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Automotive security

Automotive security refers to the branch of computer security focused on the cyber risks related to the automotive context. The increasingly high number of ECUs in vehicles and, alongside, the implementation of multiple different means of communication from and towards the vehicle in a remote and wireless manner led to the necessity of a branch of cybersecurity dedicated to the threats associated with vehicles. Not to be confused with automotive safety. == Causes == The implementation of multiple ECUs (Electronic Control Units) inside vehicles began in the early '70s thanks to the development of integrated circuits and microprocessors that made it economically feasible to produce the ECUs on a large scale. Since then the number of ECUs has increased to up to 100 per vehicle. These units nowadays control almost everything in the vehicle, from simple tasks such as activating the wipers to more safety-related ones like brake-by-wire or ABS (Anti-lock Braking System). Autonomous driving is also strongly reliant on the implementation of new, complex ECUs such as the ADAS, alongside sensors (lidars and radars) and their control units. Inside the vehicle, the ECUs are connected with each other through cabled or wireless communication networks, such as CAN bus (controller area network), MOST bus (Media Oriented System Transport), FlexRay (Automotive Network Communications Protocol) or RF (radio frequency) as in many implementations of TPMSs (tire-pressure monitoring systems). Many of these ECUs require data received through these networks that arrive from various sensors to operate and use such data to modify the behavior of the vehicle (e.g., the cruise control modifies the vehicle's speed depending on signals arriving from a button usually located on the steering wheel). Since the development of cheap wireless communication technologies such as Bluetooth, LTE, Wi-Fi, RFID and similar, automotive producers and OEMs have designed ECUs that implement such technologies with the goal of improving the experience of the driver and passengers. Safety-related systems such as the OnStar from General Motors, telematic units, communication between smartphones and the vehicle's speakers through Bluetooth, Android Auto and Apple CarPlay. == Threat model == Threat models of the automotive world are based on both real-world and theoretically possible attacks. Most real-world attacks aim at the safety of the people in and around the car, by modifying the cyber-physical capabilities of the vehicle (e.g., steering, braking, accelerating without requiring actions from the driver), while theoretical attacks have been supposed to focus also on privacy-related goals, such as obtaining GPS data on the vehicle, or capturing microphone signals and similar. Regarding the attack surfaces of the vehicle, they are usually divided in long-range, short-range, and local attack surfaces: LTE and DSRC can be considered long-range ones, while Bluetooth and Wi-Fi are usually considered short-range although still wireless. Finally, USB, OBD-II and all the attack surfaces that require physical access to the car are defined as local. An attacker that is able to implement the attack through a long-range surface is considered stronger and more dangerous than the one that requires physical access to the vehicle. In 2015 the possibility of attacks on vehicles already on the market has been proven possible by Miller and Valasek, that managed to disrupt the driving of a Jeep Cherokee while remotely connecting to it through remote wireless communication. === Controller area network attacks === The most common network used in vehicles and the one that is mainly used for safety-related communication is CAN, due to its real-time properties, simplicity, and cheapness. For this reason the majority of real-world attacks have been implemented against ECUs connected through this type of network. The majority of attacks demonstrated either against actual vehicles or in testbeds fall in one or more of the following categories: ==== Sniffing ==== Sniffing in the computer security field generally refers to the possibility of intercepting and logging packets or more generally data from a network. In the case of CAN, since it is a bus network, every node listens to all communication on the network. It is useful for the attacker to read data to learn the behavior of the other nodes of the network before implementing the actual attack. Usually, the final goal of the attacker is not to simply sniff the data on CAN, since the packets passing on this type of network are not usually valuable just to read. ==== Denial of service ==== Denial of service (DoS) in information security is usually described as an attack that has the objective of making a machine or a network unavailable. DoS attacks against ECUs connected to CAN buses can be done both against the network, by abusing the arbitration protocol used by CAN to always win the arbitration, and targeting the single ECU, by abusing the error handling protocol of CAN. In this second case the attacker flags the messages of the victim as faulty to convince the victim of being broken and therefore shut itself off the network. ==== Spoofing ==== Spoofing attacks comprise all cases in which an attacker, by falsifying data, sends messages pretending to be another node of the network. In automotive security usually spoofing attacks are divided into masquerade and replay attacks. Replay attacks are defined as all those where the attacker pretends to be the victim and sends sniffed data that the victim sent in a previous iteration of authentication. Masquerade attacks are, on the contrary, spoofing attacks where the data payload has been created by the attacker. == Real life automotive threat example == Security researchers Charlie Miller and Chris Valasek have successfully demonstrated remote access to a wide variety of vehicle controls using a Jeep Cherokee as the target. They were able to control the radio, environmental controls, windshield wipers, and certain engine and brake functions. The method used to hack the system was implementation of pre-programmed chip into the controller area network (CAN) bus. By inserting this chip into the CAN bus, he was able to send arbitrary message to CAN bus. One other thing that Miller has pointed out is the danger of the CAN bus, as it broadcasts the signal which the message can be caught by the hackers throughout the network. The control of the vehicle was all done remotely, manipulating the system without any physical interaction. Miller states that he could control any of some 1.4 million vehicles in the United States regardless of the location or distance, the only thing needed is for someone to turn on the vehicle to gain access. The work by Miller and Valasek replicated earlier work completed and published by academics in 2010 and 2011 on a different vehicle. The earlier work demonstrated the ability to compromise a vehicle remotely, over multiple wireless channels (including cellular), and the ability to remotely control critical components on the vehicle post-compromise, including the telematics unit and the car's brakes. While the earlier academic work was publicly visible, both in peer-reviewed scholarly publications and in the press, the Miller and Valesek work received even greater public visibility. == Security measures == The increasing complexity of devices and networks in the automotive context requires the application of security measures to limit the capabilities of a potential attacker. Since the early 2000 many different countermeasures have been proposed and, in some cases, applied. Following, a list of the most common security measures: Sub-networks: to limit the attacker capabilities even if he/she manages to access the vehicle from remote through a remotely connected ECU, the networks of the vehicle are divided in multiple sub-networks, and the most critical ECUs are not placed in the same sub-networks of the ECUs that can be accessed from remote. Gateways: the sub-networks are divided by secure gateways or firewalls that block messages from crossing from a sub-network to the other if they were not intended to. Intrusion Detection Systems (IDS): on each critical sub-network, one of the nodes (ECUs) connected to it has the goal of reading all data passing on the sub-network and detect messages that, given some rules, are considered malicious (made by an attacker). The arbitrary messages can be caught by the passenger by using IDS which will notify the owner regarding with unexpected message. Authentication protocols: in order to implement authentication on networks where it is not already implemented (such as CAN), it is possible to design an authentication protocol that works on the higher layers of the ISO OSI model, by using part of the data payload of a message to authenticate the message itself. Hardware Security Modules: since many ECUs are not powerful enough to keep real-time delays whi
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Image warping

Image warping is the process of digitally manipulating an image such that any shapes portrayed in the image have been significantly distorted. Warping may be used for correcting image distortion as well as for creative purposes (e.g., morphing). The same techniques are equally applicable to video. While an image can be transformed in various ways, pure warping means that points are mapped to points without changing the colors. This can be based mathematically on any function from (part of) the plane to the plane. If the function is injective the original can be reconstructed. If the function is a bijection any image can be inversely transformed. Some methods are: Images may be distorted through simulation of optical aberrations. Images may be viewed as if they had been projected onto a curved or mirrored surface. (This is often seen in ray traced images.) Images can be partitioned into image polygons and each polygon distorted. Images can be distorted using morphing. The most obvious approach to transforming a digital image is the forward mapping. This applies the transform directly to the source image, typically generating unevenly-spaced points that will then be interpolated to generate the required regularly-spaced pixels. However, for injective transforms reverse mapping is also available. This applies the inverse transform to the target pixels to find the unevenly-spaced locations in the source image that contribute to them. Estimating them from source image pixels will require interpolation of the source image. To work out what kind of warping has taken place between consecutive images, one can use optical flow estimation techniques. == Image warping toolbox == ImWIP is an open-source, image warping tool for modeling deformation and motion in digital images, which contains differentiable image warping operators, together with their exact adjoints and derivatives.
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Apps to analyse COVID-19 sounds

Apps to analyse COVID-19 sounds are mobile software applications designed to collect respiratory sounds and aid diagnosis in response to the COVID-19 pandemic. Numerous applications are in development, with different institutions and companies taking various approaches to privacy and data collection. Current efforts are aimed at gathering data. In a later stage, it is possible that sound apps will have the capacity (and ethical approvals) to provide information back to users. In order to develop and train signal analysis approaches, large datasets are required. == History == The COVID-19 outbreak was announced as a global pandemic by the World Health Organization in March 2020 and has affected a growing number of people globally. In this context, advanced artificial intelligence techniques are being considered as tools in aiding our response to global health crisis. Other COVID-19 apps which offer solutions for user tracking have been developed. At the same time a number of approaches which tries to use respiratory sounds and artificial intelligence to understand if the disease can be diagnosed have been proposed. A few studies are available as preprints (i.e. not yet peer-reviewed) documents. == Methodologies == The potential for using speech and sound analysis by artificial intelligence to help in this scenario, by surveying which types of related or contextually significant phenomena can be automatically assessed from speech or sound has been recently overviewed. These include the automatic recognition and monitoring of breathing, dry and wet coughing or sneezing sounds, speech under cold, eating behaviour, sleepiness, or pain. Additionally, the potential use-cases of intelligent speech analysis for COVID-19 diagnosed patients has also been presented. In particular, by analysing speech recordings from these patients, an audio-only-based model to automatically categorise the health state of patients from four aspects, including the severity of illness, sleep quality, fatigue, and anxiety, is constructed. This work shows promise in estimating the severity of illness. Machine learning methods have been explored to recognize and diagnose coughs from different diseases. These included a low complexity, automated recognition and diagnostic tool for screening respiratory infections that utilizes convolutional neural networks (CNNs) to detect cough within environment audio and diagnose three potential illnesses (i.e. bronchitis, bronchiolitis and pertussis) based on their unique cough audio features. A large-scale crowdsourced dataset of respiratory sounds has been collected to aid diagnosis of COVID-19: coughs and breathing sounds are sufficient to distinguish users affected by COVID-19 versus those affected by asthma or healthy controls. Behind these studies is the ambition that automated systems to screen for respiratory diseases based on voice, raw cough or other sound data would have positive medical applications in both clinical and public health arenas. == List of apps to analyse COVID-19 sounds ==
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Spatial anti-aliasing

In digital signal processing, spatial anti-aliasing is a technique for minimizing the distortion artifacts (aliasing) when representing a high-resolution image at a lower resolution. Anti-aliasing is used in digital photography, computer graphics, digital audio, and many other applications. Anti-aliasing means removing signal components that have a higher frequency than is able to be properly resolved by the recording (or sampling) device. This removal is done before (re)sampling at a lower resolution. When sampling is performed without removing this part of the signal, it causes undesirable artifacts such as black-and-white noise. In signal acquisition and audio, anti-aliasing is often done using an analog anti-aliasing filter to remove the out-of-band component of the input signal prior to sampling with an analog-to-digital converter. In digital photography, optical anti-aliasing filters made of birefringent materials smooth the signal in the spatial optical domain. The anti-aliasing filter essentially blurs the image slightly in order to reduce the resolution to or below that achievable by the digital sensor (the larger the pixel pitch, the lower the achievable resolution at the sensor level). == Examples == In computer graphics, anti-aliasing improves the appearance of "jagged" polygon edges, or "jaggies", so they are smoothed out on the screen. However, it incurs a performance cost for the graphics card and uses more video memory. The level of anti-aliasing determines how smooth polygon edges are (and how much video memory it consumes). Near the top of an image with a receding checker-board pattern, the image is difficult to recognise and often not considered aesthetically pleasing. In contrast, when anti-aliased the checker-board near the top blends into grey, which is usually the desired effect when the resolution is insufficient to show the detail. Even near the bottom of the image, the edges appear much smoother in the anti-aliased image. Multiple methods exist, including the sinc filter, which is considered a better anti-aliasing algorithm. When magnified, it can be seen how anti-aliasing interpolates the brightness of the pixels at the boundaries to produce grey pixels since the space is occupied by both black and white tiles. These help make the sinc filter antialiased image appear much smoother than the original. In a simple diamond image, anti-aliasing blends the boundary pixels; this reduces the aesthetically jarring effect of the sharp, step-like boundaries that appear in the aliased graphic. Anti-aliasing is often applied in rendering text on a computer screen, to suggest smooth contours that better emulate the appearance of text produced by conventional ink-and-paper printing. Particularly with fonts displayed on typical LCD screens, it is common to use subpixel rendering techniques like ClearType. Sub-pixel rendering requires special colour-balanced anti-aliasing filters to turn what would be severe colour distortion into barely-noticeable colour fringes. Equivalent results can be had by making individual sub-pixels addressable as if they were full pixels, and supplying a hardware-based anti-aliasing filter as is done in the OLPC XO-1 laptop's display controller. Pixel geometry affects all of this, whether the anti-aliasing and sub-pixel addressing are done in software or hardware. == Simplest approach to anti-aliasing == The most basic approach to anti-aliasing a pixel is determining what percentage of the pixel is occupied by a given region in the vector graphic - in this case a pixel-sized square, possibly transposed over several pixels - and using that percentage as the colour. A Python program producing a basic plot of a single, white-on-black anti-aliased point using the method is as follows: This method is generally best suited for simple graphics, such as basic lines or curves, and applications that would otherwise have to convert absolute coordinates to pixel-constrained coordinates, such as 3D graphics. It is a fairly fast function, but it is relatively low-quality, and gets slower as the complexity of the shape increases. For purposes requiring very high-quality graphics or very complex vector shapes, this will probably not be the best approach. Note: The plot_antialiased_point routine above cannot blindly set the colour value to the percent calculated. It must add the new value to the existing value at that location up to a maximum of 1. Otherwise, the brightness of each pixel will be equal to the darkest value calculated in time for that location which produces a very bad result. For example, if one point sets a brightness level of 0.90 for a given pixel and another point calculated later barely touches that pixel and has a brightness of 0.05, the final value set for that pixel should be 0.95, not 0.05. For more sophisticated shapes, the algorithm may be generalized as rendering the shape to a pixel grid with higher resolution than the target display surface (usually a multiple that is a power of 2 to reduce distortion), then using bicubic interpolation to determine the average intensity of each real pixel on the display surface. == Signal processing approach to anti-aliasing == In this approach, the ideal image is regarded as a signal. The image displayed on the screen is taken as samples, at each (x,y) pixel position, of a filtered version of the signal. Ideally, one would understand how the human brain would process the original signal, and provide an on-screen image that will yield the most similar response by the brain. The most widely accepted analytic tool for such problems is the Fourier transform; this decomposes a signal into basis functions of different frequencies, known as frequency components, and gives us the amplitude of each frequency component in the signal. The waves are of the form: cos ⁡ ( 2 j π x ) cos ⁡ ( 2 k π y ) {\displaystyle \ \cos(2j\pi x)\cos(2k\pi y)} where j and k are arbitrary non-negative integers. There are also frequency components involving the sine functions in one or both dimensions, but for the purpose of this discussion, the cosine will suffice. The numbers j and k together are the frequency of the component: j is the frequency in the x direction, and k is the frequency in the y direction. The goal of an anti-aliasing filter is to greatly reduce frequencies above a certain limit, known as the Nyquist frequency, so that the signal will be accurately represented by its samples, or nearly so, in accordance with the sampling theorem; there are many different choices of detailed algorithm, with different filter transfer functions. Current knowledge of human visual perception is not sufficient, in general, to say what approach will look best. == Two dimensional considerations == The previous discussion assumes that the rectangular mesh sampling is the dominant part of the problem. The filter usually considered optimal is not rotationally symmetrical, as shown in this first figure; this is because the data is sampled on a square lattice, not using a continuous image. This sampling pattern is the justification for doing signal processing along each axis, as it is traditionally done on one dimensional data. Lanczos resampling is based on convolution of the data with a discrete representation of the sinc function. If the resolution is not limited by the rectangular sampling rate of either the source or target image, then one should ideally use rotationally symmetrical filter or interpolation functions, as though the data were a two dimensional function of continuous x and y. The sinc function of the radius has too long a tail to make a good filter (it is not even square-integrable). A more appropriate analog to the one-dimensional sinc is the two-dimensional Airy disc amplitude, the 2D Fourier transform of a circular region in 2D frequency space, as opposed to a square region. One might consider a Gaussian plus enough of its second derivative to flatten the top (in the frequency domain) or sharpen it up (in the spatial domain), as shown. Functions based on the Gaussian function are natural choices, because convolution with a Gaussian gives another Gaussian whether applied to x and y or to the radius. Similarly to wavelets, another of its properties is that it is halfway between being localized in the configuration (x and y) and in the spectral (j and k) representation. As an interpolation function, a Gaussian alone seems too spread out to preserve the maximum possible detail, and thus the second derivative is added. As an example, when printing a photographic negative with plentiful processing capability and on a printer with a hexagonal pattern, there is no reason to use sinc function interpolation. Such interpolation would treat diagonal lines differently from horizontal and vertical lines, which is like a weak form of aliasing. == Practical real-time anti-aliasing approximations == There are only a handful of primitives used at the lowest level in a real-time rend
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Deep Learning Indaba

The Deep Learning Indaba is an annual conference and educational event that aims to strengthen machine learning and artificial intelligence (AI) capacity across Africa. Launched in 2017, it brings together students, researchers, industry practitioners, and policymakers from across the African continent. == History == The Deep Learning Indaba began in 2017 at the University of the Witwatersrand with over 300 participants from 23 African countries, offering tutorials in advanced AI topics and featuring notable speakers like Nando de Freitas. In 2018, it expanded to 650 delegates at Stellenbosch University, introducing parallel sessions to encourage collaboration. The 2019 edition in Nairobi, Kenya, reflected further growth, with increasing sponsorship and support from major tech companies like Google and Microsoft. === Deep Learning IndabaX ===
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Eat App

Eat App is a global restaurant technology company that provides a cloud-based management platform for restaurants, hotels, and other venues. The platform enables venues to accept online reservations seamlessly, manage tables, and enhance customer relationship management (CRM). It utilizes AI to improve operational efficiency, provides marketing automation, and helps build a comprehensive guestbook. The company also offers a consumer app and website for discovering and booking restaurant tables online. According to the company, the system has seated over 100 million guests, and the number continues to grow. Eat was founded by Nezar Kadhem and David Feuillard in 2015 and has raised $13M to date from Silicon Valley's 500 startups, Middle East Venture Partners (MEVP), Derayah VC, amongst other business angels. The company is currently operational across the world, with offices in Dubai and the United States. == Product overview == === For restaurants === Eat App’s reservation system allows for a digital record of all reservations, all guests that have previously visited the restaurant, as well as analytics on the performance of the restaurant. The table management feature simplifies traditional restaurant operations by providing a live snapshot of current status, seating optimization, and shift management. The CRM and analytics suite gathers and monitors data to build a segmented guestbook for personalized marketing and provides dashboards for data-driven decision-making. Additionally, the review feature makes it easy for restaurants to automatically collect reviews from their guests. Additionally, Eat App includes a chit printer function that seamlessly prints reservation details at host stands and a review management feature that allows restaurants to manage online reviews directly within the platform. == History == In February 2015, Eat App raised $300k from Bahrain-based business angel group TENMOU. In June 2018, Eat raised $1.2 million from Dubai-based Middle East Venture Partners (MEVP). In February 2020, Eat App raised $5 million in a Series B funding round led by 500 Startups, Derayah Venture Fund, and MEVP, with participation from a few angel investors and family members. In February 2021, Eat App launched its technology with The Emaar Hospitality Group, implementing it across over 50 restaurants in Emaar properties and hotels. The cloud-based system runs natively on iPads in each restaurant, providing Emaar staff access to reservations and guest information, and integrates with the U by Emaar loyalty app to personalize service. On September 28, 2022, Eat App announced the closing of an $11 million Series B funding round. The investment was led by Middle East Venture Partners (MEVP), 500 Startups, Derayah Venture Capital, Dallah Albaraka, Ali Zaid Al Quraishi & Brothers Company, and Rasameel Investment Company, with participation from existing investors.
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CodeCheck

CodeCheck is a mobile app that provides consumers with information about the ingredients in cosmetic products, as well as the ingredients and nutritional values of food. Users can access this information by scanning the product’s barcode with a smartphone or by using a text-based search. The app is available for iOS and Android devices in Germany, Austria, Switzerland, the United Kingdom, the United States, and the Netherlands. == History == CodeCheck was founded in 2010 as an association, online database, and app by Roman Bleichenbacher, who was then a student in Zurich. A website of the same name had already been launched in 2002, where users could enter information about ingredients, nutritional values, and manufacturers of products. The first round of financing took place in July 2014 and raised over 1.1 million Swiss francs, which coincided with the founding of CodeCheck AG. Investors included Doodle founders Myke Näf and Paul E. Sevinç. The company subsequently expanded to Austria and Germany. In the same year, Boris Manhart became CEO. CodeCheck GmbH was established in Berlin in 2016. The app became available in the United States in 2017 and in the United Kingdom in November 2019. In 2020, it was also launched in the Netherlands. Following insolvency proceedings, the app has been owned by Producto Check GmbH since 2022. == Functions == The app can be used to scan the barcode of food and cosmetic products. It then displays information about ingredients, nutritional values, manufacturers and certification labels. For many years, users were able to enter and edit product information themselves and indicate advantages and disadvantages of individual products. Since 2020, the app has placed greater emphasis on machine text recognition. The collected data is combined with substance ratings using an algorithm. These ratings are based on scientific studies and expert assessments, including those from the Consumer Advice Centre in Hamburg, Greenpeace, the WWF and the German Association for the Environment and Nature Conservation (BUND e. V.), and cannot be modified by users or manufacturers. The app also provides information on the sugar and fat content of food products. In addition, it indicates whether a product contains hormone-active substances, microplastics, palm oil, animal-derived ingredients, lactose or gluten. Since 2020, the app has displayed a climate score for food products in cooperation with the Eaternity Institute. == Financing == CodeCheck is primarily financed through native advertising and banner ads. Since 2018, the company has also offered analysis services and survey tools directly to fast-moving consumer goods (FMCG) manufacturers. In addition, access to the API is available, enabling other companies to use the product database. With the introduction of a subscription model in 2019, the CodeCheck app can be used ad-free and in offline mode. Since 2021, CodeCheck has also offered its own “Green Label” certification for manufacturers. Products are certified if at least 90 percent of their ingredients are classified as harmless. == Awards == In May 2015, the app topped the download charts for the first time, reaching 2.3 million installations. By September 2019, the app had once again reached the top of the German app charts, surpassing five million downloads.
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Projection-slice theorem

In mathematics, the projection-slice theorem, central slice theorem or Fourier slice theorem in two dimensions states that the results of the following two calculations are equal: Take a two-dimensional function f(r), project (e.g. using the Radon transform) it onto a (one-dimensional) line, and do a Fourier transform of that projection. Take that same function, but do a two-dimensional Fourier transform first, and then slice the function through its origin, parallel to the projection line. In operator terms, if F1 and F2 are the 1- and 2-dimensional Fourier transform operators mentioned above, P1 is the projection operator (which projects a 2-D function onto a 1-D line), S1 is a slice operator (which extracts a 1-D central slice from a function), then F 1 P 1 = S 1 F 2 . {\displaystyle F_{1}P_{1}=S_{1}F_{2}.} This idea can be extended to higher dimensions. This theorem is used, for example, in the analysis of medical CT scans where a "projection" is an x-ray image of an internal organ. The Fourier transforms of these images are seen to be slices through the Fourier transform of the 3-dimensional density of the internal organ, and these slices can be interpolated to build up a complete Fourier transform of that density. The inverse Fourier transform is then used to recover the 3-dimensional density of the object. This technique was first derived by Ronald N. Bracewell in 1956 for a radio-astronomy problem. == The projection-slice theorem in N dimensions == In N dimensions, the projection-slice theorem states that the Fourier transform of the projection of an N-dimensional function f(r) onto an m-dimensional linear submanifold is equal to an m-dimensional slice of the N-dimensional Fourier transform of that function consisting of an m-dimensional linear submanifold through the origin in the Fourier space which is parallel to the projection submanifold. In operator terms: F m P m = S m F N . {\displaystyle F_{m}P_{m}=S_{m}F_{N}.\,} == The generalized Fourier-slice theorem == In addition to generalizing to N dimensions, the projection-slice theorem can be further generalized with an arbitrary change of basis. For convenience of notation, we consider the change of basis to be represented as B, an N-by-N invertible matrix operating on N-dimensional column vectors. Then the generalized Fourier-slice theorem can be stated as F m P m B = S m B − T | B − T | F N {\displaystyle F_{m}P_{m}B=S_{m}{\frac {B^{-T}}{|B^{-T}|}}F_{N}} where B − T = ( B − 1 ) T {\displaystyle B^{-T}=(B^{-1})^{T}} is the transpose of the inverse of the change of basis transform. == Proof in two dimensions == The projection-slice theorem is easily proven for the case of two dimensions. Without loss of generality, we can take the projection line to be the x-axis. There is no loss of generality because if we use a shifted and rotated line, the law still applies. Using a shifted line (in y) gives the same projection and therefore the same 1D Fourier transform results. The rotated function is the Fourier pair of the rotated Fourier transform, for which the theorem again holds. If f(x, y) is a two-dimensional function, then the projection of f(x, y) onto the x axis is p(x) where p ( x ) = ∫ − ∞ ∞ f ( x , y ) d y . {\displaystyle p(x)=\int _{-\infty }^{\infty }f(x,y)\,dy.} The Fourier transform of f ( x , y ) {\displaystyle f(x,y)} is F ( k x , k y ) = ∫ − ∞ ∞ ∫ − ∞ ∞ f ( x , y ) e − 2 π i ( x k x + y k y ) d x d y . {\displaystyle F(k_{x},k_{y})=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }f(x,y)\,e^{-2\pi i(xk_{x}+yk_{y})}\,dxdy.} The slice is then s ( k x ) {\displaystyle s(k_{x})} s ( k x ) = F ( k x , 0 ) = ∫ − ∞ ∞ ∫ − ∞ ∞ f ( x , y ) e − 2 π i x k x d x d y {\displaystyle s(k_{x})=F(k_{x},0)=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }f(x,y)\,e^{-2\pi ixk_{x}}\,dxdy} = ∫ − ∞ ∞ [ ∫ − ∞ ∞ f ( x , y ) d y ] e − 2 π i x k x d x {\displaystyle =\int _{-\infty }^{\infty }\left[\int _{-\infty }^{\infty }f(x,y)\,dy\right]\,e^{-2\pi ixk_{x}}dx} = ∫ − ∞ ∞ p ( x ) e − 2 π i x k x d x {\displaystyle =\int _{-\infty }^{\infty }p(x)\,e^{-2\pi ixk_{x}}dx} which is just the Fourier transform of p(x). The proof for higher dimensions is easily generalized from the above example. == The FHA cycle == If the two-dimensional function f(r) is circularly symmetric, it may be represented as f(r), where r = |r|. In this case the projection onto any projection line will be the Abel transform of f(r). The two-dimensional Fourier transform of f(r) will be a circularly symmetric function given by the zeroth-order Hankel transform of f(r), which will therefore also represent any slice through the origin. The projection-slice theorem then states that the Fourier transform of the projection equals the slice or F 1 A 1 = H , {\displaystyle F_{1}A_{1}=H,} where A1 represents the Abel-transform operator, projecting a two-dimensional circularly symmetric function onto a one-dimensional line, F1 represents the 1-D Fourier-transform operator, and H represents the zeroth-order Hankel-transform operator. == Extension to fan beam or cone-beam CT == The projection-slice theorem is suitable for CT image reconstruction with parallel beam projections. It does not directly apply to fanbeam or conebeam CT. The theorem was extended to fan-beam and conebeam CT image reconstruction by Shuang-ren Zhao in 1995.
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Learning to rank

Learning to rank (LTR) or machine-learned ranking (MLR) is the application of machine learning, often supervised, semi-supervised or reinforcement learning, in the construction of ranking models for information retrieval and recommender systems. Training data may, for example, consist of lists of items with some partial order specified between items in each list. This order is typically induced by giving a numerical or ordinal score or a binary judgment (e.g. "relevant" or "not relevant") for each item. The goal of constructing the ranking model is to rank new, unseen lists in a similar way to rankings in the training data. == Applications == === In information retrieval === Ranking is a central part of many information retrieval problems, such as document retrieval, collaborative filtering, sentiment analysis, and online advertising. A possible architecture of a machine-learned search engine is shown in the accompanying figure. Training data consists of queries and documents matching them together with the relevance degree of each match. It may be prepared manually by human assessors (or raters, as Google calls them), who check results for some queries and determine relevance of each result. It is not feasible to check the relevance of all documents, and so typically a technique called pooling is used — only the top few documents, retrieved by some existing ranking models are checked. This technique may introduce selection bias. Alternatively, training data may be derived automatically by analyzing clickthrough logs (i.e. search results which got clicks from users), query chains, or such search engines' features as Google's (since-replaced) SearchWiki. Clickthrough logs can be biased by the tendency of users to click on the top search results on the assumption that they are already well-ranked. Training data is used by a learning algorithm to produce a ranking model which computes the relevance of documents for actual queries. Typically, users expect a search query to complete in a short time (such as a few hundred milliseconds for web search), which makes it impossible to evaluate a complex ranking model on each document in the corpus, and so a two-phase scheme is used. First, a small number of potentially relevant documents are identified using simpler retrieval models which permit fast query evaluation, such as the vector space model, Boolean model, weighted AND, or BM25. This phase is called top- k {\displaystyle k} document retrieval and many heuristics were proposed in the literature to accelerate it, such as using a document's static quality score and tiered indexes. In the second phase, a more accurate but computationally expensive machine-learned model is used to re-rank these documents. === In other areas === Learning to rank algorithms have been applied in areas other than information retrieval: In machine translation for ranking a set of hypothesized translations; In computational biology for ranking candidate 3-D structures in protein structure prediction problems; In recommender systems for identifying a ranked list of related news articles to recommend to a user after he or she has read a current news article. == Feature vectors == For the convenience of MLR algorithms, query-document pairs are usually represented by numerical vectors, which are called feature vectors. Such an approach is sometimes called bag of features and is analogous to the bag of words model and vector space model used in information retrieval for representation of documents. Components of such vectors are called features, factors or ranking signals. They may be divided into three groups (features from document retrieval are shown as examples): Query-independent or static features — those features, which depend only on the document, but not on the query. For example, PageRank or document's length. Such features can be precomputed in off-line mode during indexing. They may be used to compute document's static quality score (or static rank), which is often used to speed up search query evaluation. Query-dependent or dynamic features — those features, which depend both on the contents of the document and the query, such as TF-IDF score or other non-machine-learned ranking functions. Query-level features or query features, which depend only on the query. For example, the number of words in a query. Some examples of features, which were used in the well-known LETOR dataset: TF, TF-IDF, BM25, and language modeling scores of document's zones (title, body, anchors text, URL) for a given query; Lengths and IDF sums of document's zones; Document's PageRank, HITS ranks and their variants. Selecting and designing good features is an important area in machine learning, which is called feature engineering. == Evaluation measures == There are several measures (metrics) which are commonly used to judge how well an algorithm is doing on training data and to compare the performance of different MLR algorithms. Often a learning-to-rank problem is reformulated as an optimization problem with respect to one of these metrics. Examples of ranking quality measures: Mean average precision (MAP); DCG and NDCG; Precision@n, NDCG@n, where "@n" denotes that the metrics are evaluated only on top n documents; Mean reciprocal rank; Kendall's tau; Spearman's rho. DCG and its normalized variant NDCG are usually preferred in academic research when multiple levels of relevance are used. Other metrics such as MAP, MRR and precision, are defined only for binary judgments. Recently, there have been proposed several new evaluation metrics which claim to model user's satisfaction with search results better than the DCG metric: Expected reciprocal rank (ERR); Yandex's pfound. Both of these metrics are based on the assumption that the user is more likely to stop looking at search results after examining a more relevant document, than after a less relevant document. == Approaches == Learning to Rank approaches are often categorized using one of three approaches: pointwise (where individual documents are ranked), pairwise (where pairs of documents are ranked into a relative order), and listwise (where an entire list of documents are ordered). Tie-Yan Liu of Microsoft Research Asia has analyzed existing algorithms for learning to rank problems in his book Learning to Rank for Information Retrieval. He categorized them into three groups by their input spaces, output spaces, hypothesis spaces (the core function of the model) and loss functions: the pointwise, pairwise, and listwise approach. In practice, listwise approaches often outperform pairwise approaches and pointwise approaches. This statement was further supported by a large scale experiment on the performance of different learning-to-rank methods on a large collection of benchmark data sets. In this section, without further notice, x {\displaystyle x} denotes an object to be evaluated, for example, a document or an image, f ( x ) {\displaystyle f(x)} denotes a single-value hypothesis, h ( ⋅ ) {\displaystyle h(\cdot )} denotes a bi-variate or multi-variate function and L ( ⋅ ) {\displaystyle L(\cdot )} denotes the loss function. === Pointwise approach === In this case, it is assumed that each query-document pair in the training data has a numerical or ordinal score. Then the learning-to-rank problem can be approximated by a regression problem — given a single query-document pair, predict its score. Formally speaking, the pointwise approach aims at learning a function f ( x ) {\displaystyle f(x)} predicting the real-value or ordinal score of a document x {\displaystyle x} using the loss function L ( f ; x j , y j ) {\displaystyle L(f;x_{j},y_{j})} . A number of existing supervised machine learning algorithms can be readily used for this purpose. Ordinal regression and classification algorithms can also be used in pointwise approach when they are used to predict the score of a single query-document pair, and it takes a small, finite number of values. === Pairwise approach === In this case, the learning-to-rank problem is approximated by a classification problem — learning a binary classifier h ( x u , x v ) {\displaystyle h(x_{u},x_{v})} that can tell which document is better in a given pair of documents. The classifier shall take two documents as its input and the goal is to minimize a loss function L ( h ; x u , x v , y u , v ) {\displaystyle L(h;x_{u},x_{v},y_{u,v})} . The loss function typically reflects the number and magnitude of inversions in the induced ranking. In many cases, the binary classifier h ( x u , x v ) {\displaystyle h(x_{u},x_{v})} is implemented with a scoring function f ( x ) {\displaystyle f(x)} . As an example, RankNet adapts a probability model and defines h ( x u , x v ) {\displaystyle h(x_{u},x_{v})} as the estimated probability of the document x u {\displaystyle x_{u}} has higher quality than x v {\displaystyle x_{v}} : P u , v ( f ) = CDF ( f ( x u ) − f ( x v ) ) , {\displaystyle P_{u,v}(f)={\text{CDF}
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Audio mining

Audio mining is a technique by which the content of an audio signal can be automatically analyzed and searched. It is most commonly used in the field of automatic speech recognition, where the analysis tries to identify any speech within the audio. The term audio mining is sometimes used interchangeably with audio indexing, phonetic searching, phonetic indexing, speech indexing, audio analytics, speech analytics, word spotting, and information retrieval. Audio indexing, however, is mostly used to describe the pre-process of audio mining, in which the audio file is broken down into a searchable index of words. == History == Academic research on audio mining began in the late 1970s in schools like Carnegie Mellon University, Columbia University, the Georgia Institute of Technology, and the University of Texas. Audio data indexing and retrieval began to receive attention and demand in the early 1990s, when multimedia content started to develop and the volume of audio content significantly increased. Before audio mining became the mainstream method, written transcripts of audio content were created and manually analyzed. == Process == Audio mining is typically split into four components: audio indexing, speech processing and recognition systems, feature extraction and audio classification. The audio will typically be processed by a speech recognition system in order to identify word or phoneme units that are likely to occur in the spoken content. This information may either be used immediately in pre-defined searches for keywords or phrases (a real-time "word spotting" system), or the output of the speech recognizer may be stored in an index file. One or more audio mining index files can then be loaded at a later date in order to run searches for keywords or phrases. The results of a search will normally be in terms of hits, which are regions within files that are good matches for the chosen keywords. The user may then be able to listen to the audio corresponding to these hits in order to verify if a correct match was found. === Audio Indexing === In audio, there is the main problem of information retrieval - there is a need to locate the text documents that contain the search key. Unlike humans, a computer is not able to distinguish between the different types of audios such as speed, mood, noise, music or human speech - an effective searching method is needed. Hence, audio indexing allows efficient search for information by analyzing an entire file using speech recognition. An index of content is then produced, bearing words and their locations done through content-based audio retrieval, focusing on extracted audio features. It is done through mainly two methods: Large Vocabulary Continuous Speech Recognition (LVCSR) and Phonetic-based Indexing. ==== Large Vocabulary Continuous Speech Recognizers (LVCSR) ==== In text-based indexing or large vocabulary continuous speech recognition (LVCSR), the audio file is first broken down into recognizable phonemes. It is then run through a dictionary that can contain several hundred thousand entries and matched with words and phrases to produce a full text transcript. A user can then simply search a desired word term and the relevant portion of the audio content will be returned. If the text or word could not be found in the dictionary, the system will choose the next most similar entry it can find. The system uses a language understanding model to create a confidence level for its matches. If the confidence level be below 100 percent, the system will provide options of all the found matches. ===== Advantages and disadvantages ===== The main draw of LVCSR is its high accuracy and high searching speed. In LVCSR, statistical methods are used to predict the likelihood of different word sequences, hence the accuracy is much higher than the single word lookup of a phonetic search. If the word can be found, the probability of the word spoken is very high. Meanwhile, while initial processing of audio takes a fair bit of time, searching is quick as just a simple test to text matching is needed. On the other hand, LVCSR is susceptible to common issues of speech recognition. The inherent random nature of audio and problems of external noise all affect the accuracies of text-based indexing. Another problem with LVCSR is its over reliance on its dictionary database. LVCSR only recognizes words that are found in their dictionary databases, and these dictionaries and databases are unable to keep up with the constant evolving of new terminology, names and words. Should the dictionary not contain a word, there is no way for the system to identify or predict it. This reduces the accuracy and reliability of the system. This is named the Out-of-vocabulary (OOV) problem. Audio mining systems try to cope with OOV by continuously updating the dictionary and language model used, but the problem still remains significant and has probed a search for alternatives. Additionally, due to the need to constantly update and maintain task-based knowledge and large training databases to cope with the OOV problem, high computational costs are incurred. This makes LVCSR an expensive approach to audio mining. ==== Phonetic-based Indexing ==== Phonetic-based indexing also breaks the audio file into recognizable phonemes, but instead of converting them to a text index, they are kept as they are and analyzed to create a phonetic-based index. The process of phonetic-based indexing can be split into two phases. The first phase is indexing. It begins by converting the input media into a standard audio representation format (PCM). Then, an acoustic model is applied to the speech. This acoustic model represents characteristics of both an acoustic channel (an environment in which the speech was uttered and a transducer through which it was recorded) and a natural language (in which human beings expressed the input speech). This produces a corresponding phonetic search track, or phonetic audio track (PAT), a highly compressed representation of the phonetic content of the input media. The second phase is searching. The user's search query term is parsed into a possible phoneme string using a phonetic dictionary. Then, multiple PAT files can be scanned at high speed during a single search for likely phonetic sequences that closely match corresponding strings of phonemes in the query term. ===== Advantages and disadvantages ===== Phonetic indexing is most attractive as it is largely unaffected by linguistic issues such as unrecognized words and spelling errors. Phonetic preprocessing maintains an open vocabulary that does not require updating. That makes it particularly useful for searching specialized terminology or words in foreign languages that do not commonly appear in dictionaries. It is also more effective for searching audio files with disruptive background noise and/or unclear utterances as it can compile results based on the sounds it can discern, and should the user wish to, they can search through the options until they find the desired item. Furthermore, in contrast to LVCSR, it can process audio files very quickly as there are very few unique phonemes between languages. However, phonemes cannot be effectively indexed like an entire word, thus searching on a phonetic-based system is slow. An issue with phonetic indexing is its low accuracy. Phoneme-based searches result in more false matches than text-based indexing. This is especially prevalent for short search terms, which have a stronger likelihood of sounding similar to other words or being part of bigger words. It could also return irrelevant results from other languages. Unless the system recognizes exactly the entire word, or understands phonetic sequences of languages, it is difficult for phonetic-based indexing to return accurate findings. === Speech processing and recognition system === Deemed as the most critical and complex component of audio mining, speech recognition requires the knowledge of human speech production system and its modeling. To correspond the Human speech production system, the electrical speech production system is developed to consist of: Speech generation Speech perception Voiced & unvoiced speech Model of human speech The electrical speech production system converts acoustic signal into corresponding representation of the spoken through the acoustic models in their software where all phonemes are represented. A statistical language model aids in the process by identifying how likely words are to follow each other in certain languages. Put together with a complex probability analysis, the speech recognition system is capable of taking an unknown speech signal and transcribing it into words based on the program's dictionary. ASR (automatic speech recognition) system includes: Acoustic analysis: input sound waveform is transformed into a feature Acoustic model: establishes relationship between speech signal and phonemes, pronunciation model and lang
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Frameserver

A frameserver is any program that acts as a media source in the process called frameserving, which transfers digital video data from one computer program to another without intermediate files. The program that receives the data – the frameclient – could be any type of video application. The process is controlled by the frameclient: the frameclient requests audio/video frames and the frameserver serves them. The client can request frames in any order, allowing it to pause or jump to an arbitrary frame, just as a media player does with a file on disk. The client is most commonly a media encoder, a non-linear editing system, or a media player. == Frameservers == AviSynth VirtualDub VapourSynth Debugmode FrameServer
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Automated attendant

In telephony, an automated attendant (also auto attendant, auto-attendant, autoattendant, automatic phone menus, AA, or virtual receptionist) allows callers to be automatically transferred to an extension without the intervention of an operator/receptionist. Many AAs will also offer a simple menu system ("for sales, press 1, for service, press 2," etc.). An auto attendant may also allow a caller to reach a live operator by dialing a number, usually "0". Typically the auto attendant is included in a business's phone system such as a PBX, but some services allow businesses to use an AA without such a system. Modern AA services (which now overlap with more complicated interactive voice response or IVR systems) can route calls to mobile phones, VoIP virtual phones, other AAs/IVRs, or other locations using traditional land-line phones or voice message machines. == Feature description == Telephone callers will recognize an automated attendant system as one that greets calls incoming to an organization with a recorded greeting of the form, "Thank you for calling .... If you know your party's extension, you may dial it any time during this message." Callers who have a touch-tone (DTMF) phone can dial an extension number or, in most cases, wait for operator ("attendant") assistance. Since the telephone network does not transmit the DC signals from rotary dial telephones (except for audible clicks), callers who have rotary dial phones have to wait for assistance. On a purely technical level it could be argued that an automated attendant is a very simple kind of IVR however, in the telecom industry the terms IVR and auto attendant are generally considered distinct. An automated attendant serves a very specific purpose (replace live operator and route calls), whereas an IVR can perform all sorts of functions (telephone banking, account inquiries, etc.). An AA will often include a directory which will allow a caller to dial by name in order to find a user on a system. There is no standard format to these directories, and they can use combinations of first name, last name, or both. The following lists common routing steps that are components of an automated attendant: Transfer to extension Transfer to voicemail Play message (i.e., "our address is ...") Go to a sub-menu Repeat choices In addition, an automated attendant would be expected to have values for the following: '0' – where to go when the caller dials '0' Timeout – what to do if the caller does nothing (usually go to the same place as '0') Default mailbox – where to send calls if '0' is not answered (or is not pointing to a live person) == Background == PBXs (private branch exchanges) or PABXs (private automatic branch exchanges) are telephone systems that serve an organization that has many telephone extensions but fewer telephone lines (sometimes called "trunks") that connect that organization to the rest of the global telecommunications network. While persons within an enterprise served by a PBX can call each other by dialing their extension numbers, incoming calls, i.e., calls originating from a telephone not served by the PBX but intended for a party served by the PBX, required assistance from a switchboard operator (also called a "switchboard attendant") or a telephone service called DID ("direct inward dialing"). Direct inward dialing has advantages such as rapid connection to the destination party and disadvantages including cost, lack of identification of the called organization and use of ten-digit telephone numbers. Automated attendants provide, among many other things, a way for an external caller to be directed to an extension or department served by a PBX system without using direct inward dialing or without switchboard attendant assistance. == History == Automated attendants are not part of voicemail systems. Voice messaging (or voicemail or VM) technology has existed since the late 1970s; in the early 1980s companies provided voice-prompting systems that allowed callers to reach (route the call) to an intended party, not necessarily to leave a message. Automated attendant systems are also referred to as automated menu systems and much early work in this field was done by Michael J. Freeman, Ph.D. == Time-based routing == Many auto attendants will have options to allow for time-of-day routing, as well as weekend and holiday routing. The specifics of these features will depend entirely on the particular automated attendant, but typically there would be a normal greeting and routing steps that would take place during normal business hours, and a different greeting and routing for non-business hours.
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Embodied agent

In artificial intelligence, an embodied agent, also sometimes referred to as an interface agent, is an intelligent agent that interacts with the environment through a physical body within that environment. Agents that are represented graphically with a body, for example a human or a cartoon animal, are also called embodied agents, although they have only virtual, not physical, embodiment. A branch of artificial intelligence focuses on empowering such agents to interact autonomously with human beings and the environment. Mobile robots are one example of physically embodied agents; Ananova and Microsoft Agent are examples of graphically embodied agents. Embodied conversational agents are embodied agents (usually with a graphical front-end as opposed to a robotic body) that are capable of engaging in conversation with one another and with humans employing the same verbal and nonverbal means that humans do (such as gesture, facial expression, and so forth). == Embodied conversational agents == Embodied conversational agents are a form of intelligent user interface. Graphically embodied agents aim to unite gesture, facial expression and speech to enable face-to-face communication with users, providing a powerful means of human-computer interaction. == Advantages == Face-to-face communication allows communication protocols that give a much richer communication channel than other means of communicating. It enables pragmatic communication acts such as conversational turn-taking, facial expression of emotions, information structure and emphasis, visualization and iconic gestures, and orientation in a three-dimensional environment. This communication takes place through both verbal and non-verbal channels such as gaze, gesture, spoken intonation and body posture. Research has found that users prefer a non-verbal visual indication of an embodied system's internal state to a verbal indication, demonstrating the value of additional non-verbal communication channels. As well as this, the face-to-face communication involved in interacting with an embodied agent can be conducted alongside another task without distracting the human participants, instead improving the enjoyment of such an interaction. Furthermore, the use of an embodied presentation agent results in improved recall of the presented information. Embodied agents also provide a social dimension to the interaction. Humans willingly ascribe social awareness to computers, and thus interaction with embodied agents follows social conventions, similar to human to human interactions. This social interaction both raises the believably and perceived trustworthiness of agents, and increases the user's engagement with the system. Rickenberg and Reeves found that the presence of an embodied agent on a website increased the level of user trust in that website, but also increased users' anxiety and affected their performance, as if they were being watched by a real human. Another effect of the social aspect of agents is that presentations given by an embodied agent are perceived as being more entertaining and less difficult than similar presentations given without an agent. Research shows that perceived enjoyment, followed by perceived usefulness and ease of use, is the major factor influencing user adoption of embodied agents. A study in January 2004 by Byron Reeves at Stanford demonstrated how digital characters could "enhance online experiences" through explaining how virtual characters essentially add a sense of familiarity to the user experience and make it more approachable. This increase in likability in turn helps make the products better, which benefits both the end users and those creating the product. === Applications === The rich style of communication that characterizes human conversation makes conversational interaction with embodied conversational agents ideal for many non-traditional interaction tasks. A familiar application of graphically embodied agents is computer games; embodied agents are ideal for this setting because the richer communication style makes interacting with the agent enjoyable. Embodied conversational agents have also been used in virtual training environments, portable personal navigation guides, interactive fiction and storytelling systems, interactive online characters and automated presenters and commentators. Major virtual assistants like Siri, Amazon Alexa and Google Assistant do not come with any visual embodied representation, which is believed to limit the sense of human presence by users. The U.S. Department of Defense utilizes a software agent called SGT STAR on U.S. Army-run Web sites and Web applications for site navigation, recruitment and propaganda purposes. Sgt. Star is run by the Army Marketing and Research Group, a division operated directly from The Pentagon. Sgt. Star is based upon the ActiveSentry technology developed by Next IT, a Washington-based information technology services company. Other such bots in the Sgt. Star "family" are utilized by the Federal Bureau of Investigation and the Central Intelligence Agency for intelligence gathering purposes.
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Tensor operator

In pure and applied mathematics, quantum mechanics and computer graphics, a tensor operator generalizes the notion of operators which are scalars and vectors. A special class of these are spherical tensor operators which apply the notion of the spherical basis and spherical harmonics. The spherical basis closely relates to the description of angular momentum in quantum mechanics and spherical harmonic functions. The coordinate-free generalization of a tensor operator is known as a representation operator. == The general notion of scalar, vector, and tensor operators == In quantum mechanics, physical observables that are scalars, vectors, and tensors, must be represented by scalar, vector, and tensor operators, respectively. Whether something is a scalar, vector, or tensor depends on how it is viewed by two observers whose coordinate frames are related to each other by a rotation. Alternatively, one may ask how, for a single observer, a physical quantity transforms if the state of the system is rotated. Consider, for example, a system consisting of a molecule of mass M {\displaystyle M} , traveling with a definite center of mass momentum, p z ^ {\displaystyle p{\mathbf {\hat {z}} }} , in the z {\displaystyle z} direction. If we rotate the system by 90 ∘ {\displaystyle 90^{\circ }} about the y {\displaystyle y} axis, the momentum will change to p x ^ {\displaystyle p{\mathbf {\hat {x}} }} , which is in the x {\displaystyle x} direction. The center-of-mass kinetic energy of the molecule will, however, be unchanged at p 2 / 2 M {\displaystyle p^{2}/2M} . The kinetic energy is a scalar and the momentum is a vector, and these two quantities must be represented by a scalar and a vector operator, respectively. By the latter in particular, we mean an operator whose expected values in the initial and the rotated states are p z ^ {\displaystyle p{\mathbf {\hat {z}} }} and p x ^ {\displaystyle p{\mathbf {\hat {x}} }} . The kinetic energy on the other hand must be represented by a scalar operator, whose expected value must be the same in the initial and the rotated states. In the same way, tensor quantities must be represented by tensor operators. An example of a tensor quantity (of rank two) is the electrical quadrupole moment of the above molecule. Likewise, the octupole and hexadecapole moments would be tensors of rank three and four, respectively. Other examples of scalar operators are the total energy operator (more commonly called the Hamiltonian), the potential energy, and the dipole-dipole interaction energy of two atoms. Examples of vector operators are the momentum, the position, the orbital angular momentum, L {\displaystyle {\mathbf {L} }} , and the spin angular momentum, S {\displaystyle {\mathbf {S} }} . (Fine print: Angular momentum is a vector as far as rotations are concerned, but unlike position or momentum it does not change sign under space inversion, and when one wishes to provide this information, it is said to be a pseudovector.) Scalar, vector and tensor operators can also be formed by products of operators. For example, the scalar product L ⋅ S {\displaystyle {\mathbf {L} }\cdot {\mathbf {S} }} of the two vector operators, L {\displaystyle {\mathbf {L} }} and S {\displaystyle {\mathbf {S} }} , is a scalar operator, which figures prominently in discussions of the spin–orbit interaction. Similarly, the quadrupole moment tensor of our example molecule has the nine components Q i j = ∑ α q α ( 3 r α , i r α , j − r α 2 δ i j ) . {\displaystyle Q_{ij}=\sum _{\alpha }q_{\alpha }\left(3r_{\alpha ,i}r_{\alpha ,j}-r_{\alpha }^{2}\delta _{ij}\right).} Here, the indices i {\displaystyle i} and j {\displaystyle j} can independently take on the values 1, 2, and 3 (or x {\displaystyle x} , y {\displaystyle y} , and z {\displaystyle z} ) corresponding to the three Cartesian axes, the index α {\displaystyle \alpha } runs over all particles (electrons and nuclei) in the molecule, q α {\displaystyle q_{\alpha }} is the charge on particle α {\displaystyle \alpha } , and r α , i {\displaystyle r_{\alpha ,i}} is the i {\displaystyle i} -th component of the position of this particle. Each term in the sum is a tensor operator. In particular, the nine products r α , i r α , j {\displaystyle r_{\alpha ,i}r_{\alpha ,j}} together form a second rank tensor, formed by taking the outer product of the vector operator r α {\displaystyle {\mathbf {r} }_{\alpha }} with itself. == Rotations of quantum states == === Quantum rotation operator === The rotation operator about the unit vector n (defining the axis of rotation) through angle θ is U [ R ( θ , n ^ ) ] = exp ⁡ ( − i θ ℏ n ^ ⋅ J ) {\displaystyle U[R(\theta ,{\hat {\mathbf {n} }})]=\exp \left(-{\frac {i\theta }{\hbar }}{\hat {\mathbf {n} }}\cdot \mathbf {J} \right)} where J = (Jx, Jy, Jz) are the rotation generators (also the angular momentum matrices): J x = ℏ 2 ( 0 1 0 1 0 1 0 1 0 ) J y = ℏ 2 ( 0 i 0 − i 0 i 0 − i 0 ) J z = ℏ ( − 1 0 0 0 0 0 0 0 1 ) {\displaystyle J_{x}={\frac {\hbar }{\sqrt {2}}}{\begin{pmatrix}0&1&0\\1&0&1\\0&1&0\end{pmatrix}}\,\quad J_{y}={\frac {\hbar }{\sqrt {2}}}{\begin{pmatrix}0&i&0\\-i&0&i\\0&-i&0\end{pmatrix}}\,\quad J_{z}=\hbar {\begin{pmatrix}-1&0&0\\0&0&0\\0&0&1\end{pmatrix}}} and let R ^ = R ^ ( θ , n ^ ) {\displaystyle {\widehat {R}}={\widehat {R}}(\theta ,{\hat {\mathbf {n} }})} be a rotation matrix. According to the Rodrigues' rotation formula, the rotation operator then amounts to U [ R ( θ , n ^ ) ] = 1 1 − i sin ⁡ θ ℏ n ^ ⋅ J − 1 − cos ⁡ θ ℏ 2 ( n ^ ⋅ J ) 2 . {\displaystyle U[R(\theta ,{\hat {\mathbf {n} }})]=1\!\!1-{\frac {i\sin \theta }{\hbar }}{\hat {\mathbf {n} }}\cdot \mathbf {J} -{\frac {1-\cos \theta }{\hbar ^{2}}}({\hat {\mathbf {n} }}\cdot \mathbf {J} )^{2}.} An operator Ω ^ {\displaystyle {\widehat {\Omega }}} is invariant under a unitary transformation U if Ω ^ = U † Ω ^ U ; {\displaystyle {\widehat {\Omega }}={U}^{\dagger }{\widehat {\Omega }}U;} in this case for the rotation U ^ ( R ) {\displaystyle {\widehat {U}}(R)} , Ω ^ = U ( R ) † Ω ^ U ( R ) = exp ⁡ ( i θ ℏ n ^ ⋅ J ) Ω ^ exp ⁡ ( − i θ ℏ n ^ ⋅ J ) . {\displaystyle {\widehat {\Omega }}={U(R)}^{\dagger }{\widehat {\Omega }}U(R)=\exp \left({\frac {i\theta }{\hbar }}{\hat {\mathbf {n} }}\cdot \mathbf {J} \right){\widehat {\Omega }}\exp \left(-{\frac {i\theta }{\hbar }}{\hat {\mathbf {n} }}\cdot \mathbf {J} \right).} === Angular momentum eigenkets === The orthonormal basis set for total angular momentum is | j , m ⟩ {\displaystyle |j,m\rangle } , where j is the total angular momentum quantum number and m is the magnetic angular momentum quantum number, which takes values −j, −j + 1, ..., j − 1, j. A general state within the j subspace | ψ ⟩ = ∑ m c j m | j , m ⟩ {\displaystyle |\psi \rangle =\sum _{m}c_{jm}|j,m\rangle } rotates to a new state by: | ψ ¯ ⟩ = U ( R ) | ψ ⟩ = ∑ m c j m U ( R ) | j , m ⟩ {\displaystyle |{\bar {\psi }}\rangle =U(R)|\psi \rangle =\sum _{m}c_{jm}U(R)|j,m\rangle } Using the completeness condition: I = ∑ m ′ | j , m ′ ⟩ ⟨ j , m ′ | {\displaystyle I=\sum _{m'}|j,m'\rangle \langle j,m'|} we have | ψ ¯ ⟩ = I U ( R ) | ψ ⟩ = ∑ m m ′ c j m | j , m ′ ⟩ ⟨ j , m ′ | U ( R ) | j , m ⟩ {\displaystyle |{\bar {\psi }}\rangle =IU(R)|\psi \rangle =\sum _{mm'}c_{jm}|j,m'\rangle \langle j,m'|U(R)|j,m\rangle } Introducing the Wigner D matrix elements: D ( R ) m ′ m ( j ) = ⟨ j , m ′ | U ( R ) | j , m ⟩ {\displaystyle {D(R)}_{m'm}^{(j)}=\langle j,m'|U(R)|j,m\rangle } gives the matrix multiplication: | ψ ¯ ⟩ = ∑ m m ′ c j m D m ′ m ( j ) | j , m ′ ⟩ ⇒ | ψ ¯ ⟩ = D ( j ) | ψ ⟩ {\displaystyle |{\bar {\psi }}\rangle =\sum _{mm'}c_{jm}D_{m'm}^{(j)}|j,m'\rangle \quad \Rightarrow \quad |{\bar {\psi }}\rangle =D^{(j)}|\psi \rangle } For one basis ket: | j , m ¯ ⟩ = ∑ m ′ D ( R ) m ′ m ( j ) | j , m ′ ⟩ {\displaystyle |{\overline {j,m}}\rangle =\sum _{m'}{D(R)}_{m'm}^{(j)}|j,m'\rangle } For the case of orbital angular momentum, the eigenstates | ℓ , m ⟩ {\displaystyle |\ell ,m\rangle } of the orbital angular momentum operator L and solutions of Laplace's equation on a 3d sphere are spherical harmonics: Y ℓ m ( θ , ϕ ) = ⟨ θ , ϕ | ℓ , m ⟩ = ( 2 ℓ + 1 ) 4 π ( ℓ − m ) ! ( ℓ + m ) ! P ℓ m ( cos ⁡ θ ) e i m ϕ {\displaystyle Y_{\ell }^{m}(\theta ,\phi )=\langle \theta ,\phi |\ell ,m\rangle ={\sqrt {{(2\ell +1) \over 4\pi }{(\ell -m)! \over (\ell +m)!}}}\,P_{\ell }^{m}(\cos {\theta })\,e^{im\phi }} where Pℓm is an associated Legendre polynomial, ℓ is the orbital angular momentum quantum number, and m is the orbital magnetic quantum number which takes the values −ℓ, −ℓ + 1, ... ℓ − 1, ℓ The formalism of spherical harmonics have wide applications in applied mathematics, and are closely related to the formalism of spherical tensors, as shown below. Spherical harmonics are functions of the polar and azimuthal angles, ϕ and θ respectively, which can be conveniently collected into a unit vector n(θ, ϕ) pointing in the direction of those angles, in the Cartesian basis it is: n ^ ( θ , ϕ ) = cos ⁡ ϕ sin ⁡ θ e x + s
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Fatsecret

Fatsecret, commonly styled as fatsecret, is a mobile application, website and API that helps people achieve their weight loss goals and find accurate nutrition information. It also offers a weight loss clinic with coaching and medically supported programs. The platform powers global health apps. == History == Fatsecret was founded in 2006 in Melbourne, Australia by Lenny Moses and Rodney Moses. As of 2019, Lenny serves as the company's CEO. The company is known for its calorie counting and meal tracking app, and by April 2016, the company claimed to have 45 million users of its services. In August 2018, a premium version of its app was released. Since August 2009, the company has operated the Fatsecret Platform API, which allows access to its global food and nutrition database. Fatsecret reportedly had 900,000 downloads of its app in January 2020. In an analysis of several Health & Fitness app subcategories for the United States in January 2021, Fatsecret was reported to have the highest 30 day user retention rate of top Calorie Counter + Meal Planner for Weight Loss apps.
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