Multiscale hierarchical decomposition methods for images corrupted by multiplicative noise

Recovering images corrupted by multiplicative noise is a well known challenging task. Motivated by the success of multiscale hierarchical decomposition methods (MHDM) in image processing, we adapt a variety of both classical and new multiplicative noise removing models to the MHDM form. On the basis of previous work, we further present a tight and a refined version of the corresponding multiplicative MHDM. We discuss existence and uniqueness of solutions for the proposed models, and additionally, provide convergence properties. Moreover, we present a discrepancy principle stopping criterion which prevents recovering excess noise in the multiscale reconstruction. Through comprehensive numerical experiments and comparisons, we qualitatively and quantitatively evaluate the validity of all proposed models for denoising and deblurring images degraded by multiplicative noise. By construction, these multiplicative multiscale hierarchical decomposition methods have the added benefit of recovering many scales of an image, which can provide features of interest beyond image denoising

Variational models and numerical algorithms for selective image segmentation

This thesis deals with the numerical solution of nonlinear partial differential equations and their application in image processing. The differential equations we deal with here arise from the minimization of variational models for image restoration techniques (such as denoising) and recognition of objects techniques (such as segmentation). Image denoising is a technique aimed at restoring a digital image that has been contaminated by noise while segmentation is a fundamental task in image analysis responsible for partitioning an image as sub-regions or representing the image into something that is more meaningful and easier to analyze such as extracting one or more specific objects of interest in images based on relevant information or a desired feature. Although there has been a lot of research in the restoration of images, the performance of such methods is still poor, especially when the images have a high level of noise or when the algorithms are slow. Task of the segmentation is even more challenging problem due to the difficulty of delineating, even manually, the contours of the objects of interest. The problems are often due to low contrast, fuzzy contours, similar intensities with adjacent objects, or the objects to be extracted having no real contours. The first objective of this work is to develop fast image restoration and segmentation methods which provide better denoising and fast and robust performance for image segmentation. The contribution presented here is the development of a restarted homotopy analysis method which has been designed to be easily adaptable to various types of image processing problems. As a second research objective we propose a framework for image selective segmentation which partitions an image based on the information known in advance of the object/objects to be extracted (for example the left kidney is the target to be extracted in a CT image and the prior knowledge is a few markers in this object of interest). This kind of segmentation appears especially in medical applications. Medical experts usually estimate and manually draw the boundaries of the organ/organs based on their experience. Our aim is to introduce automatic segmentation of the object of interest as a contribution not only to the way doctors and surgeons diagnose and operate but to other fields as well. The proposed methods showed success in segmenting different objects and perform well in different types of images not only in two-dimensional but in three-dimensional images as well

High-Order Sparsity Exploiting Methods with Applications in Imaging and PDEs

abstract: High-order methods are known for their accuracy and computational performance when applied to solving partial differential equations and have widespread use in representing images compactly. Nonetheless, high-order methods have difficulty representing functions containing discontinuities or functions having slow spectral decay in the chosen basis. Certain sensing techniques such as MRI and SAR provide data in terms of Fourier coefficients, and thus prescribe a natural high-order basis. The field of compressed sensing has introduced a set of techniques based on $\ell^1$ regularization that promote sparsity and facilitate working with functions having discontinuities. In this dissertation, high-order methods and $\ell^1$ regularization are used to address three problems: reconstructing piecewise smooth functions from sparse and and noisy Fourier data, recovering edge locations in piecewise smooth functions from sparse and noisy Fourier data, and reducing time-stepping constraints when numerically solving certain time-dependent hyperbolic partial differential equations.Dissertation/ThesisDoctoral Dissertation Applied Mathematics 201

New PDE models for imaging problems and applications

Variational methods and Partial Differential Equations (PDEs) have been extensively employed for the mathematical formulation of a myriad of problems describing physical phenomena such as heat propagation, thermodynamic transformations and many more. In imaging, PDEs following variational principles are often considered. In their general form these models combine a regularisation and a data fitting term, balancing one against the other appropriately. Total variation (TV) regularisation is often used due to its edgepreserving and smoothing properties. In this thesis, we focus on the design of TV-based models for several different applications. We start considering PDE models encoding higher-order derivatives to overcome wellknown TV reconstruction drawbacks. Due to their high differential order and nonlinear nature, the computation of the numerical solution of these equations is often challenging. In this thesis, we propose directional splitting techniques and use Newton-type methods that despite these numerical hurdles render reliable and efficient computational schemes. Next, we discuss the problem of choosing the appropriate data fitting term in the case when multiple noise statistics in the data are present due, for instance, to different acquisition and transmission problems. We propose a novel variational model which encodes appropriately and consistently the different noise distributions in this case. Balancing the effect of the regularisation against the data fitting is also crucial. For this sake, we consider a learning approach which estimates the optimal ratio between the two by using training sets of examples via bilevel optimisation. Numerically, we use a combination of SemiSmooth (SSN) and quasi-Newton methods to solve the problem efficiently. Finally, we consider TV-based models in the framework of graphs for image segmentation problems. Here, spectral properties combined with matrix completion techniques are needed to overcome the computational limitations due to the large amount of image data. Further, a semi-supervised technique for the measurement of the segmented region by means of the Hough transform is proposed

Automated Remote Sensing Image Interpretation with Limited Labeled Training Data

Automated remote sensing image interpretation has been investigated for more than a decade. In early years, most work was based on the assumption that there are sufficient labeled samples to be used for training. However, ground-truth collection is a very tedious and time-consuming task and sometimes very expensive, especially in the field of remote sensing that usually relies on field surveys to collect ground truth. In recent years, as the development of advanced machine learning techniques, remote sensing image interpretation with limited ground-truth has caught the attention of researchers in the fields of both remote sensing and computer science. Three approaches that focus on different aspects of the interpretation process, i.e., feature extraction, classification, and segmentation, are proposed to deal with the limited ground truth problem. First, feature extraction techniques, which usually serve as a pre-processing step for remote sensing image classification are explored. Instead of only focusing on feature extraction, a joint feature extraction and classification framework is proposed based on ensemble local manifold learning. Second, classifiers in the case of limited labeled training data are investigated, and an enhanced ensemble learning method that outperforms state-of-the-art classification methods is proposed. Third, image segmentation techniques are investigated, with the aid of unlabeled samples and spatial information. A semi-supervised self-training method is proposed, which is capable of expanding the number of training samples by its own and hence improving classification performance iteratively. Experiments show that the proposed approaches outperform state-of-the-art techniques in terms of classification accuracy on benchmark remote sensing datasets.4 month

4D imaging in tomography and optical nanoscopy

Diese Dissertation gehört zu den Gebieten mathematische Bildverarbeitung und inverse Probleme. Ein inverses Problem ist die Aufgabe, Modellparameter anhand von gemessenen Daten zu berechnen. Solche Probleme treten in zahlreichen Anwendungen in Wissenschaft und Technik auf, z.B. in medizinischer Bildgebung, Biophysik oder Astronomie. Wir betrachten Rekonstruktionsprobleme mit Poisson Rauschen in der Tomographie und optischen Nanoskopie. Bei letzterer gilt es Bilder ausgehend von verzerrten und verrauschten Messungen zu rekonstruieren, wohingegen in der Positronen-Emissions-Tomographie die Aufgabe in der Visualisierung physiologischer Prozesse eines Patienten besteht. Standardmethoden zur 3D Bildrekonstruktion berücksichtigen keine zeitabhängigen Informationen oder Dynamik, z.B. Herzschlag oder Atmung in der Tomographie oder Zellmigration in der Mikroskopie. Diese Dissertation behandelt Modelle, Analyse und effiziente Algorithmen für 3D und 4D zeitabhängige inverse Probleme. This thesis contributes to the field of mathematical image processing and inverse problems. An inverse problem is a task, where the values of some model parameters must be computed from observed data. Such problems arise in a wide variety of applications in sciences and engineering, such as medical imaging, biophysics or astronomy. We mainly consider reconstruction problems with Poisson noise in tomography and optical nanoscopy. In the latter case, the task is to reconstruct images from blurred and noisy measurements, whereas in positron emission tomography the task is to visualize physiological processes of a patient. In 3D static image reconstruction standard methods do not incorporate time-dependent information or dynamics, e.g. heart beat or breathing in tomography or cell motion in microscopy. This thesis is a treatise on models, analysis and efficient algorithms to solve 3D and 4D time-dependent inverse problems

Nonlocal smoothing and adaptive morphology for scalar- and matrix-valued images

In this work we deal with two classic degradation processes in image analysis, namely noise contamination and incomplete data. Standard greyscale and colour photographs as well as matrix-valued images, e.g. diffusion-tensor magnetic resonance imaging, may be corrupted by Gaussian or impulse noise, and may suffer from missing data. In this thesis we develop novel reconstruction approaches to image smoothing and image completion that are applicable to both scalar- and matrix-valued images. For the image smoothing problem, we propose discrete variational methods consisting of nonlocal data and smoothness constraints that penalise general dissimilarity measures. We obtain edge-preserving filters by the joint use of such measures rich in texture content together with robust non-convex penalisers. For the image completion problem, we introduce adaptive, anisotropic morphological partial differential equations modelling the dilation and erosion processes. They adjust themselves to the local geometry to adaptively fill in missing data, complete broken directional structures and even enhance flow-like patterns in an anisotropic manner. The excellent reconstruction capabilities of the proposed techniques are tested on various synthetic and real-world data sets.In dieser Arbeit beschäftigen wir uns mit zwei klassischen Störungsquellen in der Bildanalyse, nämlich mit Rauschen und unvollständigen Daten. Klassische Grauwert- und Farb-Fotografien wie auch matrixwertige Bilder, zum Beispiel Diffusionstensor-Magnetresonanz-Aufnahmen, können durch Gauß- oder Impulsrauschen gestört werden, oder können durch fehlende Daten gestört sein. In dieser Arbeit entwickeln wir neue Rekonstruktionsverfahren zum zur Bildglättung und zur Bildvervollständigung, die sowohl auf skalar- als auch auf matrixwertige Bilddaten anwendbar sind. Zur Lösung des Bildglättungsproblems schlagen wir diskrete Variationsverfahren vor, die aus nichtlokalen Daten- und Glattheitstermen bestehen und allgemeine auf Bildausschnitten definierte Unähnlichkeitsmaße bestrafen. Kantenerhaltende Filter werden durch die gemeinsame Verwendung solcher Maße in stark texturierten Regionen zusammen mit robusten nichtkonvexen Straffunktionen möglich. Für das Problem der Datenvervollständigung führen wir adaptive anisotrope morphologische partielle Differentialgleichungen ein, die Dilatations- und Erosionsprozesse modellieren. Diese passen sich der lokalen Geometrie an, um adaptiv fehlende Daten aufzufüllen, unterbrochene gerichtet Strukturen zu schließen und sogar flussartige Strukturen anisotrop zu verstärken. Die ausgezeichneten Rekonstruktionseigenschaften der vorgestellten Techniken werden anhand verschiedener synthetischer und realer Datensätze demonstriert