AI Face Blur

AI Face Blur — independent reviews, comparisons, pricing and step-by-step guides on Aizhi.

  • Inception (deep learning architecture)

    Inception (deep learning architecture)

    Inception is a family of convolutional neural network (CNN) for computer vision, introduced by researchers at Google in 2014 as GoogLeNet (later renamed Inception v1). The series was historically important as an early CNN that separates the stem (data ingest), body (data processing), and head (prediction), an architectural design that persists in all modern CNN. == Version history == === Inception v1 === In 2014, a team at Google developed the GoogLeNet architecture, an instance of which won the ImageNet Large-Scale Visual Recognition Challenge 2014 (ILSVRC14). The name came from the LeNet of 1998, since both LeNet and GoogLeNet are CNNs. They also called it "Inception" after a "we need to go deeper" internet meme, a phrase from Inception (2010) the film. Because later, more versions were released, the original Inception architecture was renamed again as "Inception v1". The models and the code were released under Apache 2.0 license on GitHub. The Inception v1 architecture is a deep CNN composed of 22 layers. Most of these layers were "Inception modules". The original paper stated that Inception modules are a "logical culmination" of Network in Network and (Arora et al, 2014). Since Inception v1 is deep, it suffered from the vanishing gradient problem. The team solved it by using two "auxiliary classifiers", which are linear-softmax classifiers inserted at 1/3-deep and 2/3-deep within the network, and the loss function is a weighted sum of all three: L = 0.3 L a u x , 1 + 0.3 L a u x , 2 + L r e a l {\displaystyle L=0.3L_{aux,1}+0.3L_{aux,2}+L_{real}} These were removed after training was complete. This was later solved by the ResNet architecture. The architecture consists of three parts stacked on top of one another: The stem (data ingestion): The first few convolutional layers perform data preprocessing to downscale images to a smaller size. The body (data processing): The next many Inception modules perform the bulk of data processing. The head (prediction): The final fully-connected layer and softmax produces a probability distribution for image classification. This structure is used in most modern CNN architectures. === Inception v2 === Inception v2 was released in 2015, in a paper that is more famous for proposing batch normalization. It had 13.6 million parameters. It improves on Inception v1 by adding batch normalization, and removing dropout and local response normalization which they found became unnecessary when batch normalization is used. === Inception v3 === Inception v3 was released in 2016. It improves on Inception v2 by using factorized convolutions. As an example, a single 5×5 convolution can be factored into 3×3 stacked on top of another 3×3. Both has a receptive field of size 5×5. The 5×5 convolution kernel has 25 parameters, compared to just 18 in the factorized version. Thus, the 5×5 convolution is strictly more powerful than the factorized version. However, this power is not necessarily needed. Empirically, the research team found that factorized convolutions help. It also uses a form of dimension-reduction by concatenating the output from a convolutional layer and a pooling layer. As an example, a tensor of size 35 × 35 × 320 {\displaystyle 35\times 35\times 320} can be downscaled by a convolution with stride 2 to 17 × 17 × 320 {\displaystyle 17\times 17\times 320} , and by maxpooling with pool size 2 × 2 {\displaystyle 2\times 2} to 17 × 17 × 320 {\displaystyle 17\times 17\times 320} . These are then concatenated to 17 × 17 × 640 {\displaystyle 17\times 17\times 640} . Other than this, it also removed the lowest auxiliary classifier during training. They found that the auxiliary head worked as a form of regularization. They also proposed label-smoothing regularization in classification. For an image with label c {\displaystyle c} , instead of making the model to predict the probability distribution δ c = ( 0 , 0 , … , 0 , 1 ⏟ c -th entry , 0 , … , 0 ) {\displaystyle \delta _{c}=(0,0,\dots ,0,\underbrace {1} _{c{\text{-th entry}}},0,\dots ,0)} , they made the model predict the smoothed distribution ( 1 − ϵ ) δ c + ϵ / K {\displaystyle (1-\epsilon )\delta _{c}+\epsilon /K} where K {\displaystyle K} is the total number of classes. === Inception v4 === In 2017, the team released Inception v4, Inception ResNet v1, and Inception ResNet v2. Inception v4 is an incremental update with even more factorized convolutions, and other complications that were empirically found to improve benchmarks. Inception ResNet v1 and v2 are both modifications of Inception v4, where residual connections are added to each Inception module, inspired by the ResNet architecture. === Xception === Xception ("Extreme Inception") was published in 2017. It is a linear stack of depthwise separable convolution layers with residual connections. The design was proposed on the hypothesis that in a CNN, the cross-channels correlations and spatial correlations in the feature maps can be entirely decoupled. Training each network took 3 days on 60 K80 GPUs, or approximately 0.5 petaFLOP-days.

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  • Rule-based machine translation

    Rule-based machine translation

    Rule-based machine translation (RBMT) is a classical approach of machine translation systems based on linguistic information about source and target languages. Such information is retrieved from (unilingual, bilingual or multilingual) dictionaries and grammars covering the main semantic, morphological, and syntactic regularities of each language. Having input sentences, an RBMT system generates output sentences on the basis of analysis of both the source and the target languages involved. RBMT has been progressively superseded by more efficient methods, particularly neural machine translation. == History == The first RBMT systems were developed in the early 1970s. The most important steps of this evolution were the emergence of the following RBMT systems: Systran Japanese MT systems Today, other common RBMT systems include: Apertium GramTrans == Types of RBMT == There are three different types of rule-based machine translation systems: Direct Systems (Dictionary Based Machine Translation) map input to output with basic rules. Transfer RBMT Systems (Transfer Based Machine Translation) employ morphological and syntactical analysis. Interlingual RBMT Systems (Interlingua) use an abstract meaning. RBMT systems can also be characterized as the systems opposite to Example-based Systems of Machine Translation (Example Based Machine Translation), whereas Hybrid Machine Translations Systems make use of many principles derived from RBMT. == Basic principles == The main approach of RBMT systems is based on linking the structure of the given input sentence with the structure of the demanded output sentence, necessarily preserving their unique meaning. The following example can illustrate the general frame of RBMT: A girl eats an apple. Source Language = English; Demanded Target Language = German Minimally, to get a German translation of this English sentence one needs: A dictionary that will map each English word to an appropriate German word. Rules representing regular English sentence structure. Rules representing regular German sentence structure. And finally, we need rules according to which one can relate these two structures together. Accordingly, we can state the following stages of translation: 1st: getting basic part-of-speech information of each source word: a = indef.article; girl = noun; eats = verb; an = indef.article; apple = noun 2nd: getting syntactic information about the verb "to eat": NP-eat-NP; here: eat – Present Simple, 3rd Person Singular, Active Voice 3rd: parsing the source sentence: (NP an apple) = the object of eat Often only partial parsing is sufficient to get to the syntactic structure of the source sentence and to map it onto the structure of the target sentence. 4th: translate English words into German a (category = indef.article) => ein (category = indef.article) girl (category = noun) => Mädchen (category = noun) eat (category = verb) => essen (category = verb) an (category = indef. article) => ein (category = indef.article) apple (category = noun) => Apfel (category = noun) 5th: Mapping dictionary entries into appropriate inflected forms (final generation): A girl eats an apple. => Ein Mädchen isst einen Apfel. == Ontologies == An ontology is a formal representation of knowledge that includes the concepts (such as objects, processes etc.) in a domain and some relations between them. If the stored information is of linguistic nature, one can speak of a lexicon. In NLP, ontologies can be used as a source of knowledge for machine translation systems. With access to a large knowledge base, rule-based systems can be enabled to resolve many (especially lexical) ambiguities on their own. In the following classic examples, as humans, we are able to interpret the prepositional phrase according to the context because we use our world knowledge, stored in our lexicons:I saw a man/star/molecule with a microscope/telescope/binoculars.Since the syntax does not change, a traditional rule-based machine translation system may not be able to differentiate between the meanings. With a large enough ontology as a source of knowledge however, the possible interpretations of ambiguous words in a specific context can be reduced. === Building ontologies === The ontology generated for the PANGLOSS knowledge-based machine translation system in 1993 may serve as an example of how an ontology for NLP purposes can be compiled: A large-scale ontology is necessary to help parsing in the active modules of the machine translation system. In the PANGLOSS example, about 50,000 nodes were intended to be subsumed under the smaller, manually-built upper (abstract) region of the ontology. Because of its size, it had to be created automatically. The goal was to merge the two resources LDOCE online and WordNet to combine the benefits of both: concise definitions from Longman, and semantic relations allowing for semi-automatic taxonomization to the ontology from WordNet. A definition match algorithm was created to automatically merge the correct meanings of ambiguous words between the two online resources, based on the words that the definitions of those meanings have in common in LDOCE and WordNet. Using a similarity matrix, the algorithm delivered matches between meanings including a confidence factor. This algorithm alone, however, did not match all meanings correctly on its own. A second hierarchy match algorithm was therefore created which uses the taxonomic hierarchies found in WordNet (deep hierarchies) and partially in LDOCE (flat hierarchies). This works by first matching unambiguous meanings, then limiting the search space to only the respective ancestors and descendants of those matched meanings. Thus, the algorithm matched locally unambiguous meanings (for instance, while the word seal as such is ambiguous, there is only one meaning of seal in the animal subhierarchy). Both algorithms complemented each other and helped constructing a large-scale ontology for the machine translation system. The WordNet hierarchies, coupled with the matching definitions of LDOCE, were subordinated to the ontology's upper region. As a result, the PANGLOSS MT system was able to make use of this knowledge base, mainly in its generation element. == Components == The RBMT system contains: a SL morphological analyser - analyses a source language word and provides the morphological information; a SL parser - is a syntax analyser which analyses source language sentences; a translator - used to translate a source language word into the target language; a TL morphological generator - works as a generator of appropriate target language words for the given grammatica information; a TL parser - works as a composer of suitable target language sentences; Several dictionaries - more specifically a minimum of three dictionaries: a SL dictionary - needed by the source language morphological analyser for morphological analysis, a bilingual dictionary - used by the translator to translate source language words into target language words, a TL dictionary - needed by the target language morphological generator to generate target language words. The RBMT system makes use of the following: a Source Grammar for the input language which builds syntactic constructions from input sentences; a Source Lexicon which captures all of the allowable vocabulary in the domain; Source Mapping Rules which indicate how syntactic heads and grammatical functions in the source language are mapped onto domain concepts and semantic roles in the interlingua; a Domain Model/Ontology which defines the classes of domain concepts and restricts the fillers of semantic roles for each class; Target Mapping Rules which indicate how domain concepts and semantic roles in the interlingua are mapped onto syntactic heads and grammatical functions in the target language; a Target Lexicon which contains appropriate target lexemes for each domain concept; a Target Grammar for the target language which realizes target syntactic constructions as linearized output sentences. == Advantages == No bilingual texts are required. This makes it possible to create translation systems for languages that have no texts in common, or even no digitized data whatsoever. Domain independent. Rules are usually written in a domain independent manner, so the vast majority of rules will "just work" in every domain, and only a few specific cases per domain may need rules written for them. No quality ceiling. Every error can be corrected with a targeted rule, even if the trigger case is extremely rare. This is in contrast to statistical systems where infrequent forms will be washed away by default. Total control. Because all rules are hand-written, you can easily debug a rule-based system to see exactly where a given error enters the system, and why. Reusability. Because RBMT systems are generally built from a strong source language analysis that is fed to a transfer step and target language generator, the source language analysis and targe

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  • Information retrieval

    Information retrieval

    Information retrieval (IR) in computing and information science is the task of identifying and retrieving information system resources that are relevant to an information need. The information need can be specified in the form of a search query. In the case of document retrieval, queries can be based on full-text or other content-based indexing. Information retrieval is the science of searching for information in a document, searching for documents themselves, and also searching for the metadata that describes data, and for databases of texts, images, or sounds. Cross-modal retrieval implies retrieval across modalities. Automated information retrieval systems are used to reduce what has been called information overload. An IR system is a software system that provides access to books, journals, and other documents, as well as storing and managing those documents. Web search engines are the most visible IR applications. == Overview == An information retrieval process begins when a user enters a query into the system. Queries are formal statements of information needs, for example search strings in web search engines. In information retrieval, a query does not uniquely identify a single object in the collection. Instead, several objects may match the query, perhaps with different degrees of relevance. An object is an entity that is represented by information in a content collection or database. User queries are matched against the database information. However, as opposed to classical SQL queries of a database, in information retrieval the results returned may or may not match the query, so results are typically ranked. This ranking of results is a key difference of information retrieval searching compared to database searching. Depending on the application the data objects may be, for example, text documents, images, audio, mind maps or videos. Often the documents themselves are not kept or stored directly in the IR system, but are instead represented in the system by document surrogates or metadata. Most IR systems compute a numeric score on how well each object in the database matches the query, and rank the objects according to this value. The top ranking objects are then shown to the user. The process may then be iterated if the user wishes to refine the query. == History == there is ... a machine called the Univac ... whereby letters and figures are coded as a pattern of magnetic spots on a long steel tape. By this means the text of a document, preceded by its subject code symbol, can be recorded ... the machine ... automatically selects and types out those references which have been coded in any desired way at a rate of 120 words a minute The idea of using computers to search for relevant pieces of information was popularized in the article As We May Think by Vannevar Bush in 1945. It would appear that Bush was inspired by patents for a 'statistical machine' – filed by Emanuel Goldberg in the 1920s and 1930s – that searched for documents stored on film. The first description of a computer searching for information was described by Holmstrom in 1948, detailing an early mention of the Univac computer. Automated information retrieval systems were introduced in the 1950s: one even featured in the 1957 romantic comedy Desk Set. In the 1960s, the first large information retrieval research group was formed by Gerard Salton at Cornell. By the 1970s several different retrieval techniques had been shown to perform well on small text corpora such as the Cranfield collection (several thousand documents). Large-scale retrieval systems, such as the Lockheed Dialog system, came into use early in the 1970s. In 1992, the US Department of Defense along with the National Institute of Standards and Technology (NIST), cosponsored the Text Retrieval Conference (TREC) as part of the TIPSTER text program. The aim of this was to look into the information retrieval community by supplying the infrastructure that was needed for evaluation of text retrieval methodologies on a very large text collection. This catalyzed research on methods that scale to huge corpora. The introduction of web search engines has boosted the need for very large scale retrieval systems even further. By the late 1990s, the rise of the World Wide Web fundamentally transformed information retrieval. While early search engines such as AltaVista (1995) and Yahoo! (1994) offered keyword-based retrieval, they were limited in scale and ranking refinement. The breakthrough came in 1998 with the founding of Google, which introduced the PageRank algorithm, using the web's hyperlink structure to assess page importance and improve relevance ranking. During the 2000s, web search systems evolved rapidly with the integration of machine learning techniques. These systems began to incorporate user behavior data (e.g., click-through logs), query reformulation, and content-based signals to improve search accuracy and personalization. In 2009, Microsoft launched Bing, introducing features that would later incorporate semantic web technologies through the development of its Satori knowledge base. Academic analysis have highlighted Bing's semantic capabilities, including structured data use and entity recognition, as part of a broader industry shift toward improving search relevance and understanding user intent through natural language processing. A major leap occurred in 2018, when Google deployed BERT (Bidirectional Encoder Representations from Transformers) to better understand the contextual meaning of queries and documents. This marked one of the first times deep neural language models were used at scale in real-world retrieval systems. BERT's bidirectional training enabled a more refined comprehension of word relationships in context, improving the handling of natural language queries. Because of its success, transformer-based models gained traction in academic research and commercial search applications. Simultaneously, the research community began exploring neural ranking models that outperformed traditional lexical-based methods. Long-standing benchmarks such as the Text REtrieval Conference (TREC), initiated in 1992, and more recent evaluation frameworks Microsoft MARCO(MAchine Reading COmprehension) (2019) became central to training and evaluating retrieval systems across multiple tasks and domains. MS MARCO has also been adopted in the TREC Deep Learning Tracks, where it serves as a core dataset for evaluating advances in neural ranking models within a standardized benchmarking environment. As deep learning became integral to information retrieval systems, researchers began to categorize neural approaches into three broad classes: sparse, dense, and hybrid models. Sparse models, including traditional term-based methods and learned variants like SPLADE, rely on interpretable representations and inverted indexes to enable efficient exact term matching with added semantic signals. Dense models, such as dual-encoder architectures like ColBERT, use continuous vector embeddings to support semantic similarity beyond keyword overlap. Hybrid models aim to combine the advantages of both, balancing the lexical (token) precision of sparse methods with the semantic depth of dense models. This way of categorizing models balances scalability, relevance, and efficiency in retrieval systems. As IR systems increasingly rely on deep learning, concerns around bias, fairness, and explainability have also come to the picture. Research is now focused not just on relevance and efficiency, but on transparency, accountability, and user trust in retrieval algorithms. == Applications == Areas where information retrieval techniques are employed include (the entries are in alphabetical order within each category): === General applications === Digital libraries Information filtering Recommender systems Media search Blog search Image retrieval 3D retrieval Music retrieval News search Speech retrieval Video retrieval Search engines Site search Desktop search Enterprise search Federated search Mobile search Social search Web search === Domain-specific applications === Expert search finding Genomic information retrieval Geographic information retrieval Information retrieval for chemical structures Information retrieval in software engineering Legal information retrieval Vertical search === Other retrieval methods === Methods/Techniques in which information retrieval techniques are employed include: Cross-modal retrieval Adversarial information retrieval Automatic summarization Multi-document summarization Compound term processing Cross-lingual retrieval Document classification Spam filtering Question answering == Model types == In order to effectively retrieve relevant documents by IR strategies, the documents are typically transformed into a suitable representation. Each retrieval strategy incorporates a specific model for its document representation purposes. The picture on the right illustrates the relationship of som

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  • Euratlas

    Euratlas

    Euratlas is a Switzerland-based software company dedicated to elaborate digital history maps of Europe. Founded in 2001, Euratlas has created a collection of history maps of Europe from year 1 AD to year 2000 AD that present the evolution of every country from the Roman Empire to present times. The evolution includes sovereign states and their administrative subdivisions, but also unorganized peoples and dependent territories. The maps show European country borders at regular intervals of 100 years, but not year by year. This leaves out many important turning points in history. Euratlas is considered a digital humanities company, and a scholar research software used in the field of historic cartography. It is broadly known among American and European universities, who mainly use Euratlas as a research tool and as a digital library atlas. == Sequential mapping policy == This concept was first designed by the German scholar Christian Kruse (1753–1827). Kruse, well aware that historical accounts are often biased for geographical, philosophical or political reasons, created a set of sequential maps in order to give a global vision of the successive political situations. Nowadays, the majority of atlases don't use this approach, but are event-based, like the well-known Penguin Atlas of History. The sequential approach intends to make the sequence of maps more neutral and suitable for students, historians and professionals of several fields. Although, this approach has been discussed as it leaves out many important history events that are not reflected on any of the maps because of the century interval. == Geo-referenced historical data == Initially, the European maps by century were developed as vector maps. From 2006 on, they have been converted to a geographic information system (GIS) database, enabling geo-referenced data capabilities. The map information is distributed in several layers: physical (geography information layer); political information layer (supranational entities, sovereign states, administrative divisions, dependent states and autonomous peoples); and special layers for cities and uncertain borders. The software database also contains much non-geographical information about political relationships between the various kinds of territories. == Map projection == Euratlas History Maps uses a Mercator projection, with the center in Europe. The maps include the North-African coast and the Near-East, offering a complete view of the Mediterranean Basin. The European Russia plains are shown, but not Scandinavia, specially Finland, which is cropped off the map view.

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  • BevQ

    BevQ

    BevQ is a queue management mobile application developed by Faircode Technologies of Kochi, Kerala. It is provided by the Kerala State Beverages Corporation under Government of Kerala. == History == This app was released together by the Government of Kerala and the Kerala State Beverages Corporation in order to implement social distancing in the liquor stores Kerala in the case of the COVID-19 pandemic in Kerala and to reduce the congestion of people. The BevQ App was released by Faircode Technologies on 27 May 2020 on the Google Play Store. In January 2021, the app was withdrawn as bars had opened. In June 2021, there was a commitment from the Kerala CM that the App will be relaunched again. It has been reported that over 132,000 new users downloaded the app in the 48 hours after the announcement. == Achievements == The BEVQ app, which works only in the state of Kerala, beat all other Indian food and drink apps in 2020 to see the highest growth in year-on-year sessions, according to the State of Mobile 2021 report by App Annie. The app even beat the likes of Domino’s, which is used all across India. Around 300 government Liquor shops and 900 private liquor shops were enlisted in the platform. More than 200 million unique users registered in the platform. About 250,000 tokens were given out a day.

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  • Scale-space axioms

    Scale-space axioms

    In image processing and computer vision, a scale space framework can be used to represent an image as a family of gradually smoothed images. This framework is very general and a variety of scale space representations exist. A typical approach for choosing a particular type of scale space representation is to establish a set of scale-space axioms, describing basic properties of the desired scale-space representation and often chosen so as to make the representation useful in practical applications. Once established, the axioms narrow the possible scale-space representations to a smaller class, typically with only a few free parameters. A set of standard scale space axioms, discussed below, leads to the linear Gaussian scale-space, which is the most common type of scale space used in image processing and computer vision. == Scale space axioms for the linear scale-space representation == The linear scale space representation L ( x , y , t ) = ( T t f ) ( x , y ) = g ( x , y , t ) ∗ f ( x , y ) {\displaystyle L(x,y,t)=(T_{t}f)(x,y)=g(x,y,t)f(x,y)} of signal f ( x , y ) {\displaystyle f(x,y)} obtained by smoothing with the Gaussian kernel g ( x , y , t ) {\displaystyle g(x,y,t)} satisfies a number of properties 'scale-space axioms' that make it a special form of multi-scale representation: linearity T t ( a f + b h ) = a T t f + b T t h {\displaystyle T_{t}(af+bh)=aT_{t}f+bT_{t}h} where f {\displaystyle f} and h {\displaystyle h} are signals while a {\displaystyle a} and b {\displaystyle b} are constants, shift invariance T t S ( Δ x , Δ y ) f = S ( Δ x , Δ y ) T t f {\displaystyle T_{t}S_{(\Delta x,\Delta _{y})}f=S_{(\Delta x,\Delta _{y})}T_{t}f} where S ( Δ x , Δ y ) {\displaystyle S_{(\Delta x,\Delta _{y})}} denotes the shift (translation) operator ( S ( Δ x , Δ y ) f ) ( x , y ) = f ( x − Δ x , y − Δ y ) {\displaystyle (S_{(\Delta x,\Delta _{y})}f)(x,y)=f(x-\Delta x,y-\Delta y)} semi-group structure g ( x , y , t 1 ) ∗ g ( x , y , t 2 ) = g ( x , y , t 1 + t 2 ) {\displaystyle g(x,y,t_{1})g(x,y,t_{2})=g(x,y,t_{1}+t_{2})} with the associated cascade smoothing property L ( x , y , t 2 ) = g ( x , y , t 2 − t 1 ) ∗ L ( x , y , t 1 ) {\displaystyle L(x,y,t_{2})=g(x,y,t_{2}-t_{1})L(x,y,t_{1})} existence of an infinitesimal generator A {\displaystyle A} ∂ t L ( x , y , t ) = ( A L ) ( x , y , t ) {\displaystyle \partial _{t}L(x,y,t)=(AL)(x,y,t)} non-creation of local extrema (zero-crossings) in one dimension, non-enhancement of local extrema in any number of dimensions ∂ t L ( x , y , t ) ≤ 0 {\displaystyle \partial _{t}L(x,y,t)\leq 0} at spatial maxima and ∂ t L ( x , y , t ) ≥ 0 {\displaystyle \partial _{t}L(x,y,t)\geq 0} at spatial minima, rotational symmetry g ( x , y , t ) = h ( x 2 + y 2 , t ) {\displaystyle g(x,y,t)=h(x^{2}+y^{2},t)} for some function h {\displaystyle h} , scale invariance g ^ ( ω x , ω y , t ) = h ^ ( ω x φ ( t ) , ω x φ ( t ) ) {\displaystyle {\hat {g}}(\omega _{x},\omega _{y},t)={\hat {h}}({\frac {\omega _{x}}{\varphi (t)}},{\frac {\omega _{x}}{\varphi (t)}})} for some functions φ {\displaystyle \varphi } and h ^ {\displaystyle {\hat {h}}} where g ^ {\displaystyle {\hat {g}}} denotes the Fourier transform of g {\displaystyle g} , positivity g ( x , y , t ) ≥ 0 {\displaystyle g(x,y,t)\geq 0} , normalization ∫ x = − ∞ ∞ ∫ y = − ∞ ∞ g ( x , y , t ) d x d y = 1 {\displaystyle \int _{x=-\infty }^{\infty }\int _{y=-\infty }^{\infty }g(x,y,t)\,dx\,dy=1} . In fact, it can be shown that the Gaussian kernel is a unique choice given several different combinations of subsets of these scale-space axioms: most of the axioms (linearity, shift-invariance, semigroup) correspond to scaling being a semigroup of shift-invariant linear operator, which is satisfied by a number of families integral transforms, while "non-creation of local extrema" for one-dimensional signals or "non-enhancement of local extrema" for higher-dimensional signals are the crucial axioms which relate scale-spaces to smoothing (formally, parabolic partial differential equations), and hence select for the Gaussian. The Gaussian kernel is also separable in Cartesian coordinates, i.e. g ( x , y , t ) = g ( x , t ) g ( y , t ) {\displaystyle g(x,y,t)=g(x,t)\,g(y,t)} . Separability is, however, not counted as a scale-space axiom, since it is a coordinate dependent property related to issues of implementation. In addition, the requirement of separability in combination with rotational symmetry per se fixates the smoothing kernel to be a Gaussian. There exists a generalization of the Gaussian scale-space theory to more general affine and spatio-temporal scale-spaces. In addition to variabilities over scale, which original scale-space theory was designed to handle, this generalized scale-space theory also comprises other types of variabilities, including image deformations caused by viewing variations, approximated by local affine transformations, and relative motions between objects in the world and the observer, approximated by local Galilean transformations. In this theory, rotational symmetry is not imposed as a necessary scale-space axiom and is instead replaced by requirements of affine and/or Galilean covariance. The generalized scale-space theory leads to predictions about receptive field profiles in good qualitative agreement with receptive field profiles measured by cell recordings in biological vision. In the computer vision, image processing and signal processing literature there are many other multi-scale approaches, using wavelets and a variety of other kernels, that do not exploit or require the same requirements as scale space descriptions do; please see the article on related multi-scale approaches. There has also been work on discrete scale-space concepts that carry the scale-space properties over to the discrete domain; see the article on scale space implementation for examples and references.

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  • Corpus of Linguistic Acceptability

    Corpus of Linguistic Acceptability

    Corpus of Linguistic Acceptability (CoLA) is a dataset the primary purpose of which is to serve as a benchmark for evaluating the ability of artificial neural networks, including large language models, to judge the grammatical correctness of sentences. It consists of 10,657 English sentences from published linguistics literature that were manually labeled either as grammatical or ungrammatical. == Public version == The publicly available version of CoLA contains 9,594 sentences that belong to training and development sets. It excludes 1,063 sentences reserved for a held-out test set.

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  • Boyfriend Maker

    Boyfriend Maker

    Boyfriend Maker was a dating sim, romance chatbot smartphone app for iOS (iPhone) and Android devices, developed by Japanese studio 36 You Games (styled as 36You) and distributed under the freemium business model. Boyfriend Maker incorporated advanced artificial intelligence chat technology a decade before products such as ChatGPT. According to the developer's website, Boyfriend Maker is an "app that lets you interact and chat with quirky virtual boyfriends". While each virtual boyfriend has certain unique characteristics, the various instances of the boyfriend are powered by a chat engine, that (at least within a language and market) can utilise vocabulary and knowledge acquired in a chat with one user in subsequent chats with other users. == Gameplay == Users gain experience points and in-game coins. Users can customize their virtual boyfriend's appearance by selecting items such as hair, clothing, face, and a necklace. == Apple delisting and reintroduction == In late November 2012, the original iOS Boyfriend Maker app was delisted from the Apple App Store due to "ribald" chat, according to the New York Times. Boyfriend Maker was removed by Apple due to "reports of references to violent sexual acts and pedophilia". Boyfriend Maker had an age rating of 4+, even though the chat bot "responds with often strange and explicit text unsuitable for young children". User-posted chat excerpts indicate that the virtual boyfriend would sometimes transition abruptly to sexual chat in response to a seemingly innocent question. In one user-posted example, in response to the question, "what kind of wedding cake will we have" the boyfriend responds, "a good sex ima be on top of u u gonna ride oon me bitin the pillow gurrl ima fuck da shit out of u". The developer's use of the SimSimi-created third-party chat engine may be responsible for the sexual text. As the virtual boyfriend converses with human users, the SimSimi chat engine acquires vocabulary from users of the game and applies this "learned" vocabulary in chats with other users. The chat engine might also employ lines harvested from human-human chat logs, song lyrics, movies or TV shows. In April 2013, a detuned and presumably tamer version of the app, titled Boyfriend Plus, was permitted on Apple's App Store.

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  • Embodied agent

    Embodied agent

    In artificial intelligence, an embodied agent, also sometimes referred to as an interface agent, is an intelligent agent that interacts with the environment through a physical body within that environment. Agents that are represented graphically with a body, for example a human or a cartoon animal, are also called embodied agents, although they have only virtual, not physical, embodiment. A branch of artificial intelligence focuses on empowering such agents to interact autonomously with human beings and the environment. Mobile robots are one example of physically embodied agents; Ananova and Microsoft Agent are examples of graphically embodied agents. Embodied conversational agents are embodied agents (usually with a graphical front-end as opposed to a robotic body) that are capable of engaging in conversation with one another and with humans employing the same verbal and nonverbal means that humans do (such as gesture, facial expression, and so forth). == Embodied conversational agents == Embodied conversational agents are a form of intelligent user interface. Graphically embodied agents aim to unite gesture, facial expression and speech to enable face-to-face communication with users, providing a powerful means of human-computer interaction. == Advantages == Face-to-face communication allows communication protocols that give a much richer communication channel than other means of communicating. It enables pragmatic communication acts such as conversational turn-taking, facial expression of emotions, information structure and emphasis, visualization and iconic gestures, and orientation in a three-dimensional environment. This communication takes place through both verbal and non-verbal channels such as gaze, gesture, spoken intonation and body posture. Research has found that users prefer a non-verbal visual indication of an embodied system's internal state to a verbal indication, demonstrating the value of additional non-verbal communication channels. As well as this, the face-to-face communication involved in interacting with an embodied agent can be conducted alongside another task without distracting the human participants, instead improving the enjoyment of such an interaction. Furthermore, the use of an embodied presentation agent results in improved recall of the presented information. Embodied agents also provide a social dimension to the interaction. Humans willingly ascribe social awareness to computers, and thus interaction with embodied agents follows social conventions, similar to human to human interactions. This social interaction both raises the believably and perceived trustworthiness of agents, and increases the user's engagement with the system. Rickenberg and Reeves found that the presence of an embodied agent on a website increased the level of user trust in that website, but also increased users' anxiety and affected their performance, as if they were being watched by a real human. Another effect of the social aspect of agents is that presentations given by an embodied agent are perceived as being more entertaining and less difficult than similar presentations given without an agent. Research shows that perceived enjoyment, followed by perceived usefulness and ease of use, is the major factor influencing user adoption of embodied agents. A study in January 2004 by Byron Reeves at Stanford demonstrated how digital characters could "enhance online experiences" through explaining how virtual characters essentially add a sense of familiarity to the user experience and make it more approachable. This increase in likability in turn helps make the products better, which benefits both the end users and those creating the product. === Applications === The rich style of communication that characterizes human conversation makes conversational interaction with embodied conversational agents ideal for many non-traditional interaction tasks. A familiar application of graphically embodied agents is computer games; embodied agents are ideal for this setting because the richer communication style makes interacting with the agent enjoyable. Embodied conversational agents have also been used in virtual training environments, portable personal navigation guides, interactive fiction and storytelling systems, interactive online characters and automated presenters and commentators. Major virtual assistants like Siri, Amazon Alexa and Google Assistant do not come with any visual embodied representation, which is believed to limit the sense of human presence by users. The U.S. Department of Defense utilizes a software agent called SGT STAR on U.S. Army-run Web sites and Web applications for site navigation, recruitment and propaganda purposes. Sgt. Star is run by the Army Marketing and Research Group, a division operated directly from The Pentagon. Sgt. Star is based upon the ActiveSentry technology developed by Next IT, a Washington-based information technology services company. Other such bots in the Sgt. Star "family" are utilized by the Federal Bureau of Investigation and the Central Intelligence Agency for intelligence gathering purposes.

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  • Kruti

    Kruti

    Kruti is a multilingual AI agent and chatbot developed by the Indian company Ola Krutrim. It is designed to perform real-world tasks for users, such as booking taxis and ordering food, by integrating directly with various online services. It is notable for its ability to understand and respond in multiple Indian languages. Developed by a team founded by Bhavish Aggarwal, Kruti functions as an "agentic" AI, meaning it can reason, plan, and execute multi-step tasks to fulfill a user's request. The backend technology combines several open-source large language models with Ola's proprietary Krutrim V2 model. The system was developed to work primarily on smartphones, addressing the Indian market's specific needs, including language diversity and potential bandwidth constraints. Kruti was officially released in June 2025, replacing an earlier chatbot from the company that was also named Krutrim. Initially supporting 13 languages, the company plans to expand its capabilities to 22 Indian languages. == Background == Kruti is an improved version of Ola's Krutrim chatbot, which was first launched in 2023 and was intended to be replaced by Kruti. It was officially released on 12 June 2025 as an upgrade to passive chatbots, with support for text and voice in 13 Indian languages. As an agentic AI, it can execute tasks with customization and reasoning, providing adaptive answers based on user preferences and past interactions. Kruti is optimized for smartphone usage and designed to accommodate bandwidth constraints and usage patterns in India. To ensure scalability and cost-effective performance, it combines various open-source large language models with Ola's own Krutrim V2, which has 12 billion parameters. Its speech recognition is built to identify regional Indian languages, dialects, and accents. Due to its integration with numerous apps and services, Kruti is context-aware and can proactively complete tasks. Initially connected only with Ola ecosystem services, Krutrim intends to expand and incorporate various Indian services into Kruti, with the goal of adding services from Blinkit, Swiggy, and Uber with respective voice command support. On 20 June 2025, Krutrim acquired the AI platform BharatSah‘AI’yak to increase its involvement in government, education, and agriculture projects. This acquisition will allow Kruti to assist in broadening the scope of BharatSah'AI'yak's work on India-centric, vernacular retrieval-augmented generation AI bots. == Development == Kruti is designed to perform tasks with minimal user input, accepting documents, images, and text, without requiring users to switch between applications. Its agentic framework breaks queries into sub-tasks executed by multiple agents working sequentially or concurrently, with reported accuracy exceeding 90%. Kruti connects to company databases and APIs via the Model Context Protocol and presents responses as summaries, tables, or narratives adapted to user behaviour. The system supports payments via credit/debit cards and UPI. The underlying stack, which includes foundation models and AI training and inference systems, is intended to support adaptation across sectors such as healthcare, education, and finance. Ola Cabs and the Open Network for Digital Commerce have begun integrating Kruti into their platforms pending broader reliability testing.

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  • Visual servoing

    Visual servoing

    Visual servoing, also known as vision-based robot control and abbreviated VS, is a technique which uses feedback information extracted from a vision sensor (visual feedback) to control the motion of a robot. One of the earliest papers that talks about visual servoing was from the SRI International Labs in 1979. == Visual servoing taxonomy == There are two fundamental configurations of the robot end-effector (hand) and the camera: Eye-in-hand, or end-point open-loop control, where the camera is attached to the moving hand and observing the relative position of the target. Eye-to-hand, or end-point closed-loop control, where the camera is fixed in the world and observing the target and the motion of the hand. Visual Servoing control techniques are broadly classified into the following types: Image-based (IBVS) Position/pose-based (PBVS) Hybrid approach IBVS was proposed by Weiss and Sanderson. The control law is based on the error between current and desired features on the image plane, and does not involve any estimate of the pose of the target. The features may be the coordinates of visual features, lines or moments of regions. IBVS has difficulties with motions very large rotations, which has come to be called camera retreat. PBVS is a model-based technique (with a single camera). This is because the pose of the object of interest is estimated with respect to the camera and then a command is issued to the robot controller, which in turn controls the robot. In this case the image features are extracted as well, but are additionally used to estimate 3D information (pose of the object in Cartesian space), hence it is servoing in 3D. Hybrid approaches use some combination of the 2D and 3D servoing. There have been a few different approaches to hybrid servoing 2-1/2-D Servoing Motion partition-based Partitioned DOF Based == Survey == The following description of the prior work is divided into 3 parts Survey of existing visual servoing methods. Various features used and their impacts on visual servoing. Error and stability analysis of visual servoing schemes. === Survey of existing visual servoing methods === Visual servo systems, also called servoing, have been around since the early 1980s , although the term visual servo itself was only coined in 1987. Visual Servoing is, in essence, a method for robot control where the sensor used is a camera (visual sensor). Servoing consists primarily of two techniques, one involves using information from the image to directly control the degrees of freedom (DOF) of the robot, thus referred to as Image Based Visual Servoing (IBVS). While the other involves the geometric interpretation of the information extracted from the camera, such as estimating the pose of the target and parameters of the camera (assuming some basic model of the target is known). Other servoing classifications exist based on the variations in each component of a servoing system , e.g. the location of the camera, the two kinds are eye-in-hand and hand–eye configurations. Based on the control loop, the two kinds are end-point-open-loop and end-point-closed-loop. Based on whether the control is applied to the joints (or DOF) directly or as a position command to a robot controller the two types are direct servoing and dynamic look-and-move. Being one of the earliest works the authors proposed a hierarchical visual servo scheme applied to image-based servoing. The technique relies on the assumption that a good set of features can be extracted from the object of interest (e.g. edges, corners and centroids) and used as a partial model along with global models of the scene and robot. The control strategy is applied to a simulation of a two and three DOF robot arm. Feddema et al. introduced the idea of generating task trajectory with respect to the feature velocity. This is to ensure that the sensors are not rendered ineffective (stopping the feedback) for any the robot motions. The authors assume that the objects are known a priori (e.g. CAD model) and all the features can be extracted from the object. The work by Espiau et al. discusses some of the basic questions in visual servoing. The discussions concentrate on modeling of the interaction matrix, camera, visual features (points, lines, etc..). In an adaptive servoing system was proposed with a look-and-move servoing architecture. The method used optical flow along with SSD to provide a confidence metric and a stochastic controller with Kalman filtering for the control scheme. The system assumes (in the examples) that the plane of the camera and the plane of the features are parallel., discusses an approach of velocity control using the Jacobian relationship s˙ = Jv˙ . In addition the author uses Kalman filtering, assuming that the extracted position of the target have inherent errors (sensor errors). A model of the target velocity is developed and used as a feed-forward input in the control loop. Also, mentions the importance of looking into kinematic discrepancy, dynamic effects, repeatability, settling time oscillations and lag in response. Corke poses a set of very critical questions on visual servoing and tries to elaborate on their implications. The paper primarily focuses the dynamics of visual servoing. The author tries to address problems like lag and stability, while also talking about feed-forward paths in the control loop. The paper also, tries to seek justification for trajectory generation, methodology of axis control and development of performance metrics. Chaumette in provides good insight into the two major problems with IBVS. One, servoing to a local minima and second, reaching a Jacobian singularity. The author show that image points alone do not make good features due to the occurrence of singularities. The paper continues, by discussing the possible additional checks to prevent singularities namely, condition numbers of J_s and Jˆ+_s, to check the null space of ˆ J_s and J^T_s . One main point that the author highlights is the relation between local minima and unrealizable image feature motions. Over the years many hybrid techniques have been developed. These involve computing partial/complete pose from Epipolar Geometry using multiple views or multiple cameras. The values are obtained by direct estimation or through a learning or a statistical scheme. While others have used a switching approach that changes between image-based and position-based on a Lyapnov function. The early hybrid techniques that used a combination of image-based and pose-based (2D and 3D information) approaches for servoing required either a full or partial model of the object in order to extract the pose information and used a variety of techniques to extract the motion information from the image. used an affine motion model from the image motion in addition to a rough polyhedral CAD model to extract the object pose with respect to the camera to be able to servo onto the object (on the lines of PBVS). 2-1/2-D visual servoing developed by Malis et al. is a well known technique that breaks down the information required for servoing into an organized fashion which decouples rotations and translations. The papers assume that the desired pose is known a priori. The rotational information is obtained from partial pose estimation, a homography, (essentially 3D information) giving an axis of rotation and the angle (by computing the eigenvalues and eigenvectors of the homography). The translational information is obtained from the image directly by tracking a set of feature points. The only conditions being that the feature points being tracked never leave the field of view and that a depth estimate be predetermined by some off-line technique. 2-1/2-D servoing has been shown to be more stable than the techniques that preceded it. Another interesting observation with this formulation is that the authors claim that the visual Jacobian will have no singularities during the motions. The hybrid technique developed by Corke and Hutchinson, popularly called portioned approach partitions the visual (or image) Jacobian into motions (both rotations and translations) relating X and Y axes and motions related to the Z axis. outlines the technique, to break out columns of the visual Jacobian that correspond to the Z axis translation and rotation (namely, the third and sixth columns). The partitioned approach is shown to handle the Chaumette Conundrum discussed in. This technique requires a good depth estimate in order to function properly. outlines a hybrid approach where the servoing task is split into two, namely main and secondary. The main task is keep the features of interest within the field of view. While the secondary task is to mark a fixation point and use it as a reference to bring the camera to the desired pose. The technique does need a depth estimate from an off-line procedure. The paper discusses two examples for which depth estimates are obtained from robot odometry and by assuming that all

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  • VideoPoet

    VideoPoet

    VideoPoet is a large language model developed by Google Research in 2023 for video making. It can be asked to animate still images. The model accepts text, images, and videos as inputs, with a program to add feature for any input to any format generated content. VideoPoet was publicly announced on December 19, 2023. It uses an autoregressive language model.

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  • Replika

    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".

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  • Intrinsic dimension

    Intrinsic dimension

    In mathematics, the intrinsic dimension of a subset can be thought of as the minimal number of variables needed to represent the subset. The concept has widespread applications in geometry, dynamical systems, signal processing, statistics, and other fields. Due to its widespread applications and vague conceptualization, there are many different ways to define it rigorously. Consequently, the same set might have different intrinsic dimensions according to different definitions. The intrinsic dimension can be used as a lower bound of what dimension it is possible to compress a data set into through dimension reduction, but it can also be used as a measure of the complexity of the data set or signal. For a data set or signal of N variables, its intrinsic dimension M satisfies 0 ≤ M ≤ N, although estimators may yield higher values. == Exact dimension == === Differential === In differential geometry, given a differentiable manifold N and a submanifold M, the intrinsic dimension of M is its dimension. Suppose N has n dimensions and M has m dimensions, then that means around any point in M, there exists a local coordinate system ( x 1 , … , x m , x m + 1 , … , x n ) {\displaystyle (x_{1},\dots ,x_{m},x_{m+1},\dots ,x_{n})} of N, such that the manifold M is simply the subset of N defined by x m + 1 = 0 , … , x n = 0 {\displaystyle x_{m+1}=0,\dots ,x_{n}=0} . === Metric === Given a mere metric space, we can still define its intrinsic dimension. The most general case is the Hausdorff dimension, though for metric spaces occurring in practice, the box-counting dimension and the packing dimension often are identical to the Hausdorff dimension. Let X , d {\textstyle X,d} be a metric space and A ⊂ X {\textstyle A\subset X} be totally bounded. Define the covering number N ( A , ε ) = min { k : A ⊂ ⋃ i = 1 k B ( x i , ε ) } . {\displaystyle N(A,\varepsilon )=\min \left\{k:A\subset \bigcup _{i=1}^{k}B\left(x_{i},\varepsilon \right)\right\}.} The metric entropy is H ( A , ε ) = log ⁡ N ( A , ε ) {\textstyle H(A,\varepsilon )=\log N(A,\varepsilon )} (any log base). The upper and lower metric entropy dimensions are dim ¯ E A = lim sup ε ↓ 0 H ( A , ε ) log ⁡ ( 1 / ε ) , dim _ E A = lim inf ε ↓ 0 H ( A , ε ) log ⁡ ( 1 / ε ) . {\displaystyle {\overline {\dim }}_{E}A=\limsup _{\varepsilon \downarrow 0}{\frac {H(A,\varepsilon )}{\log(1/\varepsilon )}},\quad {\underline {\dim }}_{E}A=\liminf _{\varepsilon \downarrow 0}{\frac {H(A,\varepsilon )}{\log(1/\varepsilon )}}.} If they are equal, then dim E ⁡ A {\textstyle \operatorname {dim} _{E}A} is that common value, called the metric entropy dimension. The entropy dimensions are usually used in information theory, and especially coding theory, since entropy is involved in its definition. === Topological === If X {\displaystyle X} is merely a topological space, then we can still define its intrinsic dimension, using the topological dimension or Lebesgue covering dimension. An open cover of a topological space X is a family of open sets Uα such that their union is the whole space, ∪ α {\displaystyle \cup _{\alpha }} Uα = X. The order or ply of an open cover A {\displaystyle {\mathfrak {A}}} = {Uα} is the smallest number m (if it exists) for which each point of the space belongs to at most m open sets in the cover: in other words Uα1 ∩ ⋅⋅⋅ ∩ Uαm+1 = ∅ {\displaystyle \emptyset } for α1, ..., αm+1 distinct. A refinement of an open cover A {\displaystyle {\mathfrak {A}}} = {Uα} is another open cover B {\displaystyle {\mathfrak {B}}} = {Vβ}, such that each Vβ is contained in some Uα. The covering dimension of a topological space X is defined to be the minimum value of n such that every finite open cover A {\displaystyle {\mathfrak {A}}} of X has an open refinement B {\displaystyle {\mathfrak {B}}} with order n + 1. The refinement B {\displaystyle {\mathfrak {B}}} can always be chosen to be finite. Thus, if n is finite, Vβ1 ∩ ⋅⋅⋅ ∩ Vβn+2 = ∅ {\displaystyle \emptyset } for β1, ..., βn+2 distinct. If no such minimal n exists, the space is said to have infinite covering dimension. == Introductory example == Let f ( x 1 , x 2 ) {\textstyle f(x_{1},x_{2})} be a two-variable function (or signal) which is of the form f ( x 1 , x 2 ) = g ( x 1 ) {\textstyle f(x_{1},x_{2})=g(x_{1})} for some one-variable function g which is not constant. This means that f varies, in accordance to g, with the first variable or along the first coordinate. On the other hand, f is constant with respect to the second variable or along the second coordinate. It is only necessary to know the value of one, namely the first, variable in order to determine the value of f. Hence, it is a two-variable function but its intrinsic dimension is one. A slightly more complicated example is f ( x 1 , x 2 ) = g ( x 1 + x 2 ) {\textstyle f(x_{1},x_{2})=g(x_{1}+x_{2})} . f is still intrinsic one-dimensional, which can be seen by making a variable transformation y 1 = x 1 + x 2 {\textstyle y_{1}=x_{1}+x_{2}} and y 2 = x 1 − x 2 {\textstyle y_{2}=x_{1}-x_{2}} which gives f ( y 1 + y 2 2 , y 1 − y 2 2 ) = g ( y 1 ) {\textstyle f\left({\frac {y_{1}+y_{2}}{2}},{\frac {y_{1}-y_{2}}{2}}\right)=g\left(y_{1}\right)} . Since the variation in f can be described by the single variable y1 its intrinsic dimension is one. For the case that f is constant, its intrinsic dimension is zero since no variable is needed to describe variation. For the general case, when the intrinsic dimension of the two-variable function f is neither zero or one, it is two. In the literature, functions which are of intrinsic dimension zero, one, or two are sometimes referred to as i0D, i1D or i2D, respectively. == Signal processing == In signal processing of multidimensional signals, the intrinsic dimension of the signal describes how many variables are needed to generate a good approximation of the signal. For an N-variable function f, the set of variables can be represented as an N-dimensional vector x: f = f ( x ) where x = ( x 1 , … , x N ) {\textstyle f=f\left(\mathbf {x} \right){\text{ where }}\mathbf {x} =\left(x_{1},\dots ,x_{N}\right)} . If for some M-variable function g and M × N matrix A it is the case that for all x; f ( x ) = g ( A x ) , {\textstyle f(\mathbf {x} )=g(\mathbf {Ax} ),} M is the smallest number for which the above relation between f and g can be found, then the intrinsic dimension of f is M. The intrinsic dimension is a characterization of f, it is not an unambiguous characterization of g nor of A. That is, if the above relation is satisfied for some f, g, and A, it must also be satisfied for the same f and g′ and A′ given by g ′ ( y ) = g ( B y ) {\textstyle g'\left(\mathbf {y} \right)=g\left(\mathbf {By} \right)} and A ′ = B − 1 A {\textstyle \mathbf {A'} =\mathbf {B} ^{-1}\mathbf {A} } where B is a non-singular M × M matrix, since f ( x ) = g ′ ( A ′ x ) = g ( B A ′ x ) = g ( A x ) {\textstyle f\left(\mathbf {x} \right)=g'\left(\mathbf {A'x} \right)=g\left(\mathbf {BA'x} \right)=g\left(\mathbf {Ax} \right)} . == The Fourier transform of signals of low intrinsic dimension == An N variable function which has intrinsic dimension M < N has a characteristic Fourier transform. Intuitively, since this type of function is constant along one or several dimensions its Fourier transform must appear like an impulse (the Fourier transform of a constant) along the same dimension in the frequency domain. === A simple example === Let f be a two-variable function which is i1D. This means that there exists a normalized vector n ∈ R 2 {\textstyle \mathbf {n} \in \mathbb {R} ^{2}} and a one-variable function g such that f ( x ) = g ( n T x ) {\textstyle f(\mathbf {x} )=g(\mathbf {n} ^{\operatorname {T} }\mathbf {x} )} for all x ∈ R 2 {\textstyle \mathbf {x} \in \mathbb {R} ^{2}} . If F is the Fourier transform of f (both are two-variable functions) it must be the case that F ( u ) = G ( n T u ) ⋅ δ ( m T u ) {\textstyle F\left(\mathbf {u} \right)=G\left(\mathbf {n} ^{\mathrm {T} }\mathbf {u} \right)\cdot \delta \left(\mathbf {m} ^{\mathrm {T} }\mathbf {u} \right)} . Here G is the Fourier transform of g (both are one-variable functions), δ is the Dirac impulse function and m is a normalized vector in R 2 {\textstyle \mathbb {R} ^{2}} perpendicular to n. This means that F vanishes everywhere except on a line which passes through the origin of the frequency domain and is parallel to m. Along this line F varies according to G. === The general case === Let f be an N-variable function which has intrinsic dimension M, that is, there exists an M-variable function g and M × N matrix A such that f ( x ) = g ( A x ) ∀ x {\textstyle f(\mathbf {x} )=g(\mathbf {Ax} )\quad \forall \mathbf {x} } . Its Fourier transform F can then be described as follows: F vanishes everywhere except for a subspace of dimension M The subspace M is spanned by the rows of the matrix A In the subspace, F varies according to G the Fourier transform of g == Generalizations == The type of intrinsic dimension described above assume

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  • Photometric stereo

    Photometric stereo

    Photometric stereo is a technique in computer vision for estimating the surface normals of objects by observing that object under different lighting conditions (photometry). It is based on the fact that the amount of light reflected by a surface is dependent on the orientation of the surface in relation to the light source and the observer. By measuring the amount of light reflected into a camera, the space of possible surface orientations is limited. Given enough light sources from different angles, the surface orientation may be constrained to a single orientation or even overconstrained. The technique was originally introduced by Woodham in 1980. The special case where the data is a single image is known as shape from shading, and was analyzed by B. K. P. Horn in 1989. Photometric stereo has since been generalized to many other situations, including extended light sources and non-Lambertian surface finishes. Current research aims to make the method work in the presence of projected shadows, highlights, and non-uniform lighting. Photometric stereo is widely used in various fields, including archaeology, cultural heritage conservation, and quality control. It is now integrated into widely used open-source software, such as Meshroom. == Basic method == Under Woodham's original assumptions — Lambertian reflectance, known point-like distant light sources, and uniform albedo — the problem can be solved by inverting the linear equation I = L ⋅ n {\displaystyle I=L\cdot n} , where I {\displaystyle I} is a (known) vector of m {\displaystyle m} observed intensities, n {\displaystyle n} is the (unknown) surface normal, and L {\displaystyle L} is a (known) 3 × m {\displaystyle 3\times m} matrix of normalized light directions. This model can easily be extended to surfaces with non-uniform albedo, while keeping the problem linear. Taking an albedo reflectivity of k {\displaystyle k} , the formula for the reflected light intensity becomes I = k ( L ⋅ n ) . {\displaystyle I=k(L\cdot n).} If L {\displaystyle L} is square (there are exactly 3 lights) and non-singular, it can be inverted, giving L − 1 I = k n . {\displaystyle L^{-1}I=kn.} Since the normal vector is known to have length 1, k {\displaystyle k} must be the length of the vector k n {\displaystyle kn} , and n {\displaystyle n} is the normalised direction of that vector. If L {\displaystyle L} is not square (there are more than 3 lights), a generalisation of the inverse can be obtained using the Moore–Penrose pseudoinverse, by simply multiplying both sides with L T {\displaystyle L^{T}} , giving L T I = L T k ( L ⋅ n ) , {\displaystyle L^{T}I=L^{T}k(L\cdot n),} ( L T L ) − 1 L T I = k n , {\displaystyle (L^{T}L)^{-1}L^{T}I=kn,} after which the normal vector and albedo can be solved as described above. == Non-Lambertian surfaces == The classical photometric stereo problem concerns itself only with Lambertian surfaces, with perfectly diffuse reflection. This is unrealistic for many types of materials, especially metals, glass and smooth plastics, and will lead to aberrations in the resulting normal vectors. Many methods have been developed to lift this assumption. In this section, a few of these are listed. === Specular reflections === Historically, in computer graphics, the commonly used model to render surfaces started with Lambertian surfaces and progressed first to include simple specular reflections. Computer vision followed a similar course with photometric stereo. Specular reflections were among the first deviations from the Lambertian model. These are a few adaptations that have been developed. Many techniques ultimately rely on modelling the reflectance function of the surface, that is, how much light is reflected in each direction. This reflectance function has to be invertible. The reflected light intensities towards the camera is measured, and the inverse reflectance function is fit onto the measured intensities, resulting in a unique solution for the normal vector. === General BRDFs and beyond === According to the Bidirectional reflectance distribution function (BRDF) model, a surface may distribute the amount of light it receives in any outward direction. This is the most general known model for opaque surfaces. Some techniques have been developed to model (almost) general BRDFs. In practice, all of these require many light sources to obtain reliable data. These are methods in which surfaces with general BRDFs can be measured. Determine the explicit BRDF prior to scanning. To do this, a different surface is required that has the same or a very similar BRDF, of which the actual geometry (or at least the normal vectors for many points on the surface) is already known. The lights are then individually shone upon the known surface, and the amount of reflection into the camera is measured. Using this information, a look-up table can be created that maps reflected intensities for each light source to a list of possible normal vectors. This puts constraints on the possible normal vectors the surface may have, and reduces the photometric stereo problem to an interpolation between measurements. Typical known surfaces to calibrate the look-up table with are spheres for their wide variety of surface orientations. Restricting the BRDF to be symmetrical. If the BRDF is symmetrical, the direction of the light can be restricted to a cone about the direction to the camera. Which cone this is depends on the BRDF itself, the normal vector of the surface, and the measured intensity. Given enough measured intensities and the resulting light directions, these cones can be approximated and therefore the normal vectors of the surface. Some progress has been made towards modelling an even more general surfaces, such as Spatially Varying Bidirectional Distribution Functions (SVBRDF), Bidirectional surface scattering reflectance distribution functions (BSSRDF), and accounting for interreflections. However, such methods are still fairly restrictive in photometric stereo. Better results have been achieved with structured light. == Uncalibrated photometric stereo == Uncalibrated Photometric Stereo is an approach in photometric stereo that aims to reconstruct the 3D shape of an object from images captured under unknown lighting conditions. Unlike classical methods, which often assume controlled or known lighting setups, this approach removes these constraints, making it adaptable to diverse and real-world environments. The advent of deep learning has revolutionized universal PS by replacing handcrafted assumptions with data-driven models. Recent approaches leverage Transformer-based architectures and multi-scale encoder–decoder networks to directly estimate surface normals from input images. Uncalibrated Photometric Stereo is inherently an ill-posed problem, as it attempts to recover 3D shape and lighting conditions simultaneously from images alone. This leads to fundamental ambiguities in the reconstruction process, which manifest as systematic errors in the recovered geometry, including global distortions in the object's overall shape, and misinterpretation of surface orientation, where concave regions may appear convex and vice versa. To address the challenges of uncalibrated photometric stereo, hybrid methods have emerged that combine multi-view stereo and photometric stereo. These approaches leverage the strengths of both techniques, including geometric reliability and resolution.

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