also adds more consistent and accurate mapping between the two feature spaces. This is due to the unique property of the beta process model that the sparse representation can be decomposed to values and dictionary atom indicators. The proposed algorithm is able to learn sparse representations that correspond to the same dictionary atoms with the same sparsity but different values in coupled feature spaces, thus bringing consistent and accurate mapping between coupled feature spaces. Two applications, single image super-resolution and inverse halftoning, are chosen to evaluate the performance of the proposed Bayesian approach. In both cases, the Bayesian approach, either for single feature space or coupled feature spaces, outperforms state-of-the-art methods in comparative domains

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

Radial Basis Functions: Biomedical Applications and Parallelization

Radial basis function (RBF) is a real-valued function whose values depend only on the distances between an interpolation point and a set of user-specified points called centers. RBF interpolation is one of the primary methods to reconstruct functions from multi-dimensional scattered data. Its abilities to generalize arbitrary space dimensions and to provide spectral accuracy have made it particularly popular in different application areas, including but not limited to: finding numerical solutions of partial differential equations (PDEs), image processing, computer vision and graphics, deep learning and neural networks, etc. The present thesis discusses three applications of RBF interpolation in biomedical engineering areas: (1) Calcium dynamics modeling, in which we numerically solve a set of PDEs by using meshless numerical methods and RBF-based interpolation techniques; (2) Image restoration and transformation, where an image is restored from its triangular mesh representation or transformed under translation, rotation, and scaling, etc. from its original form; (3) Porous structure design, in which the RBF interpolation used to reconstruct a 3D volume containing porous structures from a set of regularly or randomly placed points inside a user-provided surface shape. All these three applications have been investigated and their effectiveness has been supported with numerous experimental results. In particular, we innovatively utilize anisotropic distance metrics to define the distance in RBF interpolation and apply them to the aforementioned second and third applications, which show significant improvement in preserving image features or capturing connected porous structures over the isotropic distance-based RBF method. Beside the algorithm designs and their applications in biomedical areas, we also explore several common parallelization techniques (including OpenMP and CUDA-based GPU programming) to accelerate the performance of the present algorithms. In particular, we analyze how parallel programming can help RBF interpolation to speed up the meshless PDE solver as well as image processing. While RBF has been widely used in various science and engineering fields, the current thesis is expected to trigger some more interest from computational scientists or students into this fast-growing area and specifically apply these techniques to biomedical problems such as the ones investigated in the present work

Digital Color Imaging

This paper surveys current technology and research in the area of digital color imaging. In order to establish the background and lay down terminology, fundamental concepts of color perception and measurement are first presented us-ing vector-space notation and terminology. Present-day color recording and reproduction systems are reviewed along with the common mathematical models used for representing these devices. Algorithms for processing color images for display and communication are surveyed, and a forecast of research trends is attempted. An extensive bibliography is provided

Hardware-accelerated algorithms in visual computing

This thesis presents new parallel algorithms which accelerate computer vision methods by the use of graphics processors (GPUs) and evaluates them with respect to their speed, scalability, and the quality of their results. It covers the fields of homogeneous and anisotropic diffusion processes, diffusion image inpainting, optic flow, and halftoning. In this turn, it compares different solvers for homogeneous diffusion and presents a novel \u27extended\u27 box filter. Moreover, it suggests to use the fast explicit diffusion scheme (FED) as an efficient and flexible solver for nonlinear and in particular for anisotropic parabolic diffusion problems on graphics hardware. For elliptic diffusion-like processes, it recommends to use cascadic FED or Fast Jacobi schemes. The presented optic flow algorithm represents one of the fastest yet very accurate techniques. Finally, it presents a novel halftoning scheme which yields state-of-the-art results for many applications in image processing and computer graphics.Diese Arbeit präsentiert neue parallele Algorithmen zur Beschleunigung von Methoden in der Bildinformatik mittels Grafikprozessoren (GPUs), und evaluiert diese im Hinblick auf Geschwindigkeit, Skalierungsverhalten, und Qualität der Resultate. Sie behandelt dabei die Gebiete der homogenen und anisotropen Diffusionsprozesse, Inpainting (Bildvervollständigung) mittels Diffusion, die Bestimmung des optischen Flusses, sowie Halbtonverfahren. Dabei werden verschiedene Löser für homogene Diffusion verglichen und ein neuer \u27erweiterter\u27 Mittelwertfilter präsentiert. Ferner wird vorgeschlagen, das schnelle explizite Diffusionsschema (FED) als effizienten und flexiblen Löser für parabolische nichtlineare und speziell anisotrope Diffusionsprozesse auf Grafikprozessoren einzusetzen. Für elliptische diffusionsartige Prozesse wird hingegen empfohlen, kaskadierte FED- oder schnelle Jacobi-Verfahren einzusetzen. Der vorgestellte Algorithmus zur Berechnung des optischen Flusses stellt eines der schnellsten und dennoch äußerst genauen Verfahren dar. Schließlich wird ein neues Halbtonverfahren präsentiert, das in vielen Bereichen der Bildverarbeitung und Computergrafik Ergebnisse produziert, die den Stand der Technik repräsentieren

Perceptual Error Optimization for {Monte Carlo} Rendering

Realistic image synthesis involves computing high-dimensional light transport integrals which in practice are numerically estimated using Monte Carlo integration. The error of this estimation manifests itself in the image as visually displeasing aliasing or noise. To ameliorate this, we develop a theoretical framework for optimizing screen-space error distribution. Our model is flexible and works for arbitrary target error power spectra. We focus on perceptual error optimization by leveraging models of the human visual system's (HVS) point spread function (PSF) from halftoning literature. This results in a specific optimization problem whose solution distributes the error as visually pleasing blue noise in image space. We develop a set of algorithms that provide a trade-off between quality and speed, showing substantial improvements over prior state of the art. We perform evaluations using both quantitative and perceptual error metrics to support our analysis, and provide extensive supplemental material to help evaluate the perceptual improvements achieved by our methods

Perceptual error optimization for Monte Carlo rendering

Realistic image synthesis involves computing high-dimensional light transport integrals which in practice are numerically estimated using Monte Carlo integration. The error of this estimation manifests itself in the image as visually displeasing aliasing or noise. To ameliorate this, we develop a theoretical framework for optimizing screen-space error distribution. Our model is flexible and works for arbitrary target error power spectra. We focus on perceptual error optimization by leveraging models of the human visual system's (HVS) point spread function (PSF) from halftoning literature. This results in a specific optimization problem whose solution distributes the error as visually pleasing blue noise in image space. We develop a set of algorithms that provide a trade-off between quality and speed, showing substantial improvements over prior state of the art. We perform evaluations using both quantitative and perceptual error metrics to support our analysis, and provide extensive supplemental material to help evaluate the perceptual improvements achieved by our methods

M-lattice, a system for signal synthesis and processing based on reaction-diffusion

Thesis (Sc. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 1994.Includes bibliographical references (leaves 148-154).by Alexander Semyon Sherstinsky.Sc.D

Sparse Modeling for Image and Vision Processing

In recent years, a large amount of multi-disciplinary research has been conducted on sparse models and their applications. In statistics and machine learning, the sparsity principle is used to perform model selection---that is, automatically selecting a simple model among a large collection of them. In signal processing, sparse coding consists of representing data with linear combinations of a few dictionary elements. Subsequently, the corresponding tools have been widely adopted by several scientific communities such as neuroscience, bioinformatics, or computer vision. The goal of this monograph is to offer a self-contained view of sparse modeling for visual recognition and image processing. More specifically, we focus on applications where the dictionary is learned and adapted to data, yielding a compact representation that has been successful in various contexts.Comment: 205 pages, to appear in Foundations and Trends in Computer Graphics and Visio

Compression, pose tracking, and halftoning

In this thesis, we discuss image compression, pose tracking, and halftoning. Although these areas seem to be unrelated at first glance, they can be connected through video coding as application scenario. Our first contribution is an image compression algorithm based on a rectangular subdivision scheme which stores only a small subsets of the image points. From these points, the remained of the image is reconstructed using partial differential equations. Afterwards, we present a pose tracking algorithm that is able to follow the 3-D position and orientation of multiple objects simultaneously. The algorithm can deal with noisy sequences, and naturally handles both occlusions between different objects, as well as occlusions occurring in kinematic chains. Our third contribution is a halftoning algorithm based on electrostatic principles, which can easily be adjusted to different settings through a number of extensions. Examples include modifications to handle varying dot sizes or hatching. In the final part of the thesis, we show how to combine our image compression, pose tracking, and halftoning algorithms to novel video compression codecs. In each of these four topics, our algorithms yield excellent results that outperform those of other state-of-the-art algorithms.In dieser Arbeit werden die auf den ersten Blick vollkommen voneinander unabhängig erscheinenden Bereiche Bildkompression, 3D-Posenschätzung und Halbtonverfahren behandelt und im Bereich der Videokompression sinnvoll zusammengeführt. Unser erster Beitrag ist ein Bildkompressionsalgorithmus, der auf einem rechteckigen Unterteilungsschema basiert. Dieser Algorithmus speichert nur eine kleine Teilmenge der im Bild vorhandenen Punkte, während die restlichen Punkte mittels partieller Differentialgleichungen rekonstruiert werden. Danach stellen wir ein Posenschätzverfahren vor, welches die 3D-Position und Ausrichtung von mehreren Objekten anhand von Bilddaten gleichzeitig verfolgen kann. Unser Verfahren funktioniert bei verrauschten Videos und im Falle von Objektüberlagerungen. Auch Verdeckungen innerhalb einer kinematischen Kette werden natürlich behandelt. Unser dritter Beitrag ist ein Halbtonverfahren, das auf elektrostatischen Prinzipien beruht. Durch eine Reihe von Erweiterungen kann dieses Verfahren flexibel an verschiedene Szenarien angepasst werden. So ist es beispielsweise möglich, verschiedene Punktgrößen zu verwenden oder Schraffuren zu erzeugen. Der letzte Teil der Arbeit zeigt, wie man unseren Bildkompressionsalgorithmus, unser Posenschätzverfahren und unser Halbtonverfahren zu neuen Videokompressionsalgorithmen kombinieren kann. Die für jeden der vier Themenbereiche entwickelten Verfahren erzielen hervorragende Resultate, welche die Ergebnisse anderer moderner Verfahren übertreffen