Signal processing with Fourier analysis, novel algorithms and applications

Fourier analysis is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions, also analogously known as sinusoidal modeling. The original idea of Fourier had a profound impact on mathematical analysis, physics and engineering because it diagonalizes time-invariant convolution operators. In the past signal processing was a topic that stayed almost exclusively in electrical engineering, where only the experts could cancel noise, compress and reconstruct signals. Nowadays it is almost ubiquitous, as everyone now deals with modern digital signals. Medical imaging, wireless communications and power systems of the future will experience more data processing conditions and wider range of applications requirements than the systems of today. Such systems will require more powerful, efficient and flexible signal processing algorithms that are well designed to handle such needs. No matter how advanced our hardware technology becomes we will still need intelligent and efficient algorithms to address the growing demands in signal processing. In this thesis, we investigate novel techniques to solve a suite of four fundamental problems in signal processing that have a wide range of applications. The relevant equations, literature of signal processing applications, analysis and final numerical algorithms/methods to solve them using Fourier analysis are discussed for different applications in the electrical engineering/computer science. The first four chapters cover the following topics of central importance in the field of signal processing: • Fast Phasor Estimation using Adaptive Signal Processing (Chapter 2) • Frequency Estimation from Nonuniform Samples (Chapter 3) • 2D Polar and 3D Spherical Polar Nonuniform Discrete Fourier Transform (Chapter 4) • Robust 3D registration using Spherical Polar Discrete Fourier Transform and Spherical Harmonics (Chapter 5) Even though each of these four methods discussed may seem completely disparate, the underlying motivation for more efficient processing by exploiting the Fourier domain signal structure remains the same. The main contribution of this thesis is the innovation in the analysis, synthesis, discretization of certain well known problems like phasor estimation, frequency estimation, computations of a particular non-uniform Fourier transform and signal registration on the transformed domain. We conduct propositions and evaluations of certain applications relevant algorithms such as, frequency estimation algorithm using non-uniform sampling, polar and spherical polar Fourier transform. The techniques proposed are also useful in the field of computer vision and medical imaging. From a practical perspective, the proposed algorithms are shown to improve the existing solutions in the respective fields where they are applied/evaluated. The formulation and final proposition is shown to have a variety of benefits. Future work with potentials in medical imaging, directional wavelets, volume rendering, video/3D object classifications, high dimensional registration are also discussed in the final chapter. Finally, in the spirit of reproducible research we release the implementation of these algorithms to the public using Github

Optimization problems in electron microscopy of single particles

The final publication is available at Springer via http://dx.doi.org/10.1007/s10479-006-0078-8Electron Microscopy is a valuable tool for the elucidation of the three-dimensional structure of macromolecular complexes. Knowledge about the macromolecular structure provides important information about its function and how it is carried out. This work addresses the issue of three-dimensional reconstruction of biological macromolecules from electron microscopy images. In particular, it focuses on a methodology known as “single-particles” and makes a thorough review of all those steps that can be expressed as an optimization problem. In spite of important advances in recent years, there are still unresolved challenges in the field that offer an excellent testbed for new and more powerful optimization techniques.We acknowledge partial support from the “Comunidad Autónoma de Madrid” through grants CAM-07B-0032-2002, GR/SAL/0653/2004 and GR/SAL/0342/2004, the “Comisión Interministerial de Ciencia yTecnologia” of Spain through grants BIO2001-1237, BIO2001-4253-E, BIO2001-4339-E, BIO2002- 10855-E, BFU2004-00217/BMC, the Spanish FIS grant (G03/185), the European Union through grants QLK2- 2000-00634, QLRI-2000-31237, QLRT-2000-0136, QLRI-2001-00015, FP6-502828 and the NIH through grant HL70472. Alberto Pascual and Roberto Marabini acknowledge support by the Spanish Ramon y Cajal Program

Medical image analysis methods for anatomical surface reconstruction using tracked 3D ultrasound

The thesis focuses on a study of techniques for acquisition and reconstruction of surface data from anatomical objects by means of tracked 3D ultrasound. In the context of the work two experimental scanning systems are developed and tested on both artificial objects and biological tissues. The first system is based on the freehand ultrasound principle and utilizes a conventional 2D ultrasound transducer coupled with an electromechanical 3D position tracker. The main properties and the basic features of this system are discussed. A number of experiments show that its accuracy in the close to ideal conditions reaches 1.2 mm RMS. The second proposed system implements the sequential triggered scanning approach. The system consists of an ultrasound machine, a workstation and a scanning body (a moving tank filled with liquid and a transducer fixation block) that performs transducer positioning and tracking functions. The system is tested on artificial and real bones. The performed experiments illustrate that it provides significantly better accuracy than the freehand ultrasound (about 0.2 mm RMS) and allows acquiring regular data with a good precision. This makes such a system a promising tool for orthopaedic and trauma surgeons during contactless X-ray-free examinations of injured extremities. The second major subject of the thesis concerns development of medical image analysis methods for 3D surface reconstruction and 2D object detection. We introduce a method based on mesh-growing surface reconstruction that is designed for noisy and sparse data received from 3D tracked ultrasound scanners. A series of experiments on synthetic and ultrasound data show an appropriate reconstruction accuracy. The reconstruction error is measured as the averaged distance between the faces of the mesh and the points from the cloud. Dependently on the initial settings of the method the error varies in range 0.04 - 0.2% for artificial data and 0.3 - 0.7 mm for ultrasound bone data. The reconstructed surfaces correctly interpolate the original point clouds and demonstrate proper smoothness. The next significant problem considered in the work is 2D object detection. Although medical object detection is not integrated into the developed scanning systems, it can be used as a possible further extension of the systems for automatic detection of specific anatomical structures. We analyse the existent object detection methods and introduce a modification of the one based on the popular Generalized Hough Transform (GHT). Unlike the original GHT, the developed method is invariant to rotation and uniform scaling, and uses an intuitive two-point parametrization. We propose several implementations of the feature-to-vote conversion function with the corresponding vote analysis principles. Special attention is devoted to a study of the hierarchical vote analysis and its probabilistic properties. We introduce a parameter space subdivision strategy that reduces the probability of vote peak omission, and show that it can be efficiently implemented in practice using the Gumbel probability distribution

Numerical Approaches for Solving the Combined Reconstruction and Registration of Digital Breast Tomosynthesis

Heavy demands on the development of medical imaging modalities for breast cancer detection have been witnessed in the last three decades in an attempt to reduce the mortality associated with the disease. Recently, Digital Breast Tomosynthesis (DBT) shows its promising in the early diagnosis when lesions are small. In particular, it offers potential benefits over X-ray mammography - the current modality of choice for breast screening - of increased sensitivity and specificity for comparable X-ray dose, speed, and cost. An important feature of DBT is that it provides a pseudo-3D image of the breast. This is of particular relevance for heterogeneous dense breasts of young women, which can inhibit detection of cancer using conventional mammography. In the same way that it is difficult to see a bird from the edge of the forest, detecting cancer in a conventional 2D mammogram is a challenging task. Three-dimensional DBT, however, enables us to step through the forest, i.e., the breast, reducing the confounding effect of superimposed tissue and so (potentially) increasing the sensitivity and specificity of cancer detection. The workflow in which DBT would be used clinically, involves two key tasks: reconstruction, to generate a 3D image of the breast, and registration, to enable images from different visits to be compared as is routinely performed by radiologists working with conventional mammograms. Conventional approaches proposed in the literature separate these steps, solving each task independently. This can be effective if reconstructing using a complete set of data. However, for ill-posed limited-angle problems such as DBT, estimating the deformation is difficult because of the significant artefacts associated with DBT reconstructions, leading to severe inaccuracies in the registration. The aim of my work is to find and evaluate methods capable of allying these two tasks, which will enhance the performance of each process as a result. Consequently, I prove that the processes of reconstruction and registration of DBT are not independent but reciprocal. This thesis proposes innovative numerical approaches combining reconstruction of a pair of temporal DBT acquisitions with their registration iteratively and simultaneously. To evaluate the performance of my methods I use synthetic images, breast MRI, and DBT simulations with in-vivo breast compressions. I show that, compared to the conventional sequential method, jointly estimating image intensities and transformation parameters gives superior results with respect to both reconstruction fidelity and registration accuracy

Directional edge and texture representations for image processing

An efficient representation for natural images is of fundamental importance in image processing and analysis. The commonly used separable transforms such as wavelets axe not best suited for images due to their inability to exploit directional regularities such as edges and oriented textural patterns; while most of the recently proposed directional schemes cannot represent these two types of features in a unified transform. This thesis focuses on the development of directional representations for images which can capture both edges and textures in a multiresolution manner. The thesis first considers the problem of extracting linear features with the multiresolution Fourier transform (MFT). Based on a previous MFT-based linear feature model, the work extends the extraction method into the situation when the image is corrupted by noise. The problem is tackled by the combination of a "Signal+Noise" frequency model, a refinement stage and a robust classification scheme. As a result, the MFT is able to perform linear feature analysis on noisy images on which previous methods failed. A new set of transforms called the multiscale polar cosine transforms (MPCT) are also proposed in order to represent textures. The MPCT can be regarded as real-valued MFT with similar basis functions of oriented sinusoids. It is shown that the transform can represent textural patches more efficiently than the conventional Fourier basis. With a directional best cosine basis, the MPCT packet (MPCPT) is shown to be an efficient representation for edges and textures, despite its high computational burden. The problem of representing edges and textures in a fixed transform with less complexity is then considered. This is achieved by applying a Gaussian frequency filter, which matches the disperson of the magnitude spectrum, on the local MFT coefficients. This is particularly effective in denoising natural images, due to its ability to preserve both types of feature. Further improvements can be made by employing the information given by the linear feature extraction process in the filter's configuration. The denoising results compare favourably against other state-of-the-art directional representations

Quantifying function in the zebrafish embryonic heart: a study on the role of timed mechanical cues

2014 Summer.Congenital heart defects are a relatively common problem, yet the cause is unknown in the large majority of cases. A significant amount of past research has shown that there is a link between altered blood-induced mechanical stress and heart development. However, very little research has been done to examine the effect of altered loading timing. During embryonic development, the heart undergoes a drastic change in morphology from its original valveless tube structure to a complete multi-chambered pump with valves. Blood flow dynamics are consequently altered significantly as well. Given the changes occurring through this period, it is hypothesized that significant and persistent decreases in heart function occur when cardiac loading is altered during certain time windows of early development. The main objectives of this work were to (1) develop a methodology to quantify heart function in the embryonic zebrafish from high-speed bright field images, (2) develop a model for temporary and noninvasive alteration of cardiac loading, and (3) apply the methodology to normal and treated embryos to determine whether certain time windows of altered loading are more impactful than others. Results indicated that altered loading during the tube and early looping stages of development produce persistent changes in heart morphology along with accompanying decreases in cardiac function. Altered loading during late cardiac looping resulted in temporary changes in function which did not persist through the latest time point measured. This work has produced extensive tools for quantifying heart function from high speed images and presents a new model for altered cardiac loading in the zebrafish. Results support the hypothesis that the heart is more sensitive to altered loading during certain windows in development. This provides new insight into how congenital defects may develop and sets the stage for future experiments investigating the effects of altered loading on heart development