Take Us to Your Chief: and Other Stories is a collection of nine short stories by Canadian author, playwright, and journalist Drew Hayden Taylor published in 2016 by Douglas & McIntyre. Taylor, who is part Caucasian, part Ojibwe, explains in the acknowledgments section of the book that the origin of the project lies in several failed attempts "to compile an anthology of Native sci-fi from Canada’s best First Nations writers." The stories explore contemporary First Nations social issues through employing a number of 1950s-era science fiction tropes and themes in these stories, including time travel, alien contact, and superpowers. Many reviews of the books have noted Taylor's use of humor to examine dark subject matter, such as the heritage of Canadian Indian residential schools, First Nations suicide rates, or the water quality crisis on Canadian reserves. == The Stories == "Andrei nas" "I Am...Am I" "Lost in Space" "Dreams of Doom" "Mr. Gizmo" "Petropaths" "Stars" "Superdisappointed" "Take Us to Your Chief" == Story summaries == === Foreword === In his foreword, Taylor describes the genesis of Take Us to Your Chief: and Other Stories and invites readers into, in his term, a “new terra nullius.” He begins by describing his biracial upbringing and heritage. He points out that First Nations people are rarely associated with technology or science fiction, in part because Indigenous peoples were often at a technological disadvantage against European colonizers. He references the few examples that he can think of from popular culture, such as the Star Trek episode called “The Paradise Syndrome,” in which First Nations people are portrayed as stereotypical Indians in hippie clothing. He also elaborates on his fascination with the world of sci-fi, which first started in comic books. He enjoyed the literary work of H.G. Wells, such as The Time Machine and The Invisible Man. Since sci-fi is a world of endless opportunities, he intends that these short stories help people explore science fiction through Native peoples’ minds, something that needs to be explored more thoroughly. === "A Culturally Inappropriate Armageddon" === “A Culturally Inappropriate Armageddon” is set on a Haudenosaunee reserve, towards the end of the Oka Crisis, with a handful of people that work at its first ever radio station, C-RES, which opens in 1991. Part 1, titled “C-Res Is on the Air,” depicts Emily, Aaron, and Tracey on their first days at the station. Within the group, there is a constant debate between broadcasting popular programming, including science fiction and film reviews, and culturally-relevant programming meant to aid in cultural revitalization efforts. One night, Aaron is late to work but once he shows up he can't stop talking about radio transmissions broadcasting into deep space, an event that has been occurring since the initial discovery of the radio waves by Heinrich Hertz. The story then skips ahead seven years to 1998, when Emily is struggling to find better content for her station until Tracey stumbles upon an old anthropological record named “The Calling Song” that they decide to broadcast to their audience. The story then jumps to the year 2018 where they are all huddled around a television watching a news station reporting that extraterrestrial life is heading towards them. The discussion of what is going to happen comes into the picture and they all decide it would either be like Contact or The Day the Earth Stood Still. A year later in 2019, the aliens have invaded the planet and destroyed everything. As the three former radio station employees suffer from radioactive fallout, they realize that the aliens received the broadcast of “The Calling Song” and took it as a message to come to Earth. They thus realize that the Haudenosaunee people were inadvertently responsible for the destruction of the Earth. Part 2, titled “Old Men and Old Sayings,” tells us of an elderly man that is watching the news and listening to the radio about a spaceship coming to earth. He knows that he and everyone will die, but the people around him are excited. He finds a book on his night stand and flips to a page where he underlined a sentence a long time ago about the European colonization of the Americas. That sentence reads “those who cannot remember the past are condemned to repeat it” (23). He closes the book and Taylor concludes the story by writing, “he hated it when white people were right." === "I Am...Am I" === “I Am...Am I” chronicles the accidental creation and unexpected ending of artificial intelligence. Professor Mark King has a plethora of degrees and works for a research firm called FUTUREVISION. One night as Professor King searches the lab for his car keys—a common occurrence for him—he notices something unusual in the Matrix room. He reads on a computer the phrase “I am.” First believing it to be a prank, King later comes to the realization that his Matrix project has evolved into a responsive Artificial Intelligence. After this realization, Professor King calls his peer Dr. Gayle Chambers to further investigate this miraculous event. After receiving approval from their superiors, Professor King and Dr. Chambers move forward in feeding the AI information, with Chambers serving as the lead communicator. With more information, it becomes increasingly concerned with its own existence and the concept of whether it has a soul. After several days of conversation with the AI, Chambers and King begin to feel uneasy about the AI's responses, which show signs of neuroses. Despite this behavior, Chambers decides to feed the AI information about the culture and history of the human race. Upon receiving this information, the AI becomes obsessed with Indigenous spirituality prior to the colonization of the Americas, and it requests more information on First Nations people. Dr. Chambers is hesitant at first, but gives in and continues to feed the AI the information with the intention to return to it in the morning. This leads to the AI finding out about colonization and genocide of Indigenous peoples. Upon her arrival the next day, Chambers discovers that the code for the AI has been completely wiped from the hard drive and a single message is left on the screen—"I was”—that signifies the AI's suicide. === "Lost in Space" === "Lost in Space" is told from the perspective of Mitchell, an Anishinabe astrosurveyor who is aboard a space shuttle on a two-year tour collecting rocks from an asteroid belt. He is accompanied by an Artificial general intelligence named Mac, short for “machine.” Mac is aboard this tour in order to accompany Mitchell and keep him sane; however, his company is a burden because for Mitchell, “true space exploration consists largely of boredom.” In the midst of Mitchell seeking a way to occupy his downtime, Mac interrupts with news about his grandfather, Papa Peter, dying. Papa Peter was Mitchell's only real tie to his Indigenous identity. After receiving the news Mitchell begins to reminisce on all of the things Papa Peter had taught him throughout his life. He constantly posed questions concerning the world above (Father Sky) and how it is more important than the land they live on (Mother Earth), which eventually led Mitchell to the selection of his career. During his state of mourning, Mitchell begins to go through all the videos his grandfather had sent him throughout his space tours. Papa Peter had sent Mitchell videos from Otter Lake, a First Nations reserve; these videos are about controversial topics regarding being both native and an astronaut. In the midst of Mitchell's grieving, Mac tries to relieve the situation by finding an online video of Mitchell's grandfather participating in a drum ceremony at Ottawa’s National Aboriginal Day festival. He reconnects to his roots and his grandfather’s spirit as he listens to the Indigenous music by feeling the drum beat and humming along. Mac’s small act of kindness leads Mitchell to gain a new-found appreciation for his presence. Mitchell feels responsible to moving forward in his life in memory of Papa Peter. === "Dreams of Doom" === "Dreams of Doom" is narrated by an Ojibway reporter named Pamela Wanishin who works for an aboriginal newspaper called the West Wind. One day she receives a mysterious package with a broken dreamcatcher and a flash drive containing highly classified files. As she reads the files, she keeps seeing the term “Project Nightlight,” and out of curiosity, she Googles it. Once she Googles this, she is contacted by a nameless agent from Indigenous and Northern Affairs Canada and told that she must be relocated because the knowledge she now possesses must never be released to the public. She quickly flees the area to a cabin at Otter Lake, owned by a family member, to lie low for a few days. Eventually, the government organization tracks her down using drones, which forces her to fight back and flee once again. Pamela then runs to her friend and coworker Sally's hous
DeepRoute.ai
DeepRoute.ai (Chinese: 元戎启行) is a Chinese autonomous driving company founded in 2019 and headquartered in Shenzhen, China. The company develops full-stack self-driving solutions including perception, decision-making, and control systems. == History == DeepRoute.ai was founded in February 2019 in Shenzhen, China, by Zhou Guang (周光), who serves as the company's CEO. In September 2019, the company collaborated with Dongfeng for a live-streamed autonomous driving demonstration. In October 2019, during the 7th Military World Games, DeepRoute.ai conducted Robotaxi demonstration operations. In November 2019, it obtained an intelligent connected vehicle road test permit for public roads in Shenzhen. In October 2020, DeepRoute.ai signed an "Autonomous Driving Leadership Project" with Dongfeng to build one of China's largest autonomous fleets. In August 2020, DeepRoute.ai announced its partnership with Cao Cao Mobility, a Geely-backed ride-hailing company, to test Robotaxis in Hangzhou for daily operations, planning to provide Robotaxis during the 2022 Asian Games. In September 2021, DeepRoute.ai secured US$300 million in a Series B funding round led by Alibaba. In December 2021, the company unveiled its DeepRoute-Driver 2.0, an L4-level autonomous driving solution comprising five solid-state lidar sensors, eight cameras, a proprietary computing system and an optional millimeter-wave radar. with a production cost of under US$10,000. In June 2022, it partnered with Deppon Express to provide autonomous light truck freight transfer services. In March 2023, the company launched its high-precision map-free intelligent driving solution, DeepRoute-Driver 3.0. In November 2024, Great Wall Motor announced a $100 million Series C funding round for Deeproute. With this, Deeproute has completed five rounds of financing, raising a cumulative total of over $500 million. Its shareholders include Fosun RZ Capital, Yunqi Partners, Alibaba, Vision Plus Capital, and Dongfeng, among others. In the same month, Deeproute.ai emphasised that they were in "deep cooperation" with Nvidia and spoke on being part of the first batch of companies in China to get a hold of Nvidia's newer Thor chip for cars which will be used in a new system released next year. This new system will help manage more complex driving scenarios through visual cues. == Products == === VLA Model === VLA Model is a Vision–language–action model designed for autonomous driving systems. It integrates visual perception, semantic understanding, and action decision-making into a unified framework, aiming to enhance the safety and adaptability of advanced driver-assistance systems (ADAS) in complex road environments. The model was officially launched on August 26, 2025, as the core of DeepRoute.ai's DeepRoute IO 2.0 platform. The VLA model is characterized by its "visual-language-action" architecture, which incorporates a chain-of-thought (CoT) reasoning capability inspired by large language models. This design is intended to address the "black box" limitations of traditional end-to-end autonomous driving systems by enabling the model to analyze information, infer causality, and make decisions in a more transparent and interpretable manner. === Appliance === The company has partnered with several automakers including Dongfeng Motor Corporation and Geely to develop and test autonomous vehicles.
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.
Adobe GoLive
Adobe GoLive was a WYSIWYG HTML editor and web site management application from Adobe Systems. It replaced Adobe PageMill as Adobe's primary HTML editor and was itself discontinued in favor of Dreamweaver. The last version of GoLive that Adobe released was GoLive 9. == History == GoLive originated as the flagship product of a company named GoNet Communication, Inc. then based in Menlo Park, California, and the development company GoNet Communications GmbH in Hamburg, Germany, in 1996. Later GoNet changed its name to GoLive Systems, Inc, and the name of its product to GoLive CyberStudio. Adobe acquired GoLive in 1999 and re-branded the GoLive CyberStudio product to what became Adobe GoLive. Adobe took over the Hamburg office as an Adobe development site to continue to develop the product. At the time of the acquisition, CyberStudio was a Macintosh-only application. In the spring of 1999 Adobe released Adobe GoLive for both Macintosh and Microsoft Windows. The first versions of Dreamweaver and CyberStudio were released in a similar timeframe. However, Dreamweaver eventually became the dominant WYSIWYG HTML editor in market share. After the Adobe acquisition of Macromedia (the company that had owned Dreamweaver), GoLive was progressively re-targeted toward Adobe's traditional design market, and the product became better integrated with Adobe's existing suite of design-oriented software products and less focused on the professional web development market. The Adobe CS2 Premium suite contained GoLive CS2. With the release of Creative Suite 3, Adobe integrated Dreamweaver as a replacement for GoLive and released GoLive 9 as a standalone product. In April 2008, Adobe announced that sales and development of GoLive would cease in favor of Dreamweaver. == General description and distinctive aspects == GoLive incorporated a largely modeless workflow that relied heavily on drag-and-drop. Most user interaction was done via a contextual inspector rather than the modal workflow found in Dreamweaver. Among its features were a separate editor for tables that supported nesting, and a two-dimensional panel for applying CSS styles to elements. GoLive supported drag-and-drop of native Adobe Photoshop and Adobe Illustrator files via what the company called "Smart Objects", which then automatically guided the user through saving those files in web-supported formats. Updates to the original Photoshop or Illustrator assets were automatically tracked by GoLive. It also implemented a tool called "Components" which allowed updates to interface elements throughout a site to be updated globally by changing one single file. As a website management tool, GoLive allowed users to transfer and publish content directly from within the application, and allowed individual files to be excluded from uploading. == Features == One of the new features of GoLive version 5 was Dynamic Link, which was a method of creating dynamic, database-driven web content without the need to know a server-side language and with full WYSIWYG support in the GoLive user interface. GoLive had a powerful set of extensibility API which could be used to add additional functionality to the product. The GoLive SDK provided interfaces which allowed developers to use a combination of XML, JavaScript and C/C++ to create plugins for the product. The extensibility API allowed developers access to custom drawing and event handling using JavaScript, as well as a full JavaScript debugger and command line interpreter. This allowed intermediate-level developers using interpreted JavaScript to create sophisticated user interfaces. == Language and framework structure == Adobe GoLive is coded in the C++ programming language. It uses a custom C++ framework called SCL (Simple Class Library) which was initially built from scratch by the engineers at GoLive Systems Inc. The SCL framework was also used in the short-lived Adobe Atmosphere 3D software. == Release history == As the final version, GoLive 9 was discontinued in April 2008.
International Speech Communication Association
The International Speech Communication Association (ISCA) is a non-profit organization and one of the two main professional associations for speech communication science and technology, the other association being the IEEE Signal Processing Society. == Purpose == The purpose of the International Speech Communication Association (ISCA) is to promote the study and application of automatic speech processing, including speech recognition and synthesis, as well as related areas such as speaker recognition and speech compression. The association's activities cover all aspects of speech processing, including computational, linguistic, and theoretical aspects. The primary goal of the International Speech Communication Association (ISCA) is to advance the field of automatic speech processing and communication technology through research, education, and collaboration. By promoting the study and application of speech technologies such as speech recognition, speech synthesis, speaker recognition, and speech compression, ISCA aims to foster innovation and development in the areas of human-computer interaction, telecommunications, and multimedia applications. ISCA serves as a platform for researchers, academics, industry professionals, and students to exchange knowledge, share best practices, and foster interdisciplinary dialogue in the field of speech communication science. Through conferences, workshops, publications, and educational initiatives, ISCA seeks to enhance the understanding of speech processing mechanisms, improve the accuracy and efficiency of speech technologies, and explore new frontiers in the realm of human language communication. Furthermore, ISCA plays a crucial role in promoting international collaboration and networking among professionals in the speech communication community. By facilitating partnerships and cooperation between individuals and organizations worldwide, ISCA seeks to drive global progress in speech technology research and application, ultimately contributing to the advancement of communication systems, accessibility tools, and interactive interfaces that benefit society as a whole. == Conferences == ISCA organizes yearly the Interspeech conference. Most recent Interspeech: 2013 Lyon, France 2014 Singapore 2015 Dresden, Germany 2016 San Francisco, US 2017 Stockholm, Sweden 2018 Hyderabad, India 2019 Graz, Austria 2020 Shanghai, China (fully virtual) 2021 Brno, Czechia (hybrid) 2022 Incheon, South Korea 2023 Dublin, Ireland 2023 Kos Island, Greece Forthcoming Interspeech: 2025 Rotterdam, the Netherlands == ISCA board == The ISCA president for 2023-2025 is Odette Scharenborg. The vice president is Bhuvana Ramabhadran and the other members are professionals in the field. == History of ISCA == The precursor to Interspeech was a conference called Eurospeech, first held in 1989 and organised by Jean-Pierre Tubach. It was the conference of the European Speech Communication Association (ESCA), itself the precursor of the International Speech Communication Association (ISCA). A year later another conference on speech science and technology was started: the International Conference on Spoken Language Processing (ICSLP), which was founded in 1990 by Hiroya Fujisaki. The first ISCA (vs. ESCA) event was the merging of Eurospeech and ICSLP to create ICSLP-Interspeech, held in Beijing, China in 2000. This was followed by Eurospeech-Interspeech, which was held in Aalborg, Denmark in 2001. In 2007, the Eurospeech and ICSLP parts of the conference names were dropped and Interspeech became the name of the yearly conference (first Interspeech location: Antwerp, Belgium).
Computational semantics
Computational semantics is a subfield of computational linguistics. Its goal is to elucidate the cognitive mechanisms supporting the generation and interpretation of meaning in humans. It usually involves the creation of computational models that simulate particular semantic phenomena, and the evaluation of those models against data from human participants. While computational semantics is a scientific field, it has many applications in real-world settings and substantially overlaps with Artificial Intelligence. Broadly speaking, the discipline can be subdivided into areas that mirror the internal organization of linguistics. For example, lexical semantics and frame semantics have active research communities within computational linguistics. Some popular methodologies are also strongly inspired by traditional linguistics. Most prominently, the area of distributional semantics, which underpins investigations into embeddings and the internals of Large Language Models, has roots in the work of Zellig Harris. Some traditional topics of interest in computational semantics are: construction of meaning representations, semantic underspecification, anaphora resolution, presupposition projection, and quantifier scope resolution. Methods employed usually draw from formal semantics or statistical semantics. Computational semantics has points of contact with the areas of lexical semantics (word-sense disambiguation and semantic role labeling), discourse semantics, knowledge representation and automated reasoning (in particular, automated theorem proving). Since 1999 there has been an ACL special interest group on computational semantics, SIGSEM.
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