Towards a Mathematical Theory of Super-Resolution

This paper develops a mathematical theory of super-resolution. Broadly speaking, super-resolution is the problem of recovering the fine details of an object---the high end of its spectrum---from coarse scale information only---from samples at the low end of the spectrum. Suppose we have many point sources at unknown locations in $[0,1]$ and with unknown complex-valued amplitudes. We only observe Fourier samples of this object up until a frequency cut-off $f_c$. We show that one can super-resolve these point sources with infinite precision---i.e. recover the exact locations and amplitudes---by solving a simple convex optimization problem, which can essentially be reformulated as a semidefinite program. This holds provided that the distance between sources is at least $2/f_c$. This result extends to higher dimensions and other models. In one dimension for instance, it is possible to recover a piecewise smooth function by resolving the discontinuity points with infinite precision as well. We also show that the theory and methods are robust to noise. In particular, in the discrete setting we develop some theoretical results explaining how the accuracy of the super-resolved signal is expected to degrade when both the noise level and the {\em super-resolution factor} vary.Comment: 48 pages, 12 figure

NUMERICAL AND LABORATORY STUDY OF SEISMIC WAVES PROPAGATION, TEMPERATURE EFFECTS AND FLUID FLOWS IN MULTILAYERED MEDIA

Steel production by continuous casting is nowadays the most efficient method and the one that yields the best quality semi-finished products. The types of steel that can be produced varies greatly depending on the composition of the mixtures, the casting powders used to prevent oxidation and reduce heat loss, the cooling rate, and many other factors. During continuous casting, heat from the molten steel must be removed in large quantities and quickly to allow the first layer of solid skin to be created, so the continuous casting moulds, i.e. large hollow tubes generally made of copper alloys, are immersed in a conveyor with a closed water circuit where water circulates at high speed and pressure. In addition to water, there are also other parameters that can be monitored to increase production quality, such as powder deposition on the casting bath and steel level control. It would be useful to have automatic systems capable of replacing manual human control, to avoid the hazardous situations obviously present in steel mills, but also to increase knowledge of the production process through the acquisition of reliable data. This research aims to experimentally explore the possibility of measuring the level of molten steel in the mould by making time-of-flight measurements in the wall of the ingot using ultrasonic transducers similar the ones used for non-destructive testing of materials. These time-of-flight measurements are then converted to temperature and determine a thermal profile along the mould wall, from which the steel level is derived using an ad-hoc constructed algorithm. The research activity was divided into the realization of a real-time hardware and software system that was eventually adopted in real production systems as well. To understand how to design an initial prototype and how to choose the key parameters of the measurement system, a numerical model was implemented to simulate Gaussian beams, which are used to approximate the propagation of ultrasonic beams in even heterogeneous media, as in this case. The results obtained, both from numerical simulations and laboratory tests, made it possible to implement a first measurement tool that adopted a technique already known in the literature but innovative in the sense of application to an industrial context such as continuous castingSteel production by continuous casting is nowadays the most efficient method and the one that yields the best quality semi-finished products. The types of steel that can be produced varies greatly depending on the composition of the mixtures, the casting powders used to prevent oxidation and reduce heat loss, the cooling rate, and many other factors. During continuous casting, heat from the molten steel must be removed in large quantities and quickly to allow the first layer of solid skin to be created, so the continuous casting moulds, i.e. large hollow tubes generally made of copper alloys, are immersed in a conveyor with a closed water circuit where water circulates at high speed and pressure. In addition to water, there are also other parameters that can be monitored to increase production quality, such as powder deposition on the casting bath and steel level control. It would be useful to have automatic systems capable of replacing manual human control, to avoid the hazardous situations obviously present in steel mills, but also to increase knowledge of the production process through the acquisition of reliable data. This research aims to experimentally explore the possibility of measuring the level of molten steel in the mould by making time-of-flight measurements in the wall of the ingot using ultrasonic transducers similar the ones used for non-destructive testing of materials. These time-of-flight measurements are then converted to temperature and determine a thermal profile along the mould wall, from which the steel level is derived using an ad-hoc constructed algorithm. The research activity was divided into the realization of a real-time hardware and software system that was eventually adopted in real production systems as well. To understand how to design an initial prototype and how to choose the key parameters of the measurement system, a numerical model was implemented to simulate Gaussian beams, which are used to approximate the propagation of ultrasonic beams in even heterogeneous media, as in this case. The results obtained, both from numerical simulations and laboratory tests, made it possible to implement a first measurement tool that adopted a technique already known in the literature but innovative in the sense of application to an industrial context such as continuous castin

GPS-based CERN-LNGS time link for Borexino

We describe the design, the equipment, and the calibration of a new GPS based time link between CERN and the Borexino experiment at the Gran Sasso Laboratory in Italy. This system has been installed and operated in Borexino since March 2012, and used for a precise measurement of CNGS muon neutrinos speed in May 2012. The result of the measurement will be reported in a different letter.Comment: 13 pages, 11 figure

Spatial Vector Microwave Measurement

V této práci je představena nová interferometrická měřicí metoda pro měření koeficientu přenosu mezi dvěma anténami. Jestliže je přenos mezi anténami realizován odrazem od nějakého předmětu, lze metodu využít např. pro mikrovlnné zobrazování. Navržený systém obsahuje referenční větev obsahující anténu, která přímo ozařuje přijímací anténu a testovací větev, kde anténa ozařuje testovaný objekt. Elektromagnetická vlna z testovacího kanálu je od testovacího objektu odražena do přijímací antény, kde interferuje s vlnou z referenční větve. Pro jednoznačné získání fázového posunu mezi referenční a testovací vlnou jsou provedena postupně minimálně dvě interferometrická měření, kdy je v referenčním kanálu nastaven vhodný fázový posun a amplituda přenosu. Při měření můžeme provést více nezávislých interferometrických měření a vzniklá redundance může být využita ke zmenšení nejistot měření. Dále byl popsán způsob geometrické representace měření, který umožňuje názorně odhadnout nejisty měření. Nejistoty měření byly určeny i na základě numerické Monte Carlo metody. Navržená konfigurace byla ověřena jak přesným měřením za použití vektorového analyzátoru pro ověření nejistot měření, tak původní konfigurací pro ověření funkčnosti celého konceptu. Navrženou metodou bylo provedeno mikrovlnné zobrazování metodou inverzní syntetické apertury a byla tak ověřena použitelnost navrženého systému.This work presents a new interferometric measuring method for measuring the transmission coefficient between two antennas. If the transmission between the antennas is realized by a reflection from an object, the method can be used, e.g., for microwave imaging. The proposed system contains a reference branch containing an antenna that directly irradiates the receiving antenna and a test branch where the antenna irradiates the object under test. The electromagnetic wave from the test channel is reflected from the test object into the receiving antenna where it interferes with the wave from the reference channel. To achieve a unambiguous phase shift between the reference and test waves, at least two interferometric measurements are performed sequentially, with a suitable phase shift and the amplitude of the transmission being set in the reference channel. We can perform more independent interferometric measurements while redundancy can be used to reduce measurement uncertainty. Furthermore, a method of geometric representation of the measurement has been described which makes it possible to clearly estimate the measurement uncertainty. Measurement uncertainties were determined by the numerical Monte Carlo method. The proposed configuration has been verified by accurate measurements using a vector analyzer to verify measurement uncertainties, and the original configuration to verify the functionality of the entire concept. Microwave imaging using the inverse synthetic aperture method was performed to verify the usability of the proposed system

Wave modelling - the state of the art

This paper is the product of the wave modelling community and it tries to make a picture of the present situation in this branch of science, exploring the previous and the most recent results and looking ahead towards the solution of the problems we presently face. Both theory and applications are considered. The many faces of the subject imply separate discussions. This is reflected into the single sections, seven of them, each dealing with a specific topic, the whole providing a broad and solid overview of the present state of the art. After an introduction framing the problem and the approach we followed, we deal in sequence with the following subjects: (Section) 2, generation by wind; 3, nonlinear interactions in deep water; 4, white-capping dissipation; 5, nonlinear interactions in shallow water; 6, dissipation at the sea bottom; 7, wave propagation; 8, numerics. The two final sections, 9 and 10, summarize the present situation from a general point of view and try to look at the future developments

Absolute flux density calibrations of radio sources: 2.3 GHz

A detailed description of a NASA/JPL Deep Space Network program to improve S-band gain calibrations of large aperture antennas is reported. The program is considered unique in at least three ways; first, absolute gain calibrations of high quality suppressed-sidelobe dual mode horns first provide a high accuracy foundation to the foundation to the program. Second, a very careful transfer calibration technique using an artificial far-field coherent-wave source was used to accurately obtain the gain of one large (26 m) aperture. Third, using the calibrated large aperture directly, the absolute flux density of five selected galactic and extragalactic natural radio sources was determined with an absolute accuracy better than 2 percent, now quoted at the familiar 1 sigma confidence level. The follow-on considerations to apply these results to an operational network of ground antennas are discussed. It is concluded that absolute gain accuracies within + or - 0.30 to 0.40 db are possible, depending primarily on the repeatability (scatter) in the field data from Deep Space Network user stations

The nature of relaxation processes revealed by the action signals of phase modulated light fields

We introduce a generalized theoretical approach to study action signals induced by the absorption of two-photons from two phase modulated laser beams and subject it to experimental testing for two types of photoactive samples, solution of rhodamine 6G and GaP photodiode. In our experiment, the phases of the laser beams are modulated at the frequencies f1 and f2, respectively. The action signals, such as photoluminescence and photocurrent, which result from the absorption of two photons, are isolated at frequencies m f (f=|f1-f2|, m=0,1,2...). We demonstrate that the ratio of the amplitudes of the secondary (m=2) and the primary (m=1) signals is sensitive to the type of relaxation process taken place in the system and thus can be used for its identification. Such sensitivity originates from cumulative effects of non-equilibrated state of the system between the light pulses. When the cumulative effects are small, i.e. the relaxation time is much shorter then the laser repetition rate or the laser intensity is high enough to dominate the system behavior, the ratio achieves its reference value 1:4 (the signature of two-photon absorption). In the intermediate regimes the ratio changes rapidly with the growth of intensity from zero value in case of second order relaxation process, while it demonstrates slow monotonic decrease for linear relaxation. In the article we also determine the value of the recombination rate in a GaP photodiode by using the above approach

SparsePak: A Formatted Fiber Field-Unit for The WIYN Telescope Bench Spectrograph. II. On-Sky Performance

We present a performance analysis of SparsePak and the WIYN Bench Spectrograph for precision studies of stellar and ionized gas kinematics of external galaxies. We focus on spectrograph configurations with echelle and low-order gratings yielding spectral resolutions of ~10000 between 500-900nm. These configurations are of general relevance to the spectrograph performance. Benchmarks include spectral resolution, sampling, vignetting, scattered light, and an estimate of the system absolute throughput. Comparisons are made to other, existing, fiber feeds on the WIYN Bench Spectrograph. Vignetting and relative throughput are found to agree with a geometric model of the optical system. An aperture-correction protocol for spectrophotometric standard-star calibrations has been established using independent WIYN imaging data and the unique capabilities of the SparsePak fiber array. The WIYN point-spread-function is well-fit by a Moffat profile with a constant power-law outer slope of index -4.4. We use SparsePak commissioning data to debunk a long-standing myth concerning sky-subtraction with fibers: By properly treating the multi-fiber data as a ``long-slit'' it is possible to achieve precision sky subtraction with a signal-to-noise performance as good or better than conventional long-slit spectroscopy. No beam-switching is required, and hence the method is efficient. Finally, we give several examples of science measurements which SparsePak now makes routine. These include H$\alpha$ velocity fields of low surface-brightness disks, gas and stellar velocity-fields of nearly face-on disks, and stellar absorption-line profiles of galaxy disks at spectral resolutions of ~24,000.Comment: To appear in ApJSupp (Feb 2005); 19 pages text; 7 tables; 27 figures (embedded); high-resolution version at http://www.astro.wisc.edu/~mab/publications/spkII_pre.pd