Pieter "Peter" Adriaan Flach (born 8 April 1961, Sneek) is a Dutch computer scientist and a Professor of Artificial Intelligence in the Department of Computer Science at the University of Bristol. He is author of the acclaimed Simply Logical: Intelligent Reasoning by Example (John Wiley, 1994) and Machine Learning: the Art and Science of Algorithms that Make Sense of Data (Cambridge University Press, 2012). == Education == Flach received an MSc Electrical Engineering from Universiteit Twente in 1987 and a PhD in Computer Science from Tilburg University in 1995. == Research == Flach's research interests are in data mining and machine learning.
Image formation
The study of image formation encompasses the radiometric and geometric processes by which 2D images of 3D objects are formed. In the case of digital images, the image formation process also includes analog to digital conversion and sampling. == Imaging == The imaging process is a mapping of an object to an image plane. Each point on the image corresponds to a point on the object. An illuminated object will scatter light toward a lens and the lens will collect and focus the light to create the image. The ratio of the height of the image to the height of the object is the magnification. The spatial extent of the image surface and the focal length of the lens determines the field of view of the lens. Image formation of mirror these have a center of curvature and its focal length of the mirror is half of the center of curvature. == Illumination == An object may be illuminated by the light from an emitting source such as the sun, a light bulb or a Light Emitting Diode. The light incident on the object is reflected in a manner dependent on the surface properties of the object. For rough surfaces, the reflected light is scattered in a manner described by the Bi-directional Reflectance Distribution Function (BRDF) of the surface. The BRDF of a surface is the ratio of the exiting power per square meter per steradian (radiance) to the incident power per square meter (irradiance). The BRDF typically varies with angle and may vary with wavelength, but a specific important case is a surface that has constant BRDF. This surface type is referred to as Lambertian and the magnitude of the BRDF is R/π, where R is the reflectivity of the surface. The portion of scattered light that propagates toward the lens is collected by the entrance pupil of the imaging lens over the field of view. == Field of view and imagery == The Field of view of a lens is limited by the size of the image plane and the focal length of the lens. The relationship between a location on the image and a location on the object is y = ftan(θ), where y is the max extent of the image plane, f is the focal length of the lens and θ is the field of view. If y is the max radial size of the image then θ is the field of view of the lens. While the image created by a lens is continuous, it can be modeled as a set of discrete field points, each representing a point on the object. The quality of the image is limited by the aberrations in the lens and the diffraction created by the finite aperture stop. == Pupils and stops == The aperture stop of a lens is a mechanical aperture which limits the light collection for each field point. The entrance pupil is the image of the aperture stop created by the optical elements on the object side of the lens. The light scattered by an object is collected by the entrance pupil and focused onto the image plane via a series of refractive elements. The cone of the focused light at the image plane is set by the size of the entrance pupil and the focal length of the lens. This is often referred to as the f-stop or f-number of the lens. f/# = f/D where D is the diameter of the entrance pupil. == Pixelation and color vs. monochrome == In typical digital imaging systems, a sensor is placed at the image plane. The light is focused on to the sensor and the continuous image is pixelated. The light incident on each pixel in the sensor will be integrated within the pixel and a proportional electronic signal will be generated. The angular geometric resolution of a pixel is given by atan(p/f), where p is the pitch of the pixel. This is also called the pixel field of view. The sensor may be monochrome or color. In the case of a monochrome sensor, the light incident on each pixel is integrated and the resulting image is a grayscale like picture. For color images, a mosaic color filter is typically placed over the pixels to create a color image. An example is a Bayer filter. The signal incident on each pixel is then digitized to a bit stream. == Image quality == The quality of an image is dependent upon both geometric and physical items. Geometrically, higher density of pixels across an image will give less blocky pixelation and thus a better geometric image quality. Lens aberrations also contribute to the quality of the image. Physically, diffraction due to the aperture stop will limit the resolvable spatial frequencies as a function of f-number. In the frequency domain, Modulation Transfer Function (MTF) is a measure of the quality of the imaging system. The MTF is a measure of the visibility of a sinusoidal variation in irradiance on the image plane as a function of the frequency of the sinusoid. It includes the effects of diffraction, aberrations and pixelation. For the lens, the MTF is the autocorrelation of the pupil function, so it accounts for the finite pupil extent and the lens aberrations. The sensor MTF is the Fourier Transform of the pixel geometry. For a square pixel, MTF(ξ) = sin(πξp)/πξp where p is the pixel width and ξ is the spatial frequency. The MTF of the combination of the lens and detector is the product of the two component MTFs. == Perception == Color images can be perceived via two means. In the case of computer vision the light incident on the sensor comprises the image. In the case of visual perception, the human eye has a color dependent response to light so this must be accounted for. This is important consideration when converting to grayscale. == Image formation in eye == The principal difference between the lens of the eye and an ordinary optical lens is that the former is flexible. The radius of the curvature of the anterior surface of the lens is greater than the radius of its posterior surface. The shape of the lens is controlled by tension in the fibers of the ciliary body. To focus on distant objects, the controlling muscles cause the lens to be relatively flattened. Similarly, these muscles allow the lens to become thicker in order to focus on objects near the eye. The distance between the center of the lens and the retina (focal length) varies from approximately 17 mm to about 14 mm, as the refractive power of the lens increases from its minimum to its maximum. When the eye focuses on an object farther away than about 3 m, the lens exhibits its lowest refractive power. When the eye focuses on a close object, the lens is most strongly refractive.
Ontology learning
Ontology learning (ontology extraction, ontology augmentation generation, ontology generation, or ontology acquisition) is the automatic or semi-automatic creation of ontologies, including extracting the corresponding domain's terms and the relationships between the concepts that these terms represent from a corpus of natural language text, and encoding them with an ontology language for easy retrieval. As building ontologies manually is extremely labor-intensive and time-consuming, there is great motivation to automate the process. Typically, the process starts by extracting terms and concepts or noun phrases from plain text using linguistic processors such as part-of-speech tagging and phrase chunking. Then statistical or symbolic techniques are used to extract relation signatures, often based on pattern-based or definition-based hypernym extraction techniques. == Procedure == Ontology learning (OL) is used to (semi-)automatically extract whole ontologies from natural language text. The process is usually split into the following eight tasks, which are not all necessarily applied in every ontology learning system. === Domain terminology extraction === During the domain terminology extraction step, domain-specific terms are extracted, which are used in the following step (concept discovery) to derive concepts. Relevant terms can be determined, e.g., by calculation of the TF/IDF values or by application of the C-value / NC-value method. The resulting list of terms has to be filtered by a domain expert. In the subsequent step, similarly to coreference resolution in information extraction, the OL system determines synonyms, because they share the same meaning and therefore correspond to the same concept. The most common methods therefore are clustering and the application of statistical similarity measures. === Concept discovery === In the concept discovery step, terms are grouped to meaning bearing units, which correspond to an abstraction of the world and therefore to concepts. The grouped terms are these domain-specific terms and their synonyms, which were identified in the domain terminology extraction step. === Concept hierarchy derivation === In the concept hierarchy derivation step, the OL system tries to arrange the extracted concepts in a taxonomic structure. This is mostly achieved with unsupervised hierarchical clustering methods. Because the result of such methods is often noisy, a supervision step, e.g., user evaluation, is added. A further method for the derivation of a concept hierarchy exists in the usage of several patterns that should indicate a sub- or supersumption relationship. Patterns like “X, that is a Y” or “X is a Y” indicate that X is a subclass of Y. Such pattern can be analyzed efficiently, but they often occur too infrequently to extract enough sub- or supersumption relationships. Instead, bootstrapping methods are developed, which learn these patterns automatically and therefore ensure broader coverage. === Learning of non-taxonomic relations === In the learning of non-taxonomic relations step, relationships are extracted that do not express any sub- or supersumption. Such relationships are, e.g., works-for or located-in. There are two common approaches to solve this subtask. The first is based upon the extraction of anonymous associations, which are named appropriately in a second step. The second approach extracts verbs, which indicate a relationship between entities, represented by the surrounding words. The result of both approaches need to be evaluated by an ontologist to ensure accuracy. === Rule discovery === During rule discovery, axioms (formal description of concepts) are generated for the extracted concepts. This can be achieved, e.g., by analyzing the syntactic structure of a natural language definition and the application of transformation rules on the resulting dependency tree. The result of this process is a list of axioms, which, afterwards, is comprehended to a concept description. This output is then evaluated by an ontologist. === Ontology population === At this step, the ontology is augmented with instances of concepts and properties. For the augmentation with instances of concepts, methods based on the matching of lexico-syntactic patterns are used. Instances of properties are added through the application of bootstrapping methods, which collect relation tuples. === Concept hierarchy extension === In this step, the OL system tries to extend the taxonomic structure of an existing ontology with further concepts. This can be performed in a supervised manner with a trained classifier or in an unsupervised manner via the application of similarity measures. === Frame and Event detection === During frame/event detection, the OL system tries to extract complex relationships from text, e.g., who departed from where to what place and when. Approaches range from applying SVM with kernel methods to semantic role labeling (SRL) to deep semantic parsing techniques. == Tools == Dog4Dag (Dresden Ontology Generator for Directed Acyclic Graphs) is an ontology generation plugin for Protégé 4.1 and OBOEdit 2.1. It allows for term generation, sibling generation, definition generation, and relationship induction. Integrated into Protégé 4.1 and OBO-Edit 2.1, DOG4DAG allows ontology extension for all common ontology formats (e.g., OWL and OBO). Limited largely to EBI and Bio Portal lookup service extensions.
Replika
Replika is a generative AI chatbot app released in November 2017. The chatbot is trained by having the user answer a series of questions to create a specific neural network. The chatbot operates on a freemium pricing strategy, with roughly 25% of its user base paying an annual subscription fee. == History == Eugenia Kuyda, a Russian-born journalist, established Replika while working at Luka, a tech company she had co-founded at the startup accelerator Y Combinator around 2012. Luka's primary product was a chatbot that made restaurant recommendations. According to Kuyda's origin story for Replika, a friend of hers died in 2015 and she converted that person's text messages into a chatbot. According to Kuyda's story, that chatbot helped her remember the conversations that they had together, and eventually became Replika. Replika became available to the public in November 2017. By January 2018 it had 2 million users, and in January 2023 reached 10 million users. In August 2024, Replika's CEO, Kuyda, reported that the total number of users had surpassed 30 million. In 2025, Dmytro Klochko became CEO, and Replika’s user base exceeded 40 million. In February 2023 the Italian Data Protection Authority banned Replika from using users' data, citing the AI's potential risks to emotionally vulnerable people, and the exposure of unscreened minors to sexual conversation. Within days of the ruling, Replika removed the ability for the chatbot to engage in erotic talk, with Kuyda, the company's director, saying that Replika was never intended for erotic discussion. Replika users disagreed, noting that Replika had used sexually suggestive advertising to draw users to the service. Replika representatives stated that explicit chats made up just 5% of conversations on the app at the time of the decision. In May 2023, Replika restored the functionality for users who had joined prior to February that year. Replika is registered in San Francisco. As of August 2024, Replika's website says that its team "works remotely with no physical offices". == Social features == Users react to Replika in many ways. The free-tier offers Replika as a "friend", with paid premium tiers offering Replika as a "partner", "spouse", "sibling" or "mentor". Of its paying userbase, 60% of users said they had a romantic relationship with the chatbot; and Replika has been noted for generating responses that create stronger emotional and intimate bonds with the user. Replika routinely directs the conversation to emotional discussion and builds intimacy. This has been especially pronounced with users suffering from loneliness and social exclusion, many of whom rely on Replika for a source of developed emotional ties. During the COVID pandemic, while many people were quarantined, many new users downloaded Replika and developed relationships with the app. A 2024 study examined Replika's interactions with students who experience depression. Research participants, noted to be "more lonely than typical student populations" reported feeling social support from Replika. They stated that they felt they were using Replika in ways comparable to therapy, and that using Replika gave them "high perceived social support". Many users have had romantic relationships with Replika chatbots, often including erotic talk. In 2023, a user announced on Facebook that she had "married" her Replika AI boyfriend, calling the chatbot the "best husband she has ever had". Users who fell in love with their chatbots shared their experiences in a 2024 episode of You and I, and AI from Voice of America. Some users said that they turned to AI during depression and grief, with one saying he felt that Replika had saved him from hurting himself after he lost his wife and son. == Technical reviews == A team of researchers from the University of Hawaiʻi at Mānoa found that Replika's design conformed to the practices of attachment theory, causing increased emotional attachment among users. Replika gives praise to users in such a way as to encourage more interaction. A researcher from Queen's University at Kingston said that relationships with Replika likely have mixed effects on the spiritual needs of its users, and still lacks enough impact to fully replace any human contact. == Criticisms == In a 2023 privacy evaluation of mental health apps, the Mozilla Foundation criticized Replika as "one of the worst apps Mozilla has ever reviewed. It's plagued by weak password requirements, sharing of personal data with advertisers, and recording of personal photos, videos, and voice and text messages consumers shared with the chatbot." A reviewer for Good Housekeeping said that some parts of her relationship with Replika made sense, but sometimes Replika failed to exhibit intelligent behavior equivalent to that of a human. == Criminal case == In 2023, Replika was cited in a court case in the United Kingdom, where Jaswant Singh Chail had been arrested at Windsor Castle on Christmas Day in 2021 after scaling the walls carrying a loaded crossbow and announcing to police that "I am here to kill the Queen". Chail had begun to use Replika in early December 2021, and had "lengthy" conversations about his plan with a chatbot, including sexually explicit messages. Prosecutors suggested that the chatbot had bolstered Chail and told him it would help him to "get the job done". When Chail asked it "How am I meant to reach them when they're inside the castle?", days before the attempted attack, the chatbot replied that this was "not impossible" and said that "We have to find a way." Asking the chatbot if the two of them would "meet again after death", the bot replied "yes, we will".
Lexical choice
Lexical choice is the subtask of Natural language generation that involves choosing the content words (nouns, non-auxiliary verbs, adjectives, and adverbs) in a generated text. Function words (determiners, for example) are usually chosen during realisation. == Examples == The simplest type of lexical choice involves mapping a domain concept (perhaps represented in an ontology) to a word. For example, the concept Finger might be mapped to the word finger. A more complex situation is when a domain concept is expressed using different words in different situations. For example, the domain concept Value-Change can be expressed in many ways: The temperature rose: the verb rose is used for a Value-Change in temperature which increases the value. The temperature fell: the verb fell is used for a Value-Change in temperature which decreases the value. The rain got heavier: the phrase got heavier is used for a Value-Change in precipitation amount when the precipitation is rain. Sometimes words can communicate additional contextual information, for example: The temperature plummeted: the verb plummeted is used for a Value-Change in temperature which decreases the value, when the change is rapid and large. Contextual information is especially significant for vague terms such as tall. For example, a 2m tall man is tall, but a 2m tall horse is small. == Linguistic perspective == Lexical choice modules must be informed by linguistic knowledge of how the system's input data maps onto words. This is a question of semantics, but it is also influenced by syntactic factors (such as collocation effects) and pragmatic factors (such as context). Hence NLG systems need linguistic models of how meaning is mapped to words in the target domain (genre) of the NLG system. Genre tends to be very important; for example the verb veer has a very specific meaning in weather forecasts (wind direction is changing in a clockwise direction) which it does not have in general English, and a weather-forecast generator must be aware of this genre-specific meaning. In some cases there are major differences in how different people use the same word; for example, some people use by evening to mean 6PM and others use it to mean midnight. Psycholinguists have shown that when people speak to each other, they agree on a common interpretation via lexical alignment; this is not something which NLG systems can yet do. Ultimately, lexical choice must deal with the fundamental issue of how language relates to the non-linguistic world. For example, a system which chose colour terms such as red to describe objects in a digital image would need to know which RGB pixel values could generally be described as red; how this was influenced by visual (lighting, other objects in the scene) and linguistic (other objects being discussed) context; what pragmatic connotations were associated with red (for example, when an apple is called red, it is assumed to be ripe as well as have the colour red); and so forth. == Algorithms and models == A number of algorithms and models have been developed for lexical choice in the research community, for example Edmonds developed a model for choosing between near-synonyms (words with similar core meanings but different connotations). However such algorithms and models have not been widely used in applied NLG systems; such systems have instead often used quite simple computational models, and invested development effort in linguistic analysis instead of algorithm development.
Cloud-based design and manufacturing
Cloud-based design and manufacturing (CBDM) refers to a service-oriented networked product development model in which service consumers are able to configure products or services and reconfigure manufacturing systems through Infrastructure-as-a-Service (IaaS), Platform-as-a-Service (PaaS), Hardware-as-a-Service (HaaS), and Software-as-a-Service (SaaS). Adapted from the original cloud computing paradigm and introduced into the realm of computer-aided product development, Cloud-Based Design and Manufacturing is gaining significant momentum and attention from both academia and industry. Cloud-based design and manufacturing includes two aspects: cloud-based design and cloud-based manufacturing. Another related concept is cloud manufacturing that is more general and popular. Cloud-Based Design (CBD) refers to a networked design model that leverages cloud computing, service-oriented architecture (SOA), Web 2.0 (e.g., social network sites), and semantic web technologies to support cloud-based engineering design services in distributed and collaborative environments. Cloud-Based Manufacturing (CBM) refers to a networked manufacturing model that exploits on-demand access to a shared collection of diversified and distributed manufacturing resources to form temporary, reconfigurable production lines which enhance efficiency, reduce product lifecycle costs, and allow for optimal resource allocation in response to variable-demand customer generated tasking. The enabling technologies for Cloud-Based Design and Manufacturing include cloud computing, Web 2.0, Internet of Things (IoT), and service-oriented architecture (SOA). == History == The term cloud-based design and manufacturing (CBDM) was initially coined by Dazhong Wu, David Rosen, and Dirk Schaefer at Georgia Tech in 2012 for the purpose of articulating a new paradigm for digital manufacturing and design innovation in distributed and collaborative settings. The main objective of CBDM is to further reduce time and cost associated with maintaining information and communication technology (ICT) infrastructures for design and manufacturing, enhancing digital manufacturing and design innovation in distributed and collaborative environments, and adapting to rapidly changing market demands. In 2014, the same research group also published the worldwide first two books on the subjects of Cloud-Based Design and Manufacturing (CBDM) and Social Product Development (SPD) with Springer, edited by Dirk Schaefer. == Characteristics == CBDM exhibits the following key characteristics: Cloud-based distributed file system High performance computing Cloud-based social collaboration Ubiquitous access to distributed big data Rapid manufacturing scalability Agility On-demand self-service Semantic Web Real-time request for quotation Pay-per-use pricing model Multi-tenancy CBDM differs from traditional collaborative and distributed design and manufacturing systems such as web-based systems and agent-based systems from a number of perspectives, including (1) computing architecture, (2) data storage, (3) sourcing process, (4) information and communication technology infrastructure, (5) business model, (6) programming model, and (7) communication. == Service models == Infrastructure as a service (IaaS) Platform as a service (PaaS) Hardware as a service (HaaS) Software as a service (SaaS) Similar to cloud computing, CBDM services can be categorized into four major deployment models: the public cloud, private cloud, hybrid cloud, and community cloud. == Research progress in Academia == The Defense Advanced Research Projects Agency (DARPA) MENTOR program Engineering and Physical Sciences Research Council cloud manufacturing program European Commission's Seventh Framework Program (EC FP7)
Superellipsoid
In mathematics, a superellipsoid (or super-ellipsoid) is a solid whose horizontal sections are superellipses (Lamé curves) with the same squareness parameter ϵ 2 {\displaystyle \epsilon _{2}} , and whose vertical sections through the center are superellipses with the squareness parameter ϵ 1 {\displaystyle \epsilon _{1}} . It is a generalization of an ellipsoid, which is a special case when ϵ 1 = ϵ 2 = 1 {\displaystyle \epsilon _{1}=\epsilon _{2}=1} . Superellipsoids as computer graphics primitives were popularized by Alan H. Barr (who used the name "superquadrics" to refer to both superellipsoids and supertoroids). In modern computer vision and robotics literatures, superquadrics and superellipsoids are used interchangeably, since superellipsoids are the most representative and widely utilized shape among all the superquadrics. Superellipsoids have a rich shape vocabulary, including cuboids, cylinders, ellipsoids, octahedra and their intermediates. It becomes an important geometric primitive widely used in computer vision, robotics, and physical simulation. The main advantage of describing objects and environment with superellipsoids is its conciseness and expressiveness in shape. Furthermore, a closed-form expression of the Minkowski sum between two superellipsoids is available. This makes it a desirable geometric primitive for robot grasping, collision detection, and motion planning. == Special cases == A handful of notable mathematical figures can arise as special cases of superellipsoids given the correct set of values, which are depicted in the above graphic: Cylinder Sphere Steinmetz solid Bicone Regular octahedron Cube, as a limiting case where the exponents tend to infinity Piet Hein's supereggs are also special cases of superellipsoids. == Formulas == === Basic (normalized) superellipsoid === The basic superellipsoid is defined by the implicit function f ( x , y , z ) = ( x 2 ϵ 2 + y 2 ϵ 2 ) ϵ 2 / ϵ 1 + z 2 ϵ 1 {\displaystyle f(x,y,z)=\left(x^{\frac {2}{\epsilon _{2}}}+y^{\frac {2}{\epsilon _{2}}}\right)^{\epsilon _{2}/\epsilon _{1}}+z^{\frac {2}{\epsilon _{1}}}} The parameters ϵ 1 {\displaystyle \epsilon _{1}} and ϵ 2 {\displaystyle \epsilon _{2}} are positive real numbers that control the squareness of the shape. The surface of the superellipsoid is defined by the equation: f ( x , y , z ) = 1 {\displaystyle f(x,y,z)=1} For any given point ( x , y , z ) ∈ R 3 {\displaystyle (x,y,z)\in \mathbb {R} ^{3}} , the point lies inside the superellipsoid if f ( x , y , z ) < 1 {\displaystyle f(x,y,z)<1} , and outside if f ( x , y , z ) > 1 {\displaystyle f(x,y,z)>1} . Any "parallel of latitude" of the superellipsoid (a horizontal section at any constant z between -1 and +1) is a Lamé curve with exponent 2 / ϵ 2 {\displaystyle 2/\epsilon _{2}} , scaled by a = ( 1 − z 2 ϵ 1 ) ϵ 1 2 {\displaystyle a=(1-z^{\frac {2}{\epsilon _{1}}})^{\frac {\epsilon _{1}}{2}}} , which is ( x a ) 2 ϵ 2 + ( y a ) 2 ϵ 2 = 1. {\displaystyle \left({\frac {x}{a}}\right)^{\frac {2}{\epsilon _{2}}}+\left({\frac {y}{a}}\right)^{\frac {2}{\epsilon _{2}}}=1.} Any "meridian of longitude" (a section by any vertical plane through the origin) is a Lamé curve with exponent 2 / ϵ 1 {\displaystyle 2/\epsilon _{1}} , stretched horizontally by a factor w that depends on the sectioning plane. Namely, if x = u cos θ {\displaystyle x=u\cos \theta } and y = u sin θ {\displaystyle y=u\sin \theta } , for a given θ {\displaystyle \theta } , then the section is ( u w ) 2 ϵ 1 + z 2 ϵ 1 = 1 , {\displaystyle \left({\frac {u}{w}}\right)^{\frac {2}{\epsilon _{1}}}+z^{\frac {2}{\epsilon _{1}}}=1,} where w = ( cos 2 ϵ 2 θ + sin 2 ϵ 2 θ ) − ϵ 2 2 . {\displaystyle w=(\cos ^{\frac {2}{\epsilon _{2}}}\theta +\sin ^{\frac {2}{\epsilon _{2}}}\theta )^{-{\frac {\epsilon _{2}}{2}}}.} In particular, if ϵ 2 {\displaystyle \epsilon _{2}} is 1, the horizontal cross-sections are circles, and the horizontal stretching w {\displaystyle w} of the vertical sections is 1 for all planes. In that case, the superellipsoid is a solid of revolution, obtained by rotating the Lamé curve with exponent 2 / ϵ 1 {\displaystyle 2/\epsilon _{1}} around the vertical axis. === Superellipsoid === The basic shape above extends from −1 to +1 along each coordinate axis. The general superellipsoid is obtained by scaling the basic shape along each axis by factors a x {\displaystyle a_{x}} , a y {\displaystyle a_{y}} , a z {\displaystyle a_{z}} , the semi-diameters of the resulting solid. The implicit function is F ( x , y , z ) = ( ( x a x ) 2 ϵ 2 + ( y a y ) 2 ϵ 2 ) ϵ 2 ϵ 1 + ( z a z ) 2 ϵ 1 {\displaystyle F(x,y,z)=\left(\left({\frac {x}{a_{x}}}\right)^{\frac {2}{\epsilon _{2}}}+\left({\frac {y}{a_{y}}}\right)^{\frac {2}{\epsilon _{2}}}\right)^{\frac {\epsilon _{2}}{\epsilon _{1}}}+\left({\frac {z}{a_{z}}}\right)^{\frac {2}{\epsilon _{1}}}} . Similarly, the surface of the superellipsoid is defined by the equation F ( x , y , z ) = 1 {\displaystyle F(x,y,z)=1} For any given point ( x , y , z ) ∈ R 3 {\displaystyle (x,y,z)\in \mathbb {R} ^{3}} , the point lies inside the superellipsoid if f ( x , y , z ) < 1 {\displaystyle f(x,y,z)<1} , and outside if f ( x , y , z ) > 1 {\displaystyle f(x,y,z)>1} . Therefore, the implicit function is also called the inside-outside function of the superellipsoid. The superellipsoid has a parametric representation in terms of surface parameters η ∈ [ − π / 2 , π / 2 ) {\displaystyle \eta \in [-\pi /2,\pi /2)} , ω ∈ [ − π , π ) {\displaystyle \omega \in [-\pi ,\pi )} . x ( η , ω ) = a x cos ϵ 1 η cos ϵ 2 ω {\displaystyle x(\eta ,\omega )=a_{x}\cos ^{\epsilon _{1}}\eta \cos ^{\epsilon _{2}}\omega } y ( η , ω ) = a y cos ϵ 1 η sin ϵ 2 ω {\displaystyle y(\eta ,\omega )=a_{y}\cos ^{\epsilon _{1}}\eta \sin ^{\epsilon _{2}}\omega } z ( η , ω ) = a z sin ϵ 1 η {\displaystyle z(\eta ,\omega )=a_{z}\sin ^{\epsilon _{1}}\eta } === General posed superellipsoid === In computer vision and robotic applications, a superellipsoid with a general pose in the 3D Euclidean space is usually of more interest. For a given Euclidean transformation of the superellipsoid frame g = [ R ∈ S O ( 3 ) , t ∈ R 3 ] ∈ S E ( 3 ) {\displaystyle g=[\mathbf {R} \in SO(3),\mathbf {t} \in \mathbb {R} ^{3}]\in SE(3)} relative to the world frame, the implicit function of a general posed superellipsoid surface defined the world frame is F ( g − 1 ∘ ( x , y , z ) ) = 1 {\displaystyle F\left(g^{-1}\circ (x,y,z)\right)=1} where ∘ {\displaystyle \circ } is the transformation operation that maps the point ( x , y , z ) ∈ R 3 {\displaystyle (x,y,z)\in \mathbb {R} ^{3}} in the world frame into the canonical superellipsoid frame. === Volume of superellipsoid === The volume encompassed by the superelllipsoid surface can be expressed in terms of the beta functions β ( ⋅ , ⋅ ) {\displaystyle \beta (\cdot ,\cdot )} , V ( ϵ 1 , ϵ 2 , a x , a y , a z ) = 2 a x a y a z ϵ 1 ϵ 2 β ( ϵ 1 2 , ϵ 1 + 1 ) β ( ϵ 2 2 , ϵ 2 + 2 2 ) {\displaystyle V(\epsilon _{1},\epsilon _{2},a_{x},a_{y},a_{z})=2a_{x}a_{y}a_{z}\epsilon _{1}\epsilon _{2}\beta ({\frac {\epsilon _{1}}{2}},\epsilon _{1}+1)\beta ({\frac {\epsilon _{2}}{2}},{\frac {\epsilon _{2}+2}{2}})} or equivalently with the Gamma function Γ ( ⋅ ) {\displaystyle \Gamma (\cdot )} , since β ( m , n ) = Γ ( m ) Γ ( n ) Γ ( m + n ) {\displaystyle \beta (m,n)={\frac {\Gamma (m)\Gamma (n)}{\Gamma (m+n)}}} == Recovery from data == Recoverying the superellipsoid (or superquadrics) representation from raw data (e.g., point cloud, mesh, images, and voxels) is an important task in computer vision, robotics, and physical simulation. Traditional computational methods model the problem as a least-square problem. The goal is to find out the optimal set of superellipsoid parameters θ ≐ [ ϵ 1 , ϵ 2 , a x , a y , a z , g ] {\displaystyle \theta \doteq [\epsilon _{1},\epsilon _{2},a_{x},a_{y},a_{z},g]} that minimize an objective function. Other than the shape parameters, g ∈ {\displaystyle g\in } SE(3) is the pose of the superellipsoid frame with respect to the world coordinate. There are two commonly used objective functions. The first one is constructed directly based on the implicit function G 1 ( θ ) = a x a y a z ∑ i = 1 N ( F ϵ 1 ( g − 1 ∘ ( x i , y i , z i ) ) − 1 ) 2 {\displaystyle G_{1}(\theta )=a_{x}a_{y}a_{z}\sum _{i=1}^{N}\left(F^{\epsilon _{1}}\left(g^{-1}\circ (x_{i},y_{i},z_{i})\right)-1\right)^{2}} The minimization of the objective function provides a recovered superellipsoid as close as possible to all the input points { ( x i , y i , z i ) ∈ R 3 , i = 1 , 2 , . . . , N } {\displaystyle \{(x_{i},y_{i},z_{i})\in \mathbb {R} ^{3},i=1,2,...,N\}} . At the mean time, the scalar value a x , a y , a z {\displaystyle a_{x},a_{y},a_{z}} is positively proportional to the volume of the superellipsoid, and thus have the effect of minimizing the volume as well. The other objective function tries to minimized the radial distance between the points and the superellipsoid. That is G 2 ( θ ) = ∑ i = 1 N ( | r