The representation of the Holocaust on social media has been a subject of scholarly inquiry and media attention. == Selfies at Holocaust memorial sites == Some visitors take selfies at Holocaust memorials, which has been the subject of controversy. In 2018, Rhian Sugden, a British model, received criticism after posting a selfie at the Memorial to the Murdered Jews of Europe in Berlin with the caption "ET phone home". She later removed the caption, but defended taking the photograph. Other celebrities have also been criticised for photographs at the Berlin memorial, including Indian actress Priyanka Chopra and US politician Pete Buttigieg, whose husband posted a photograph of him at the memorial on a personal social media account. The Israeli artist and satirist Shahak Shapira set up the website yolocaust.de in 2017 to expose people who take inappropriate selfies at the Holocaust memorial in Berlin. Shapira went through thousands of selfies posted to social media sites such as Facebook, Instagram, Tinder, and Grindr, choosing the twelve that he found most offensive. When the images were moused over, the website replaces the memorial backdrop with black and white images of Nazi victims. "Yolocaust" is a portmanteau of "Holocaust" and YOLO, an acronym for "you only live once". The website went viral, receiving 1.2 million views in the first 24 hours after its launch. Shapira honored requests to take down all of the photographs, which he had used without permission, and the website remains with only a textual documentation of the project. In an analysis of comments by Internet users on the project, Christoph Bareither estimated that 75% were positive. However, the memorial's architect, Peter Eisenman, criticized the website. In his 2018 book Postcards from Auschwitz, Grinnell professor Daniel P. Reynolds defends the practice of selfie-taking at Holocaust sites. In 2019, the Auschwitz-Birkenau State Museum requested that visitors not take inappropriate selfies, although the museum's staff acknowledged that other visitors take selfies in a thoughtful and respectful manner, which they did not criticize. In an academic paper, Gemma Commane and Rebekah Potton analyze the use of Instagram to share tourist photographs at Holocaust sites and conclude that "Instagram encourages conversation and empathy, keeping the Holocaust visible in youth discourses". According to their analysis, most images are tagged with respectful hashtags such as #tragic, #remembrance, and #sadness. The Auschwitz museum has an official Instagram account, auschwitzmemorial, which it uses to share selected appropriate Instagram posts. However, the image feed for the hashtag "Auschwitz" includes potentially offensive images such as an image of "Nazi Vs. Jews #beerpong". This image, according to the authors, expresses "mockery and contempt" for Holocaust victims. They also document offensive memes using images of Holocaust atrocities and shared on Instagram. Some social media users post in order to criticize what they see as inappropriate behavior at Holocaust sites, with one commenting, "Taking photos posing next to razor wire, selfies with victim's hair in the background, and even group shots in front of the crematoria had to be seen to be believed." == Assessment of tourism == Social media posts have been used by researchers to analyze the phenomenon of Holocaust-related tourism. == Social media groups == People have created groups on Facebook to discuss issues related to the Holocaust. One paper analyses two such groups, "The Holocaust and My Family" and "The Descendants of the Victims and Survivors of the Holocaust" in which people engage in collective trauma processing. == Eva.stories == In 2019, Israeli high-tech entrepreneur Mati Kochavi created a fictitious Instagram account for Eva Heyman, a Hungarian-Jewish girl who was murdered in Auschwitz concentration camp. The project met with mixed reception. Israeli prime minister Benjamin Netanyahu praised the project, saying that it "exposes the immense tragedy of our people through the story of one girl". == Holocaust denial == The issue of Holocaust denial on social media has also attracted attention. In October 2020, Facebook reversed its policy and banned Holocaust denial from the platform. Founder Mark Zuckerberg had previously argued that such content should not be banned on freedom of speech grounds.
Structured-light 3D scanner
A structured-light 3D scanner is a device used to capture the three-dimensional shape of an object by projecting light patterns, such as grids or stripes, onto its surface. The deformation of these patterns is recorded by cameras and processed using specialized algorithms to generate a detailed 3D model. Structured-light 3D scanning is widely employed in fields such as industrial design, quality control, cultural heritage preservation, augmented reality gaming, and medical imaging. Compared to laser-based 3D scanning, structured-light scanners use non-coherent light sources, such as LEDs or projectors, which enable faster data acquisition and eliminate potential safety concerns associated with lasers. However, the accuracy of structured-light scanning can be influenced by external factors, including ambient lighting conditions and the reflective properties of the scanned object. == Principle == Projecting a narrow band of light onto a three-dimensional surface creates a line of illumination that appears distorted when viewed from perspectives other than that of the projector. This distortion can be analyzed to reconstruct the geometry of the surface, a technique known as light sectioning. Projecting patterns composed of multiple stripes or arbitrary fringes simultaneously enables the acquisition of numerous data points at once, improving scanning speed. While various structured light projection techniques exist, parallel stripe patterns are among the most commonly used. By analyzing the displacement of these stripes, the three-dimensional coordinates of surface details can be accurately determined. === Generation of light patterns === Two major methods of stripe pattern generation have been established: Laser interference and projection. The laser interference method works with two wide planar laser beam fronts. Their interference results in regular, equidistant line patterns. Different pattern sizes can be obtained by changing the angle between these beams. The method allows for the exact and easy generation of very fine patterns with unlimited depth of field. Disadvantages are high cost of implementation, difficulties providing the ideal beam geometry, and laser typical effects like speckle noise and the possible self interference with beam parts reflected from objects. Typically, there is no means of modulating individual stripes, such as with Gray codes. The projection method uses incoherent light and basically works like a video projector. Patterns are usually generated by passing light through a digital spatial light modulator, typically based on one of the three currently most widespread digital projection technologies, transmissive liquid crystal, reflective liquid crystal on silicon (LCOS) or digital light processing (DLP; moving micro mirror) modulators, which have various comparative advantages and disadvantages for this application. Other methods of projection could be and have been used, however. Patterns generated by digital display projectors have small discontinuities due to the pixel boundaries in the displays. Sufficiently small boundaries however can practically be neglected as they are evened out by the slightest defocus. A typical measuring assembly consists of one projector and at least one camera. For many applications, two cameras on opposite sides of the projector have been established as useful. Invisible (or imperceptible) structured light uses structured light without interfering with other computer vision tasks for which the projected pattern will be confusing. Example methods include the use of infrared light or of extremely high framerates alternating between two exact opposite patterns. === Calibration === Geometric distortions by optics and perspective must be compensated by a calibration of the measuring equipment, using special calibration patterns and surfaces. A mathematical model is used for describing the imaging properties of projector and cameras. Essentially based on the simple geometric properties of a pinhole camera, the model also has to take into account the geometric distortions and optical aberration of projector and camera lenses. The parameters of the camera as well as its orientation in space can be determined by a series of calibration measurements, using photogrammetric bundle adjustment. === Analysis of stripe patterns === There are several depth cues contained in the observed stripe patterns. The displacement of any single stripe can directly be converted into 3D coordinates. For this purpose, the individual stripe has to be identified, which can for example be accomplished by tracing or counting stripes (pattern recognition method). Another common method projects alternating stripe patterns, resulting in binary Gray code sequences identifying the number of each individual stripe hitting the object. An important depth cue also results from the varying stripe widths along the object surface. Stripe width is a function of the steepness of a surface part, i.e. the first derivative of the elevation. Stripe frequency and phase deliver similar cues and can be analyzed by a Fourier transform. Finally, the wavelet transform has recently been discussed for the same purpose. In many practical implementations, series of measurements combining pattern recognition, Gray codes and Fourier transform are obtained for a complete and unambiguous reconstruction of shapes. Another method also belonging to the area of fringe projection has been demonstrated, utilizing the depth of field of the camera. It is also possible to use projected patterns primarily as a means of structure insertion into scenes, for an essentially photogrammetric acquisition. === Precision and range === The optical resolution of fringe projection methods depends on the width of the stripes used and their optical quality. It is also limited by the wavelength of light. An extreme reduction of stripe width proves inefficient due to limitations in depth of field, camera resolution and display resolution. Therefore, the phase shift method has been widely established: A number of at least 3, typically about 10 exposures are taken with slightly shifted stripes. The first theoretical deductions of this method relied on stripes with a sine wave shaped intensity modulation, but the methods work with "rectangular" modulated stripes, as delivered from LCD or DLP displays as well. By phase shifting, surface detail of e.g. 1/10 the stripe pitch can be resolved. Current optical stripe pattern profilometry hence allows for detail resolutions down to the wavelength of light, below 1 micrometer in practice or, with larger stripe patterns, to approx. 1/10 of the stripe width. Concerning level accuracy, interpolating over several pixels of the acquired camera image can yield a reliable height resolution and also accuracy, down to 1/50 pixel. Arbitrarily large objects can be measured with accordingly large stripe patterns and setups. Practical applications are documented involving objects several meters in size. Typical accuracy figures are: Planarity of a 2-foot (0.61 m) wide surface, to 10 micrometres (0.00039 in). Shape of a motor combustion chamber to 2 micrometres (7.9×10−5 in) (elevation), yielding a volume accuracy 10 times better than with volumetric dosing. Shape of an object 2 inches (51 mm) large, to about 1 micrometre (3.9×10−5 in) Radius of a blade edge of e.g. 10 micrometres (0.00039 in), to ±0.4 μm === Navigation === As the method can measure shapes from only one perspective at a time, complete 3D shapes have to be combined from different measurements in different angles. This can be accomplished by attaching marker points to the object and combining perspectives afterwards by matching these markers. The process can be automated, by mounting the object on a motorized turntable on robotic inspection cell, or CNC positioning device. Markers can as well be applied on a positioning device instead of the object itself. The 3D data gathered can be used to retrieve CAD (computer aided design) data and models from existing components (reverse engineering), hand formed samples or sculptures, natural objects or artifacts. === Challenges === As with all optical methods, reflective or transparent surfaces raise difficulties. Reflections cause light to be reflected either away from the camera or right into its optics. In both cases, the dynamic range of the camera can be exceeded. Transparent or semi-transparent surfaces also cause major difficulties. In these cases, coating the surfaces with a thin opaque lacquer just for measuring purposes is a common practice. A recent method handles highly reflective and specular objects by inserting a 1-dimensional diffuser between the light source (e.g., projector) and the object to be scanned. Alternative optical techniques have been proposed for handling perfectly transparent and specular objects. Double reflections and inter-reflections can cause the stripe pattern to be overlaid with unwanted ligh
Agent mining
Agent mining is a research field that combines two areas of computer science: multiagent systems and data mining. It explores how intelligent computer agents can work together to discover, analyze, and learn from large amounts of data more effectively than traditional methods. == Historical context == The interaction and the integration between multiagent systems and data mining have a long history. The very early work on agent mining focused on agent-based knowledge discovery, agent-based distributed data mining, and agent-based distributed machine learning, and using data mining to enhance agent intelligence. The International Workshop on Agents and Data Mining Interaction has been held for more than 10 times, co-located with the International Conference on Autonomous Agents and Multi-Agent Systems. Several proceedings are available from Springer Lecture Notes in Computer Science.
International Olympiad in Artificial Intelligence
The International Olympiad in Artificial Intelligence (IOAI) is an annual International Science Olympiad in the field of artificial intelligence (AI) for secondary education students under the age of 20. The first IOAI was held in Burgas, Bulgaria, in 2024. Each country or territory may send up to two teams, each consisting of up to four students supported by one leader. Participants are selected through a multi-stage National Olympiad in Artificial Intelligence (NOAI) and/or a Regional Olympiad such as the NAOAI or APOAI. Participants at the IOAI compete on an individual basis. As of 2025, there were 61 countries and territories participating in the IOAI. Three hundred students participated in IOAI 2025. As of 2026, 130 countries and territories are accredited for participation in the IOAI. == Competition Structure == The IOAI consists of three contests: the Individual Contest, the Team Challenge, and the GAITE contest. Medals are awarded based solely on the Individual Contest. === Individual Contest === The Individual Contest is the main competition of the IOAI in which contestants compete individually on separate computers and are not permitted to communicate during the contest. Medals are awarded solely on the basis of the total score from the two-day Individual Contest. The Individual Contest consists of two on-site contest days (six hours per day), preceded by an at-home practice round and an on-site practice session. In IOAI 2025, three at-home problems were released for preparation approximately one month before the on-site contest. Results from this at-home round do not affect final results. The first on-site contest day (Individual Contest 1) comprises three tasks as extensions and continuations of the at-home tasks, while the second day (Individual Contest 2) comprises two or three tasks which are novel and different from the at-home tasks. The Individual Contest tasks span various AI domains such as machine learning, natural language processing, and computer vision. The IOAI 2025 contest rules describe tasks as requiring typical machine-learning workflows, including writing code, fitting models on training data, and running inference on test data, using identical local machines and GPU resources (minimum 24 GB RAM). Tasks, datasets, and submissions are handled through a contest platform (Bohrium), including a web-based Jupyter notebook environment for GPU access. Internet access is restricted to a whitelist of documentation sites and an integrated compact large language model accessible within the platform. The use of external APIs are prohibited unless a task explicitly allows them. In IOAI 2025, each contest task was scored up to 100 points and could include multiple subtasks. Scores are normalized using a baseline solution and a maximum score derived from either a Scientific Committee solution or the best contestant submission. Contestants can view only their own scores during the contest; a live scoreboard may be available publicly outside the contest hall but is not permitted to be viewed by contestants during the contest. For non-English-speaking teams, the IOAI hold a translation session beginning three hours before each contest day in which team leaders review and may amend machine-translated task statements; translations must match the English original and are published after the contest. The IOAI committee also enforces quarantine restrictions during these translation sessions, where neither contestants or team leaders may not use cell phones, laptops, and other communication devices. === Team Challenge === The Team Challenge is a team-based component of the IOAI. The results of this part do not affect the distribution of medals. The IOAI 2025 rules describe it as a “creative and AI-oriented challenge” in which a team's contestants sit together and cooperate, with the format varying by year. In IOAI 2024, teams worked with existing AI image and video generation tools to produce a visual result. In IOAI 2025, teams were assigned to program a robot to complete various tasks. === GAITE Contest === The GAITE (Global AI Talent Empowerment) contest is a simplified version of the individual contest with a separate scoreboard, where participants may ask for hints. It is designed for countries and territories with limited International Science Olympiads history, and it awards alternative prizes instead of medals. == Awards Distribution == The top 50% of the participants in the individual contest receive gold, silver and bronze medals in ratio of 1:2:3, respectively. The top three individuals receive honorary trophies. As in other International Science Olympiads, if an individual is in the top 50% on one of the days, but does not receive a medal, they receive an honorary mention during the awards ceremony. The GAITE contest has similar cutoff logic, but receives a reward instead of a medal. The top three teams in the Team Challenge receive trophies. == National selection and regional competitions == National delegations are selected through country-level qualification processes referred to as National Olympiads in Artificial Intelligence (NOAI) or equivalent, which are widely known for their low success rates. Although the total number of participants worldwide is not published, available data indicate exceptionally competitive national pools; for example, Brazil reports over 716,000 competitors, while Russia reports more than 72,000. In addition, Regional Olympiads (for example, APOAI or NAOAI) provide continent-level competition and preparation platforms in most regions. === National Selection (National Olympiads in Artificial Intelligence) === Participating countries and territories select their students for the IOAI through a National Olympiad in Artificial Intelligence (NOAI) or an equivalent process. The names of these selection processes differ by country, but almost all of them (excluding newer countries participating in the GAITE contest) have in common that the process comprises multiple and/or extremely rigorous selection stages. United States / Canada – The USA–North America AI Olympiad (USAAIO) is a three-round process including an invitational in-person round and a subsequent selection camp, after which a national delegation is selected for IOAI. Russia – The Russian Olympiad in Artificial Intelligence is organized as a multi-stage process (training, qualification, main round, final). Organizers reported 72,316 registrations for the training round and 52,260 registrations for the qualifying round in one season, with tasks spanning mathematics, algorithms/programming, and machine learning; 977 students were disqualified following plagiarism checks. Japan – Japan's national selection consists of multiple stages, beginning with the Japan Olympiad in Artificial Intelligence (JOAI), a large-scale Kaggle-style competition. High-performing participants advance through additional assessment stages, including written solution reports and technical interviews. From this process, eight students are selected for the APOAI team, with four ultimately chosen to represent Japan at the IOAI. Brazil – Brazil's National Olympiad in Artificial Intelligence (ONIA) is conducted as a large competition which consists of progressive rounds of evaluation. It identifies 28 top students from over 716,000 competitors, four of which are selected for the IOAI. The competition is held in four phases across two cycles, including a two-step third phase and a final training-and-evaluation phase that selects a four-student national team. Singapore – Singapore's national Olympiad consists of two rounds: an online preliminary round (300 MCQs in 3 hours) selects the top 150 performers to advance to the final assessment, which includes both theory questions and Python programming tasks. Additional training and selection may follow the finals for top performers. Poland – The Polish AI Olympiad adopts a two-stage structure: an open online first stage (at-home tasks) and a second-stage competitive camp with 30 selected participants competing for a four-person IOAI team. France – The Olympiades Françaises d'Intelligence Artificielle (OFIA), organized by France-IOI, follow a three-stage structure consisting of an open online qualification round, a second selection round, and a multi-day national training camp and final in Paris. Bangladesh – The Bangladesh AI Olympiad (BdAIO) selects competitors in three rounds: the online preliminary round, the national finals, and the team selection camp. In 2025, 406 participants competed in the national finals. Norway – The Norwrgian AI Olympiad (NOKI) is a three-stage selection system; however, unlike other countries, its first two rounds are shared with the Norwegian Informatics Olympiad. The national Olympiad reports 1,180 participants in the first round. Hong Kong – The national Olympiad reported more than 800 preliminary-round entrants, narrowing through multiple rounds to 25 finalists, with a subsequent
Opposition to AI data centers
Since 2024, dozens of local community-led protest campaigns have emerged in opposition to AI data centers. == Motivations == Organized opposition to AI data centers has been driven by concerns about energy use, energy costs, noise pollution, air pollution, and water waste. Opposition sentiment is widespread with a Gallup poll conducted in March 2026 showing that 70% of respondents oppose the construction of new AI data centers in their neighborhood. == Impact == In 2025, local opposition to AI data centers led to the delay or cancellation of projects totalling US$156 billion. == Specific protests and outcomes in the United States == According to Data Center Watch, there are has been a wave of dozens of protests against AI data centers since 2022. Below is a non-exhaustive list of some notable examples. === Goodyear and Buckeye, Arizona: Tract AI Data Center Proposal === In Goodyear and Buckeye, Arizona, a $14 billion project by developer Tract was withdrawn after local authorities blocked necessary rezoning in response to pressure from resident organizers. Opponest stiff resistance due to concerns over building heights, noise pollution, and the potential strain on local utilities. However, the company announced a revised project near the Buckeye airport in August 2024, with the backing of local officials and the mayor. === Peculiar, Missouri: Diode Ventures Harper Road Technology Park Proposal === In Peculiar, Missouri, residents from the group "Peaceful Peculiar" organized to stop a data center proposal from Diode Ventures called Harper Road Technology Park. Citing concerns around noise and light pollution, health, environmental impacts, jobs, property values, and energy use, organizers attended local planning and zoning meetings in large numbers and lobbied councilors to reject the proposal. Ultimately, the city council unanimously rejected the proposal in September 2024. === Chesterton, Indiana: Provident Realty Advisors Proposal === In Chesterton, Indiana, the Texas-based company Provident Reality Advisors applied for a $1.3 billion construction of a data center complex on the Brassie Golf Club property. Provident Realty Advisors wanted to purchase the 200 acres owned by PPM Chesterton LLC in 2024 order to build a data center complex, with eight buildings and an end user of a hyperscaler. The Town Council of Chesterton released a statement saying that they would never support this project, at least not at the scale and location it was planned for. They cited fears of added noise for locals, electrical or water management concerns, the intrusiveness of a data center built next to houses, and more. Provident released a statement shortly after rescinding their plan, because it was clear than the town of Chesterton would not support them. === Cascade Locks, Oregon: Roundhouse Digital Infrastructure Proposal === Startup data center developer Roundhouse Digital Infrastructure had planned to build out a 10-megawatt data center using a vacant industrial building and nearby 10-acre site in the Port of Cascade Locks, Oregon. After significant organized community opposition, the project was abandoned. === Forth Worth, Texas: WUSF 5 Rock Creek East Proposal === In September 2024, the City Council of Fort Worth, Texas approved a zoning change that would allow construction of a data center. In responses, neighbors mounted opposition citing concerns about traffic, light pollution, energy consumption, water use, and noise issues if the data center were to be built. In response to extensive public comments opposing a tax break for the data center, a city councilor withdrew his motion to approve the tax break. As of April, 2026, the future of the project is still uncertain. === Santa Clara, California: GI Partners Proposal === GI partners sought to build a new AI data center in Santa Clara, California, which is already home to many data centers, by acquiring a conditional permit use that would have allowed the developer to knock down a property and replace it with a data center. To obtain this permit they were required to go before members of the Planning Commission. Ultimately, the project was delayed with the Planning Commission requiring GI partners to do more public outreach. === Virginia === ==== Richmond: DC Blox Proposal ==== After residents organized to lobby the municipal government to block the proposal to avoid noise pollution and higher energy use, commissioners denied the company's permit. ==== Catlett Station: Headwaters Site Proposal ==== In Catlett, Virginia, developer Headwaters proposed construction of a data center complex just north of the town in 2020. In response, a residents' organization called "Protect Catlett" was formed to oppose the project. Arguments against the data center involved its impacts on water and power availability, its noise as a residential disturbance, and its destruction of historic and community heritage buildings. Arguments in favor cited job creation and $20 million in local tax revenue if the project were to go through. Protect Catlett utilized town halls and public comments to mobilize opposition to the project. They also dedicated time to educating other residents about the project's negative impacts and canvassing door-to-door in order to garner even more opposition to the project. Ultimately, after fervent opposition from most town residents, the project was canceled by the town and the developer. ==== Culpeper County: Culpeper Acquisitions Proposal ==== Culpeper Acquisitions, LLC, proposed a massive $12 billion data center project in Culpeper County, Virginia, designed to feature 4.6 million square feet of space across nine multi-story buildings. Coalition to Save Culpeper (C2SC) is an activist organization formed to resist the development of the project. C2SC has been active on many fronts including, messaging on social media, reaching out to local officials, and organizing meetings to bring community members with aligned interests together. Ultimately, the project was delayed due to unanimous denial by the Culpeper County Planning Commission on June 12, 2024, which was driven by intense opposition from C2SC. C2SC was successful in their mission largely because they were able to get so many people from the community behind it, and put enough pressure on local officials to take action. ==== Midlothian: Province Group Proposal ==== In late October 2025, the Powhatan County Board of Supervisors in Virginia voted unanimously to approve the $3 billion data center, despite the county's Planning Commission having unanimously recommended denial several days earlier. The reasoning behind their support for the center is that it will generate substantial tax revenue, reducing the county's reliance on residential property taxes. This appeal of lowering residential property taxes is the major selling point for the center's development. The developer, California-based Province Group, incentivized the Board by being agreeable to its conditions for building the center. The center is still on track for development, but faces local resistance, though little information is available on specific groups opposing it. ==== Warrenton: Amazon Proposal ==== Citizens for Farquier County (CFFC) advocates to "preserve the natural, historic and agricultural resources" of their county. Historically, this has meant opposing the building of a dam or lights in front of fast food stores. This group has recently mobilized in opposition of a plan to build data centers for Amazon. They first filed a suit to stop the construction in 2023 and it has been in litigation ever since. The case hinges on opposition to a 2021 zoning amendment which allowed data centers to be built in town. CFFC's lawyer, Dale Mullen, argues that this amendment violates state law, which requires such amendments to state their "public purpose". They argue that the permit for the Amazon data center was "void from the beginning". The CFFC also organized to vote out town council members who approved the first data center and were up for reelection, replacing them with candidates who opposed the data center. In May 2025, after attending town council meetings to speak out against the data center, the planning commission voted 4–1 to remove the zoning amendment allowing data center construction in town, citing public opposition. Currently, CFFC is advocating along with Piedmont Environmental Group, for phasing out data center tax breaks at the state level. ==== France: Marseille opposition ==== In France, local opposition materialised in response to proposed data centre developments, especially in and around the city of Marseille. Opposition came from activists, such as "Clouds Were Under Our Feet" group, residents ,and local politicians. Issues raised related to energy use, environmental impact, and limited local benefits (such as the creation of a few jobs only). == Legislation in the United States == Legal limits and moratoriums on the construction of new d
Foveated imaging
Foveated imaging is a digital image processing technique in which the image resolution, or amount of detail, varies across the image according to one or more "fixation points". A fixation point indicates the highest resolution region of the image and corresponds to the center of the eye's retina, the fovea. The location of a fixation point may be specified in many ways. For example, when viewing an image on a computer monitor, one may specify a fixation using a pointing device, like a computer mouse. Eye trackers which precisely measure the eye's position and movement are also commonly used to determine fixation points in perception experiments. When the display is manipulated with the use of an eye tracker, this is known as a gaze contingent display. Fixations may also be determined automatically using computer algorithms. Some common applications of foveated imaging include imaging sensor hardware and image compression. For descriptions of these and other applications, see the list below. Miniaturized foveated imaging systems can be realized by high-resolution 3D printing of multi-lens objectives directly on a CMOS (Complementary metal-oxide-semiconductor) chip. Foveated imaging is also commonly referred to as space variant imaging or gaze contingent imaging. == Applications == === Compression === Contrast sensitivity falls off dramatically as one moves from the center of the retina to the periphery. In lossy image compression, one may take advantage of this fact in order to compactly encode images. If one knows the viewer's approximate point of gaze, one may reduce the amount of information contained in the image as the distance from the point of gaze increases. Because the fall-off in the eye's resolution is dramatic, the potential reduction in display information can be substantial. Also, foveation encoding may be applied to the image before other types of image compression are applied and therefore can result in a multiplicative reduction. === Foveated sensors === Foveated sensors are multiresolution hardware devices that allow image data to be collected with higher resolution concentrated at a fixation point. An advantage to using foveated sensor hardware is that the image collection and encoding can occur much faster than in a system that post-processes a high resolution image in software. === Simulation === Foveated imaging has been used to simulate visual fields with arbitrary spatial resolution. For example, one may present video containing a blurred region representing a scotoma. By using an eye-tracker and holding the blurred region fixed relative to the viewer's gaze, the viewer will have a visual experience similar to that of a person with an actual scotoma. === Video gaming === Foveated rendering is a rendering optimization technique which uses an eye tracker integrated with a virtual reality headset to reduce the rendering workload by greatly reducing the image quality in the peripheral vision (outside of the zone gazed by the fovea).. However, other than the near-eye displays (e.g., virtual reality headset), foveated rendering is also suitable for large high-resolution display walls, desktop monitor, and even for smart phones. Over the time different foveated rendering techniques are proposed, for instance, adaptive resolution, geometric simplification, shading simplification and chromatic degradation, spatio-temporal deterioration . If we consider the variable sample distribution of physically-based rendering under the shader (e.g., hit/miss etc.), then this degradation strategies are applied on overall foveated rendering. At the CES 2016, SensoMotoric Instruments (SMI) demoed a new 250 Hz eye tracking system and a working foveated rendering solution. It resulted from a partnership with camera sensor manufacturer Omnivision who provided the camera hardware for the new system. The Apple Vision Pro mixed reality headset features dynamic foveated rendering provided by its visionOS operating system. === Quality assessment === Foveated imaging may be useful in providing a subjective image quality measure. Traditional image quality measures, such as peak signal-to-noise ratio, are typically performed on fixed resolution images and do not take into account some aspects of the human visual system, like the change in spatial resolution across the retina. A foveated quality index may therefore more accurately determine image quality as perceived by humans. === Image database retrieval === In databases that contain very high resolution images, such as a satellite image database, it may be desirable to interactively retrieve images in order to reduce retrieval time. Foveated imaging allows one to scan low resolution images and retrieve only high resolution portions as they are needed. This is sometimes called progressive transmission. == Example images ==
Possibility theory
Possibility theory is a mathematical theory for dealing with certain types of uncertainty and is an alternative to probability theory. It uses measures of possibility and necessity between 0 and 1, ranging from impossible to possible and unnecessary to necessary, respectively. Professor Lotfi Zadeh first introduced possibility theory in 1978 as an extension of his theory of fuzzy sets and fuzzy logic. Didier Dubois and Henri Prade further contributed to its development. Earlier, in the 1950s, economist G. L. S. Shackle proposed the min/max algebra to describe degrees of potential surprise. == Formalization of possibility == For simplicity, assume that the universe of discourse Ω is a finite set. A possibility measure is a function Π {\displaystyle \Pi } from 2 Ω {\displaystyle 2^{\Omega }} to [0, 1] such that: Axiom 1: Π ( ∅ ) = 0 {\displaystyle \Pi (\varnothing )=0} Axiom 2: Π ( Ω ) = 1 {\displaystyle \Pi (\Omega )=1} Axiom 3: Π ( U ∪ V ) = max ( Π ( U ) , Π ( V ) ) {\displaystyle \Pi (U\cup V)=\max \left(\Pi (U),\Pi (V)\right)} for any disjoint subsets U {\displaystyle U} and V {\displaystyle V} . It follows that, like probability on finite probability spaces, the possibility measure is determined by its behavior on singletons: Π ( U ) = max ω ∈ U Π ( { ω } ) . {\displaystyle \Pi (U)=\max _{\omega \in U}\Pi (\{\omega \}).} Axiom 1 can be interpreted as the assumption that Ω is an exhaustive description of future states of the world, because it means that no belief weight is given to elements outside Ω. Axiom 2 could be interpreted as the assumption that the evidence from which Π {\displaystyle \Pi } was constructed is free of any contradiction. Technically, it implies that there is at least one element in Ω with possibility 1. Axiom 3 corresponds to the additivity axiom in probabilities. However, there is an important practical difference. Possibility theory is computationally more convenient because Axioms 1–3 imply that: Π ( U ∪ V ) = max ( Π ( U ) , Π ( V ) ) {\displaystyle \Pi (U\cup V)=\max \left(\Pi (U),\Pi (V)\right)} for any subsets U {\displaystyle U} and V {\displaystyle V} . Because one can know the possibility of the union from the possibility of each component, it can be said that possibility is compositional with respect to the union operator. Note however that it is not compositional with respect to the intersection operator. Generally: Π ( U ∩ V ) ≤ min ( Π ( U ) , Π ( V ) ) ≤ max ( Π ( U ) , Π ( V ) ) . {\displaystyle \Pi (U\cap V)\leq \min \left(\Pi (U),\Pi (V)\right)\leq \max \left(\Pi (U),\Pi (V)\right).} When Ω is not finite, Axiom 3 can be replaced by: For all index sets I {\displaystyle I} , if the subsets U i , i ∈ I {\displaystyle U_{i,\,i\in I}} are pairwise disjoint, Π ( ⋃ i ∈ I U i ) = sup i ∈ I Π ( U i ) . {\displaystyle \Pi \left(\bigcup _{i\in I}U_{i}\right)=\sup _{i\in I}\Pi (U_{i}).} == Necessity == Whereas probability theory uses a single number, the probability, to describe how likely an event is to occur, possibility theory uses two concepts, the possibility and the necessity of the event. For any set U {\displaystyle U} , the necessity measure is defined by N ( U ) = 1 − Π ( U ¯ ) {\displaystyle N(U)=1-\Pi ({\overline {U}})} . In the above formula, U ¯ {\displaystyle {\overline {U}}} denotes the complement of U {\displaystyle U} , that is the elements of Ω {\displaystyle \Omega } that do not belong to U {\displaystyle U} . It is straightforward to show that: N ( U ) ≤ Π ( U ) {\displaystyle N(U)\leq \Pi (U)} for any U {\displaystyle U} and that: N ( U ∩ V ) = min ( N ( U ) , N ( V ) ) {\displaystyle N(U\cap V)=\min(N(U),N(V))} . Note that contrary to probability theory, possibility is not self-dual. That is, for any event U {\displaystyle U} , we only have the inequality: Π ( U ) + Π ( U ¯ ) ≥ 1 {\displaystyle \Pi (U)+\Pi ({\overline {U}})\geq 1} However, the following duality rule holds: For any event U {\displaystyle U} , either Π ( U ) = 1 {\displaystyle \Pi (U)=1} , or N ( U ) = 0 {\displaystyle N(U)=0} Accordingly, beliefs about an event can be represented by a number and a bit. == Interpretation == There are four cases that can be interpreted as follows: N ( U ) = 1 {\displaystyle N(U)=1} means that U {\displaystyle U} is necessary. U {\displaystyle U} is certainly true. It implies that Π ( U ) = 1 {\displaystyle \Pi (U)=1} . Π ( U ) = 0 {\displaystyle \Pi (U)=0} means that U {\displaystyle U} is impossible. U {\displaystyle U} is certainly false. It implies that N ( U ) = 0 {\displaystyle N(U)=0} . Π ( U ) = 1 {\displaystyle \Pi (U)=1} means that U {\displaystyle U} is possible. I would not be surprised at all if U {\displaystyle U} occurs. It leaves N ( U ) {\displaystyle N(U)} unconstrained. N ( U ) = 0 {\displaystyle N(U)=0} means that U {\displaystyle U} is unnecessary. I would not be surprised at all if U {\displaystyle U} does not occur. It leaves Π ( U ) {\displaystyle \Pi (U)} unconstrained. The intersection of the last two cases is N ( U ) = 0 {\displaystyle N(U)=0} and Π ( U ) = 1 {\displaystyle \Pi (U)=1} meaning that I believe nothing at all about U {\displaystyle U} . Because it allows for indeterminacy like this, possibility theory relates to the graduation of a many-valued logic, such as intuitionistic logic, rather than the classical two-valued logic. Note that unlike possibility, fuzzy logic is compositional with respect to both the union and the intersection operator. The relationship with fuzzy theory can be explained with the following classic example. Fuzzy logic: When a bottle is half full, it can be said that the level of truth of the proposition "The bottle is full" is 0.5. The word "full" is seen as a fuzzy predicate describing the amount of liquid in the bottle. Possibility theory: There is one bottle, either completely full or totally empty. The proposition "the possibility level that the bottle is full is 0.5" describes a degree of belief. One way to interpret 0.5 in that proposition is to define its meaning as: I am ready to bet that it's empty as long as the odds are even (1:1) or better, and I would not bet at any rate that it's full. == Possibility theory as an imprecise probability theory == There is an extensive formal correspondence between probability and possibility theories, where the addition operator corresponds to the maximum operator. A possibility measure can be seen as a consonant plausibility measure in the Dempster–Shafer theory of evidence. The operators of possibility theory can be seen as a hyper-cautious version of the operators of the transferable belief model, a modern development of the theory of evidence. Possibility can be seen as an upper probability: any possibility distribution defines a unique credal set of admissible probability distributions by K = { P ∣ ∀ S P ( S ) ≤ Π ( S ) } . {\displaystyle K=\{\,P\mid \forall S\ P(S)\leq \Pi (S)\,\}.} This allows one to study possibility theory using the tools of imprecise probabilities. == Necessity logic == We call generalized possibility every function satisfying Axiom 1 and Axiom 3. We call generalized necessity the dual of a generalized possibility. The generalized necessities are related to a very simple and interesting fuzzy logic called necessity logic. In the deduction apparatus of necessity logic the logical axioms are the usual classical tautologies. Also, there is only a fuzzy inference rule extending the usual modus ponens. Such a rule says that if α and α → β are proved at degree λ and μ, respectively, then we can assert β at degree min{λ,μ}. It is easy to see that the theories of such a logic are the generalized necessities and that the completely consistent theories coincide with the necessities (see for example Gerla 2001).