Speech synthesis is the artificial production of human speech. A computer system used for this purpose is called a speech synthesizer, and can be implemented in software or hardware products. A text-to-speech (TTS) system converts normal language text into speech; other systems render symbolic linguistic representations like phonetic transcriptions into speech. The reverse process is speech recognition. Synthesized speech can be created by concatenating pieces of recorded speech that are stored in a database. Systems differ in the size of the stored speech units; a system that stores phones or diphones provides the largest output range, but may lack clarity. For specific usage domains, the storage of entire words or sentences allows for high-quality output. Alternatively, a synthesizer can incorporate a model of the vocal tract and other human voice characteristics to create a completely "synthetic" voice output. The quality of a speech synthesizer is judged by its similarity to the human voice and by its ability to be understood clearly. An intelligible text-to-speech program allows people with visual impairments or reading disabilities to listen to written words on a home computer. The earliest computer operating system to have included a speech synthesizer was Unix in 1974, through the Unix speak utility. In 2000, Microsoft Sam was the default text-to-speech voice synthesizer used by the narrator accessibility feature, which shipped with all Windows 2000 operating systems, and subsequent Windows XP systems. A text-to-speech system (or "engine") is composed of two parts: a front-end and a back-end. The front-end has two major tasks. First, it converts raw text containing symbols like numbers and abbreviations into the equivalent of written-out words. This process is often called text normalization, pre-processing, or tokenization. The front-end then assigns phonetic transcriptions to each word, and divides and marks the text into prosodic units, like phrases, clauses, and sentences. The process of assigning phonetic transcriptions to words is called text-to-phoneme or grapheme-to-phoneme conversion. Phonetic transcriptions and prosody information together make up the symbolic linguistic representation that is output by the front-end. The back-end—often referred to as the synthesizer—then converts the symbolic linguistic representation into sound. In certain systems, this part includes the computation of the target prosody (pitch contour, phoneme durations), which is then imposed on the output speech. == History == Long before the invention of electronic signal processing, some people tried to build machines to emulate human speech. There were also legends of the existence of "Brazen Heads", such as those involving Pope Silvester II (d. 1003 AD), Albertus Magnus (1198–1280), and Roger Bacon (1214–1294). In 1779, the German-Danish scientist Christian Gottlieb Kratzenstein won the first prize in a competition announced by the Russian Imperial Academy of Sciences and Arts for models he built of the human vocal tract that could produce the five long vowel sounds (in International Phonetic Alphabet notation: [aː], [eː], [iː], [oː] and [uː]). There followed the bellows-operated "acoustic-mechanical speech machine" of Wolfgang von Kempelen of Pressburg, Hungary, described in a 1791 paper. This machine added models of the tongue and lips, enabling it to produce consonants as well as vowels. In 1837, Charles Wheatstone produced a "speaking machine" based on von Kempelen's design, and in 1846, Joseph Faber exhibited the "Euphonia". In 1923, Paget resurrected Wheatstone's design. In the 1930s, Bell Labs developed the vocoder, which automatically analyzed speech into its fundamental tones and resonances. From his work on the vocoder, Homer Dudley developed a keyboard-operated voice-synthesizer called The Voder (Voice Demonstrator), which he exhibited at the 1939 New York World's Fair. Franklin S. Cooper and his colleagues at Haskins Laboratories built the pattern playback in the late 1940s and completed it in 1950. There were several different versions of this hardware device; only one currently survives. The machine converts pictures of the acoustic patterns of speech in the form of a spectrogram back into sound. Using this device, Alvin Liberman and colleagues discovered acoustic cues for the perception of phonetic segments (consonants and vowels). === Electronic devices === The first computer-based speech-synthesis systems originated in the late 1950s. Noriko Umeda et al. developed the first general English text-to-speech system in 1968, at the Electrotechnical Laboratory in Japan. In 1961, physicist John Larry Kelly, Jr and his colleague Louis Gerstman used an IBM 704 computer to synthesize speech, an event among the most prominent in the history of Bell Labs. Kelly's voice recorder synthesizer (vocoder) recreated the song "Daisy Bell", with musical accompaniment from Max Mathews. Coincidentally, Arthur C. Clarke was visiting his friend and colleague John Pierce at the Bell Labs Murray Hill facility. Clarke was so impressed by the demonstration that he used it in the climactic scene of his screenplay for his novel 2001: A Space Odyssey, where the HAL 9000 computer sings the same song as astronaut Dave Bowman puts it to sleep. Despite the success of purely electronic speech synthesis, research into mechanical speech-synthesizers continues. Linear predictive coding (LPC), a form of speech coding, began development with the work of Fumitada Itakura of Nagoya University and Shuzo Saito of Nippon Telegraph and Telephone (NTT) in 1966. Further developments in LPC technology were made by Bishnu S. Atal and Manfred R. Schroeder at Bell Labs during the 1970s. LPC was later the basis for early speech synthesizer chips, such as the Texas Instruments LPC Speech Chips used in the Speak & Spell toys from 1978. In 1975, Fumitada Itakura developed the line spectral pairs (LSP) method for high-compression speech coding, while at NTT. From 1975 to 1981, Itakura studied problems in speech analysis and synthesis based on the LSP method. In 1980, his team developed an LSP-based speech synthesizer chip. LSP is an important technology for speech synthesis and coding, and in the 1990s was adopted by almost all international speech coding standards as an essential component, contributing to the enhancement of digital speech communication over mobile channels and the internet. In 1975, MUSA was released, and was one of the first Speech Synthesis systems. It consisted of a stand-alone computer hardware and a specialized software that enabled it to read Italian. A second version, released in 1978, was also able to sing Italian in an "a cappella" style. Dominant systems in the 1980s and 1990s were the DECtalk system, based largely on the work of Dennis Klatt at MIT, and the Bell Labs system; the latter was one of the first multilingual language-independent systems, making extensive use of natural language processing methods. Handheld electronics featuring speech synthesis began emerging in the 1970s. One of the first was the Telesensory Systems Inc. (TSI) Speech+ portable calculator for the blind in 1976. Other devices had primarily educational purposes, such as the Speak & Spell toy produced by Texas Instruments in 1978. Fidelity released a speaking version of its electronic chess computer in 1979. The first video game to feature speech synthesis was the 1980 shoot 'em up arcade game, Stratovox (known in Japan as Speak & Rescue), from Sun Electronics. The first personal computer game with speech synthesis was Manbiki Shoujo (Shoplifting Girl), released in 1980 for the PET 2001, for which the game's developer, Hiroshi Suzuki, developed a "zero cross" programming technique to produce a synthesized speech waveform. Another early example, the arcade version of Berzerk, also dates from 1980. The Milton Bradley Company produced the first multi-player electronic game using voice synthesis, Milton, in the same year. In 1976, Computalker Consultants released their CT-1 Speech Synthesizer. Designed by D. Lloyd Rice and Jim Cooper, it was an analog synthesizer built to work with microcomputers using the S-100 bus standard. Synthesized voices typically sounded male until 1990, when Ann Syrdal, at AT&T Bell Laboratories, created a female voice. Ray Kurzweil predicted in 2005 that as the cost-performance ratio caused speech synthesizers to become cheaper and more accessible, more people would benefit from the use of text-to-speech programs. === Artificial intelligence === In September 2016, DeepMind released WaveNet, which demonstrated that deep learning models are capable of modeling raw waveforms and generating speech from acoustic features like spectrograms or mel-spectrograms, starting the field of deep learning speech synthesis. Although WaveNet was initially considered to be computationally expensive and slow to be used in consumer products at the time, a year after its
Fuse Services Framework
Fuse Services Framework is an open source SOAP and REST web services platform based on Apache CXF for use in enterprise IT organizations. It is productized and supported by the Fuse group at FuseSource Corp. Fuse Services Framework service-enables new and existing systems for use in enterprise SOA infrastructure. Fuse Services Framework is a pluggable, small-footprint engine that creates high performance, secure and robust services in minutes using front-end programming APIs like JAX-WS and JAX-RS. It supports multiple transports and bindings and is extensible so developers can add bindings for additional message formats so all systems can work together without having to communicate through a centralized server. Fuse Services Framework is now a part of Red Hat JBoss Fuse. Fabric8 is a free Apache 2.0 Licensed upstream community for the JBoss Fuse product from Red Hat.
Stanhope Demonstrator
The Stanhope Demonstrator was the first machine to solve problems in logic. It was designed by Charles Stanhope, 3rd Earl Stanhope to demonstrate consequences in logic symbolically. The first model was constructed in 1775. It consisted of two slides coloured red and gray mounted in a square brass frame. This could be used to demonstrate the solution to a syllogistic type of problem in which objects might have two different properties and the question was how many would have both properties. Scales marked zero to ten were used to set the numbers or proportions of objects with the two properties. This form of inference anticipated the numerically definite syllogism which Augustus De Morgan laid out in his book, Formal Logic, in 1847. == Construction == The device was a brass plate about four inches square which was mounted on a piece of mahogany which was three-quarters of an inch thick. There was an opening with a depression in the wood about one and a half inches square and half an inch deep. This opening was called the holon, meaning "whole", and represented the full set of objects under consideration. A slide of red translucent glass could be inserted from the right across the holon. A slide of gray wood could be slid under the red slide. When the device was used for the "Rule for the Logic of Certainty", the gray slider was inserted from the left. When it was used for the "Rule for the Logic of Probability", the gray slider was inserted from above. The red and the gray sliders represented the two affirmative propositions which were being combined. Stanhope called these ho and los. At least four of the devices with this square style were built. In 1879, Robert Harley wrote that he had one which he had been given by Stanhope's great-grandson, Arthur, who had kept one. The other two were owned by Henry Prevost Babbage – the son of Charles Babbage, who continued his work on the Analytical Engine. One of the devices was donated to the Science Museum, London by the last Earl in 1953. Other styles, such as circular models, were constructed, but these were less convenient.
Deepfake
Deepfakes (a portmanteau of 'deep learning' and 'fake') are images, videos, or audio that have been edited or generated using artificial intelligence, AI-based tools or audio-video editing software. They may depict real or fictional people and are considered a form of synthetic media, that is media that is usually created by artificial intelligence systems by combining various media elements into a new media artifact. While the act of creating fake content is not new, deepfakes uniquely leverage machine learning and artificial intelligence techniques, including facial recognition algorithms and artificial neural networks such as variational autoencoders and generative adversarial networks (GANs). In turn, the field of image forensics has worked to develop techniques to detect manipulated images. Deepfakes have garnered widespread attention for their potential use in creating child sexual abuse material, celebrity pornographic videos, revenge porn, fake news, hoaxes, bullying, and financial fraud. Academics have raised concerns about the potential for deepfakes to promote disinformation and hate speech, as well as interfere with elections. In response, the information technology industry and governments have proposed recommendations and methods to detect and mitigate their use. Academic research has also delved deeper into the factors driving deepfake engagement online as well as potential countermeasures to malicious application of deepfakes. From traditional entertainment to gaming, deepfake technology has evolved to be increasingly convincing and available to the public, allowing for the disruption of the entertainment and media industries. == History == Photo manipulation was developed in the 19th century and soon applied to motion pictures. Technology steadily improved during the 20th century, and more quickly with the advent of digital video. Deepfake technology has been developed by researchers at academic institutions beginning in the 1990s, and later by amateurs in online communities. More recently, the methods have been adopted by industry. The development of generative adversarial networks (GANs) in the mid-2010s represented a key technical turning point in the evolution of deepfakes. GANs allowed for the creation of highly realistic fake images and videos by training competing neural networks, achieving a much improved visual fidelity over previous methods of creating the content using rules or by using autoencoders, and formed the basis for modern deepfake methods. === Academic research === Academic research related to deepfakes is split between the field of computer vision, a sub-field of computer science, which develops techniques for creating and identifying deepfakes, and humanities and social science approaches that study the social, ethical, aesthetic implications as well as journalistic and informational implications of deepfakes. As deepfakes have risen in prominence in popularity with innovations provided by AI tools, significant research has gone into detection methods and defining the factors driving engagement with deepfakes on the internet. Deepfakes have been shown to appear on social media platforms and other parts of the internet for purposes ranging from entertainment and education related to deepfakes to misinformation to elicit strong reactions. There are gaps in research related to the propagation of deepfakes on social media. Negativity and emotional response are the primary driving factors for users sharing deepfakes. === Social science and humanities approaches to deepfakes === In cinema studies, deepfakes illustrate how "the human face is emerging as a central object of ambivalence in the digital age". Video artists have used deepfakes to "playfully rewrite film history by retrofitting canonical cinema with new star performers". Film scholar Christopher Holliday analyses how altering the gender and race of performers in familiar movie scenes destabilizes gender classifications and categories. The concept of "queering" deepfakes is also discussed in Oliver M. Gingrich's discussion of media artworks that use deepfakes to reframe gender, including British artist Jake Elwes' Zizi: Queering the Dataset, an artwork that uses deepfakes of drag queens to intentionally play with gender. The aesthetic potentials of deepfakes are also beginning to be explored. Theatre historian John Fletcher notes that early demonstrations of deepfakes are presented as performances, and situates these in the context of theater, discussing "some of the more troubling paradigm shifts" that deepfakes represent as a performance genre. While most English-language academic studies of deepfakes focus on the Western anxieties about disinformation and pornography, digital anthropologist Gabriele de Seta has analyzed the Chinese reception of deepfakes, which are known as huanlian, which translates to "changing faces". The Chinese term does not contain the "fake" of the English deepfake, and de Seta argues that this cultural context may explain why the Chinese response has centered on practical regulatory measures to "fraud risks, image rights, economic profit, and ethical imbalances". === Computer science research on deepfakes === A landmark early project was the "Video Rewrite" program, published in 1997. The program modified existing video footage of a person speaking to depict that person mouthing the words from a different audio track. It was the first system to fully automate this kind of facial reanimation, and it did so using machine learning techniques to make connections between the sounds produced by a video's subject and the shape of the subject's face. Contemporary academic projects have focused on creating more realistic videos and improving deepfake techniques. The "Synthesizing Obama" program, published in 2017, modifies video footage of former president Barack Obama to depict him mouthing the words contained in a separate audio track. The project lists as a main research contribution to its photorealistic technique for synthesizing mouth shapes from audio. The "Face2Face" program, published in 2016, modifies video footage of a person's face to depict them mimicking another person's facial expressions. The project highlights its primary research contribution as the development of the first method for re-enacting facial expressions in real time using a camera that does not capture depth, enabling the technique to work with common consumer cameras. Researchers have also shown that deepfakes are expanding into other domains such as medical imagery. In this work, it was shown how an attacker can automatically inject or remove lung cancer in a patient's 3D CT scan. The result was so convincing that it fooled three radiologists and a state-of-the-art lung cancer detection AI. To demonstrate the threat, the authors successfully performed the attack on a hospital in a White hat penetration test. A survey of deepfakes, published in May 2020, provides a timeline of how the creation and detection of deepfakes have advanced over the last few years. The survey identifies that researchers have been focusing on resolving the following challenges of deepfake creation: Generalization. High-quality deepfakes are often achieved by training on hours of footage of the target. This challenge is to minimize the amount of training data and the time to train the model required to produce quality images and to enable the execution of trained models on new identities (unseen during training). Paired Training. Training a supervised model can produce high-quality results, but requires data pairing. This is the process of finding examples of inputs and their desired outputs for the model to learn from. Data pairing is laborious and impractical when training on multiple identities and facial behaviors. Some solutions include self-supervised training (using frames from the same video), the use of unpaired networks such as Cycle-GAN, or the manipulation of network embeddings. Identity leakage. This is where the identity of the driver (i.e., the actor controlling the face in a reenactment) is partially transferred to the generated face. Some solutions proposed include attention mechanisms, few-shot learning, disentanglement, boundary conversions, and skip connections. Occlusions. When part of the face is obstructed with a hand, hair, glasses, or any other item then artifacts can occur. A common occlusion is a closed mouth which hides the inside of the mouth and the teeth. Some solutions include image segmentation during training and in-painting. Temporal coherence. In videos containing deepfakes, artifacts such as flickering and jitter can occur because the network has no context of the preceding frames. Some researchers provide this context or use novel temporal coherence losses to help improve realism. As the technology improves, the interference is diminishing. Overall, deepfakes are expected to have several implications in media and society, med
Minne Atairu
Minne Atairu is a Nigerian interdisciplinary artist, a recipient of the 2021 Global South Award Lumen Prize for Art and Technology. She generates synthetic Benin Bronzes through recombination of historical fragments, sculptures, texts, images, and sounds. == Early life and education == Atairu was born in Benin, Nigeria. She holds a bachelor's degree in art history from the University of Maiduguri in Maiduguri, Nigeria; a master's degree in museum studies from the George Washington University in Washington, D.C.; and a doctorate in art education from Teachers College, Columbia University in New York City. Her academic research integrates artificial intelligence, art/museum education and hip-hop based education. == Works == Atairu's artmaking involves using artificial intelligence (AI; such as StyleGAN, GPT-3) to make artwork. She uses tools such as Midjourney and Blender software to develop her works. === Mami Wata === Her first work is a Yoruba goddess called Mami Wata where she used Midjourney in generating the images. === To the Hand === For her 2023 installation To the Hand at The Shed arts center, she worked with Blender to convert text into 3D-printed sculptures made of corn starch or sugarcane infused with bronze. The rings of ground terra-cotta that surround the sculpture represent the walls and deep moats of Benin. == Publications == Atairu, Minne (February 1, 2024). "Reimagining Benin Bronzes using generative adversarial networks". AI & Society. 39 (1): 91–102. doi:10.1007/s00146-023-01761-7. ISSN 1435-5655.
PenTile matrix family
PenTile matrix is a family of patented subpixel matrix schemes used in electronic device displays. PenTile is a trademark of Samsung. PenTile matrices are used in AMOLED and LCD displays. These subpixel layouts are specifically designed to operate with proprietary algorithms for subpixel rendering embedded in the display driver, allowing plug and play compatibility with conventional RGB (Red-Green-Blue) stripe panels. == Overview == "PenTile Matrix" (a neologism from penta-, meaning "five" in Greek and tile) describes the geometric layout of the prototypical subpixel arrangement developed in the early 1990s. The layout consists of a quincunx comprising two red subpixels, two green subpixels, and one central blue subpixel in each unit cell. It was inspired by biomimicry of the human retina, which has nearly equal numbers of L and M type cone cells, but significantly fewer S cones. As the S cones are primarily responsible for perceiving blue colors, which do not appreciably affect the perception of luminance, reducing the number of blue subpixels with respect to the red and green subpixels in a display does not reduce the image quality. However, the layout may cause color leakage image distortion, which can be reduced by filters. In some cases the layout causes reduced moiré and blockiness compared to conventional RGB layouts. The PenTile layout is specifically designed to work with and be dependent upon subpixel rendering that uses only one and a quarter subpixel per pixel, on average, to render an image. That is, that any given input pixel is mapped to either a red-centered logical pixel, or a green-centered logical pixel. === History === PenTile was invented by Candice H. Brown Elliott, for which she was awarded the Society for Information Display's Otto Schade Prize in 2014. The technology was licensed by the company Clairvoyante from 2000 until 2008, during which time several prototype PenTile displays were developed by a number of Asian liquid crystal display (LCD) manufacturers. In March 2008, Samsung Electronics acquired Clairvoyante's PenTile IP assets. Samsung then funded a new company, Nouvoyance, Inc. to continue development of the PenTile technology. == PenTile RGBG == PenTile RGBG layout used in AMOLED and plasma displays uses green pixels interleaved with alternating red and blue pixels. The human eye is most sensitive to green, especially for high resolution luminance information. The green subpixels are mapped to input pixels on a one-to-one basis. The red and blue subpixels are subsampled, reconstructing the chroma signal at a lower resolution. The luminance signal is processed using adaptive subpixel rendering filters to optimize reconstruction of high spatial frequencies from the input image, wherein the green subpixels provide the majority of the reconstruction. The red and blue subpixels are capable of reconstructing the horizontal and vertical spatial frequencies, but not the highest of the diagonal. Diagonal high spatial frequency information in the red and blue channels of the input image are transferred to the green subpixels for image reconstruction. Thus the RG-BG scheme creates a color display with one third fewer subpixels than a traditional RGB-RGB scheme but with the same measured luminance display resolution. This is similar to the Bayer filter commonly used in digital cameras. === Devices === As of 2021, "almost all" OLED screens in portable consumer devices use some form of Pentile subpixel layout. == PenTile RGBW == PenTile RGBW technology, used in LCD, adds an extra subpixel to the traditional red, green and blue subpixels that is a clear area without color filtering material and with the only purpose of letting backlight come through, hence W for white. This makes it possible to produce a brighter image compared to an RGB-matrix while using the same amount of power, or produce an equally bright image while using less power. The PenTile RGBW layout uses each red, green, blue and white subpixel to present high-resolution luminance information to the human eyes' red-sensing and green-sensing cone cells, while using the combined effect of all the color subpixels to present lower-resolution chroma (color) information to all three cone cell types. Combined, this optimizes the match of display technology to the biological mechanisms of human vision. The layout uses one third fewer subpixels for the same resolution as the RGB stripe (RGB-RGB) layout, in spite of having four color primaries instead of the conventional three, using subpixel rendering combined with metamer rendering. Metamer rendering optimizes the energy distribution between the white subpixel and the combined red, green, and blue subpixels: W <> RGB, to improve image sharpness. The display driver chip has an RGB to RGBW color vector space converter and gamut mapping algorithm, followed by metamer and subpixel rendering algorithms. In order to maintain saturated color quality, to avoid simultaneous contrast error between saturated colors and peak white brightness, while simultaneously reducing backlight power requirements, the display backlight brightness is under control of the PenTile driver engine. When the image is mostly desaturated colors, those near white or grey, the backlight brightness is significantly reduced, often to less than 50% peak, while the LCD levels are increased to compensate. When the image has very bright saturated colors, the backlight brightness is maintained at higher levels. The PenTile RGBW also has an optional high-brightness mode that doubles the brightness of the desaturated color image areas, such as black-and-white text, for improved outdoor viewability. === Devices === Motorola MC65 Motorola ES55 Motorola ES400 Motorola Atrix 4G Samsung Galaxy Note 10.1 2014 version Lenovo Yoga 2 Pro Lenovo Yoga 3 Pro HP ENVY TouchSmart 14-k022tx Sleekbook MSI GS60 Ghost Pro 4K Lenovo IdeaPad Y50 4K Asus ZenBook UX303LN 4K Asus ZenBook Pro UX501JW LG UH7500/6500/6100 LG ThinQ G7/G7+ Oculus Quest 1 == Controversy == An ongoing controversy regarding the definition or measurement of resolution of color subpixelated flat panel displays led many people to question the resolution claims of PenTile display products. Journalists have noted that in "just about every flat-panel TV in existence, each pixel is composed of one red, one green, and one blue subpixel (RGB), all of uniform size". In traditional flat-panel screens, the resolution is defined by the number of red, green, and blue subpixels, in groups of three, in an array in each axis. As a result, each pixel or group of subpixels can render any colour on the screen, regardless of neighbouring pixels. This is not the case with PenTile screens. The Video Electronics Standards Association (VESA) method of measuring and defining resolution in color displays is to measure the contrast of line pairs, requiring a minimum of 50% Michelson contrast for displays intended for rendering text. The developers of PenTile displays use this VESA criterion for contrast of line pairs to calculate the resolutions specified. In the RGBG layout the alternate red and blue subpixels are 'shared' or sub-sampled with neighboring pixels. Due to the one third lower subpixel density on PenTile displays the pixel structure may be more visible when compared to RGB stripe displays with the same pixel density. The loss of subpixels for a given resolution specification has led some journalists to describe the use of PenTile as "shady practice" and "sort of cheating". For a given size and resolution specification, the PenTile screen can appear grainy, pixelated, speckled, with blurred text on some saturated colors and backgrounds when compared to RGB stripe color. This effect is understood to be caused by the restriction of the number of subpixels that may participate in the image reconstruction when colors are highly saturated to primaries. In the RGBW case, this is caused as the W subpixel will not be available in order to maintain the saturated color. In the RGBG case, this effect will occur when the color boundary is primarily red or blue, as the fully populated (one green per pixel) sub-pixel cannot contribute. For all other cases, text and especially full color images are effectively reconstructed. == Advantages and disadvantages == The PenTile layout reduces the number of subpixels needed to create a specified resolution. Consequently it is possible to achieve an HD resolution on a PenTile AMOLED screen at lower cost than other technologies, and most reviewers note that "300 ppi" (as per VESA - not full pixels) resolution displays (such as Samsung Galaxy S III) make the PenTile effect less obvious than lower resolution PenTile displays (Droid Razr). The second advantage is lower power consumption: the HTC One S's use of a PenTile display makes it more energy efficient and thinner than equivalent LCD screens, giving it better battery life than the HTC One X's IPS LCD. A PenTile AMOLED screen is also
Fuzzy differential inclusion
Fuzzy differential inclusion is the extension of differential inclusion to fuzzy sets introduced by Lotfi A. Zadeh. x ′ ( t ) ∈ [ f ( t , x ( t ) ) ] α {\displaystyle x'(t)\in [f(t,x(t))]^{\alpha }} with x ( 0 ) ∈ [ x 0 ] α {\displaystyle x(0)\in [x_{0}]^{\alpha }} Suppose f ( t , x ( t ) ) {\displaystyle f(t,x(t))} is a fuzzy valued continuous function on Euclidean space. Then it is the collection of all normal, upper semi-continuous, convex, compactly supported fuzzy subsets of R n {\displaystyle \mathbb {R} ^{n}} . == Second order differential == The second order differential is x ″ ( t ) ∈ [ k x ] α {\displaystyle x''(t)\in [kx]^{\alpha }} where k ∈ [ K ] α {\displaystyle k\in [K]^{\alpha }} , K {\displaystyle K} is trapezoidal fuzzy number ( − 1 , − 1 / 2 , 0 , 1 / 2 ) {\displaystyle (-1,-1/2,0,1/2)} , and x 0 {\displaystyle x_{0}} is a trianglular fuzzy number (-1,0,1). == Applications == Fuzzy differential inclusion (FDI) has applications in Cybernetics Artificial intelligence, Neural network, Medical imaging Robotics Atmospheric dispersion modeling Weather forecasting Cyclone Pattern recognition Population biology