Applications of Dynamic Mode Decomposition and Sparse Reconstruction in the Data-Driven Dynamic Analysis of Physical Systems

Recent advancements in data collection methods and equipment have resulted in a huge increase in the amount of data collected by observing various types of physical phenomena. Regardless of the amount of data collected, it is well known for many physical systems, the so-called information rank of the collected data is much lower than the rank of the data itself. This usually means the data may be represented sparsely in terms of a properly-chosen basis. This realization has led to methods for storing large amounts of data through compression by sacrificing negligible data quality. More importantly, with the advent of compressed sensing techniques, using an appropriate representation basis and sampling technique, it is now possible to sample data far below the Shannon-Nyquist limit thus speeding up data acquisition and also reducing the complexity of data-acquisition hardware. In this research, we explore the application of various modern data analysis techniques such as proper orthogonal decomposition (POD), dynamic mode decomposition (DMD), compressed sensing, and Kalman filter and smoother in the data-driven analysis of dynamic systems with many degrees of freedom. This research has resulted in four novel methods. The first method is developed for denoising and spatial resolution enhancement of 4D-Flow MRI data based on POD and sparse reconstruction. The second method combines DMD and compressed sensing and takes discrete cosine transform (DCT) as the representation basis for dynamic denoising and gappy data reconstruction in 2D. The third method is a fast and parameter-free dynamic denoising method which combines a reduced-order model (ROM), a Kalman filter and smoother, and a DMD-based forward model. The fourth method is developed for reconstructing a 2D incompressible flow field by taking sparse measurements from the Fourier domain. As the reconstruction basis, a custom divergence-free set of basis vectors are derived and implemented

Advanced Image Acquisition, Processing Techniques and Applications

"Advanced Image Acquisition, Processing Techniques and Applications" is the first book of a series that provides image processing principles and practical software implementation on a broad range of applications. The book integrates material from leading researchers on Applied Digital Image Acquisition and Processing. An important feature of the book is its emphasis on software tools and scientific computing in order to enhance results and arrive at problem solution

Recent Advances in Signal Processing

The signal processing task is a very critical issue in the majority of new technological inventions and challenges in a variety of applications in both science and engineering fields. Classical signal processing techniques have largely worked with mathematical models that are linear, local, stationary, and Gaussian. They have always favored closed-form tractability over real-world accuracy. These constraints were imposed by the lack of powerful computing tools. During the last few decades, signal processing theories, developments, and applications have matured rapidly and now include tools from many areas of mathematics, computer science, physics, and engineering. This book is targeted primarily toward both students and researchers who want to be exposed to a wide variety of signal processing techniques and algorithms. It includes 27 chapters that can be categorized into five different areas depending on the application at hand. These five categories are ordered to address image processing, speech processing, communication systems, time-series analysis, and educational packages respectively. The book has the advantage of providing a collection of applications that are completely independent and self-contained; thus, the interested reader can choose any chapter and skip to another without losing continuity

Reduction of Limited Angle Artifacts in Medical Tomography via Image Reconstruction

Artifacts are unwanted effects in tomographic images that do not reflect the nature of the object. Their widespread occurrence makes their reduction and if possible removal an important subject in the development of tomographic image reconstruction algorithms. Limited angle artifacts are caused by the limited angular measurements, constraining the available tomographic information. This thesis focuses on reducing these artifacts via image reconstruction in two cases of incomplete measurements from: (1) the gaps left after the removal of high density objects such as dental fillings, screws and implants in computed tomography (CT) and (2) partial ring scanner configurations in positron emission tomography (PET). In order to include knowledge about the measurement and noise, prior terms were used within the reconstruction methods. Careful consideration was given to the trade-off between image blurring and noise reduction upon reconstruction of low-dose measurements.Development of reconstruction methods is an incremental process starting with testing on simple phantoms towards more clinically relevant ones by modeling the respective physical processes involved. In this work, phantoms were constructed to ensure that the proposed reconstruction methods addressed to the limited angle problem. The reconstructed images were assessed qualitatively and quantitatively in terms of noise reduction, edge sharpness and contrast recovery.Maximum a posteriori (MAP) estimation with median root prior (MRP) was selected for the reconstruction of limited angle measurements. MAP with MRP successfully reduced the artifacts caused by limited angle data in various datasets, tested with the reconstruction of both list-mode and projection data. In all cases, its performance was found to be superior to conventional reconstruction methods such as total-variation (TV) prior, maximum likelihood expectation maximization (MLEM) and filtered backprojection (FBP). MAP with MRP was also more robust with respect to parameter selection than MAP with TV prior.This thesis demonstrates the wide-range applicability of MAP with MRP in medical tomography, especially in low-dose imaging. Furthermore, we emphasize the importance of developing and testing reconstruction methods with application-specific phantoms, together with the properties and limitations of the measurements in mind