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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.
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Can one hear the shape of a room?'' question, and we answer it with a qualified
yes''. Even a single impulse response uniquely describes a convex polyhedral room, whereas a more practical algorithm to reconstruct the room's geometry uses only first-order echoes and a few microphones. Next, we show how different problems of localization benefit from echoes. The first one is multiple indoor sound source localization. Assuming the room is known, we show that discretizing the Helmholtz equation yields a system of sparse reconstruction problems linked by the common sparsity pattern. By exploiting the full bandwidth of the sources, we show that it is possible to localize multiple unknown sound sources using only a single microphone. We then look at indoor localization with known pulses from the geometric echo perspective introduced previously. Echo sorting enables localization in non-convex rooms without a line-of-sight path, and localization with a single omni-directional sensor, which is impossible without echoes. A closely related problem is microphone position calibration; we show that echoes can help even without assuming that the room is known. Using echoes, we can localize arbitrary numbers of microphones at unknown locations in an unknown room using only one source at an unknown location---for example a finger snap---and get the room's geometry as a byproduct. Our study of source localization outgrew the initial form factor when we looked at source localization with spherical microphone arrays. Spherical signals appear well beyond spherical microphone arrays; for example, any signal defined on Earth's surface lives on a sphere. This resulted in the first slight departure from the main theme: We develop the theory and algorithms for sampling sparse signals on the sphere using finite rate-of-innovation principles and apply it to various signal processing problems on the sphere.