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Visual analytics

Visual analytics is a multidisciplinary science and technology field that emerged from information visualization and scientific visualization. It focuses on how analytical reasoning can be facilitated by interactive visual interfaces. == Overview == Visual analytics is "the science of analytical reasoning facilitated by interactive visual interfaces." It can address problems whose size, complexity, and need for closely coupled human and machine analysis may make them otherwise intractable. Visual analytics advances scientific and technological development across multiple domains, including analytical reasoning, human–computer interaction, data transformations, visual representation for computation and analysis, analytic reporting, and the transition of new technologies into practice. As a research agenda, visual analytics brings together several scientific and technical communities from computer science, information visualization, cognitive and perceptual sciences, interactive design, graphic design, and social sciences. Visual analytics integrates new computational and theory-based tools with innovative interactive techniques and visual representations to enable human-information discourse. The design of the tools and techniques is based on cognitive, design, and perceptual principles. This science of analytical reasoning provides the reasoning framework upon which one can build both strategic and tactical visual analytics technologies for threat analysis, prevention, and response. Analytical reasoning is central to the analyst's task of applying human judgments to reach conclusions from a combination of evidence and assumptions. Visual analytics has some overlapping goals and techniques with information visualization and scientific visualization. There is currently no clear consensus on the boundaries between these fields, but broadly speaking the three areas can be distinguished as follows: Scientific visualization deals with data that has a natural geometric structure (e.g., MRI data, wind flows). Information visualization handles abstract data structures such as trees or graphs. Visual analytics is especially concerned with coupling interactive visual representations with underlying analytical processes (e.g., statistical procedures, data mining techniques) such that high-level, complex activities can be effectively performed (e.g., sense making, reasoning, decision making). Visual analytics seeks to marry techniques from information visualization with techniques from computational transformation and analysis of data. Information visualization forms part of the direct interface between user and machine, amplifying human cognitive capabilities in six basic ways: by increasing cognitive resources, such as by using a visual resource to expand human working memory, by reducing search, such as by representing a large amount of data in a small space, by enhancing the recognition of patterns, such as when information is organized in space by its time relationships, by supporting the easy perceptual inference of relationships that are otherwise more difficult to induce, by perceptual monitoring of a large number of potential events, and by providing a manipulable medium that, unlike static diagrams, enables the exploration of a space of parameter values These capabilities of information visualization, combined with computational data analysis, can be applied to analytic reasoning to support the sense-making process. == History == As an interdisciplinary approach, visual analytics has its roots in information visualization, cognitive sciences, and computer science. The term and scope of the field was defined in the early 2000s through researchers such as Jim Thomas, Kristin A. Cook, John Stasko, Pak Chung Wong, Daniel A. Keim and David S. Ebert. As a reaction to the September 11, 2001 attacks the United States Department of Homeland Security was established in late 2002, combining dozens of previously separated government agencies. Building upon earlier work on visual data mining by Daniel A. Keim starting in the late 1990s, this simultaneously lead to the development of a research agenda for visual analytics. As part of these efforts the National Visualization and Analytics Center (NVAC) at Pacific Northwest National Laboratory was established in 2004, whose charter was to develop system to mitigate information overload after the September 11, 2001 attacks in the intelligence community. Their research work determined core challenges, posed open research questions, and positioned visual analytics as a new research domain, in particular through the 2005 research agenda Illuminating the Path. In 2006, the IEEE VIS community led by Pak Chung Wong and Daniel A. Keim launched the annual IEEE Conference on Visual Analytics Science and Technology (VAST), providing a dedicated venue for research into visual analytics, which in 2020 merged to form the IEEE Visualization conference. In 2008, scope and challenges of visual analytics were conceptually defined by Daniel A. Keim and Jim Thomas in their influential book about visual data mining. The domain was further refined as part of the European Commissions FP7 VisMaster program in the late 2000s. == Topics == === Scope === Visual analytics is a multidisciplinary field that includes the following focus areas: Analytical reasoning techniques that enable users to obtain deep insights that directly support assessment, planning, and decision making Data representations and transformations that convert all types of conflicting and dynamic data in ways that support visualization and analysis Techniques to support production, presentation, and dissemination of the results of an analysis to communicate information in the appropriate context to a variety of audiences. Visual representations and interaction techniques that take advantage of the human eye's broad bandwidth pathway into the mind to allow users to see, explore, and understand large amounts of information at once. === Analytical reasoning techniques === Analytical reasoning techniques are the method by which users obtain deep insights that directly support situation assessment, planning, and decision making. Visual analytics must facilitate high-quality human judgment with a limited investment of the analysts’ time. Visual analytics tools must enable diverse analytical tasks such as: Understanding past and present situations quickly, as well as the trends and events that have produced current conditions Identifying possible alternative futures and their warning signs Monitoring current events for emergence of warning signs as well as unexpected events Determining indicators of the intent of an action or an individual Supporting the decision maker in times of crisis. These tasks will be conducted through a combination of individual and collaborative analysis, often under extreme time pressure. Visual analytics must enable hypothesis-based and scenario-based analytical techniques, providing support for the analyst to reason based on the available evidence. === Data representations === Data representations are structured forms suitable for computer-based transformations. These structures must exist in the original data or be derivable from the data themselves. They must retain the information and knowledge content and the related context within the original data to the greatest degree possible. The structures of underlying data representations are generally neither accessible nor intuitive to the user of the visual analytics tool. They are frequently more complex in nature than the original data and are not necessarily smaller in size than the original data. The structures of the data representations may contain hundreds or thousands of dimensions and be unintelligible to a person, but they must be transformable into lower-dimensional representations for visualization and analysis. === Theories of visualization === Theories of visualization include: Jacques Bertin's Semiology of Graphics (1967) Nelson Goodman's Languages of Art (1977) Jock D. Mackinlay's Automated design of optimal visualization (APT) (1986) Leland Wilkinson's Grammar of Graphics (1998) Hadley Wickham's Layered Grammar of Graphics (2010) === Visual representations === Visual representations translate data into a visible form that highlights important features, including commonalities and anomalies. These visual representations make it easy for users to perceive salient aspects of their data quickly. Augmenting the cognitive reasoning process with perceptual reasoning through visual representations permits the analytical reasoning process to become faster and more focused. == Process == The input for the data sets used in the visual analytics process are heterogeneous data sources (i.e., the internet, newspapers, books, scientific experiments, expert systems). From these rich sources, the data sets S = S1, ..., Sm are chosen, whereas each Si , i ∈ (1, ..., m) consists of attrib
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Hebbian theory

Hebbian theory is a neuropsychological theory claiming that an increase in synaptic efficacy arises from a presynaptic cell's repeated and persistent stimulation of a postsynaptic cell. It is an attempt to explain synaptic plasticity, the adaptation of neurons during the learning process. Hebbian theory was introduced by Donald Hebb in his 1949 book The Organization of Behavior. The theory is also called Hebb's rule, Hebb's law, Hebb's postulate, and cell assembly theory. Hebb states it as follows: Let us assume that the persistence or repetition of a reverberatory activity (or "trace") tends to induce lasting cellular changes that add to its stability. ... When an axon of cell A is near enough to excite a cell B and repeatedly or persistently takes part in firing it, some growth process or metabolic change takes place in one or both cells such that A's efficiency, as one of the cells firing B, is increased. The theory is often summarized as "Neurons that fire together, wire together." However, Hebb emphasized that cell A needs to "take part in firing" cell B, and such causality can occur only if cell A fires just before, not at the same time as, cell B. This aspect of causation in Hebb's work foreshadowed what is now known about spike-timing-dependent plasticity, which requires temporal precedence. Hebbian theory attempts to explain associative or Hebbian learning, in which simultaneous activation of cells leads to pronounced increases in synaptic strength between those cells. It also provides a biological basis for errorless learning methods for education and memory rehabilitation. In the study of neural networks in cognitive function, it is often regarded as the neuronal basis of unsupervised learning. == Engrams, cell assembly theory, and learning == Hebbian theory provides an explanation for how neurons might connect to become engrams, which may be stored in overlapping cell assemblies, or groups of neurons that encode specific information. Initially created as a way to explain recurrent activity in specific groups of cortical neurons, Hebb's theories on the form and function of cell assemblies can be understood from the following: The general idea is an old one, that any two cells or systems of cells that are repeatedly active at the same time will tend to become 'associated' so that activity in one facilitates activity in the other. Hebb also wrote: When one cell repeatedly assists in firing another, the axon of the first cell develops synaptic knobs (or enlarges them if they already exist) in contact with the soma of the second cell. D. Alan Allport posits additional ideas regarding cell assembly theory and its role in forming engrams using the concept of auto-association, or the brain's ability to retrieve information based on a partial cue, described as follows: If the inputs to a system cause the same pattern of activity to occur repeatedly, the set of active elements constituting that pattern will become increasingly strongly inter-associated. That is, each element will tend to turn on every other element and (with negative weights) to turn off the elements that do not form part of the pattern. To put it another way, the pattern as a whole will become 'auto-associated'. We may call a learned (auto-associated) pattern an engram. Research conducted in the laboratory of Nobel laureate Eric Kandel has provided evidence supporting the role of Hebbian learning mechanisms at synapses in the marine gastropod Aplysia californica. Because synapses in the peripheral nervous system of marine invertebrates are much easier to control in experiments, Kandel's research found that Hebbian long-term potentiation along with activity-dependent presynaptic facilitation are both necessary for synaptic plasticity and classical conditioning in Aplysia californica. While research on invertebrates has established fundamental mechanisms of learning and memory, much of the work on long-lasting synaptic changes between vertebrate neurons involves the use of non-physiological experimental stimulation of brain cells. However, some of the physiologically relevant synapse modification mechanisms that have been studied in vertebrate brains do seem to be examples of Hebbian processes. One such review indicates that long-lasting changes in synaptic strengths can be induced by physiologically relevant synaptic activity using both Hebbian and non-Hebbian mechanisms. == Principles == In artificial neurons and artificial neural networks, Hebb's principle can be described as a method of determining how to alter the weights between model neurons. The weight between two neurons increases if the two neurons activate simultaneously, and reduces if they activate separately. Nodes that tend to be either both positive or both negative at the same time have strong positive weights, while those that tend to be opposite have strong negative weights. The following is a formulaic description of Hebbian learning (many other descriptions are possible): w i j = x i x j , {\displaystyle \,w_{ij}=x_{i}x_{j},} where w i j {\displaystyle w_{ij}} is the weight of the connection from neuron j {\displaystyle j} to neuron i {\displaystyle i} , and x i {\displaystyle x_{i}} is the input for neuron i {\displaystyle i} . This is an example of pattern learning, where weights are updated after every training example. In a Hopfield network, connections w i j {\displaystyle w_{ij}} are set to zero if i = j {\displaystyle i=j} (no reflexive connections allowed). With binary neurons (activations either 0 or 1), connections would be set to 1 if the connected neurons have the same activation for a pattern. When several training patterns are used, the expression becomes an average of the individuals: w i j = 1 p ∑ k = 1 p x i k x j k , {\displaystyle w_{ij}={\frac {1}{p}}\sum _{k=1}^{p}x_{i}^{k}x_{j}^{k},} where w i j {\displaystyle w_{ij}} is the weight of the connection from neuron j {\displaystyle j} to neuron i {\displaystyle i} , p {\displaystyle p} is the number of training patterns and x i k {\displaystyle x_{i}^{k}} the k {\displaystyle k} -th input for neuron i {\displaystyle i} . This is learning by epoch, with weights updated after all the training examples are presented and is last term applicable to both discrete and continuous training sets. Again, in a Hopfield network, connections w i j {\displaystyle w_{ij}} are set to zero if i = j {\displaystyle i=j} (no reflexive connections). A variation of Hebbian learning that takes into account phenomena such as blocking and other neural learning phenomena is the mathematical model of Harry Klopf. Klopf's model assumes that parts of a system with simple adaptive mechanisms can underlie more complex systems with more advanced adaptive behavior, such as neural networks. == Relationship to unsupervised learning, stability, and generalization == Because of the simple nature of Hebbian learning, based only on the coincidence of pre- and post-synaptic activity, it may not be intuitively clear why this form of plasticity leads to meaningful learning. However, it can be shown that Hebbian plasticity does pick up the statistical properties of the input in a way that can be categorized as unsupervised learning. This can be mathematically shown in a simplified example. Let us work under the simplifying assumption of a single rate-based neuron of rate y ( t ) {\displaystyle y(t)} , whose inputs have rates x 1 ( t ) . . . x N ( t ) {\displaystyle x_{1}(t)...x_{N}(t)} . The response of the neuron y ( t ) {\displaystyle y(t)} is usually described as a linear combination of its input, ∑ i w i x i {\displaystyle \sum _{i}w_{i}x_{i}} , followed by a response function f {\displaystyle f} : y = f ( ∑ i = 1 N w i x i ) . {\displaystyle y=f\left(\sum _{i=1}^{N}w_{i}x_{i}\right).} As defined in the previous sections, Hebbian plasticity describes the evolution in time of the synaptic weight w {\displaystyle w} : d w i d t = η x i y . {\displaystyle {\frac {dw_{i}}{dt}}=\eta x_{i}y.} Assuming, for simplicity, an identity response function f ( a ) = a {\displaystyle f(a)=a} , we can write d w i d t = η x i ∑ j = 1 N w j x j {\displaystyle {\frac {dw_{i}}{dt}}=\eta x_{i}\sum _{j=1}^{N}w_{j}x_{j}} or in matrix form: d w d t = η x x T w . {\displaystyle {\frac {d\mathbf {w} }{dt}}=\eta \mathbf {x} \mathbf {x} ^{T}\mathbf {w} .} As in the previous chapter, if training by epoch is done an average ⟨ … ⟩ {\displaystyle \langle \dots \rangle } over discrete or continuous (time) training set of x {\displaystyle \mathbf {x} } can be done: d w d t = ⟨ η x x T w ⟩ = η ⟨ x x T ⟩ w = η C w . {\displaystyle {\frac {d\mathbf {w} }{dt}}=\langle \eta \mathbf {x} \mathbf {x} ^{T}\mathbf {w} \rangle =\eta \langle \mathbf {x} \mathbf {x} ^{T}\rangle \mathbf {w} =\eta C\mathbf {w} .} where C = ⟨ x x T ⟩ {\displaystyle C=\langle \,\mathbf {x} \mathbf {x} ^{T}\rangle } is the correlation matrix of the input under the additional assumption that ⟨ x ⟩ = 0 {\displaystyle \langle \mathbf
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The Best Free AI Video Generator for Beginners

Trying to pick the best AI video generator? An AI video generator is software that uses machine learning to help you get more done — it scales effortlessly from a single task to thousands. The best picks balance beginner-friendly simplicity with the depth power users need, and they ship updates often. Whether you are a beginner or a pro, the right AI video generator slots into your workflow and pays for itself fast. Read on for hands-on impressions, pricing tiers, and the standout features that matter.
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State complexity

State complexity is an area of theoretical computer science dealing with the size of abstract automata, such as different kinds of finite automata. The classical result in the area is that simulating an n {\displaystyle n} -state nondeterministic finite automaton by a deterministic finite automaton requires exactly 2 n {\displaystyle 2^{n}} states in the worst case. == Transformation between variants of finite automata == Finite automata can be deterministic and nondeterministic, one-way (DFA, NFA) and two-way (2DFA, 2NFA). Other related classes are unambiguous (UFA), self-verifying (SVFA) and alternating (AFA) finite automata. These automata can also be two-way (2UFA, 2SVFA, 2AFA). All these machines can accept exactly the regular languages. However, the size of different types of automata necessary to accept the same language (measured in the number of their states) may be different. For any two types of finite automata, the state complexity tradeoff between them is an integer function f {\displaystyle f} where f ( n ) {\displaystyle f(n)} is the least number of states in automata of the second type sufficient to recognize every language recognized by an n {\displaystyle n} -state automaton of the first type. The following results are known. NFA to DFA: 2 n {\displaystyle 2^{n}} states. This is the subset construction by Rabin and Scott, proved optimal by Lupanov. UFA to DFA: 2 n {\displaystyle 2^{n}} states, see Leung, An earlier lower bound by Schmidt was smaller. NFA to UFA: 2 n − 1 {\displaystyle 2^{n}-1} states, see Leung. There was an earlier smaller lower bound by Schmidt. SVFA to DFA: Θ ( 3 n / 3 ) {\displaystyle \Theta (3^{n/3})} states, see Jirásková and Pighizzini 2DFA to DFA: n ( n n − ( n − 1 ) n ) {\displaystyle n(n^{n}-(n-1)^{n})} states, see Kapoutsis. Earlier construction by Shepherdson used more states, and an earlier lower bound by Moore was smaller. 2DFA to NFA: ( 2 n n + 1 ) = O ( 4 n n ) {\displaystyle {\binom {2n}{n+1}}=O({\frac {4^{n}}{\sqrt {n}}})} , see Kapoutsis. Earlier construction by Birget used more states. 2NFA to NFA: ( 2 n n + 1 ) {\displaystyle {\binom {2n}{n+1}}} , see Kapoutsis. 2NFA to NFA accepting the complement: O ( 4 n ) {\displaystyle O(4^{n})} states, see Vardi. AFA to DFA: 2 2 n {\displaystyle 2^{2^{n}}} states, see Chandra, Kozen and Stockmeyer. AFA to NFA: 2 n {\displaystyle 2^{n}} states, see Fellah, Jürgensen and Yu. 2AFA to DFA: 2 n 2 n {\displaystyle 2^{n2^{n}}} , see Ladner, Lipton and Stockmeyer. 2AFA to NFA: 2 Θ ( n log ⁡ n ) {\displaystyle 2^{\Theta (n\log n)}} , see Geffert and Okhotin. === The 2DFA vs. 2NFA problem and logarithmic space === It is an open problem whether all 2NFAs can be converted to 2DFAs with polynomially many states, i.e. whether there is a polynomial p ( n ) {\displaystyle p(n)} such that for every n {\displaystyle n} -state 2NFA there exists a p ( n ) {\displaystyle p(n)} -state 2DFA. The problem was raised by Sakoda and Sipser, who compared it to the P vs. NP problem in the computational complexity theory. Berman and Lingas discovered a formal relation between this problem and the L vs. NL open problem. This relation was further elaborated by Kapoutsis. == State complexity of operations for finite automata == Given a binary regularity-preserving operation on languages ∘ {\displaystyle \circ } and a family of automata X (DFA, NFA, etc.), the state complexity of ∘ {\displaystyle \circ } is an integer function f ( m , n ) {\displaystyle f(m,n)} such that for each m-state X-automaton A and n-state X-automaton B there is an f ( m , n ) {\displaystyle f(m,n)} -state X-automaton for L ( A ) ∘ L ( B ) {\displaystyle L(A)\circ L(B)} , and for all integers m, n there is an m-state X-automaton A and an n-state X-automaton B such that every X-automaton for L ( A ) ∘ L ( B ) {\displaystyle L(A)\circ L(B)} must have at least f ( m , n ) {\displaystyle f(m,n)} states. Analogous definition applies for operations with any number of arguments. The first results on state complexity of operations for DFAs were published by Maslov and by Yu, Zhuang and Salomaa. Holzer and Kutrib pioneered the state complexity of operations on NFA. The known results for basic operations are listed below. === Union === If language L 1 {\displaystyle L_{1}} requires m states and language L 2 {\displaystyle L_{2}} requires n states, how many states does L 1 ∪ L 2 {\displaystyle L_{1}\cup L_{2}} require? DFA: m n {\displaystyle mn} states, see Maslov and Yu, Zhuang and Salomaa. NFA: m + n + 1 {\displaystyle m+n+1} states, see Holzer and Kutrib. UFA: at least min ( n , m ) Ω ( log ⁡ ( min ( n , m ) ) ) {\displaystyle \min(n,m)^{\Omega (\log(\min(n,m)))}} ; between m n + m + n {\displaystyle mn+m+n} and m + n m 2 0.79 m {\displaystyle m+nm2^{0.79m}} states, see Jirásek, Jirásková and Šebej. SVFA: m n {\displaystyle mn} states, see Jirásek, Jirásková and Szabari. 2DFA: between m + n {\displaystyle m+n} and 4 m + n + 4 {\displaystyle 4m+n+4} states, see Kunc and Okhotin. 2NFA: m + n {\displaystyle m+n} states, see Kunc and Okhotin. === Intersection === How many states does L 1 ∩ L 2 {\displaystyle L_{1}\cap L_{2}} require? DFA: m n {\displaystyle mn} states, see Maslov and Yu, Zhuang and Salomaa. NFA: m n {\displaystyle mn} states, see Holzer and Kutrib. UFA: m n {\displaystyle mn} states, see Jirásek, Jirásková and Šebej. SVFA: m n {\displaystyle mn} states, see Jirásek, Jirásková and Szabari. 2DFA: between m + n {\displaystyle m+n} and m + n + 1 {\displaystyle m+n+1} states, see Kunc and Okhotin. 2NFA: between m + n {\displaystyle m+n} and m + n + 1 {\displaystyle m+n+1} states, see Kunc and Okhotin. === Complementation === If language L requires n states then how many states does its complement require? DFA: n {\displaystyle n} states, by exchanging accepting and rejecting states. NFA: 2 n {\displaystyle 2^{n}} states, see Birget. or Jirásková UFA: at least n Ω ~ ( log ⁡ n ) {\displaystyle n^{{\tilde {\Omega }}(\log n)}} states, see Göös, Kiefer and Yuan, (this follows an earlier bound by Raskin); and at most n + 1 ⋅ 2 0.5 n {\displaystyle {\sqrt {n+1}}\cdot 2^{0.5n}} states, see Indzhev and Kiefer. SVFA: n {\displaystyle n} states, by exchanging accepting and rejecting states. 2DFA: at least n {\displaystyle n} and at most 4 n {\displaystyle 4n} states, see Geffert, Mereghetti and Pighizzini. === Concatenation === How many states does L 1 L 2 = { w 1 w 2 ∣ w 1 ∈ L 1 , w 2 ∈ L 2 } {\displaystyle L_{1}L_{2}=\{w_{1}w_{2}\mid w_{1}\in L_{1},w_{2}\in L_{2}\}} require? DFA: m ⋅ 2 n − 2 n − 1 {\displaystyle m\cdot 2^{n}-2^{n-1}} states, see Maslov and Yu, Zhuang and Salomaa. NFA: m + n {\displaystyle m+n} states, see Holzer and Kutrib. UFA: 3 4 2 m + n − 1 {\displaystyle {\frac {3}{4}}2^{m+n}-1} states, see Jirásek, Jirásková and Šebej. SVFA: Θ ( 3 n / 3 2 m ) {\displaystyle \Theta (3^{n/3}2^{m})} states, see Jirásek, Jirásková and Szabari. 2DFA: at least 2 Ω ( n ) log ⁡ m {\displaystyle {\frac {2^{\Omega (n)}}{\log m}}} and at most 2 m m + 1 ⋅ 2 n n + 1 {\displaystyle 2m^{m+1}\cdot 2^{n^{n+1}}} states, see Jirásková and Okhotin. === Kleene star === DFA: 3 4 2 n {\displaystyle {\frac {3}{4}}2^{n}} states, see Maslov and Yu, Zhuang and Salomaa. NFA: n + 1 {\displaystyle n+1} states, see Holzer and Kutrib. UFA: 3 4 2 n {\displaystyle {\frac {3}{4}}2^{n}} states, see Jirásek, Jirásková and Šebej. SVFA: 3 4 2 n {\displaystyle {\frac {3}{4}}2^{n}} states, see Jirásek, Jirásková and Szabari. 2DFA: at least 1 n 2 n 2 − 1 {\displaystyle {\frac {1}{n}}2^{{\frac {n}{2}}-1}} and at most 2 O ( n n + 1 ) {\displaystyle 2^{O(n^{n+1})}} states, see Jirásková and Okhotin. === Reversal === DFA: 2 n {\displaystyle 2^{n}} states, see Mirkin, Leiss, and Yu, Zhuang and Salomaa. NFA: n + 1 {\displaystyle n+1} states, see Holzer and Kutrib. UFA: n {\displaystyle n} states. SVFA: 2 n + 1 {\displaystyle 2n+1} states, see Jirásek, Jirásková and Szabari. 2DFA: between n + 1 {\displaystyle n+1} and n + 2 {\displaystyle n+2} states, see Jirásková and Okhotin. == Finite automata over a unary alphabet == State complexity of finite automata with a one-letter (unary) alphabet, pioneered by Chrobak, is different from the multi-letter case. Let g ( n ) = e Θ ( n ln ⁡ n ) {\displaystyle g(n)=e^{\Theta ({\sqrt {n\ln n}})}} be Landau's function. === Transformation between models === For a one-letter alphabet, transformations between different types of finite automata are sometimes more efficient than in the general case. NFA to DFA: g ( n ) + O ( n 2 ) {\displaystyle g(n)+O(n^{2})} states, see Chrobak. 2DFA to DFA: g ( n ) + O ( n ) {\displaystyle g(n)+O(n)} states, see Chrobak and Kunc and Okhotin. 2NFA to DFA: O ( g ( n ) ) {\displaystyle O(g(n))} states, see Mereghetti and Pighizzini. and Geffert, Mereghetti and Pighizzini. NFA to 2DFA: at most O ( n 2 ) {\displaystyle O(n^{2})} states, see Chrobak. 2NFA to 2DFA: at most n O ( log ⁡ n ) {\displaystyle n^{O(\log n)}} states, proved by implementing the method of Savitch's theorem, see
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Lessac Technologies

Lessac Technologies, Inc. (LTI) is an American firm which develops voice synthesis software, licenses technology and sells synthesized novels as MP3 files. The firm currently has seven patents granted and three more pending for its automated methods of converting digital text into human-sounding speech, more accurately recognizing human speech and outputting the text representing the words and phrases of said speech, along with recognizing the speaker's emotional state. The LTI technology is partly based on the work of the late Arthur Lessac, a Professor of Theater at the State University of New York and the creator of Lessac Kinesensic Training, and LTI has licensed exclusive rights to exploit Arthur Lessac's copyrighted works in the fields of speech synthesis and speech recognition. Based on the view that music is speech and speech is music, Lessac's work and books focused on body and speech energies and how they go together. Arthur Lessac's textual annotation system, which was originally developed to assist actors, singers, and orators in marking up scripts to prepare for performance, is adapted in LTI's speech synthesis system as the basic representation of the speech to be synthesized (Lessemes), in contrast to many other systems which use a phonetic representation. LTI's software has two major components: (1) a linguistic front-end that converts plain text to a sequence of prosodic and phonosensory graphic symbols (Lessemes) based on Arthur Lessac's annotation system, which specify the speech units to be synthesized; (2) a signal-processing back-end that takes the Lessemes as acoustic data and produces human-sounding synthesized speech as output, using unit selection and concatenation. LTI's text-to-speech system came in second in the world-wide Blizzard Challenge 2011 and 2012. The first-place team in 2011 also employed LTI's "front-end" technology, but with its own back-end. The Blizzard Challenge, conducted by the Language Technologies Institute of Carnegie Mellon University, was devised as a way to evaluate speech synthesis techniques by having different research groups build voices from the same voice-actor recordings, and comparing the results through listening tests. LTI was founded in 2000 by H. Donald Wilson (chairman), a lawyer, LexisNexis entrepreneur and business associate of Arthur Lessac; and Gary A. Marple (chief inventor), after Marple suggested that Arthur Lessac's kinesensic voice training might be applicable to computational linguistics. After Wilson's death in 2006, his nephew John Reichenbach became the firm's CEO.
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AI Content Generators: Free vs Paid (2026)

Shopping for the best AI content generator? An AI content generator is software that uses machine learning to help you get more done — it keeps getting smarter as the underlying models improve. Pricing, accuracy, and the size of the model behind the tool are the three factors that most affect daily usefulness. Whether you are a beginner or a pro, the right AI content generator slots into your workflow and pays for itself fast. We tested the leading options and ranked them by quality, value, and ease of use.
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Margaret Mitchell (scientist)

Margaret Mitchell is a computer scientist who works on algorithmic bias and fairness in machine learning. She is most well known for her work on automatically removing undesired biases concerning demographic groups from machine learning models, as well as more transparent reporting of their intended use. == Education == Mitchell obtained a bachelor's degree in linguistics from Reed College, Portland, Oregon, in 2005. After having worked as a research assistant at the OGI School of Science and Engineering for two years, she subsequently obtained a Master's in Computational Linguistics from the University of Washington in 2009. She enrolled in a PhD program at the University of Aberdeen, where she wrote a doctoral thesis on the topic of Generating Reference to Visible Objects, graduating in 2013. == Career and research == Mitchell is best known for her work on fairness in machine learning and methods for mitigating algorithmic bias. This includes her work on introducing the concept of 'Model Cards' for more transparent model reporting, and methods for debiasing machine learning models using adversarial learning. Margaret Mitchell created the framework for recognizing and avoiding biases by testing with a variable for the group of interest, predictor and an adversary. In 2012, Mitchell joined the Human Language Technology Center of Excellence at Johns Hopkins University as a postdoctoral researcher, before taking up a position at Microsoft Research in 2013. At Microsoft, Mitchell was the research lead of the Seeing AI project, an app that offers support for the visually impaired by narrating texts and images. In November 2016, she became a senior research scientist at Google Research and Machine intelligence. While at Google, she founded and co-led the Ethical Artificial Intelligence team together with Timnit Gebru. In May 2018, she represented Google in the Partnership on AI. In February 2018, she gave a TED talk on "How we can build AI to help humans, not hurt us". In January 2021, after Timnit Gebru's termination from Google, Mitchell reportedly used a script to search through her corporate account and download emails that allegedly documented discriminatory incidents involving Gebru. An automated system locked Mitchell's account in response. In response to media attention Google claimed that she "exfiltrated thousands of files and shared them with multiple external accounts". After a five-week investigation, Mitchell was fired. Prior to her dismissal, Mitchell had been a vocal advocate for diversity at Google, and had voiced concerns about research censorship at the company. In late 2021, she joined AI start-up Hugging Face. Mitchell is a co-founder of Widening NLP, a special interest group within the Association for Computational Linguistics (ACL) seeking to increase the proportion of women and minorities working in natural language processing; and Computational Linguistics and Clinical Psychology, an annual workshop within the ACL that brings together clinicians and computational linguists to advance the state of the art in clinical psychology.
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Dan Hendrycks

Dan Hendrycks (born 1994 or 1995) is an American machine learning researcher. He serves as the director of the Center for AI Safety, a nonprofit research organization based in San Francisco, California. == Early life and education == Hendrycks was raised in a Christian evangelical household in Marshfield, Missouri. He received a B.S. from the University of Chicago in 2018 and a Ph.D. from the University of California, Berkeley in Computer Science in 2022. == Career and research == Hendrycks' research focuses on topics that include machine learning safety, machine ethics, and robustness. He credits his participation in the effective altruism (EA) movement-linked 80,000 Hours program for his career focus towards AI safety, though denies being an advocate for EA. Hendrycks is the main author of the research paper that introduced the activation function GELU in 2016, and of the paper that introduced the language model benchmark MMLU (Massive Multitask Language Understanding) in 2020. In February 2022, Hendrycks co-authored recommendations for the US National Institute of Standards and Technology (NIST) to inform the management of risks from artificial intelligence. In September 2022, Hendrycks wrote a paper providing a framework for analyzing the impact of AI research on societal risks. He later published a paper in March 2023 examining how natural selection and competitive pressures could shape the goals of artificial agents. This was followed by "An Overview of Catastrophic AI Risks", which discusses four categories of risks: malicious use, AI race dynamics, organizational risks, and rogue AI agents. Hendrycks is the safety adviser of xAI, an AI startup company founded by Elon Musk in 2023. To avoid any potential conflicts of interest, he receives a symbolic one-dollar salary and holds no company equity. In November 2024, he also joined Scale AI as an advisor collecting a one-dollar salary. Hendrycks is the creator of Humanity's Last Exam, a benchmark for evaluating the capabilities of large language models, which he developed in collaboration with Scale AI. In 2024, Hendrycks published the textbook Introduction to AI Safety, Ethics, and Society, based on courseware he had previously developed. == Selected publications == Hendrycks, Dan; Gimpel, Kevin (2020-07-08). "Gaussian Error Linear Units (GELUs)". arXiv:1606.08415 [cs.LG]. Hendrycks, Dan; Gimpel, Kevin (2018-10-03). "A Baseline for Detecting Misclassified and Out-of-Distribution Examples in Neural Networks". International Conference on Learning Representations 2017. arXiv:1610.02136. Hendrycks, Dan; Mazeika, Mantas; Dietterich, Thomas (2019-01-28). "Deep Anomaly Detection with Outlier Exposure". International Conference on Learning Representations 2019. arXiv:1812.04606. Hendrycks, Dan; Mazeika, Mantas; Zou, Andy (2021-10-25). "What Would Jiminy Cricket Do? Towards Agents That Behave Morally". Conference on Neural Information Processing Systems 2021. arXiv:2110.13136.
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Photoanalysis

Photoanalysis (or photo analysis) refers to the study of pictures to compile various types of data, for example, to measure the size distribution of virtually anything that can be captured by photo. Photoanalysis technology has changed the way mines and mills quantify fragmented material. Images are an effective way to document conditions before, after, and even during blasting activities. The technology is advancing at a high rate, and lenses, storage media memory, light sensitivity and resolution have been improving steadily. Today's digital cameras and camcorders include high-resolution optics, compact size, automatic time and date stamps, good battery life, shutters to freeze motion, and computers to autofocus and eliminate jitter using image stabilization. == Mining == Photoanalysis in mining operations can provide an automated system that forewarns a company of potential problems with materials, leading to economies and reduced damage caused from over-sized materials. It can also help determine the effectiveness of blasts. A company can use this technology to monitor materials moving on a conveyor belt in an underground environment, to measure piles left over from a blast, and even measure the amount of material being carried by dump trucks or vessels to a destination. Photoanalysis is being used on SAG mills worldwide to control the size of rock being crushed. Companies are using this technology to determine the size of particles being processed in the SAG Mill.[1] Archived 2009-05-23 at the Wayback Machine Having oversize material entering the SAG mill makes an operation less efficient, costing companies money in electrical and maintenance costs. Photoanalysis technology can eliminate unwanted material before it enters the mill, keeping rock crushing costs low. == Forestry == Wood chip size can affect the overall quality of a product. With automated photoanalysis systems, companies can remove any unwanted wrong-size particles without stopping their mill process. Photoanalysis can affect how efficiently forestry companies operate. In mills worldwide, photoanalysis technology is improving the use of lumber products, cutting back on the amount of trees being used to operate, and saving companies money through quality control optimization.[2] With the current downturn in the North American forestry industry, operators are looking at making their mills more efficient and effective when processing materials. Photoanalysis technology helps identify any weaknesses in the process by continuously monitoring different sections of an operation. == Agriculture == Agricultural companies can, using photoanalysis, monitor conveyor belts of food without contaminating the product by touching it. Other benefits of photoanalysis systems include: Automated removal of any unwanted material on food conveyor Improved quality control for the most important parts of the agricultural process Pinpoint accuracy that helps the efficiency and effectiveness of product handling techniques The importance of photoanalysis technology is being noticed by the agricultural industry as it identifies any unwanted materials going through the process. In an example, if a mouse is on a conveyor of corn, photoanalysis technology would be able to identify the unwanted object and remove it before it contaminates the whole process. == Origins of photoanalysis technology == Photoanalysis technology was created by using the Waterloo Image Enhancement Process in the 1980s. After further development of the imaging process with explosives producer DuPont, engineers Tom Palangio and Takis Katsabanis began selling photoanalysis software commercially. They later renamed the process WipFrag, standing for Waterloo Image Process Fragmentation Today, photoanalysis technology has evolved into stabilized and portable systems that can automatically capture and analyze results instantly. Thousands of these products are currently being used around the world to measure fragmented material. == Photoanalysis equipment photos == == Fragmentation analysis == Fragmentation analysis is becoming a popular term in mining, agricultural and forestry industries. With the majority of money in these industries directed towards the proper sizing of materials, companies are using fragmentation analysis to determine various factors within an operation.[3] The two main ways a company keeps track of fragmented material are through manual and automated sieving procedures. Manual sieving involves extracting a sample of material to analyze the size distribution. The results can be tabulated within two days. Automated sieving is an advanced way of sieving materials running through a process. Without having to extract the material, photoanalysis can take place, allowing for immediate results with pinpoint accuracy. == Blast Fragmentation Software == Operators are using fragmentation analysis to determine the effectiveness of various blasts. With automated sieving technology, workers can track the success of these blasts and receive instant results. Companies are using these results to determine what blasting method yielded the best results for their specific operation. The common variables associated with blast optimization are the provided Particle Size Distribution (PSD) from a shovel fragmentation system, geology including rock type and fracturing, and energy factor. By using photoanalysis the fragmented materials can be monitored, offering pinpoint accuracy and allowing mine operators to make adjustments to future blasting procedures. See Optical Granulometry to view the automated sieving process. == Pre-crushing analysis == Maintenance costs can be significantly reduced if an operation focuses on the fragmentation of the particles passing through their process. Automated sieving systems can detect and help remove any oversize material before it enters the crusher and causes maintenance problems. It also helps determine the effectiveness of the mining process prior to crushing; the sizing of material is always a critical part of operations in the mining, forestry and agricultural industries. Having an analysis taking place at every major point in an operation allows for the proper tracking of material being processed. Engineers can then determine what part of the process needs improving based solely on the size of material. == Post-crushing analysis == Measuring how effective industrial crushers are, can help save a company millions of dollars in energy costs on an annual basis. There are two components that affect a typical crusher: the size of the material inputted, and the speed at which the crusher is moving. If the user can find a perfect balance between these two components, the materials will be crushed to the right size in the shortest time possible. Meeting the material standards set by governments and large companies can be hard. Having a post-crushing analysis taking place ensures that no oversize material gets shipped; eliminating the chance of getting fined for not meeting industry specifications.
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Best AI Headshot Generators in 2026

In search of the best AI headshot generator? An AI headshot generator is software that uses machine learning to help you get more done — it turns a rough idea into a polished result in seconds. When choosing one, weigh output quality, pricing, export formats, and how well it fits the tools you already use. Whether you are a beginner or a pro, the right AI headshot generator slots into your workflow and pays for itself fast. Below we compare features, pricing, and real output so you can choose with confidence.
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Sasha Luccioni

Alexandra Sasha Luccioni (née Vorobyova; born 1990) is a computer scientist specializing in the intersection of artificial intelligence (AI) and climate change. Her work focuses on quantifying the environmental impact of AI technologies and promoting sustainable practices in machine learning development. == Early life and education == Alexandra Sasha Vorobyova was born in the Ukrainian Soviet Socialist Republic in 1990. When she was four years old, her family relocated to Ontario, Canada. Her interest in science is influenced by her family's history; her mother, grandmother, and great-grandmother all pursued careers in scientific fields. Luccioni earned a B.A. in language science from University of Paris III: Sorbonne Nouvelle in 2010. Subsequently, she completed a M.S. in cognitive science, with a minor in natural language processing, at École normale supérieure in Paris in 2012. Luccioni obtained her PhD in cognitive computing from Université du Québec à Montréal (UQAM) in 2018. == Career == Luccioni began her professional career at Nuance Communications in 2017, where she focused on natural language processing (NLP) and machine learning (ML) techniques to enhance conversational agents. She then joined Morgan Stanley’s AI/ML Center of Excellence in 2018, working on explainable artificial intelligence (AI) and decision-making systems. In 2019, she became a postdoctoral researcher at Université de Montréal and Mila, collaborating with computer scientist Yoshua Bengio on a project titled This Climate Does Not Exist. This initiative used generative adversarial networks to visualize the effects of climate change. During this time, she also contributed to integrating fairness and accountability into machine learning education at Mila. Luccioni briefly worked with the United Nations Global Pulse in 2021, developing tools to monitor COVID-19 misinformation. Later that year, she joined Hugging Face as a research scientist. Her role includes quantifying the carbon footprint of AI systems, co-chairing the carbon working group in the Big Science project, and advancing responsible machine learning practices. She helped create "CodeCarbon," an open-source software tool that estimates the carbon emissions produced during the training and operation of machine learning models. In addition to her research, she has developed tools to measure the environmental impact of AI models, communicated findings through media engagements, and presented at international conferences, including a TED Talk. In 2024, she was listed on BBC 100 Women and Time 100 AI.
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Additive smoothing

In statistics, additive smoothing, also called Laplace smoothing or Lidstone smoothing, is a technique used to smooth count data, eliminating issues caused by certain values having 0 occurrences. Given a set of observation counts x = ⟨ x 1 , x 2 , … , x d ⟩ {\displaystyle \mathbf {x} =\langle x_{1},x_{2},\ldots ,x_{d}\rangle } from a d {\displaystyle d} -dimensional multinomial distribution with N {\displaystyle N} trials, a "smoothed" version of the counts gives the estimator θ ^ i = x i + α N + α d ( i = 1 , … , d ) , {\displaystyle {\hat {\theta }}_{i}={\frac {x_{i}+\alpha }{N+\alpha d}}\qquad (i=1,\ldots ,d),} where the smoothed count x ^ i = N θ ^ i {\displaystyle {\hat {x}}_{i}=N{\hat {\theta }}_{i}} , and the "pseudocount" α > 0 is a smoothing parameter, with α = 0 corresponding to no smoothing (this parameter is explained in § Pseudocount below). Additive smoothing is a type of shrinkage estimator, as the resulting estimate will be between the empirical probability (relative frequency) x i / N {\displaystyle x_{i}/N} and the uniform probability 1 / d . {\displaystyle 1/d.} Common choices for α are 0 (no smoothing), +1⁄2 (the Jeffreys prior), or 1 (Laplace's rule of succession), but the parameter may also be set empirically based on the observed data. From a Bayesian point of view, this corresponds to the expected value of the posterior distribution, using a symmetric Dirichlet distribution with parameter α as a prior distribution. In the special case where the number of categories is 2, this is equivalent to using a beta distribution as the conjugate prior for the parameters of the binomial distribution. == History == Laplace came up with this smoothing technique when he tried to estimate the chance that the sun will rise tomorrow. His rationale was that even given a large sample of days with the rising sun, we still can not be completely sure that the sun will still rise tomorrow (known as the sunrise problem). == Pseudocount == A pseudocount is an amount (not generally an integer, despite its name) added to the number of observed cases in order to change the expected probability in a model of those data, when not known to be zero. It is so named because, roughly speaking, a pseudo-count of value α {\displaystyle \alpha } weighs into the posterior distribution similarly to each category having an additional count of α {\displaystyle \alpha } . If the number of occurrences of each item i {\displaystyle i} is x i {\displaystyle x_{i}} out of N {\displaystyle N} samples, the empirical probability of event i {\displaystyle i} is p i , empirical = x i N , {\displaystyle p_{i,{\text{empirical}}}={\frac {x_{i}}{N}},} but the posterior probability when additively smoothed is p i , α -smoothed = x i + α N + α d , {\displaystyle p_{i,\alpha {\text{-smoothed}}}={\frac {x_{i}+\alpha }{N+\alpha d}},} as if to increase each count x i {\displaystyle x_{i}} by α {\displaystyle \alpha } a priori. Depending on the prior knowledge, which is sometimes a subjective value, a pseudocount may have any non-negative finite value. It may only be zero (or the possibility ignored) if impossible by definition, such as the possibility of a decimal digit of π being a letter, or a physical possibility that would be rejected and so not counted, such as a computer printing a letter when a valid program for π is run, or excluded and not counted because of no interest, such as if only interested in the zeros and ones. Generally, there is also a possibility that no value may be computable or observable in a finite time (see the halting problem). But at least one possibility must have a non-zero pseudocount, otherwise no prediction could be computed before the first observation. The relative values of pseudocounts represent the relative prior expected probabilities of their possibilities. The sum of the pseudocounts, which may be very large, represents the estimated weight of the prior knowledge compared with all the actual observations (one for each) when determining the expected probability. In any observed data set or sample there is the possibility, especially with low-probability events and with small data sets, of a possible event not occurring. Its observed frequency is therefore zero, apparently implying a probability of zero. This oversimplification is inaccurate and often unhelpful, particularly in probability-based machine learning techniques such as artificial neural networks and hidden Markov models. By artificially adjusting the probability of rare (but not impossible) events so those probabilities are not exactly zero, zero-frequency problems are avoided. Also see Cromwell's rule. === Choice of pseudocount === ==== Weakly informative prior ==== One common approach is to add 1 to each observed number of events, including the zero-count possibilities. This is sometimes called Laplace's rule of succession. This approach is equivalent to assuming a uniform prior distribution over the probabilities for each possible event (spanning the simplex where each probability is between 0 and 1, and they all sum to 1). Using the Jeffreys prior approach, a pseudocount of one half should be added to each possible outcome. Pseudocounts should be set to one or one-half only when there is no prior knowledge at all – see the principle of indifference. However, given appropriate prior knowledge, the sum should be adjusted in proportion to the expectation that the prior probabilities should be considered correct, despite evidence to the contrary – see further analysis. Higher values are appropriate inasmuch as there is prior knowledge of the true values (for a mint-condition coin, say); lower values inasmuch as there is prior knowledge that there is probable bias, but of unknown degree (for a bent coin, say). ==== Frequentist interval ==== One way to motivate pseudocounts, particularly for binomial data, is via a formula for the midpoint of an interval estimate, particularly a binomial proportion confidence interval. The best-known is due to Edwin Bidwell Wilson, in Wilson (1927): the midpoint of the Wilson score interval corresponding to ⁠ z {\displaystyle z} ⁠ standard deviations on either side is n S + z n + 2 z {\displaystyle {\frac {n_{S}+z}{n+2z}}} Taking z = 2 {\displaystyle z=2} standard deviations to approximate a 95% confidence interval (⁠ z ≈ 1.96 {\displaystyle z\approx 1.96} ⁠) yields pseudocount of 2 for each outcome, so 4 in total, colloquially known as the "plus four rule": n S + 2 n + 4 {\displaystyle {\frac {n_{S}+2}{n+4}}} This is also the midpoint of the Agresti–Coull interval (Agresti & Coull 1998). ==== Known incidence rates ==== Often the bias of an unknown trial population is tested against a control population with known parameters (incidence rates) μ = ⟨ μ 1 , μ 2 , … , μ d ⟩ . {\displaystyle {\boldsymbol {\mu }}=\langle \mu _{1},\mu _{2},\ldots ,\mu _{d}\rangle .} In this case the uniform probability 1 / d {\displaystyle 1/d} should be replaced by the known incidence rate of the control population μ i {\displaystyle \mu _{i}} to calculate the smoothed estimator: θ ^ i = x i + μ i α d N + α d ( i = 1 , … , d ) . {\displaystyle {\hat {\theta }}_{i}={\frac {x_{i}+\mu _{i}\alpha d}{N+\alpha d}}\qquad (i=1,\ldots ,d).} As a consistency check, if the empirical estimator happens to equal the incidence rate, i.e. μ i = x i / N , {\displaystyle \mu _{i}=x_{i}/N,} the smoothed estimator is independent of α {\displaystyle \alpha } and also equals the incidence rate. == Applications == === Classification === Additive smoothing is commonly a component of naive Bayes classifiers. === Statistical language modelling === In a bag of words model of natural language processing and information retrieval, the data consists of the number of occurrences of each word in a document. Additive smoothing allows the assignment of non-zero probabilities to words which do not occur in the sample. Studies have shown that additive smoothing is more effective than other probability smoothing methods in several retrieval tasks such as language-model-based pseudo-relevance feedback and recommender systems.
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Image texture

An image texture is the small-scale structure perceived on an image, based on the spatial arrangement of color or intensities. It can be quantified by a set of metrics calculated in image processing. Image texture metrics give us information about the whole image or selected regions. Image textures can be artificially created or found in natural scenes captured in an image. Image textures are one way that can be used to help in segmentation or classification of images. For more accurate segmentation the most useful features are spatial frequency and an average grey level. To analyze an image texture in computer graphics, there are two ways to approach the issue: structured approach and statistical approach. == Structured approach == A structured approach sees an image texture as a set of primitive texels in some regular or repeated pattern. This works well when analyzing artificial textures. To obtain a structured description a characterization of the spatial relationship of the texels is gathered by using Voronoi tessellation of the texels. == Statistical approach == A statistical approach sees an image texture as a quantitative measure of the arrangement of intensities in a region. In general this approach is easier to compute and is more widely used, since natural textures are made of patterns of irregular subelements. === Edge detection === The use of edge detection is to determine the number of edge pixels in a specified region, helps determine a characteristic of texture complexity. After edges have been found the direction of the edges can also be applied as a characteristic of texture and can be useful in determining patterns in the texture. These directions can be represented as an average or in a histogram. Consider a region with N pixels. the gradient-based edge detector is applied to this region by producing two outputs for each pixel p: the gradient magnitude Mag(p) and the gradient direction Dir(p). The edgeness per unit area can be defined by F e d g e n e s s = | { p | M a g ( p ) > T } | N {\displaystyle F_{edgeness}={\frac {|\{p|Mag(p)>T\}|}{N}}} for some threshold T. To include orientation with edgeness histograms for both gradient magnitude and gradient direction can be used. Hmag(R) denotes the normalized histogram of gradient magnitudes of region R, and Hdir(R) denotes the normalized histogram of gradient orientations of region R. Both are normalized according to the size NR Then F m a g , d i r = ( H m a g ( R ) , H d i r ( R ) ) {\displaystyle F_{mag,dir}=(H_{mag}(R),H_{dir}(R))} is a quantitative texture description of region R. === Co-occurrence matrices === The co-occurrence matrix captures numerical features of a texture using spatial relations of similar gray tones. Numerical features computed from the co-occurrence matrix can be used to represent, compare, and classify textures. The following are a subset of standard features derivable from a normalized co-occurrence matrix: A n g u l a r 2 n d M o m e n t = ∑ i ∑ j p [ i , j ] 2 C o n t r a s t = ∑ i = 1 N g ∑ j = 1 N g n 2 p [ i , j ] , where | i − j | = n C o r r e l a t i o n = ∑ i = 1 N g ∑ j = 1 N g ( i j ) p [ i , j ] − μ x μ y σ x σ y E n t r o p y = − ∑ i ∑ j p [ i , j ] l n ( p [ i , j ] ) {\displaystyle {\begin{aligned}Angular{\text{ }}2nd{\text{ }}Moment&=\sum _{i}\sum _{j}p[i,j]^{2}\\Contrast&=\sum _{i=1}^{Ng}\sum _{j=1}^{Ng}n^{2}p[i,j]{\text{, where }}|i-j|=n\\Correlation&={\frac {\sum _{i=1}^{Ng}\sum _{j=1}^{Ng}(ij)p[i,j]-\mu _{x}\mu _{y}}{\sigma _{x}\sigma _{y}}}\\Entropy&=-\sum _{i}\sum _{j}p[i,j]ln(p[i,j])\\\end{aligned}}} where p [ i , j ] {\displaystyle p[i,j]} is the [ i , j ] {\displaystyle [i,j]} th entry in a gray-tone spatial dependence matrix, and Ng is the number of distinct gray-levels in the quantized image. One negative aspect of the co-occurrence matrix is that the extracted features do not necessarily correspond to visual perception. It is used in dentistry for the objective evaluation of lesions [DOI: 10.1155/2020/8831161], treatment efficacy [DOI: 10.3390/ma13163614; DOI: 10.11607/jomi.5686; DOI: 10.3390/ma13173854; DOI: 10.3390/ma13132935] and bone reconstruction during healing [DOI: 10.5114/aoms.2013.33557; DOI: 10.1259/dmfr/22185098; EID: 2-s2.0-81455161223; DOI: 10.3390/ma13163649]. === Laws texture energy measures === Another approach is to use local masks to detect various types of texture features. Laws originally used four vectors representing texture features to create sixteen 2D masks from the outer products of the pairs of vectors. The four vectors and relevant features were as follows: L5 = [ +1 +4 6 +4 +1 ] (Level) E5 = [ -1 -2 0 +2 +1 ] (Edge) S5 = [ -1 0 2 0 -1 ] (Spot) R5 = [ +1 -4 6 -4 +1 ] (Ripple) To these 4, a fifth is sometimes added: W5 = [ -1 +2 0 -2 +1 ] (Wave) From Laws' 4 vectors, 16 5x5 "energy maps" are then filtered down to 9 in order to remove certain symmetric pairs. For instance, L5E5 measures vertical edge content and E5L5 measures horizontal edge content. The average of these two measures is the "edginess" of the content. The resulting 9 maps used by Laws are as follows: L5E5/E5L5 L5R5/R5L5 E5S5/S5E5 S5S5 R5R5 L5S5/S5L5 E5E5 E5R5/R5E5 S5R5/R5S5 Running each of these nine maps over an image to create a new image of the value of the origin ([2,2]) results in 9 "energy maps," or conceptually an image with each pixel associated with a vector of 9 texture attributes. === Autocorrelation and power spectrum === The autocorrelation function of an image can be used to detect repetitive patterns of textures. == Texture segmentation == The use of image texture can be used as a description for regions into segments. There are two main types of segmentation based on image texture, region based and boundary based. Though image texture is not a perfect measure for segmentation it is used along with other measures, such as color, that helps solve segmenting in image. === Region based === Attempts to group or cluster pixels based on texture properties. === Boundary based === Attempts to group or cluster pixels based on edges between pixels that come from different texture properties.
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AI Code Generators: Free vs Paid (2026)

Looking for the best AI code generator? An AI code generator is software that uses machine learning to help you get more done — it can save you hours every week by automating repetitive work. Most options offer a generous free tier, with paid plans unlocking higher limits, faster processing, and team features. Whether you are a beginner or a pro, the right AI code generator slots into your workflow and pays for itself fast. Read on for hands-on impressions, pricing tiers, and the standout features that matter.
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Top 10 AI Background Removers Compared (2026)

Curious about the best AI background remover? An AI background remover is software that uses machine learning to help you get more done — it combines speed, accuracy, and an interface that just works. Hands-on testing shows real-world results vary, so a short free trial is the smartest way to decide. Whether you are a beginner or a pro, the right AI background remover slots into your workflow and pays for itself fast. This guide breaks down the top picks, their pros and cons, and who each one is best for.
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