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

Algorithms for LiDAR Based Traffic Tracking: Development and Demonstration

The current state of the art of traffic tracking is based on the use of video, and requires extensive manual intervention for it to work, including hours of painstaking human examination of videos frame by frame which also make the acquisition of data extremely expensive. Fundamentally, this is because we do not have observability of the actual scene from a camera which captures a 2D projection of the 3D world. Even if video were to be automated, it would involve such algorithms as RANSACK for outlier elimination while matching features across frames or across multiple cameras. This results in algorithms without stationary relationships between input and output statistics, i.e., between sensing resolution and error and estimated positions and velocities. LiDAR directly provides 3D point clouds, giving a one-one mapping between the scene from the physical world and data. However, available eye-safe lidars have been developed for autonomous vehicles, and provide only sparse point clouds when used for longer range data acquisition. Our experimental results use the Velodyne HDL 64E lidar. The sparse nature of data points returned by the Velodyne LiDAR rendered most of the algorithms for object identification and tracking using 3D point clouds at the point cloud library (PCL), a leading multi-agency open source research initiative focused on 3D point cloud processing ineffective for our work. Hence I developed a comprehensive set of algorithms developed to identify and remove background; detect objects through clustering of remaining points; associate detected objects across frames, track the detected objects, and estimate the dimension of objects. Two different complementary algorithms based on, surface equation (in 3D Cartesian coordinates) and LiDAR spherical coordinates were developed for background identification and removal. Delaunay triangulation based clustering is performed to identify objects. Kalman filter and Hungarian assignment algorithm are used in tandem to track multiple objects simultaneously. A novel bounding box algorithm was devised taking advantage of the way LiDAR scans the environment to predict the orientation and estimate dimension of objects. Trajectory analysis is performed to identify and split any wrong associations, join trajectories belonging to same object and stitch partial trajectories. Finally, the results are stored in a format usable by various transportation or traffic engineering applications. The algorithms were tested by peers with data collected at three intersections. Detection rate and counting accuracy are above 95% which is on par with commercial video solutions that employ humans to varying degrees. While prototyping for the algorithms was done it MATLAB, preliminary tests of conversion to C++ showed that the developed algorithms can be executed in real time on standard computer hardware

Cloud To Cloud Registration For 3d Point Data

The vast potential of digital representation of objects by large collections of 3D points is being recognized on a global scale and has given rise to the popularity of point cloud data (PCD). 3D imaging sensors provide a means for quickly capturing dense and accurate geospatial information that represent the 3D geometry of objects in a digital environment. Due to spatial and temporal constraints, it is quite common that two or more sets of PCD are obtained to provide full 3D analysis. It is therefore quite essential that all the PCD are referenced to a homogeneous coordinate frame of reference. This homogeneity in coordinates is achieved through a point cloud registration task and it involves determining a set of transformation parameters and applying those parameters to transform one dataset into another reference frame or to a global reference frame. The registration task typically involves the use of targets or other geometric features that are recognizable in the different sets of PCD. The recognition of these features usually involves the use of imagery, either intensity images or true-color images or both. In this dissertation, cloud-to-cloud registration, which is also called surface matching or surface registration is investigated as an alternative registration method, which has potential for improved automation and accuracy. The challenge in cloud-to-cloud registration lies in the fact that PCD are usually unstructured and possess little semantics. Two novel techniques were developed in this dissertation, one for the pairwise registration of PCD and the other for the global registration of PCD. The developed algorithms were evaluated by comparing with popular approaches and improvements in registration accuracy up to four fold were obtained. The improvement obtained may be attributed to some of the novel considerations introduced in this dissertation. The main novel idea is the simultaneous consideration of the stochastic properties of a pair of scans via the symmetric correspondence

Common Arc Method for Diffraction Pattern Orientation

Very short pulses of x-ray free-electron lasers opened the way to obtain diffraction signal from single particles beyond the radiation dose limit. For 3D structure reconstruction many patterns are recorded in the object's unknown orientation. We describe a method for orientation of continuous diffraction patterns of non-periodic objects, utilizing intensity correlations in the curved intersections of the corresponding Ewald spheres, hence named Common Arc orientation. Present implementation of the algorithm optionally takes into account the Friedel law, handles missing data and is capable to determine the point group of symmetric objects. Its performance is demonstrated on simulated diffraction datasets and verification of the results indicates high orientation accuracy even at low signal levels. The Common Arc method fills a gap in the wide palette of the orientation methods.Comment: 16 pages, 10 figure