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Person# Golnooshsadat Elhami

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Information

Information is an abstract concept that refers to that which has the power to inform. At the most fundamental level, information pertains to the interpretation (perhaps formally) of that which may be

Simulation

A simulation is the imitation of the operation of a real-world process or system over time. Simulations require the use of models; the model represents the key characteristics or behaviors of the se

Problem solving

Problem solving is the process of achieving a goal by overcoming obstacles, a frequent part of most activities. Problems in need of solutions range from simple personal tasks (e.g. how to turn on an a

People doing similar research (113)

Sampling has always been at the heart of signal processing providing a bridge between the analogue world and discrete representations of it, as our ability to process data in continuous space is quite limited. Furthermore, sampling plays a key part in understanding how to efficiently capture, store and process signals. Shannon's sampling theorem states that if the original signal is known to have a limited bandwidth, we can retrieve the signal from uniformly-spaced samples, provided that the sampling rate is greater than twice the highest frequency in the signal. Here, we see two key attributes: prior knowledge on the original signal (limited bandwidth) and a constrained sampling setup (uniform samples at a particular rate). In this thesis, we make weaker assumptions on the sampling setup by assuming that some information, such as the sample positions, is lost. We show that under proper prior knowledge, we can reconstruct the signal from its samples uniquely or up to some equivalence class. We start by the problem of linear sampling of discrete signals, where the sample values are known, but their order is lost. In general, the original signal is impossible to retrieve from the samples, but we show that by taking out symmetry from the sampling vectors, we can reconstruct the signal uniquely. We provide an efficient algorithm to find the sample orders and thus reconstruct the original signal. We also study the problem of reconstructing a continuous signal from samples taken at unknown locations. The lost sample locations take away any hope of uniquely retrieving the signal without prior knowledge. We show that this problem is equivalent to reconstructing a composite of functions from uniformly spaced samples. Then we provide an efficient algorithm that can recover bandlimited signals warped by a linear function uniquely given enough sampling frequency. We then investigate a problem, dubbed shape from bandwidth, where we have uniform samples from a picture (projection) of an unknown surface that is painted with an unknown texture. The goal is to reconstruct the shape of the surface from these samples. We show that having prior knowledge of the texture bandwidth provides us with enough information to reconstruct the surface from its picture. We provide reconstruction algorithms for both orthogonal and central projections and provide equivalence classes of solutions in each case. Next, in two consecutive chapters, using techniques from geometrical signal processing, we offer new designs for 3-D barcodes, whose information can be retrieved from a single projection using penetrating waves from an unknown direction. Because of the unknown scan direction, the correct correspondence of the samples to the information bits in the barcodes is lost. In this case, we use the known shape of the barcode as prior knowledge to estimate the unknown scan direction from the samples, and then transform the reconstruction as a linear problem that can be solved efficiently. Finally, we cover the theory of coordinate difference matrices (CDMs): matrices that store mutual differences between coordinates of points (sensors, microphones, molecules, etc.) in space. We show how we can leverage specific properties of these matrices, such as their low rank, as prior knowledge in order to reconstruct the position of the points in space using CDMs. We use our reconstruction algorithm to solve many real-life signal processing problems.

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In many problems such as phase retrieval, molecular biology, source localization, and sensor array calibration, one can measure vector differences between pairs of points and attempt to recover the position of these points; this class of problems is called vector geometry problems (VGPs). A closely related field studies distance geometry problems (DGPs), where only the Euclidean distance between pairs of points is available. This has been extensively studied in the literature and is often associated with Euclidean distance matrices (EDMs). Although similar to DGPs, VGPs have received little attention in the literature; our goal is to fill in this gap and introduce a framework to solve VGPs. Inspired by EDM-related approaches, we arrange the differences in what we call a coordinate difference matrix (CDM) and introduce a methodology to reconstruct a set of points from CDM entries. We first propose a reconstruction scheme in 1D and then generalize it to higher dimensions. We show that our algorithm is optimal in the least-squares sense, even when we have only access to partial measurements. In addition, we provide necessary and sufficient conditions on the number and structure of measurements needed for a successful reconstruction, as well as a comparison with EDMs. In particular we show that compared to EDMs, CDMs are simpler objects, both from an algorithmic and a theoretical point of view. Therefore, CDMs should be favored over EDMs whenever vector differences are available. In the presence of noise, we provide a statistical analysis of the reconstruction error. Finally, we apply the established knowledge to five practical problems to demonstrate the versatility of this theory and showcase the wide range of applications covered by the CDM framework.

Golnooshsadat Elhami, Adam James Scholefield, Martin Vetterli

We consider three-dimensional cubic barcodes, consisting of smaller cubes, each built from one of two possible materials and carry one bit of information. To retrieve the information stored in the barcode, we measure a 2-D projection of the barcode using a penetrating wave such as X-rays, either using parallel-beam or cone-beam scanners from an unknown direction. We derive a theoretical representation of this scanning process and show that for a known barcode pose with respect to the scanner, the projection operator is linear and can be easily inverted. Moreover, we provide a method to estimate the unknown pose of the barcode from a single 2-D scan. We also propose coding schemes to correct errors and ambiguities in the reconstruction process. Finally, we test our designed barcode and reconstruction algorithms with several simulations, as well as a real-world barcode acquired with an X-ray cone-beam scanner, as a proof of concept.

2021