Inferential Theory of Learning (ITL) is an area of machine learning which describes inferential processes performed by learning agents. ITL has been continuously developed by Ryszard S. Michalski, starting in the 1980s. The first known publication of ITL was in 1983. In the ITL learning process is viewed as a search (inference) through hypotheses space guided by a specific goal. The results of learning need to be stored. Stored information will later be used by the learner for future inferences. Inferences are split into multiple categories including conclusive, deduction, and induction. In order for an inference to be considered complete it was required that all categories must be taken into account. This is how the ITL varies from other machine learning theories like Computational Learning Theory and Statistical Learning Theory; which both use singular forms of inference. == Usage == The most relevant published usage of ITL was in scientific journal published in 2012 and used ITL as a way to describe how agent-based learning works. According to the journal "The Inferential Theory of Learning (ITL) provides an elegant way of describing learning processes by agents".
Wadhwani Institute for Artificial Intelligence
Wadhwani AI, based in Mumbai, Maharashtra, is an independent, non-profit institute. Founded in 2018, it is dedicated to developing Artificial intelligence solutions for social good. Their mission is to build AI-based innovations and solutions for underserved communities in developing countries, for a wide range of domains including agriculture, education, financial inclusion, healthcare, and infrastructure. == History and funding == The institute was founded with a $30 million philanthropic effort by the Wadhwani brothers, Romesh Wadhwani and Sunil Wadhwani. The institute was inaugurated and dedicated to the nation by Narendra Modi, the 14th Prime Minister of India. In 2019, the institute received a $2 million grant from Google.org to create technologies to help reduce crop losses in cotton farming, through integrated pest management. The United States Agency for International Development awarded $2 million to the institute in 2020 to develop tools, using mathematical modeling techniques and digital technologies such as artificial intelligence and machine learning, to forecast COVID-19 disease patterns, estimate resources needed, and plan interventions. == Collaboration == With assistance from Google, the Ministry of Agriculture and Farmers' Welfare and the Wadhwani AI developed Krishi 24/7, the first AI-powered automated agricultural news monitoring and analysis tool. Through better decision-making, Krishi 24/7 will support the identification of valuable news, provide timely notifications, and respond quickly to safeguard farmers' interests and advance sustainable agricultural growth. The application converts news articles into English after scanning them in several languages. It ensures that the ministry is informed in a timely manner about pertinent occurrences that are published online by extracting key information from news items, including the headline, crop name, event type, date, location, severity, summary, and source link. The National Center for Disease Control has effectively implemented a comparable automated surveillance and analysis tool for disease outbreaks.
Teknomo–Fernandez algorithm
The Teknomo–Fernandez algorithm (TF algorithm), is an efficient algorithm for generating the background image of a given video sequence. By assuming that the background image is shown in the majority of the video, the algorithm is able to generate a good background image of a video in O ( R ) {\displaystyle O(R)} -time using only a small number of binary operations and Boolean bit operations, which require a small amount of memory and has built-in operators found in many programming languages such as C, C++, and Java. == History == People tracking from videos usually involves some form of background subtraction to segment foreground from background. Once foreground images are extracted, then desired algorithms (such as those for motion tracking, object tracking, and facial recognition) may be executed using these images. However, background subtraction requires that the background image is already available and unfortunately, this is not always the case. Traditionally, the background image is searched for manually or automatically from the video images when there are no objects. More recently, automatic background generation through object detection, medial filtering, medoid filtering, approximated median filtering, linear predictive filter, non-parametric model, Kalman filter, and adaptive smoothening have been suggested; however, most of these methods have high computational complexity and are resource-intensive. The Teknomo–Fernandez algorithm is also an automatic background generation algorithm. Its advantage, however, is its computational speed of only O ( R ) {\displaystyle O(R)} -time, depending on the resolution R {\displaystyle R} of an image and its accuracy gained within a manageable number of frames. Only at least three frames from a video is needed to produce the background image assuming that for every pixel position, the background occurs in the majority of the videos. Furthermore, it can be performed for both grayscale and colored videos. == Assumptions == The camera is stationary. The light of the environment changes only slowly relative to the motions of the people in the scene. The number of people does not occupy the scene for most of the time at the same place. Generally, however, the algorithm will certainly work whenever the following single important assumption holds: For each pixel position, the majority of the pixel values in the entire video contain the pixel value of the actual background image (at that position).As long as each part of the background is shown in the majority of the video, the entire background image needs not to appear in any of its frames. The algorithm is expected to work accurately. == Background image generation == === Equations === For three frames of image sequence x 1 {\displaystyle x_{1}} , x 2 {\displaystyle x_{2}} , and x 3 {\displaystyle x_{3}} , the background image B {\displaystyle B} is obtained using B = x 3 ( x 1 ⊕ x 2 ) + x 1 x 2 {\displaystyle B=x_{3}(x_{1}\oplus x_{2})+x_{1}x_{2}} where ⊕ {\displaystyle \oplus } denotes the exclusive disjunctive bit operator. The Boolean mode function S {\displaystyle S} of the table occurs when the number of 1 entries is larger than half of the number of images such that S = { 1 , if ∑ i = 1 n x i ≥ ⌈ n 2 + 1 ⌉ , and n ≥ 3 0 , otherwise {\displaystyle S={\begin{cases}1,&{\text{if }}\sum _{i=1}^{n}x_{i}\geq \left\lceil {\frac {n}{2}}+1\right\rceil ,{\text{ and }}n\geq 3\\0,&{\text{otherwise}}\end{cases}}} For three images, the background image B {\displaystyle B} can be taken as the value x ¯ 1 x 2 x 3 + x 1 x ¯ 2 x 3 + x 1 x 2 x ¯ 3 + x 1 x 2 x 3 {\displaystyle {\bar {x}}_{1}x_{2}x_{3}+x_{1}{\bar {x}}_{2}x_{3}+x_{1}x_{2}{\bar {x}}_{3}+x_{1}x_{2}x_{3}} === Background generation algorithm === At the first level, three frames are selected at random from the image sequence to produce a background image by combining them using the first equation. This yields a better background image at the second level. The procedure is repeated until desired level L {\displaystyle L} . == Theoretical accuracy == At level ℓ {\displaystyle \ell } , the probability p ℓ {\displaystyle p_{\ell }} that the modal bit predicted is the actual modal bit is represented by the equation p ℓ = ( p ℓ − 1 ) 3 + 3 ( p ℓ − 1 ) 2 ( 1 − p ℓ − 1 ) {\displaystyle p_{\ell }=(p_{\ell -1})^{3}+3(p_{\ell -1})^{2}(1-p_{\ell -1})} . The table below gives the computed probability values across several levels using some specific initial probabilities. It can be observed that even if the modal bit at the considered position is at a low 60% of the frames, the probability of accurate modal bit determination is already more than 99% at 6 levels. == Space complexity == The space requirement of the Teknomo–Fernandez algorithm is given by the function O ( R F + R 3 L ) {\displaystyle O(RF+R3^{L})} , depending on the resolution R {\displaystyle R} of the image, the number F {\displaystyle F} of frames in the video, and the desired number L {\displaystyle L} of levels. However, the fact that L {\displaystyle L} will probably not exceed 6 reduces the space complexity to O ( R F ) {\displaystyle O(RF)} . == Time complexity == The entire algorithm runs in O ( R ) {\displaystyle O(R)} -time, only depending on the resolution of the image. Computing the modal bit for each bit can be done in O ( 1 ) {\displaystyle O(1)} -time while the computation of the resulting image from the three given images can be done in O ( R ) {\displaystyle O(R)} -time. The number of the images to be processed in L {\displaystyle L} levels is O ( 3 L ) {\displaystyle O(3^{L})} . However, since L ≤ 6 {\displaystyle L\leq 6} , then this is actually O ( 1 ) {\displaystyle O(1)} , thus the algorithm runs in O ( R ) {\displaystyle O(R)} . == Variants == A variant of the Teknomo–Fernandez algorithm that incorporates the Monte-Carlo method named CRF has been developed. Two different configurations of CRF were implemented: CRF9,2 and CRF81,1. Experiments on some colored video sequences showed that the CRF configurations outperform the TF algorithm in terms of accuracy. However, the TF algorithm remains more efficient in terms of processing time. == Applications == Object detection Face detection Face recognition Pedestrian detection Video surveillance Motion capture Human-computer interaction Content-based video coding Traffic monitoring Real-time gesture recognition
Braina
Braina is a virtual assistant and speech-to-text dictation application for Microsoft Windows developed by Brainasoft. Braina uses natural language interface, speech synthesis, and speech recognition technology to interact with its users and allows them to use natural language sentences to perform various tasks on a computer. The name Braina is a short form of "Brain Artificial". Braina is marketed as a Microsoft Copilot alternative. It provides a voice interface for several locally run and cloud large language models, including the latest LLMs from providers such as OpenAI, Anthropic, Google, xAI, Meta, Mistral, etc; while improving data privacy. Braina also allows responses from its in-house large language models like Braina Swift and Braina Pinnacle. It has an "Artificial Brain" feature that provides persistent memory support for supported LLMs. == Features == Braina provides is able to carry out various tasks on a computer, including automation. Braina can take commands inputted through typing or through dictation to store reminders, find information online, perform mathematical operations, open files, generate images from text, transcribe speech, and control open windows or programs. Braina adapts to user behavior over time with a goal of better anticipating needs. === Speech-to-text dictation === Braina Pro can type spoken words into an active window at the location of a user's cursor. Its speech recognition technology supports more than 100 languages and dialects and is able to isolate the recognition of a user's voice from disturbing environmental factors such as background noise, other human voices, or external devices. Braina can also be taught to dictate uncommon legal, medical, and scientific terms. Users can also teach Braina uncommon names and vocabulary. Users can edit or correct dictated text without using a keyboard or mouse by giving built-in voice commands. === Text-to-speech === Braina can read aloud selected texts, such as e-books. === Custom commands and automation === Braina can automate computer tasks. It lets users create custom voice commands to perform tasks such as opening files, programs, websites, or emails, as well as executing keyboard or mouse macros. === Transcription === Braina can transcribe media file formats such as WAV, MP3, and MP4 into text. === Notes and reminders === Braina can store and recall notes and reminders. These can include scheduled or unscheduled commands, checklist items, alarms, chat conversations, memos, website snippets, bookmarks, contacts. === Image and Video generation === Braina can generate AI images and videos from text and image inputs using generative cloud AI models. These include Black Forest Labs' FLUX.2, Google's Veo, Imagen, and Nano Banana Pro, Kuaishou's Kling, Alibaba's Wan, ByteDance's Seedance and Seedream, MiniMax's Hailuo, OpenAI's GPT Image, and Tongyi Lab's Z Image Turbo. == Platforms == In addition to the desktop version for Windows operating systems, Braina is also available for the iOS and Android operating systems. The mobile version of Braina has a feature allowing remote management of a Windows PC connected via Wi-Fi. == Distributions == Braina is distributed in multiple modes. These include Braina Lite, a freeware version with limitations, and premium versions Braina Pro, Pro Plus, and Pro Ultra. Some additional features in the Pro version include dictation, custom vocabulary, video transcription, automation, custom voice commands, and persistent LLM memory. == Reception == TechRadar has consistently listed Braina as one of the best dictation and virtual assistant apps between 2015 and 2024.
Powerset (company)
Powerset was an American company based in San Francisco, California, that, in 2006, was developing a natural language search engine for the Internet. On July 1, 2008, Powerset was acquired by Microsoft for an estimated $100 million (~$143 million in 2024). Powerset was working on building a natural language search engine that could find targeted answers to user questions (as opposed to keyword based search). For example, when confronted with a question like "Which U.S. state has the highest income tax?", conventional search engines ignore the question phrasing and instead do a search on the keywords "state", "highest", "income", and "tax". Powerset on the other hand, attempts to use natural language processing to understand the nature of the question and return pages containing the answer. The company was in the process of "building a natural language search engine that reads and understands every sentence on the Web". The company has licensed natural language technology from PARC, the former Xerox Palo Alto Research Center. On May 11, 2008, the company unveiled a tool for searching a fixed subset of English Wikipedia using conversational phrases rather than keywords. Acquisition by Microsoft: One significant milestone in Powerset's history was its acquisition by Microsoft on July 1, 2008, for an estimated $100 million. This acquisition was part of Microsoft's broader strategy to enhance its search capabilities and compete more effectively with other search engine providers, particularly Google. Natural Language Search Engine: Powerset's primary focus was on developing a natural language search engine capable of understanding and interpreting user queries in a more human-like manner. Instead of simply matching keywords, Powerset aimed to comprehend the meaning behind the words, allowing for more accurate and contextually relevant search results. Technology and Partnerships: Powerset had licensed natural language technology from PARC, the Xerox Palo Alto Research Center. This technology likely played a crucial role in the development of Powerset's NLP capabilities. Wikipedia Search Tool: In May 2008, Powerset unveiled a search tool that allowed users to search a fixed subset of English Wikipedia using conversational phrases rather than traditional keywords. This demonstrated the potential of Powerset's NLP technology in providing more precise and relevant search results. == Powerlabs == In a form of beta testing, Powerset opened an online community called Powerlabs on September 17, 2007. Business Week said: "The company hopes the site will marshal thousands of people to help build and improve its search engine before it goes public next year." Said The New York Times: "[Powerset Labs] goes far beyond the 'alpha' or 'beta' testing involved in most software projects, when users put a new product through rigorous testing to find its flaws. Powerset doesn’t have a product yet, but rather a collection of promising natural language technologies, which are the fruit of years of research at Xerox PARC." Powerlabs' initial search results are taken from Wikipedia. == Notable people == Barney Pell (born March 18, 1968, in Hollywood, California) was co-founder and CEO of Powerset. Pell received his Bachelor of Science degree in symbolic systems from Stanford University in 1989, where he graduated Phi Beta Kappa and was a National Merit Scholar. Pell received a PhD in computer science from Cambridge University in 1993, where he was a Marshall Scholar. He has worked at NASA, as chief strategist and vice president of business development at StockMaster.com (acquired by Red Herring in March, 2000) and at Whizbang! Labs. Prior to joining Powerset, Pell was an Entrepreneur-in-Residence at Mayfield Fund, a venture capital firm in Silicon Valley. Pell is also a founder of Moon Express, Inc., a U.S. company awarded a $10M commercial lunar contract by NASA and a competitor in the Google Lunar X PRIZE. Steve Newcomb was the COO and co-founder of Powerset. Prior to joining Powerset, he was a co-founder of Loudfire, General Manager at Promptu, and was on the board of directors at Jaxtr. He left Powerset in October 2007 to form Virgance, a social startup incubator. Lorenzo Thione (born in Como, Italy) was the product architect and co-founder of Powerset. Prior to joining Powerset, he worked at FXPAL in natural language processing and related research fields. Thione earned his master's degree in software engineering from the University of Texas at Austin. Ronald Kaplan, former manager of research in Natural Language Theory and Technology at PARC, served as the company's CTO and CSO. Ryan Ferrier is a member of the founding team of Powerset. He managed personnel and internal operations. After 2008 he went on to co-found Serious Business, which made Facebook applications and was later bought by Zynga. Another Powerset alumnus, Alex Le, became CTO of Serious Business and went on to become an executive producer at Zynga when it bought the company. Siqi Chen founded a stealth startup in mobile computing after leaving Powerset. Tom Preston-Werner worked at Powerset and left after the acquisition to found GitHub. == Investors == Powerset attracted a wide range of investors, many of whom had considerable experience in the venture capital field. The company received $12.5 million (~$18.2 million in 2024) in Series A funding during November 2007, co-led by the venture capital firms Foundation Capital and The Founders Fund. Among the better-known investors: Esther Dyson, founding chairman of ICANN, founder of the newsletter Release 1.0 and editor at Cnet Peter Thiel, founder and former CEO of PayPal Luke Nosek, founder of PayPal Todd Parker. Managing Partner, Hidden River Ventures Reid Hoffman, executive vice president of PayPal and founder of LinkedIn First Round Capital, seed-stage venture firm
Teechart
TeeChart is a charting library for programmers, developed and managed by Steema Software of Girona, Catalonia, Spain. It is available as commercial and non-commercial software. TeeChart has been included in most Delphi and C++Builder products since 1997, and TeeChart Standard currently is part of Embarcadero RAD Studio 13 Florence. TeeChart Pro version is a commercial product that offers shareware releases for all of its formats. The TeeChart Charting Library offers charts, maps and gauges in versions for Delphi VCL/FMX, ActiveX, C# for Microsoft Visual Studio .NET. Full source code has always been available for all versions except the ActiveX version. TeeChart's user interface is translated into 38 languages. == History == The first version of TeeChart was authored in 1995 by David Berneda, co-founder of Steema, using the Borland Delphi Visual Component Library programming environment and TeeChart was first released as a shareware version and made available via Compuserve in the same year. It was written in the first version of Delphi VCL, as a 16-bit Charting Library named TeeChart version 1. The next version of TeeChart was released as a 32-bit library (Delphi 2 supported 32-bit compilation) but was badged as TeeChart VCL v3 to coincide with Borland's naming convention for inclusion on the toolbox palette of Borland Delphi v3 in 1997 and with C++ Builder v3 in 1998. It has been on the Delphi/C++ Builder toolbox palette ever since. The current version is Embarcadero RAD Studio 13 Florence. TeeChart's first ActiveX version named "version 3" too, to match the VCL version's nomenclature, was released in 1998. The version was optimised to work with Microsoft's Visual Studio v97 and v6.0 developer suites that include Visual Basic and Microsoft Visual C++ programming languages. Support for new programming environments followed with TeeChart's first native C# version for Microsoft Visual Studio .NET released in 2002 and TeeChart.Lite for .NET, a free charting component, released for Visual Studio.NET in 2003 and supporting too, Mono (programming). Steema Software released the first native TeeChart Java (programming language) version in 2006 and TeeChart's first native PHP version was released in 2009 and published as open-source in June 2010. Mobile versions of TeeChart, for Android (operating system) devices and Windows Phone 7 devices were released during the first half of 2011. In 2012 TeeChart extended functionality to iPhone/iPad and BlackBerry OS devices and a new JavaScript version was released in the same year to support HTML5 Canvas. In 2013 Steema launched TeeChart for .NET Chart for Windows Store applications and included support for Microsoft's Windows Phone 8 mobile platform. TeeChart for Xamarin.Forms written with 100% C# code and cross-platform support for .NET desktops, Windows Phone, iOS and Android was released in 2014. Also since 2014 Webforms charts now offers HTML5 interactivity. Steema launched TeeChart for Avalonia (software framework) in 2022 and in 2023 .NET_MAUI support was added to the TeeChart for .NET. == Usage == TeeChart is a general purpose charting component designed for use in differing ambits, offering a wide range of aesthetics to chart data. Generally TeeCharts published in the field, in areas where large amounts of data must be interpreted regularly, remain by designer choice in their simplest form to maximize the "data-ink ratio". Sloan Digital Sky Survey, SDSS Web Services' use for charting "Scientific .. plotting of online data" at The Virtual Observatory Spectrum Services reflects that approach. The SDSS chart authors choose to represent data using TeeChart's standard 2D line display. Speed is also a factor when choosing how to most effectively plot data. Realtime data, at frequencies of up to tens or hundreds of data points or more per second, require the most processor economic approach to charting. Computer processing time dedicated to the plotting of data needs to be as lightweight as possible, freeing-up computer tasks "to achieve real-time data acquisition, display and analysis". A critical and stated aspect of many data visualisation applications is the ability to offer interactivity to the user; NASA's document, the Orbital Debris Engineering Model Model ORDEM 3.0 - User's Guide, 2014, states that "The user may manipulate the graphs to zoom, pan, and copy to the clipboard and export to various file types" and Computer and Computing Technologies in Agriculture II, Volume 1, Daoliang, Li; Chunjiang, Zhao (2009), also using TeeChart, states "the properties at any point in the chart can be viewed moving the mouse over it". Writing about control education, Juha Lindfors states "The desired charting functionality (such as zooming and scaling) is achieved..". Charting applications have become increasingly 'onlined', made available either to a wider public or to a territorially remote userbase via networked applications. The World Wide Web (the Web) has become "by far, the most popular Internet protocol" to disseminate online applications. Most major IDEs now offer environments for web application developede aimed at browser hosted applications. Charting components, TeeChart among them, have adapted to provide models that work within a browser environment, often using static images and scripted layering techniques such as Ajax (programming) to offer a level of interactivity, improve response times and hide apparent delay from the user. Options to enrich client, browser-side processing flexibility are exploited by TeeChart libraries via modules that offer 'micro-environments' within the browser, such as the long established ActiveX technology, Adobe Flash, Microsoft Silverlight or Java Applets. Serverside environments offer too, a means to interact with browser based script to dynamically respond to charting requests. Joomla and CodeIgniter are host environments for TeeChart PHP and an example of an Embarcadero IntraWeb VCL designed application using TeeChart, is documented here. == Programmer reference == The Code Project includes a demo that uses TeeChart.Lite, called 'Self-Organizing Feature Maps (Kohonen maps)' written by Bashir Magomedovl and SourceForge includes a Database Stress and Monitor that also uses TeeChart.Lite. Books and information sources that include substantial sections about working with the Delphi version of TeeChart include "Mastering Delphi 6" by Marco Cantù, "C++ Builder 5 developer's guide", a video Delphi Tutorial on charting JPEG compression and support forums and reference pages at TeeChart Support Forums. Non-English language document sources include, in Czech "Myslíme v jazyku Delphi 7: knihovna zkušeného programátora" by Marco Cantù, and Chinese, Delphi 6, Delphi, and Delphi 5.
Multi-scale approaches
The scale space representation of a signal obtained by Gaussian smoothing satisfies a number of special properties, scale-space axioms, which make it into a special form of multi-scale representation. There are, however, also other types of "multi-scale approaches" in the areas of computer vision, image processing and signal processing, in particular the notion of wavelets. The purpose of this article is to describe a few of these approaches: == Scale-space theory for one-dimensional signals == For one-dimensional signals, there exists quite a well-developed theory for continuous and discrete kernels that guarantee that new local extrema or zero-crossings cannot be created by a convolution operation. For continuous signals, it holds that all scale-space kernels can be decomposed into the following sets of primitive smoothing kernels: the Gaussian kernel : g ( x , t ) = 1 2 π t exp ( − x 2 / 2 t ) {\displaystyle g(x,t)={\frac {1}{\sqrt {2\pi t}}}\exp({-x^{2}/2t})} where t > 0 {\displaystyle t>0} , truncated exponential kernels (filters with one real pole in the s-plane): h ( x ) = exp ( − a x ) {\displaystyle h(x)=\exp({-ax})} if x ≥ 0 {\displaystyle x\geq 0} and 0 otherwise where a > 0 {\displaystyle a>0} h ( x ) = exp ( b x ) {\displaystyle h(x)=\exp({bx})} if x ≤ 0 {\displaystyle x\leq 0} and 0 otherwise where b > 0 {\displaystyle b>0} , translations, rescalings. For discrete signals, we can, up to trivial translations and rescalings, decompose any discrete scale-space kernel into the following primitive operations: the discrete Gaussian kernel T ( n , t ) = I n ( α t ) {\displaystyle T(n,t)=I_{n}(\alpha t)} where α , t > 0 {\displaystyle \alpha ,t>0} where I n {\displaystyle I_{n}} are the modified Bessel functions of integer order, generalized binomial kernels corresponding to linear smoothing of the form f o u t ( x ) = p f i n ( x ) + q f i n ( x − 1 ) {\displaystyle f_{out}(x)=pf_{in}(x)+qf_{in}(x-1)} where p , q > 0 {\displaystyle p,q>0} f o u t ( x ) = p f i n ( x ) + q f i n ( x + 1 ) {\displaystyle f_{out}(x)=pf_{in}(x)+qf_{in}(x+1)} where p , q > 0 {\displaystyle p,q>0} , first-order recursive filters corresponding to linear smoothing of the form f o u t ( x ) = f i n ( x ) + α f o u t ( x − 1 ) {\displaystyle f_{out}(x)=f_{in}(x)+\alpha f_{out}(x-1)} where α > 0 {\displaystyle \alpha >0} f o u t ( x ) = f i n ( x ) + β f o u t ( x + 1 ) {\displaystyle f_{out}(x)=f_{in}(x)+\beta f_{out}(x+1)} where β > 0 {\displaystyle \beta >0} , the one-sided Poisson kernel p ( n , t ) = e − t t n n ! {\displaystyle p(n,t)=e^{-t}{\frac {t^{n}}{n!}}} for n ≥ 0 {\displaystyle n\geq 0} where t ≥ 0 {\displaystyle t\geq 0} p ( n , t ) = e − t t − n ( − n ) ! {\displaystyle p(n,t)=e^{-t}{\frac {t^{-n}}{(-n)!}}} for n ≤ 0 {\displaystyle n\leq 0} where t ≥ 0 {\displaystyle t\geq 0} . From this classification, it is apparent that we require a continuous semi-group structure, there are only three classes of scale-space kernels with a continuous scale parameter; the Gaussian kernel which forms the scale-space of continuous signals, the discrete Gaussian kernel which forms the scale-space of discrete signals and the time-causal Poisson kernel that forms a temporal scale-space over discrete time. If we on the other hand sacrifice the continuous semi-group structure, there are more options: For discrete signals, the use of generalized binomial kernels provides a formal basis for defining the smoothing operation in a pyramid. For temporal data, the one-sided truncated exponential kernels and the first-order recursive filters provide a way to define time-causal scale-spaces that allow for efficient numerical implementation and respect causality over time without access to the future. The first-order recursive filters also provide a framework for defining recursive approximations to the Gaussian kernel that in a weaker sense preserve some of the scale-space properties.