A distributed transaction operates within a distributed environment, typically involving multiple nodes across a network depending on the location of the data. A key aspect of distributed transactions is atomicity, which ensures that the transaction is completed in its entirety or not executed at all. It's essential to note that distributed transactions are not limited to databases. The Open Group, a vendor consortium, proposed the X/Open Distributed Transaction Processing Model (X/Open XA), which became a de facto standard for the behavior of transaction model components. Databases are common transactional resources and, often, transactions span a couple of such databases. In this case, a distributed transaction can be seen as a database transaction that must be synchronized (or provide ACID properties) among multiple participating databases which are distributed among different physical locations. The isolation property (the I of ACID) poses a special challenge for multi database transactions, since the (global) serializability property could be violated, even if each database provides it (see also global serializability). In practice most commercial database systems use strong strict two-phase locking (SS2PL) for concurrency control, which ensures global serializability, if all the participating databases employ it. A common algorithm for ensuring correct completion of a distributed transaction is the two-phase commit (2PC). This algorithm is usually applied for updates able to commit in a short period of time, ranging from couple of milliseconds to couple of minutes. There are also long-lived distributed transactions, for example a transaction to book a trip, which consists of booking a flight, a rental car and a hotel. Since booking the flight might take up to a day to get a confirmation, two-phase commit is not applicable here, it will lock the resources for this long. In this case more sophisticated techniques that involve multiple undo levels are used. The way you can undo the hotel booking by calling a desk and cancelling the reservation, a system can be designed to undo certain operations (unless they are irreversibly finished). In practice, long-lived distributed transactions are implemented in systems based on web services. Usually these transactions utilize principles of compensating transactions, Optimism and Isolation Without Locking. The X/Open standard does not cover long-lived distributed transactions. Several technologies, including Jakarta Enterprise Beans and Microsoft Transaction Server fully support distributed transaction standards. == Synchronization == In event-driven architectures, distributed transactions can be synchronized through using request–response paradigm and it can be implemented in two ways: Creating two separate queues: one for requests and the other for replies. The event producer must wait until it receives the response. Creating one dedicated ephemeral queue for each request.
ELMo
ELMo (embeddings from language model) is a word embedding method for representing a sequence of words as a corresponding sequence of vectors. It was created by researchers at the Allen Institute for Artificial Intelligence, and University of Washington and first released in February 2018. It is a bidirectional LSTM which takes character-level as inputs and produces word-level embeddings, trained on a corpus of about 30 million sentences and 1 billion words. The architecture of ELMo accomplishes a contextual understanding of tokens. Deep contextualized word representation is useful for many natural language processing tasks, such as coreference resolution and polysemy resolution. ELMo was historically important as a pioneer of self-supervised generative pretraining followed by fine-tuning, where a large model is trained to reproduce a large corpus, then the large model is augmented with additional task-specific weights and fine-tuned on supervised task data. It was an instrumental step in the evolution towards transformer-based language modelling. == Architecture == ELMo is a multilayered bidirectional LSTM on top of a token embedding layer. The output of all LSTMs concatenated together consists of the token embedding. The input text sequence is first mapped by an embedding layer into a sequence of vectors. Then two parts are run in parallel over it. The forward part is a 2-layered LSTM with 4096 units and 512 dimension projections, and a residual connection from the first to second layer. The backward part has the same architecture, but processes the sequence back-to-front. The outputs from all 5 components (embedding layer, two forward LSTM layers, and two backward LSTM layers) are concatenated and multiplied by a linear matrix ("projection matrix") to produce a 512-dimensional representation per input token. ELMo was pretrained on a text corpus of 1 billion words. The forward part is trained by repeatedly predicting the next token, and the backward part is trained by repeatedly predicting the previous token. After the ELMo model is pretrained, its parameters are frozen, except for the projection matrix, which can be fine-tuned to minimize loss on specific language tasks. This is an early example of the pretraining-fine-tune paradigm. The original paper demonstrated this by improving state of the art on six benchmark NLP tasks. === Contextual word representation === The architecture of ELMo accomplishes a contextual understanding of tokens. For example, the first forward LSTM of ELMo would process each input token in the context of all previous tokens, and the first backward LSTM would process each token in the context of all subsequent tokens. The second forward LSTM would then incorporate those to further contextualize each token. Deep contextualized word representation is useful for many natural language processing tasks, such as coreference resolution and polysemy resolution. For example, consider the sentenceShe went to the bank to withdraw money.In order to represent the token "bank", the model must resolve its polysemy in context. The first forward LSTM would process "bank" in the context of "She went to the", which would allow it to represent the word to be a location that the subject is going towards. The first backward LSTM would process "bank" in the context of "to withdraw money", which would allow it to disambiguate the word as referring to a financial institution. The second forward LSTM can then process "bank" using the representation vector provided by the first backward LSTM, thus allowing it to represent it to be a financial institution that the subject is going towards. == Historical context == ELMo is one link in a historical evolution of language modelling. Consider a simple problem of document classification, where we want to assign a label (e.g., "spam", "not spam", "politics", "sports") to a given piece of text. The simplest approach is the "bag of words" approach, where each word in the document is treated independently, and its frequency is used as a feature for classification. This was computationally cheap but ignored the order of words and their context within the sentence. GloVe and Word2Vec built upon this by learning fixed vector representations (embeddings) for words based on their co-occurrence patterns in large text corpora. Like BERT (but unlike "bag of words" such as Word2Vec and GloVe), ELMo word embeddings are context-sensitive, producing different representations for words that share the same spelling. It was trained on a corpus of about 30 million sentences and 1 billion words. Previously, bidirectional LSTM was used for contextualized word representation. ELMo applied the idea to a large scale, achieving state of the art performance. After the 2017 publication of Transformer architecture, the architecture of ELMo was changed from a multilayered bidirectional LSTM to a Transformer encoder, giving rise to BERT. BERT has a similar pretrain-fine-tune workflow, but uses a Transformer with implications for more parallelizable training.
LanguageWare
LanguageWare is a natural language processing (NLP) technology developed by IBM, which allows applications to process natural language text. It comprises a set of Java libraries that provide a range of NLP functions: language identification, text segmentation/tokenization, normalization, entity and relationship extraction, and semantic analysis and disambiguation. The analysis engine uses a finite-state machine approach at multiple levels, which aids its performance characteristics while maintaining a reasonably small footprint. The behaviour of the system is driven by a set of configurable lexico-semantic resources which describe the characteristics and domain of the processed language. A default set of resources comes as part of LanguageWare and these describe the native language characteristics, such as morphology, and the basic vocabulary for the language. Supplemental resources have been created that capture additional vocabularies, terminologies, rules and grammars, which may be generic to the language or specific to one or more domains. A set of Eclipse-based customization tooling, LanguageWare Resource Workbench, is available on IBM's alphaWorks site, and allows domain knowledge to be compiled into these resources and thereby incorporated into the analysis process. LanguageWare can be deployed as a set of UIMA-compliant annotators, Eclipse plug-ins or Web Services.
Speech segmentation
Speech segmentation is the process of identifying the boundaries between words, syllables, or phonemes in spoken natural languages. The term applies both to the mental processes used by humans, and to artificial processes of natural language processing. In the field of automatic pronunciation assessment, the process of segmenting an utterance against expected word(s) is called forced alignment. Speech segmentation is a subfield of general speech perception and an important subproblem of the technologically focused field of speech recognition, and cannot be adequately solved in isolation. As in most natural language processing problems, one must take into account context, grammar, and semantics, and even so the result is often a probabilistic division (statistically based on likelihood) rather than a categorical one. Though it seems that coarticulation—a phenomenon which may happen between adjacent words just as easily as within a single word—presents the main challenge in speech segmentation across languages, some other problems and strategies employed in solving those problems can be seen in the following sections. This problem overlaps to some extent with the problem of text segmentation that occurs in some languages which are traditionally written without inter-word spaces, like Chinese and Japanese, compared to writing systems which indicate speech segmentation between words by a word divider, such as the space. However, even for those languages, text segmentation is often much easier than speech segmentation, because the written language usually has little interference between adjacent words, and often contains additional clues not present in speech (such as the use of Chinese characters for word stems in Japanese). == Lexical recognition == In natural languages, the meaning of a complex spoken sentence can be understood by decomposing it into smaller lexical segments (roughly, the words of the language), associating a meaning to each segment, and combining those meanings according to the grammar rules of the language. Though lexical recognition is not thought to be used by infants in their first year, due to their highly limited vocabularies, it is one of the major processes involved in speech segmentation for adults. Three main models of lexical recognition exist in current research: first, whole-word access, which argues that words have a whole-word representation in the lexicon; second, decomposition, which argues that morphologically complex words are broken down into their morphemes (roots, stems, inflections, etc.) and then interpreted and; third, the view that whole-word and decomposition models are both used, but that the whole-word model provides some computational advantages and is therefore dominant in lexical recognition. To give an example, in a whole-word model, the word "cats" might be stored and searched for by letter, first "c", then "ca", "cat", and finally "cats". The same word, in a decompositional model, would likely be stored under the root word "cat" and could be searched for after removing the "s" suffix. "Falling", similarly, would be stored as "fall" and suffixed with the "ing" inflection. Though proponents of the decompositional model recognize that a morpheme-by-morpheme analysis may require significantly more computation, they argue that the unpacking of morphological information is necessary for other processes (such as syntactic structure) which may occur parallel to lexical searches. As a whole, research into systems of human lexical recognition is limited due to little experimental evidence that fully discriminates between the three main models. In any case, lexical recognition likely contributes significantly to speech segmentation through the contextual clues it provides, given that it is a heavily probabilistic system—based on the statistical likelihood of certain words or constituents occurring together. For example, one can imagine a situation where a person might say "I bought my dog at a ____ shop" and the missing word's vowel is pronounced as in "net", "sweat", or "pet". While the probability of "netshop" is extremely low, since "netshop" isn't currently a compound or phrase in English, and "sweatshop" also seems contextually improbable, "pet shop" is a good fit because it is a common phrase and is also related to the word "dog". Moreover, an utterance can have different meanings depending on how it is split into words. A popular example, often quoted in the field, is the phrase "How to wreck a nice beach", which sounds very similar to "How to recognize speech". As this example shows, proper lexical segmentation depends on context and semantics which draws on the whole of human knowledge and experience, and would thus require advanced pattern recognition and artificial intelligence technologies to be implemented on a computer. Lexical recognition is of particular value in the field of computer speech recognition, since the ability to build and search a network of semantically connected ideas would greatly increase the effectiveness of speech-recognition software. Statistical models can be used to segment and align recorded speech to words or phones. Applications include automatic lip-synch timing for cartoon animation, follow-the-bouncing-ball video sub-titling, and linguistic research. Automatic segmentation and alignment software is commercially available. == Phonotactic cues == For most spoken languages, the boundaries between lexical units are difficult to identify; phonotactics are one answer to this issue. One might expect that the inter-word spaces used by many written languages like English or Spanish would correspond to pauses in their spoken version, but that is true only in very slow speech, when the speaker deliberately inserts those pauses. In normal speech, one typically finds many consecutive words being said with no pauses between them, and often the final sounds of one word blend smoothly or fuse with the initial sounds of the next word. The notion that speech is produced like writing, as a sequence of distinct vowels and consonants, may be a relic of alphabetic heritage for some language communities. In fact, the way vowels are produced depends on the surrounding consonants just as consonants are affected by surrounding vowels; this is called coarticulation. For example, in the word "kit", the [k] is farther forward than when we say 'caught'. But also, the vowel in "kick" is phonetically different from the vowel in "kit", though we normally do not hear this. In addition, there are language-specific changes which occur in casual speech which makes it quite different from spelling. For example, in English, the phrase "hit you" could often be more appropriately spelled "hitcha". From a decompositional perspective, in many cases, phonotactics play a part in letting speakers know where to draw word boundaries. In English, the word "strawberry" is perceived by speakers as consisting (phonetically) of two parts: "straw" and "berry". Other interpretations such as "stra" and "wberry" are inhibited by English phonotactics, which does not allow the cluster "wb" word-initially. Other such examples are "day/dream" and "mile/stone" which are unlikely to be interpreted as "da/ydream" or "mil/estone" due to the phonotactic probability or improbability of certain clusters. The sentence "Five women left", which could be phonetically transcribed as [faɪvwɪmɘnlɛft], is marked since neither /vw/ in /faɪvwɪmɘn/ nor /nl/ in /wɪmɘnlɛft/ are allowed as syllable onsets or codas in English phonotactics. These phonotactic cues often allow speakers to easily distinguish the boundaries in words. Vowel harmony in languages like Finnish can also serve to provide phonotactic cues. While the system does not allow front vowels and back vowels to exist together within one morpheme, compounds allow two morphemes to maintain their own vowel harmony while coexisting in a word. Therefore, in compounds such as "selkä/ongelma" ('back problem') where vowel harmony is distinct between two constituents in a compound, the boundary will be wherever the switch in harmony takes place—between the "ä" and the "ö" in this case. Still, there are instances where phonotactics may not aid in segmentation. Words with unclear clusters or uncontrasted vowel harmony as in "opinto/uudistus" ('student reform') do not offer phonotactic clues as to how they are segmented. From the perspective of the whole-word model, however, these words are thought be stored as full words, so the constituent parts would not necessarily be relevant to lexical recognition. == In infants and non-natives == Infants are one major focus of research in speech segmentation. Since infants have not yet acquired a lexicon capable of providing extensive contextual clues or probability-based word searches within their first year, as mentioned above, they must often rely primarily upon phonotactic and rhythmic cues (with prosody being the dominant cue), all
CMU Pronouncing Dictionary
The CMU Pronouncing Dictionary (also known as CMUdict) is an open-source pronouncing dictionary originally created by the Speech Group at Carnegie Mellon University (CMU) for use in speech recognition research. CMUdict provides a mapping orthographic/phonetic for English words in their North American pronunciations. It is commonly used to generate representations for speech recognition (ASR), e.g. the CMU Sphinx system, and speech synthesis (TTS), e.g. the Festival system. CMUdict can be used as a training corpus for building statistical grapheme-to-phoneme (g2p) models that will generate pronunciations for words not yet included in the dictionary. The most recent release is 0.7b; it contains over 134,000 entries. An interactive lookup version is available. == Database format == The database is distributed as a plain text file with one entry to a line in the format "WORD
Boris FX
Boris FX is a visual effects, video editing, photography, and audio software plug-in developer based in Miami, Florida, USA. The developer is known for its flagship products, Continuum (formerly Boris Continuum Complete/BCC), Sapphire, Mocha, and Silhouette. Boris FX creates plug-in tools for feature film, broadcast television, and multimedia post-production workflows. The plug-ins are compatible with various NLEs, including Adobe After Effects and Premiere Pro, Avid Media Composer, Apple Final Cut Pro, and OFX hosts such as Autodesk Flame, Foundry Nuke, Blackmagic Design DaVinci Resolve and Fusion, and VEGAS Pro. Boris FX has incorporated artificial intelligence into its software, introducing features for noise reduction, rotoscoping, upscaling, and masking. The company has acquired technologies via mergers and acquisitions from Imagineer Systems, GenArts, Silhouette FX, Digital Film Tools, CrumplePop and Andersson Technologies to expand its visual effects, editing, photography, and audio tools. == History == Boris FX was founded in 1995 by Boris Yamnitsky. The former Media 100 engineer (a member of the original Media 100 launch team in 1993) released “Boris FX,” the first plug-in-based digital video effects (DVE) for Adobe Premiere and Media 100, in 1995. The plug-in won Best of Show at Apple Macworld in Boston, MA that same year. The Boris FX Suite includes a range of visual effects and post-production tools, such as Sapphire, Continuum, Mocha Pro, Silhouette, SynthEyes, CrumplePop, Optics, and Particle Illusion. == Media 100 == In October 2005, Yamnitsky acquired Media 100 the company that launched his plug-in career. Boris FX had a long relationship with Media 100 which bundled Boris RED software as its main titling and compositing solution. Media 100's video editing software is available as freeware for macOS. == Continuum == Continuum is a visual effect and compositing plugin suite that includes a library of over 300 effects and more than 40 transitions, including tools for image restoration, compositing, titling, particle generation, and stylized effects, along with features such as lens flares, lighting effects, and cinematic color grading presets. A key component of Continuum is its integration with the Mocha planar tracking and masking system, enabling advanced tracking and rotoscoping within the effects. The suite also includes Particle Illusion, a real-time particle generator used for creating visual effects such as explosions, smoke, and abstract motion graphics, as well as Primatte Studio, a chroma keying and compositing toolset for green screen and blue screen workflows. Continuum supports GPU acceleration and offers compatibility with HDR and 360/VR content. Regular updates introduce new effects, presets, and performance enhancements to expand its capabilities. In October 2018, Continuum relaunched Particle Illusion, a Mocha Essentials workflow with magnetic edge-snapping, and updates to Title Studio. In October 2019, Continuum introduced Corner Pin Studio with built-in Mocha tracking for quick screen replacement and inserts, 6 stylized transitions, and 4 creative effects. In October 2020, Continuum released an update that included over 80 GPU-accelerated effects such as film stocks, color grades, optical filter simulations, and a digital gobo library. The update also introduced a custom FX Editor interface, real-time particles, and more than 1,000 drag-and-drop presets. In November 2021, it added multi-frame rendering for After Effects, native Apple M1 support, fluid dynamics in Particle Illusion, and 60 color-grade presets. In October 2022, the software introduced 10 additional transitions, a revised Particle Illusion workflow, an atmospheric glow effect, and more than 250 curated presets. Continuum plugins have been used in television, streaming, and film projects, including A Black Lady Sketch Show (HBO/HBO Max), Star Trek: Discovery (CBS), Andor (Disney+), The Curse of Oak Island (History Channel), Keeping up with the Kardashians (E!), This Old House (PBS), Ms. Marvel (Disney+), MasterChef (Fox), WipeOut (TBS), The Boys (Prime Video), and The Today Show (NBC). == Mocha Pro == In December 2014, Boris FX merged with Imagineer Systems, the UK-based developer of the Academy Award-winning planar motion tracking software, Mocha Pro. Mocha Pro's features include planar tracking (motion tracking), rotoscoping, image stabilization, 3D camera tracking, and object removal. In June 2016, Mocha released (v5) which introduced Mocha Pro's tools as plug-ins for Adobe After Effects and Premiere Pro, Avid Media Composer, and OFX hosts Foundry's NUKE, Blackmagic Design Fusion, VEGAS Pro, and HitFilm. A simplified version, Mocha AE, is included with Adobe After Effects Creative Cloud and has been bundled with the software since CS4. A similar version is also available with HitFilm Pro from FXhome and VEGAS Pro. Mocha's tracking SDK is integrated into other visual effects tools, including SAM Quantel Pablo Rio, Silhouette FX, CoreMelt, and Motion VFX. Mocha Pro has been used in various film and television productions, including Birdman, Black Swan, the Harry Potter series, The Hobbit, Star Wars, The Mandalorian, Star Trek: Discovery, and The Umbrella Academy. It has also been employed in projects such as Gone Girl, The Hunger Games: Mockingjay – Part 1, Game of Thrones, and House of Cards. == Sapphire == GenArts, founded by Karl Sims in 1996, developed visual effects plug-ins that were used by studios and post-production facilities. In September 2016, Boris FX merged with former competitor, GenArts, Inc., developer of Sapphire high-end visual effects plug-ins, to expand its suite of motion graphics and VFX tools. The merger brought Sapphire alongside Boris Continuum Complete (BCC) and Mocha Pro, integrating these tools for film and television post-production. The Sapphire suite includes a library of over 270 effects and transitions, organized into categories such as lighting, stylization, distortions, textures, and transitions. Commonly used effects include glows, lens flares, film looks, and blurs. The plug-ins are designed to be GPU-accelerated, allowing for improved rendering performance and real-time previews in supported host applications. A central feature of Sapphire is the Builder tool, a node-based workspace that allows users to create custom effects and transitions by combining multiple Sapphire plug-ins. This enables a high level of creative flexibility and reusability, making it a popular tool for both editors and VFX artists. Sapphire also integrates with Mocha, Boris FX's planar tracking and masking system, allowing for advanced control of visual elements within an effect. In October 2017, Boris FX released its first new version of Sapphire since the GenArts acquisition. Sapphire (v11) now includes integrated Mocha tracking and masking tools. Sapphire is available for Adobe, Avid, the Autodesk Flame family, and OFX hosts including Blackmagic DaVinci Resolve and Fusion, and Foundry's NUKE. As part of the merger, Boris FX acquired the rights to Particle Illusion. In 2018, Boris FX reintroduced the product to the larger NLE/Compositing market. Sapphire's plug-ins transitioned from C to C++ to improve performance and support higher-resolution visual effects. This update enhanced floating-point calculations, compatibility with film editing APIs, and integration with NVIDIA's CUDA for faster rendering. The plug-ins have been used in various films, including Avatar, the Harry Potter and the Prisoner of Azkaban, Iron Man, The Lord of the Rings, The Matrix trilogy, Titanic, and X-Men. == Particle Illusion == As part of the merger with GenArts in 2016, Boris FX acquired the rights to the Particle Illusion (formerly particleIllusion) product, a storied particle system from the original developer Alan Lorence, the founder of Wondertouch. In 2018, Boris FX released a redesigned version of the product to a larger NLE/compositing market as part of Continuum (2019). The new Particle Illusion plug-in supports Adobe, Avid, and many OFX hosts. == Silhouette == In September 2019, Boris FX merged with SilhouetteFX, Academy Award-winning developer of Silhouette, a high-end digital paint, advanced rotoscoping, motion tracking, and node-based compositing application for visual effects in film post-production. The acquisition integrated Silhouette's advanced rotoscoping and paint technology, recognized by the Academy of Motion Pictures, into Boris FX's suite of products, alongside Sapphire, Continuum, and Mocha Pro. In May 2021, Boris FX released Silhouette 2021, the first version of Silhouette released by Boris FX to function both as a standalone application and as a plug-in for Adobe, Autodesk, Nuke, and other OFX hosts. Silhouette has been used in the visual effects of films such as Avatar, Avengers: Infinity War, Blade Runner 2049, Ex Machina, and Interstellar. == Optics == In June 2020, Boris FX launched Optics, its first plugin deve
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