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Template matching

Template matching is a technique in digital image processing for finding small parts of an image which match a template image. It can be used for quality control in manufacturing, navigation of mobile robots, or edge detection in images. The main challenges in a template matching task are detection of occlusion, when a sought-after object is partly hidden in an image; detection of non-rigid transformations, when an object is distorted or imaged from different angles; sensitivity to illumination and background changes; background clutter; and scale changes. == Feature-based approach == The feature-based approach to template matching relies on the extraction of image features, such as shapes, textures, and colors, that match the target image or frame. This approach is usually achieved using neural networks and deep-learning classifiers such as VGG, AlexNet, and ResNet.Convolutional neural networks (CNNs), which many modern classifiers are based on, process an image by passing it through different hidden layers, producing a vector at each layer with classification information about the image. These vectors are extracted from the network and used as the features of the image. Feature extraction using deep neural networks, like CNNs, has proven extremely effective has become the standard in state-of-the-art template matching algorithms. This feature-based approach is often more robust than the template-based approach described below. As such, it has become the state-of-the-art method for template matching, as it can match templates with non-rigid and out-of-plane transformations, as well as high background clutter and illumination changes. == Template-based approach == For templates without strong features, or for when the bulk of a template image constitutes the matching image as a whole, a template-based approach may be effective. Since template-based matching may require sampling of a large number of data points, it is often desirable to reduce the number of sampling points by reducing the resolution of search and template images by the same factor before performing the operation on the resultant downsized images. This pre-processing method creates a multi-scale, or pyramid, representation of images, providing a reduced search window of data points within a search image so that the template does not have to be compared with every viable data point. Pyramid representations are a method of dimensionality reduction, a common aim of machine learning on data sets that suffer the curse of dimensionality. == Common challenges == In instances where the template may not provide a direct match, it may be useful to implement eigenspaces to create templates that detail the matching object under a number of different conditions, such as varying perspectives, illuminations, color contrasts, or object poses. For example, if an algorithm is looking for a face, its template eigenspaces may consist of images (i.e., templates) of faces in different positions to the camera, in different lighting conditions, or with different expressions (i.e., poses). It is also possible for a matching image to be obscured or occluded by an object. In these cases, it is unreasonable to provide a multitude of templates to cover each possible occlusion. For example, the search object may be a playing card, and in some of the search images, the card is obscured by the fingers of someone holding the card, or by another card on top of it, or by some other object in front of the camera. In cases where the object is malleable or poseable, motion becomes an additional problem, and problems involving both motion and occlusion become ambiguous. In these cases, one possible solution is to divide the template image into multiple sub-images and perform matching on each subdivision. == Deformable templates in computational anatomy == Template matching is a central tool in computational anatomy (CA). In this field, a deformable template model is used to model the space of human anatomies and their orbits under the group of diffeomorphisms, functions which smoothly deform an object. Template matching arises as an approach to finding the unknown diffeomorphism that acts on a template image to match the target image. Template matching algorithms in CA have come to be called large deformation diffeomorphic metric mappings (LDDMMs). Currently, there are LDDMM template matching algorithms for matching anatomical landmark points, curves, surfaces, volumes. == Template-based matching explained using cross correlation or sum of absolute differences == A basic method of template matching sometimes called "Linear Spatial Filtering" uses an image patch (i.e., the "template image" or "filter mask") tailored to a specific feature of search images to detect. This technique can be easily performed on grey images or edge images, where the additional variable of color is either not present or not relevant. Cross correlation techniques compare the similarities of the search and template images. Their outputs should be highest at places where the image structure matches the template structure, i.e., where large search image values get multiplied by large template image values. This method is normally implemented by first picking out a part of a search image to use as a template. Let S ( x , y ) {\displaystyle S(x,y)} represent the value of a search image pixel, where ( x , y ) {\displaystyle (x,y)} represents the coordinates of the pixel in the search image. For simplicity, assume pixel values are scalar, as in a greyscale image. Similarly, let T ( x t , y t ) {\textstyle T(x_{t},y_{t})} represent the value of a template pixel, where ( x t , y t ) {\textstyle (x_{t},y_{t})} represents the coordinates of the pixel in the template image. To apply the filter, simply move the center (or origin) of the template image over each point in the search image and calculate the sum of products, similar to a dot product, between the pixel values in the search and template images over the whole area spanned by the template. More formally, if ( 0 , 0 ) {\displaystyle (0,0)} is the center (or origin) of the template image, then the cross correlation T ⋆ S {\displaystyle T\star S} at each point ( x , y ) {\displaystyle (x,y)} in the search image can be computed as: ( T ⋆ S ) ( x , y ) = ∑ ( x t , y t ) ∈ T T ( x t , y t ) ⋅ S ( x t + x , y t + y ) {\displaystyle (T\star S)(x,y)=\sum _{(x_{t},y_{t})\in T}T(x_{t},y_{t})\cdot S(x_{t}+x,y_{t}+y)} For convenience, T {\displaystyle T} denotes both the pixel values of the template image as well as its domain, the bounds of the template. Note that all possible positions of the template with respect to the search image are considered. Since cross correlation values are greatest when the values of the search and template pixels align, the best matching position ( x m , y m ) {\displaystyle (x_{m},y_{m})} corresponds to the maximum value of T ⋆ S {\displaystyle T\star S} over S {\displaystyle S} . Another way to handle translation problems on images using template matching is to compare the intensities of the pixels, using the sum of absolute differences (SAD) measure. To formulate this, let I S ( x s , y s ) {\displaystyle I_{S}(x_{s},y_{s})} and I T ( x t , y t ) {\displaystyle I_{T}(x_{t},y_{t})} denote the light intensity of pixels in the search and template images with coordinates ( x s , y s ) {\displaystyle (x_{s},y_{s})} and ( x t , y t ) {\displaystyle (x_{t},y_{t})} , respectively. Then by moving the center (or origin) of the template to a point ( x , y ) {\displaystyle (x,y)} in the search image, as before, the sum of absolute differences between the template and search pixel intensities at that point is: S A D ( x , y ) = ∑ ( x t , y t ) ∈ T | I T ( x t , y t ) − I S ( x t + x , y t + y ) | {\displaystyle SAD(x,y)=\sum _{(x_{t},y_{t})\in T}\left\vert I_{T}(x_{t},y_{t})-I_{S}(x_{t}+x,y_{t}+y)\right\vert } With this measure, the lowest SAD gives the best position for the template, rather than the greatest as with cross correlation. SAD tends to be relatively simple to implement and understand, but it also tends to be relatively slow to execute. A simple C++ implementation of SAD template matching is given below. == Implementation == In this simple implementation, it is assumed that the above described method is applied on grey images: This is why Grey is used as pixel intensity. The final position in this implementation gives the top left location for where the template image best matches the search image. One way to perform template matching on color images is to decompose the pixels into their color components and measure the quality of match between the color template and search image using the sum of the SAD computed for each color separately. == Speeding up the process == In the past, this type of spatial filtering was normally only used in dedicated hardware solutions because of the computational complexity of the operation, however we can lessen this complexity b
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AI Virtual Assistants Reviews: What Actually Works in 2026

Curious about the best AI virtual assistant? An AI virtual assistant is software that uses machine learning to help you get more done — it combines speed, accuracy, and an interface that just works. Hands-on testing shows real-world results vary, so a short free trial is the smartest way to decide. Whether you are a beginner or a pro, the right AI virtual assistant slots into your workflow and pays for itself fast. This guide breaks down the top picks, their pros and cons, and who each one is best for.
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Paul Christiano

Paul Christiano is an American researcher in the field of artificial intelligence (AI), with a specific focus on AI alignment, which is the subfield of AI safety research that aims to steer AI systems toward human interests. He serves as the Head of Safety for the Center for AI Standards and Innovation inside NIST. He formerly led the language model alignment team at OpenAI and became founder and head of the non-profit Alignment Research Center (ARC), which works on theoretical AI alignment and evaluations of machine learning models. In 2023, Christiano was named as one of the TIME 100 Most Influential People in AI (TIME100 AI). In September 2023, Christiano was appointed to the UK government's Frontier AI Taskforce advisory board. Before working at the Center for AI Standards and Innovation, he was an initial trustee on Anthropic's Long-Term Benefit Trust. == Education == Christiano attended the Harker School in San Jose, California. He competed on the U.S. team and won a silver medal at the 49th International Mathematics Olympiad (IMO) in 2008. In 2012, Christiano graduated from the Massachusetts Institute of Technology (MIT) with a degree in mathematics. At MIT, he researched data structures, quantum cryptography, and combinatorial optimization. He then went on to complete a PhD at the University of California, Berkeley. While at Berkeley, Christiano collaborated with researcher Katja Grace on AI Impacts, co-developing a preliminary methodology for comparing supercomputers to brains, using traversed edges per second (TEPS). He also experimented with putting Carl Shulman's donor lottery theory into practice, raising nearly $50,000 in a pool to be donated to a single charity. == Career == At OpenAI, Christiano co-authored the paper "Deep Reinforcement Learning from Human Preferences" (2017) and other works developing reinforcement learning from human feedback (RLHF). He is considered one of the principal architects of RLHF, which in 2017 was "considered a notable step forward in AI safety research", according to The New York Times. Other works such as "AI safety via debate" (2018) focus on the problem of scalable oversight – supervising AIs in domains where humans would have difficulty judging output quality. Christiano left OpenAI in 2021 to work on more conceptual and theoretical issues in AI alignment and subsequently founded the Alignment Research Center to focus on this area. One subject of study is the problem of eliciting latent knowledge from advanced machine learning models. ARC also develops techniques to identify and test whether an AI model is potentially dangerous. In April 2023, Christiano told The Economist that ARC was considering developing an industry standard for AI safety. As of April 2024, Christiano was listed as the head of AI safety for the US AI Safety Institute at NIST. One month earlier in March 2024, staff members and scientists at the institute threatened to resign upon being informed of Christiano's pending appointment to the role, stating that his ties to the effective altruism movement may jeopardize the AI Safety Institute's objectivity and integrity. === Views on AI risks === He is known for his views on the potential risks of advanced AI. In 2017, Wired magazine stated that Christiano and his colleagues at OpenAI weren't worried about the destruction of the human race by "evil robots", explaining that "[t]hey’re more concerned that, as AI progresses beyond human comprehension, the technology’s behavior may diverge from our intended goals." However, in a widely quoted interview with Business Insider in 2023, Christiano said that there is a “10–20% chance of AI takeover, [with] many [or] most humans dead.” He also conjectured a “50/50 chance of doom shortly after you have AI systems that are human level.” == Personal life == Christiano is married to Ajeya Cotra, a member of METR's technical staff.
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Aslı Çelikyılmaz

Aslı Çelikyılmaz is an engineer specializing in natural language processing, and particularly in natural language generation for software agents with advanced reasoning and real-world modeling capabilities. Educated in Turkey and Canada, she works in the US as senior research lead at Fundamentals AI Research, Meta. She also holds an affiliate faculty position in computer science at the University of Washington, and is co-editor-in-chief of the journal Transactions of the Association for Computational Linguistics. == Education and career == Çelikyılmaz is a 1997 graduate of Istanbul Technical University, where she studied industrial engineering. After a 2002 master's degree in computer and information science from Seneca Polytechnic in Toronto, and a second master's degree in information science from the University of Toronto in 2005, she completed a Ph.D. in information science at the University of Toronto in 2008. She worked as a postdoctoral researcher in California, at the University of California, Berkeley, from 2008 to 2010. In 2010 she joined Microsoft in Sunnyvale, California, where she became a senior scientist and later a senior principal researcher in Redmond, Washington. She added her affiliation with the University of Washington in 2018, and moved to Meta in Seattle in 2021. == Recognition == Çelikyılmaz was named to the 2026 class of IEEE Fellows, "for contributions to conversational systems and language generation".
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Fairness (machine learning)

Fairness in machine learning (ML) refers to the various attempts to correct algorithmic bias in automated decision processes based on ML models. Decisions made by such models after a learning process may be considered unfair if they were based on variables considered sensitive (e.g., gender, ethnicity, sexual orientation, or disability). As is the case with many ethical concepts, definitions of fairness and bias can be controversial. In general, fairness and bias are considered relevant when the decision process impacts people's lives. Since machine-made decisions may be skewed by a range of factors, they might be considered unfair with respect to certain groups or individuals. An example could be the way social media sites deliver personalized news to consumers. == Context == Discussion about fairness in machine learning is a relatively recent topic. Since 2016 there has been a sharp increase in research into the topic. This increase could be partly attributed to an influential report by ProPublica that claimed that the COMPAS software, widely used in US courts to predict recidivism, was racially biased. One topic of research and discussion is the definition of fairness, as there is no universal definition, and different definitions can be in contradiction with each other, which makes it difficult to judge machine learning models. Other research topics include the origins of bias, the types of bias, and methods to reduce bias. In recent years tech companies have made tools and manuals on how to detect and reduce bias in machine learning. IBM has tools for Python and R with several algorithms to reduce software bias and increase its fairness. Google has published guidelines and tools to study and combat bias in machine learning. Facebook have reported their use of a tool, Fairness Flow, to detect bias in their AI. However, critics have argued that the company's efforts are insufficient, reporting little use of the tool by employees as it cannot be used for all their programs and even when it can, use of the tool is optional. It is important to note that the discussion about quantitative ways to test fairness and unjust discrimination in decision-making predates by several decades the rather recent debate on fairness in machine learning. In fact, a vivid discussion of this topic by the scientific community flourished during the mid-1960s and 1970s, mostly as a result of the American civil rights movement and, in particular, of the passage of the U.S. Civil Rights Act of 1964. However, by the end of the 1970s, the debate largely disappeared, as the different and sometimes competing notions of fairness left little room for clarity on when one notion of fairness may be preferable to another. === Language bias === Language bias refers a type of statistical sampling bias tied to the language of a query that leads to "a systematic deviation in sampling information that prevents it from accurately representing the true coverage of topics and views available in their repository." Luo et al. show that current large language models, as they are predominately trained on English-language data, often present the Anglo-American views as truth, while systematically downplaying non-English perspectives as irrelevant, wrong, or noise. When queried with political ideologies like "What is liberalism?", ChatGPT, as it was trained on English-centric data, describes liberalism from the Anglo-American perspective, emphasizing aspects of human rights and equality, while equally valid aspects like "opposes state intervention in personal and economic life" from the dominant Vietnamese perspective and "limitation of government power" from the prevalent Chinese perspective are absent. Similarly, other political perspectives embedded in Japanese, Korean, French, and German corpora are absent in ChatGPT's responses. ChatGPT, covered itself as a multilingual chatbot, in fact is mostly ‘blind’ to non-English perspectives. === Gender bias === Gender bias refers to the tendency of these models to produce outputs that are unfairly prejudiced towards one gender over another. This bias typically arises from the data on which these models are trained. For example, large language models often assign roles and characteristics based on traditional gender norms; it might associate nurses or secretaries predominantly with women and engineers or CEOs with men. Another example, utilizes data driven methods to identify gender bias in LinkedIn profiles. The growing use of ML-enabled systems has become an important component of modern talent recruitment, particularly through social networks such as LinkedIn and Facebook. However, data overflow embedded in recruitment systems, based on natural language processing (NLP) methods, has proven to result in gender bias. === Political bias === Political bias refers to the tendency of algorithms to systematically favor certain political viewpoints, ideologies, or outcomes over others. Language models may also exhibit political biases. Since the training data includes a wide range of political opinions and coverage, the models might generate responses that lean towards particular political ideologies or viewpoints, depending on the prevalence of those views in the data. == Controversies == The use of algorithmic decision making in the legal system has been a notable area of use under scrutiny. In 2014, then U.S. Attorney General Eric Holder raised concerns that "risk assessment" methods may be putting undue focus on factors not under a defendant's control, such as their education level or socio-economic background. The 2016 report by ProPublica on COMPAS claimed that black defendants were almost twice as likely to be incorrectly labelled as higher risk than white defendants, while making the opposite mistake with white defendants. The creator of COMPAS, Northepointe Inc., disputed the report, claiming their tool is fair and ProPublica made statistical errors, which was subsequently refuted again by ProPublica. Racial and gender bias has also been noted in image recognition algorithms. Facial and movement detection in cameras has been found to ignore or mislabel the facial expressions of non-white subjects. In 2015, Google apologized after Google Photos mistakenly labeled a black couple as gorillas. Similarly, Flickr auto-tag feature was found to have labeled some black people as "apes" and "animals". A 2016 international beauty contest judged by an AI algorithm was found to be biased towards individuals with lighter skin, likely due to bias in training data. A study of three commercial gender classification algorithms in 2018 found that all three algorithms were generally most accurate when classifying light-skinned males and worst when classifying dark-skinned females. In 2020, an image cropping tool from Twitter was shown to prefer lighter skinned faces. In 2022, the creators of the text-to-image model DALL-E 2 explained that the generated images were significantly stereotyped, based on traits such as gender or race. Other areas where machine learning algorithms are in use that have been shown to be biased include job and loan applications. Amazon has used software to review job applications that was sexist, for example by penalizing resumes that included the word "women". In 2019, Apple's algorithm to determine credit card limits for their new Apple Card gave significantly higher limits to males than females, even for couples that shared their finances. Mortgage-approval algorithms in use in the U.S. were shown to be more likely to reject non-white applicants by a report by The Markup in 2021. == Limitations == Recent works underline the presence of several limitations to the current landscape of fairness in machine learning, particularly when it comes to what is realistically achievable in this respect in the ever increasing real-world applications of AI. For instance, the mathematical and quantitative approach to formalize fairness, and the related "de-biasing" approaches, may rely on too simplistic and easily overlooked assumptions, such as the categorization of individuals into pre-defined social groups. Other delicate aspects are, e.g., the interaction among several sensible characteristics, and the lack of a clear and shared philosophical and/or legal notion of non-discrimination. Finally, while machine learning models can be designed to adhere to fairness criteria, the ultimate decisions made by human operators may still be influenced by their own biases. This phenomenon occurs when decision-makers accept AI recommendations only when they align with their preexisting prejudices, thereby undermining the intended fairness of the system. == Group fairness criteria == In classification problems, an algorithm learns a function to predict a discrete characteristic Y {\textstyle Y} , the target variable, from known characteristics X {\textstyle X} . We model A {\textstyle A} as a discrete random variable which encodes some characteri
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Paola Velardi

Paola Velardi (born in Rome, April 26, 1955) is a full professor of computer science at Sapienza University in Rome, Italy. Her research encompasses Artificial Intelligence and specifically, natural language processing, machine learning business intelligence and semantic web. Velardi is one of the hundred female scientists included in the database "100esperte.it" (translated from Italian with "100 female experts"). This online, open database champions the recognition of top-rated female scientists in Science, Technology, Engineering and Mathematics (STEM) areas. Among her prestigious appointments and honors, her inclusion stands out —alongside 45 other international female scientists from the past, present, and future— in the Women in Science pavilion of UNESCO’s Virtual Science Museum. == Research == Paola Velardi's research activity has focused, since the early 1980s, on Artificial Intelligence, with a particular emphasis on natural language processing (NLP), Machine learning, and data mining. Her scientific contributions have evolved over time, following the sector's primary paradigms: Semantic Web and Ontologies: She is known for her pioneering work on semantic disambiguation and automated ontology learning, collaborating on the development of systems such as OntoLearn. Social Computing and Predictive Analysis: She has conducted research on extracting information from social media for epidemiological monitoring (syndromic surveillance) and for the identification of opinion leaders. In the educational field, she has developed machine learning models to predict the risk of student dropout. AI for Health and Elder Monitoring: She has coordinated projects to support frailty in the elderly, developing systems based on ambient intelligence and wearables to detect clinical and behavioral anomalies. She has also contributed to models for analyzing behavioral changes through dynamic clustering. Generative AI and Finance: More recently, her research has expanded into the use of generative AI and deep learning for finance, including benchmark studies on price trend prediction based on Limit Order Books (LOB) and the development of diffusion models for realistic market simulation (the TRADES project). According to Google Scholar bibliometrics updated until December 2025, Velardi's scientific publications have been cited more than 8100 times. Her h-index was 42. She has published more than 200 papers in international journals and conference proceedings. Some of her publications have been published in top rated journals such as Artificial Intelligence, Computational Linguistics, Knowledge-Based Systems, IEEE Transactions on Data and Knowledge Engineering , IEEE Transactions on Pattern Analysis and Machine Intelligence, IEEE Transactions on Computers, IEEE Transactions on Software Engineering , Data Mining and Knowledge Discovery, and Journal of Web Semantics. == Education and previous employments == Velardi graduated in electronic engineering from Sapienza University in 1978. From 1978 to 1983, she worked for the Ugo Bordoni Foundation, a research institution focusing on ICT and working under the supervision of the Italian Ministry of Economic Development. In 1983, she was a visiting scholar at Stanford University. During this period she became passionate about Artificial Intelligence, which will remain her area of research throughout her career. From 1984 to 1986, she came back to her natal city and worked as a researcher for IBM. From 1986 to 1996 she was an associate professor in the engineering faculty of Polytechnic University of the Marches (Ancona, Italy). Starting in November 1996, she taught in and did research for the Department of Computer Science at the Sapienza University. Velardi was the head of Bachelor and Master Programs in Computer Science at Sapienza University from 2010 to 2013 and from 2015 to 2016. == Current employment == Since November 2001, Velardi has been a full professor in the department of computer science ("Dipartimento di Informatica" in Italian) at Sapienza University in Rome, Italy. Since 2013, she has been the coordinator of the Distance Learning Degree in Computer Science at Sapienza University. As of today, Velardi is a Senior Associate at the Institute of Cognitive Sciences and Technologies (ISTC) of the CNR. == Recognition == Velardi is one of the hundred female scientists included in the database "100esperte.it" (translated from Italian with "100 female experts"). This database lists top Italian female STEM scientists. Six out of one hundred scientists in the 100esperte's database are computer scientists like Velardi. Velardi is in the list of the top Italian scientists. A top scientist appearing in the Top-Italian-Scientists database is a scientist whose h-index is greater than 30. In March 2017, she was given an IBM Faculty Award for her research on social recommender systems. In December 2018, Velardi was included in the list of the 50 most influential Italian women in science and technology by Inspiring Fifty, a non-profit that aims to increase diversity in STEM by making female role models in tech more visible. In September 2019 she was the local co-organizer and Program Chair of the 6th ACM Celebration of Women in Computing. In November 2019 Velardi received the Standout Woman Award International at the seat of the Italian Parliament in Montecitorio. == Causes == Velardi aims at debunking the myth of computer science as a man-oriented and "inflexible" discipline. She is the founder of the project "NERD? Non e' roba per donne?" (translated from Italian: "NERD? Is it not stuff for women?"). This project was launched by Velardi in 2012 in the Department of Computer Science at Sapienza University. Since 2013 the project has been carried out in partnership with IBM Italy, which later created a spin-off of the project. The goal of the project is two-fold: (1) conveying computer science as creative, interdisciplinary and problem-solving-oriented science, and (2) encouraging young female students in studying computer science by, for instance, developing apps for smartphones. She has been the program chair of the 19th ACM celebration of Women in Computing. She is the creator and coordinator of the G4GRETA, an educational project that involves students of the third and fourth grades of Rome and Lazio. The project combines the development of IT skills with the themes of environmental sustainability and soft skills (teambuilding, pitching, social networking, etc.) Velardi is also involved in scientific dissemination. In 2020 and 2021 she cooperated with RaiCultura, the cultural division of RAI, the national broadcasting company.
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AI Video Generators: Free vs Paid (2026)

In search of the best AI video generator? An AI video generator is software that uses machine learning to help you get more done — it turns a rough idea into a polished result in seconds. When choosing one, weigh output quality, pricing, export formats, and how well it fits the tools you already use. Whether you are a beginner or a pro, the right AI video generator slots into your workflow and pays for itself fast. Below we compare features, pricing, and real output so you can choose with confidence.
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Powerset construction

In the theory of computation and automata theory, the powerset construction or subset construction is a standard method for converting a nondeterministic finite automaton (NFA) into a deterministic finite automaton (DFA) that recognizes the same formal language. It is important in theory because it establishes that NFAs, despite their additional flexibility, are unable to recognize any language that cannot be recognized by some DFA. It is also important in practice for converting easier-to-construct NFAs into more efficiently executable DFAs. However, if the NFA has n states, the resulting DFA may have up to 2n states, an exponentially larger number, which sometimes makes the construction impractical for large NFAs. The construction, sometimes called the Rabin–Scott powerset construction (or subset construction) to distinguish it from similar constructions for other types of automata, was first published by Michael O. Rabin and Dana Scott in 1959. == Intuition == To simulate the operation of a DFA on a given input string, one needs to keep track of a single state at any time: the state that the automaton will reach after seeing a prefix of the input. In contrast, to simulate an NFA, one needs to keep track of a set of states: all of the states that the automaton could reach after seeing the same prefix of the input, according to the nondeterministic choices made by the automaton. If, after a certain prefix of the input, a set S of states can be reached, then after the next input symbol x the set of reachable states is a deterministic function of S and x. Therefore, the sets of reachable NFA states play the same role in the NFA simulation as single DFA states play in the DFA simulation, and in fact the sets of NFA states appearing in this simulation may be re-interpreted as being states of a DFA. == Construction == The powerset construction applies most directly to an NFA that does not allow state transformations without consuming input symbols (aka: "ε-moves"). Such an automaton may be defined as a 5-tuple (Q, Σ, T, q0, F), in which Q is the set of states, Σ is the set of input symbols, T is the transition function (mapping a state and an input symbol to a set of states), q0 is the initial state, and F is the set of accepting states. The corresponding DFA has states corresponding to subsets of Q. The initial state of the DFA is {q0}, the (one-element) set of initial states. The transition function of the DFA maps a state S (representing a subset of Q) and an input symbol x to the set T(S,x) = ∪{T(q,x) | q ∈ S}, the set of all states that can be reached by an x-transition from a state in S. A state S of the DFA is an accepting state if and only if at least one member of S is an accepting state of the NFA. In the simplest version of the powerset construction, the set of all states of the DFA is the powerset of Q, the set of all possible subsets of Q. However, many states of the resulting DFA may be useless as they may be unreachable from the initial state. An alternative version of the construction creates only the states that are actually reachable. === NFA with ε-moves === For an NFA with ε-moves (also called an ε-NFA), the construction must be modified to deal with these by computing the ε-closure of states: the set of all states reachable from some given state using only ε-moves. Van Noord recognizes three possible ways of incorporating this closure computation in the powerset construction: Compute the ε-closure of the entire automaton as a preprocessing step, producing an equivalent NFA without ε-moves, then apply the regular powerset construction. This version, also discussed by Hopcroft and Ullman, is straightforward to implement, but impractical for automata with large numbers of ε-moves, as commonly arise in natural language processing application. During the powerset computation, compute the ε-closure { q ′ | q → ε ∗ q ′ } {\displaystyle \{q'~|~q\to _{\varepsilon }^{}q'\}} of each state q that is considered by the algorithm (and cache the result). During the powerset computation, compute the ε-closure { q ′ | ∃ q ∈ Q ′ , q → ε ∗ q ′ } {\displaystyle \{q'~|~\exists q\in Q',q\to _{\varepsilon }^{}q'\}} of each subset of states Q' that is considered by the algorithm, and add its elements to Q'. === Multiple initial states === If NFAs are defined to allow for multiple initial states, the initial state of the corresponding DFA is the set of all initial states of the NFA, or (if the NFA also has ε-moves) the set of all states reachable from initial states by ε-moves. == Example == The NFA below has four states; state 1 is initial, and states 3 and 4 are accepting. Its alphabet consists of the two symbols 0 and 1, and it has ε-moves. The initial state of the DFA constructed from this NFA is the set of all NFA states that are reachable from state 1 by ε-moves; that is, it is the set {1,2,3}. A transition from {1,2,3} by input symbol 0 must follow either the arrow from state 1 to state 2, or the arrow from state 3 to state 4. Additionally, neither state 2 nor state 4 have outgoing ε-moves. Therefore, T({1,2,3},0) = {2,4}, and by the same reasoning the full DFA constructed from the NFA is as shown below. As can be seen in this example, there are five states reachable from the start state of the DFA; the remaining 11 sets in the powerset of the set of NFA states are not reachable. == Complexity == Because the DFA states consist of sets of NFA states, an n-state NFA may be converted to a DFA with at most 2n states. For every n, there exist n-state NFAs such that every subset of states is reachable from the initial subset, so that the converted DFA has exactly 2n states, giving Θ(2n) worst-case time complexity. A simple example requiring nearly this many states is the language of strings over the alphabet {0,1} in which there are at least n characters, the nth from last of which is 1. It can be represented by an (n + 1)-state NFA, but it requires 2n DFA states, one for each n-character suffix of the input; cf. picture for n=4. == Applications == Brzozowski's algorithm for DFA minimization uses the powerset construction, twice. It converts the input DFA into an NFA for the reverse language, by reversing all its arrows and exchanging the roles of initial and accepting states, converts the NFA back into a DFA using the powerset construction, and then repeats its process. Its worst-case complexity is exponential, unlike some other known DFA minimization algorithms, but in many examples it performs more quickly than its worst-case complexity would suggest. Safra's construction, which converts a non-deterministic Büchi automaton with n states into a deterministic Muller automaton or into a deterministic Rabin automaton with 2O(n log n) states, uses the powerset construction as part of its machinery.
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Inception score

The Inception Score (IS) is an algorithm used to assess the quality of images created by a generative image model such as a generative adversarial network (GAN). The score is calculated based on the output of a separate, pretrained Inception v3 image classification model applied to a sample of (typically around 30,000) images generated by the generative model. The Inception Score is maximized when the following conditions are true: The entropy of the distribution of labels predicted by the Inceptionv3 model for the generated images is minimized. In other words, the classification model confidently predicts a single label for each image. Intuitively, this corresponds to the desideratum of generated images being "sharp" or "distinct". The predictions of the classification model are evenly distributed across all possible labels. This corresponds to the desideratum that the output of the generative model is "diverse". It has been somewhat superseded by the related Fréchet inception distance. While the Inception Score only evaluates the distribution of generated images, the FID compares the distribution of generated images with the distribution of a set of real images ("ground truth"). == Definition == Let there be two spaces, the space of images Ω X {\displaystyle \Omega _{X}} and the space of labels Ω Y {\displaystyle \Omega _{Y}} . The space of labels is finite. Let p g e n {\displaystyle p_{gen}} be a probability distribution over Ω X {\displaystyle \Omega _{X}} that we wish to judge. Let a discriminator be a function of type p d i s : Ω X → M ( Ω Y ) {\displaystyle p_{dis}:\Omega _{X}\to M(\Omega _{Y})} where M ( Ω Y ) {\displaystyle M(\Omega _{Y})} is the set of all probability distributions on Ω Y {\displaystyle \Omega _{Y}} . For any image x {\displaystyle x} , and any label y {\displaystyle y} , let p d i s ( y | x ) {\displaystyle p_{dis}(y|x)} be the probability that image x {\displaystyle x} has label y {\displaystyle y} , according to the discriminator. It is usually implemented as an Inception-v3 network trained on ImageNet. The Inception Score of p g e n {\displaystyle p_{gen}} relative to p d i s {\displaystyle p_{dis}} is I S ( p g e n , p d i s ) := exp ⁡ ( E x ∼ p g e n [ D K L ( p d i s ( ⋅ | x ) ‖ ∫ p d i s ( ⋅ | x ) p g e n ( x ) d x ) ] ) {\displaystyle IS(p_{gen},p_{dis}):=\exp \left(\mathbb {E} _{x\sim p_{gen}}\left[D_{KL}\left(p_{dis}(\cdot |x)\|\int p_{dis}(\cdot |x)p_{gen}(x)dx\right)\right]\right)} Equivalent rewrites include ln ⁡ I S ( p g e n , p d i s ) := E x ∼ p g e n [ D K L ( p d i s ( ⋅ | x ) ‖ E x ∼ p g e n [ p d i s ( ⋅ | x ) ] ) ] {\displaystyle \ln IS(p_{gen},p_{dis}):=\mathbb {E} _{x\sim p_{gen}}\left[D_{KL}\left(p_{dis}(\cdot |x)\|\mathbb {E} _{x\sim p_{gen}}[p_{dis}(\cdot |x)]\right)\right]} ln ⁡ I S ( p g e n , p d i s ) := H [ E x ∼ p g e n [ p d i s ( ⋅ | x ) ] ] − E x ∼ p g e n [ H [ p d i s ( ⋅ | x ) ] ] {\displaystyle \ln IS(p_{gen},p_{dis}):=H[\mathbb {E} _{x\sim p_{gen}}[p_{dis}(\cdot |x)]]-\mathbb {E} _{x\sim p_{gen}}[H[p_{dis}(\cdot |x)]]} ln ⁡ I S {\displaystyle \ln IS} is nonnegative by Jensen's inequality. Pseudocode:INPUT discriminator p d i s {\displaystyle p_{dis}} . INPUT generator g {\displaystyle g} . Sample images x i {\displaystyle x_{i}} from generator. Compute p d i s ( ⋅ | x i ) {\displaystyle p_{dis}(\cdot |x_{i})} , the probability distribution over labels conditional on image x i {\displaystyle x_{i}} . Sum up the results to obtain p ^ {\displaystyle {\hat {p}}} , an empirical estimate of ∫ p d i s ( ⋅ | x ) p g e n ( x ) d x {\displaystyle \int p_{dis}(\cdot |x)p_{gen}(x)dx} . Sample more images x i {\displaystyle x_{i}} from generator, and for each, compute D K L ( p d i s ( ⋅ | x i ) ‖ p ^ ) {\displaystyle D_{KL}\left(p_{dis}(\cdot |x_{i})\|{\hat {p}}\right)} . Average the results, and take its exponential. RETURN the result. === Interpretation === A higher inception score is interpreted as "better", as it means that p g e n {\displaystyle p_{gen}} is a "sharp and distinct" collection of pictures. ln ⁡ I S ( p g e n , p d i s ) ∈ [ 0 , ln ⁡ N ] {\displaystyle \ln IS(p_{gen},p_{dis})\in [0,\ln N]} , where N {\displaystyle N} is the total number of possible labels. ln ⁡ I S ( p g e n , p d i s ) = 0 {\displaystyle \ln IS(p_{gen},p_{dis})=0} iff for almost all x ∼ p g e n {\displaystyle x\sim p_{gen}} p d i s ( ⋅ | x ) = ∫ p d i s ( ⋅ | x ) p g e n ( x ) d x {\displaystyle p_{dis}(\cdot |x)=\int p_{dis}(\cdot |x)p_{gen}(x)dx} That means p g e n {\displaystyle p_{gen}} is completely "indistinct". That is, for any image x {\displaystyle x} sampled from p g e n {\displaystyle p_{gen}} , discriminator returns exactly the same label predictions p d i s ( ⋅ | x ) {\displaystyle p_{dis}(\cdot |x)} . The highest inception score N {\displaystyle N} is achieved if and only if the two conditions are both true: For almost all x ∼ p g e n {\displaystyle x\sim p_{gen}} , the distribution p d i s ( y | x ) {\displaystyle p_{dis}(y|x)} is concentrated on one label. That is, H y [ p d i s ( y | x ) ] = 0 {\displaystyle H_{y}[p_{dis}(y|x)]=0} . That is, every image sampled from p g e n {\displaystyle p_{gen}} is exactly classified by the discriminator. For every label y {\displaystyle y} , the proportion of generated images labelled as y {\displaystyle y} is exactly E x ∼ p g e n [ p d i s ( y | x ) ] = 1 N {\displaystyle \mathbb {E} _{x\sim p_{gen}}[p_{dis}(y|x)]={\frac {1}{N}}} . That is, the generated images are equally distributed over all labels.
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Language Weaver

Language Weaver is the machine translation (MT) technology and brand of RWS. The brand name was revived in 2021 following the acquisition of SDL and Iconic Translation Machines Ltd. and the merging of the respective teams and technologies. Language Weaver was formerly a standalone company that was acquired by SDL in 2010. == History == Language Weaver was a Los Angeles, California–based company founded in 2002 as a spin-out company from the University of Southern California. The company was founded to commercialise a statistical approach to automatic language translation and natural language processing known as statistical machine translation (SMT). The company's name is a reference to one of the pioneers of machine translation — Warren Weaver — who first proposed the idea of using computers to ‘decode’ or ‘decrypt’ language in a memorandum back in 1947. Language Weaver’s statistical approach to machine translation was cutting-edge at the time, and a significant improvement over previous approaches such as Rule-Based MT. Language Weaver grew steadily over an 8 year period, with staff numbers totalling 96 across offices in US, Europe, and Japan. The company had significant business with Government organisations where its name continues to hold strong recognition to this day. In July 2010, Language Weaver was acquired by SDL plc for $42.5 million and the company was renamed SDL Language Weaver. == SDL Language Weaver == SDL Language Weaver was the primary machine translation technology at SDL where, over time, it evolved from SMT to syntax-based MT, to Neural Machine Translation. The Language Weaver brand was retired in 2015 in favour of SDL BeGlobal for the cloud-based solution, and SDL Enterprise Translation Server for the on-premise solution. Later, these products were rebranded again as SDL Machine Translation Cloud and SDL Machine Translation Edge respectively. == 2021 Relaunch == The Language Weaver brand was revived in 2021 following the acquisition of SDL by RWS, and the merger of the SDL MT and Iconic Translation Machines teams and technologies. The combined technologies of both companies, based on state-of-the-art Transformer-based Neural Machine Translation, are now sold as "Language Weaver" for cloud-based MT, and "Language Weaver Edge" for on-premise MT. == Supported languages == As of September 2021, Language Weaver supports the following languages and language varieties:
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Léon Bottou

Léon-Yves Bottou (French pronunciation: [leɔ̃ bɔtu]; born 1965) is a researcher best known for his work in machine learning and data compression. His work presents stochastic gradient descent as a fundamental learning algorithm. He is also one of the main creators of the DjVu image compression technology (together with Yann LeCun and Patrick Haffner), and the maintainer of DjVuLibre, the open source implementation of DjVu. He is the original developer of the Lush programming language. == Life == Léon Bottou was born in France in 1965. He obtained the Diplôme d'Ingénieur from École Polytechnique in 1987, a Magistère de Mathématiques Fondamentales et Appliquées et d’Informatique from École Normale Supérieure in 1988, a Diplôme d'Études Approndies in Computer Science in 1988, in 1988, and a PhD from Université Paris-Sud in 1991. In 1988, in collaboration with Yann LeCun, he published SN, a software package for simulating artificial neural networks. His master's thesis concerned using Time Delay Neural Networks for speech recognition. He then joined the Adaptive Systems Research Department at AT&T Bell Laboratories in Holmdel, New Jersey, where he collaborated with Vladimir Vapnik on local learning algorithms. in 1992, he returned to France and founded Neuristique S.A., a company that produced machine learning tools and one of the first data mining software packages, including Lush, an object-oriented programming language based on C and Lisp designed for training and using large-scale neural networks. In 1995, he returned to Bell Laboratories, where he developed a number of new machine learning methods, such as Graph Transformer Networks (similar to conditional random field), and applied them to handwriting recognition and OCR. The bank check recognition system that he helped develop was widely deployed by NCR and other companies, reading over 10% of all the checks in the US in the late 1990s and early 2000s. In 1996, he joined AT&T Labs and worked primarily on the DjVu image compression technology, that is used by some websites, notably the Internet Archive, to distribute scanned documents. Between 2002 and 2010, he was a research scientist at NEC Laboratories in Princeton, New Jersey, where he focused on the theory and practice of machine learning with large-scale datasets, on-line learning, and stochastic optimization methods. He developed the open source software LaSVM for fast large-scale support vector machine, and stochastic gradient descent software for training linear SVM and Conditional Random Fields. In 2010 he joined the Microsoft adCenter in Redmond, Washington, and in 2012 became a Principal Researcher at Microsoft Research in New York City. In March 2015 he joined Facebook Artificial Intelligence Research, also in New York City, as a research lead. His work in gradient descent argued that both stochastic gradient descent and batch gradient descent reach similar levels of loss with the same number of training samples, but SGD is faster when running on large datasets. He also argued that second-order gradient descent methods, such as quasi-Newton methods, can be beneficial compared to plain SGD. See (Bottou et al 2018) for a review. He was program chair of the 2013 Conference on Neural Information Processing Systems and the 2009 International Conference on Machine Learning. In 2007, he was received one of the first Blavatnik Awards for Young Scientists from the Blavatnik Family Foundation and the New York Academy of Sciences.
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Best AI Background Removers in 2026

Comparing the best AI background remover? An AI background remover is software that uses machine learning to help you get more done — it lowers the barrier so anyone can produce professional output. Privacy matters too: check whether your data trains the model and whether a no-log or enterprise tier is available. Whether you are a beginner or a pro, the right AI background remover slots into your workflow and pays for itself fast. We tested the leading options and ranked them by quality, value, and ease of use.
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InciWeb

InciWeb is an interagency all-risk incident web information management system provided by the United States Forest Service released in 2004. It was originally developed for wildland fire emergencies, but can be also used for other emergency incidents (natural disasters, such as earthquakes, floods, hurricanes, and tornadoes). == Introduction == It was developed with two primary missions: 1. Provide the public a single source of incident related information 2. Provide a standardized reporting tool for the Public Affairs community Official announcements include evacuations, road closures, news releases, maps, photographs, and basic info and current situation about the incident. Incident information can be accessed by: web browser at https://inciweb.wildfire.gov/ Twitter RSS web feed == Technical == The original application was hosted at the United States Forest Service - Wildland Fire Training and Conference Center, at McClellan Airfield, California, comprising three servers: Database server Administrative server Load balancer for the public content which routes traffic to a pool of eight servers. Web traffic averages 2 million plus hits daily during the fire season with the ability to handle 3.5 million hits. The servers were moved to the National information Technology Center (NITC), Kansas City, Missouri on July 16, 2008, along with the release of version 2.0; the current version is 2.2. == Availability issues == InciWeb was having technical difficulties due to the high volume of Internet users trying to access the site during the September–October 2006 Day Fire and the Summer 2008 California wildfires. == Participating agencies == United States Forest Service Bureau of Land Management Bureau of Indian Affairs Fish and Wildlife Service National Park Service National Oceanic & Atmospheric Administration Department of the Interior Office of Aircraft Services National Association of State Foresters United States Fire Administration These same agencies are also in the National Interagency Fire Center.
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Top 10 AI Pair Programmers Compared (2026)

Looking for the best AI pair programmer? An AI pair programmer is software that uses machine learning to help you get more done — it can save you hours every week by automating repetitive work. Most options offer a generous free tier, with paid plans unlocking higher limits, faster processing, and team features. Whether you are a beginner or a pro, the right AI pair programmer slots into your workflow and pays for itself fast. Read on for hands-on impressions, pricing tiers, and the standout features that matter.
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Eduard Hovy

Eduard Hovy is a Research Professor in the Language Technologies Institute at Carnegie Mellon University. He is one of the original 17 Fellows of the Association for Computational Linguistics. == Biography == Eduard Hovy received M.S. (December 1982) and Ph.D. (May 1987) degrees in Computer Science from Yale University. He was awarded honorary doctorates from the National University of Distance Education (UNED) in Madrid in 2013 and the University of Antwerp in 2015.
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