Correspondence analysis (CA) is a multivariate statistical technique proposed by Herman Otto Hartley (Hirschfeld) and later developed by Jean-Paul Benzécri. It is conceptually similar to principal component analysis, but applies to categorical rather than continuous data. In a manner similar to principal component analysis, it provides a means of displaying or summarising a set of data in two-dimensional graphical form. Its aim is to display in a biplot any structure hidden in the multivariate setting of the data table. As such it is a technique from the field of multivariate ordination. Since the variant of CA described here can be applied either with a focus on the rows or on the columns it should in fact be called simple (symmetric) correspondence analysis. It is traditionally applied to the contingency table of a pair of nominal variables where each cell contains either a count or a zero value. If more than two categorical variables are to be summarized, a variant called multiple correspondence analysis should be chosen instead. CA may also be applied to binary data given the presence/absence coding represents simplified count data i.e. a 1 describes a positive count and 0 stands for a count of zero. Depending on the scores used CA preserves the chi-square distance between either the rows or the columns of the table. Because CA is a descriptive technique, it can be applied to tables regardless of a significant chi-squared test. Although the χ 2 {\displaystyle \chi ^{2}} statistic used in inferential statistics and the chi-square distance are computationally related they should not be confused since the latter works as a multivariate statistical distance measure in CA while the χ 2 {\displaystyle \chi ^{2}} statistic is in fact a scalar not a metric. == Details == Like principal components analysis, correspondence analysis creates orthogonal components (or axes) and, for each item in a table i.e. for each row, a set of scores (sometimes called factor scores, see Factor analysis). Correspondence analysis is performed on the data table, conceived as matrix C of size m × n where m is the number of rows and n is the number of columns. In the following mathematical description of the method capital letters in italics refer to a matrix while letters in italics refer to vectors. Understanding the following computations requires knowledge of matrix algebra. === Preprocessing === Before proceeding to the central computational step of the algorithm, the values in matrix C have to be transformed. First compute a set of weights for the columns and the rows (sometimes called masses), where row and column weights are given by the row and column vectors, respectively: w m = 1 n C C 1 , w n = 1 n C 1 T C . {\displaystyle w_{m}={\frac {1}{n_{C}}}C\mathbf {1} ,\quad w_{n}={\frac {1}{n_{C}}}\mathbf {1} ^{T}C.} Here n C = ∑ i = 1 n ∑ j = 1 m C i j {\displaystyle n_{C}=\sum _{i=1}^{n}\sum _{j=1}^{m}C_{ij}} is the sum of all cell values in matrix C, or short the sum of C, and 1 {\displaystyle \mathbf {1} } is a column vector of ones with the appropriate dimension. Put in simple words, w m {\displaystyle w_{m}} is just a vector whose elements are the row sums of C divided by the sum of C, and w n {\displaystyle w_{n}} is a vector whose elements are the column sums of C divided by the sum of C. The weights are transformed into diagonal matrices W m = diag ( 1 / w m ) {\displaystyle W_{m}=\operatorname {diag} (1/{\sqrt {w_{m}}})} and W n = diag ( 1 / w n ) {\displaystyle W_{n}=\operatorname {diag} (1/{\sqrt {w_{n}}})} where the diagonal elements of W n {\displaystyle W_{n}} are 1 / w n {\displaystyle 1/{\sqrt {w_{n}}}} and those of W m {\displaystyle W_{m}} are 1 / w m {\displaystyle 1/{\sqrt {w_{m}}}} respectively i.e. the vector elements are the inverses of the square roots of the masses. The off-diagonal elements are all 0. Next, compute matrix P {\displaystyle P} by dividing C {\displaystyle C} by its sum P = 1 n C C . {\displaystyle P={\frac {1}{n_{C}}}C.} In simple words, Matrix P {\displaystyle P} is just the data matrix (contingency table or binary table) transformed into portions i.e. each cell value is just the cell portion of the sum of the whole table. Finally, compute matrix S {\displaystyle S} , sometimes called the matrix of standardized residuals, by matrix multiplication as S = W m ( P − w m w n ) W n {\displaystyle S=W_{m}(P-w_{m}w_{n})W_{n}} Note, the vectors w m {\displaystyle w_{m}} and w n {\displaystyle w_{n}} are combined in an outer product resulting in a matrix of the same dimensions as P {\displaystyle P} . In words the formula reads: matrix outer ( w m , w n ) {\displaystyle \operatorname {outer} (w_{m},w_{n})} is subtracted from matrix P {\displaystyle P} and the resulting matrix is scaled (weighted) by the diagonal matrices W m {\displaystyle W_{m}} and W n {\displaystyle W_{n}} . Multiplying the resulting matrix by the diagonal matrices is equivalent to multiply the i-th row (or column) of it by the i-th element of the diagonal of W m {\displaystyle W_{m}} or W n {\displaystyle W_{n}} , respectively. === Interpretation of preprocessing === The vectors w m {\displaystyle w_{m}} and w n {\displaystyle w_{n}} are the row and column masses or the marginal probabilities for the rows and columns, respectively. Subtracting matrix outer ( w m , w n ) {\displaystyle \operatorname {outer} (w_{m},w_{n})} from matrix P {\displaystyle P} is the matrix algebra version of double centering the data. Multiplying this difference by the diagonal weighting matrices results in a matrix containing weighted deviations from the origin of a vector space. This origin is defined by matrix outer ( w m , w n ) {\displaystyle \operatorname {outer} (w_{m},w_{n})} . In fact matrix outer ( w m , w n ) {\displaystyle \operatorname {outer} (w_{m},w_{n})} is identical with the matrix of expected frequencies in the chi-squared test. Therefore S {\displaystyle S} is computationally related to the independence model used in that test. But since CA is not an inferential method the term independence model is inappropriate here. === Orthogonal components === The table S {\displaystyle S} is then decomposed by a singular value decomposition as S = U Σ V ∗ {\displaystyle S=U\Sigma V^{}\,} where U {\displaystyle U} and V {\displaystyle V} are the left and right singular vectors of S {\displaystyle S} and Σ {\displaystyle \Sigma } is a square diagonal matrix with the singular values σ i {\displaystyle \sigma _{i}} of S {\displaystyle S} on the diagonal. Σ {\displaystyle \Sigma } is of dimension p ≤ ( min ( m , n ) − 1 ) {\displaystyle p\leq (\min(m,n)-1)} hence U {\displaystyle U} is of dimension m×p and V {\displaystyle V} is of n×p. As orthonormal vectors U {\displaystyle U} and V {\displaystyle V} fulfill U ∗ U = V ∗ V = I {\displaystyle U^{}U=V^{}V=I} . In other words, the multivariate information that is contained in C {\displaystyle C} as well as in S {\displaystyle S} is now distributed across two (coordinate) matrices U {\displaystyle U} and V {\displaystyle V} and a diagonal (scaling) matrix Σ {\displaystyle \Sigma } . The vector space defined by them has as number of dimensions p, that is the smaller of the two values, number of rows and number of columns, minus 1. === Inertia === While a principal component analysis may be said to decompose the (co)variance, and hence its measure of success is the amount of (co-)variance covered by the first few PCA axes - measured in eigenvalue -, a CA works with a weighted (co-)variance which is called inertia. The sum of the squared singular values is the total inertia I {\displaystyle \mathrm {I} } of the data table, computed as I = ∑ i = 1 p σ i 2 . {\displaystyle \mathrm {I} =\sum _{i=1}^{p}\sigma _{i}^{2}.} The total inertia I {\displaystyle \mathrm {I} } of the data table can also computed directly from S {\displaystyle S} as I = ∑ i = 1 n ∑ j = 1 m s i j 2 . {\displaystyle \mathrm {I} =\sum _{i=1}^{n}\sum _{j=1}^{m}s_{ij}^{2}.} The amount of inertia covered by the i-th set of singular vectors is ι i {\displaystyle \iota _{i}} , the principal inertia. The higher the portion of inertia covered by the first few singular vectors i.e. the larger the sum of the principal inertiae in comparison to the total inertia, the more successful a CA is. Therefore, all principal inertia values are expressed as portion ϵ i {\displaystyle \epsilon _{i}} of the total inertia ϵ i = σ i 2 / ∑ i = 1 p σ i 2 {\displaystyle \epsilon _{i}=\sigma _{i}^{2}/\sum _{i=1}^{p}\sigma _{i}^{2}} and are presented in the form of a scree plot. In fact a scree plot is just a bar plot of all principal inertia portions ϵ i {\displaystyle \epsilon _{i}} . === Coordinates === To transform the singular vectors to coordinates which preserve the chi-square distances between rows or columns an additional weighting step is necessary. The resulting coordinates are called principal coordinates in CA text books. If principal coordinates are used for
Non-local means
Non-local means is an algorithm in image processing for image denoising. Unlike "local mean" filters, which take the mean value of a group of pixels surrounding a target pixel to smooth the image, non-local means filtering takes a mean of all pixels in the image, weighted by how similar these pixels are to the target pixel. This results in much greater post-filtering clarity, and less loss of detail in the image compared with local mean algorithms. If compared with other well-known denoising techniques, non-local means adds "method noise" (i.e. error in the denoising process) which looks more like white noise, which is desirable because it is typically less disturbing in the denoised product. Recently non-local means has been extended to other image processing applications such as deinterlacing, view interpolation, and depth maps regularization. == Definition == Suppose Ω {\displaystyle \Omega } is the area of an image, and p {\displaystyle p} and q {\displaystyle q} are two points within the image. Then, the algorithm is: u ( p ) = 1 C ( p ) ∫ Ω v ( q ) f ( p , q ) d q . {\displaystyle u(p)={1 \over C(p)}\int _{\Omega }v(q)f(p,q)\,\mathrm {d} q.} where u ( p ) {\displaystyle u(p)} is the filtered value of the image at point p {\displaystyle p} , v ( q ) {\displaystyle v(q)} is the unfiltered value of the image at point q {\displaystyle q} , f ( p , q ) {\displaystyle f(p,q)} is the weighting function, and the integral is evaluated ∀ q ∈ Ω {\displaystyle \forall q\in \Omega } . C ( p ) {\displaystyle C(p)} is a normalizing factor, given by C ( p ) = ∫ Ω f ( p , q ) d q . {\displaystyle C(p)=\int _{\Omega }f(p,q)\,\mathrm {d} q.} == Common weighting functions == The purpose of the weighting function, f ( p , q ) {\displaystyle f(p,q)} , is to determine how closely related the image at the point p {\displaystyle p} is to the image at the point q {\displaystyle q} . It can take many forms. === Gaussian === The Gaussian weighting function sets up a normal distribution with a mean, μ = B ( p ) {\displaystyle \mu =B(p)} and a variable standard deviation: f ( p , q ) = e − | B ( q ) − B ( p ) | 2 h 2 {\displaystyle f(p,q)=e^{-{{\left\vert B(q)-B(p)\right\vert ^{2}} \over h^{2}}}} where h {\displaystyle h} is the filtering parameter (i.e., standard deviation) and B ( p ) {\displaystyle B(p)} is the local mean value of the image point values surrounding p {\displaystyle p} . == Discrete algorithm == For an image, Ω {\displaystyle \Omega } , with discrete pixels, a discrete algorithm is required. u ( p ) = 1 C ( p ) ∑ q ∈ Ω v ( q ) f ( p , q ) {\displaystyle u(p)={1 \over C(p)}\sum _{q\in \Omega }v(q)f(p,q)} where, once again, v ( q ) {\displaystyle v(q)} is the unfiltered value of the image at point q {\displaystyle q} . C ( p ) {\displaystyle C(p)} is given by: C ( p ) = ∑ q ∈ Ω f ( p , q ) {\displaystyle C(p)=\sum _{q\in \Omega }f(p,q)} Then, for a Gaussian weighting function, f ( p , q ) = e − | B ( q ) 2 − B ( p ) 2 | h 2 {\displaystyle f(p,q)=e^{-{{\left\vert B(q)^{2}-B(p)^{2}\right\vert } \over h^{2}}}} where B ( p ) {\displaystyle B(p)} is given by: B ( p ) = 1 | R ( p ) | ∑ i ∈ R ( p ) v ( i ) {\displaystyle B(p)={1 \over |R(p)|}\sum _{i\in R(p)}v(i)} where R ( p ) ⊆ Ω {\displaystyle R(p)\subseteq \Omega } and is a square region of pixels surrounding p {\displaystyle p} and | R ( p ) | {\displaystyle |R(p)|} is the number of pixels in the region R {\displaystyle R} . == Efficient implementation == The computational complexity of the non-local means algorithm is quadratic in the number of pixels in the image, making it particularly expensive to apply directly. Several techniques were proposed to speed up execution. One simple variant consists of restricting the computation of the mean for each pixel to a search window centred on the pixel itself, instead of the whole image. Another approximation uses summed-area tables and fast Fourier transform to calculate the similarity window between two pixels, speeding up the algorithm by a factor of 50 while preserving comparable quality of the result.
Content determination
Content determination is the subtask of natural language generation (NLG) that involves deciding on the information to be communicated in a generated text. It is closely related to the task of document structuring. == Example == Consider an NLG system which summarises information about sick babies. Suppose this system has four pieces of information it can communicate The baby is being given morphine via an IV drop The baby's heart rate shows bradycardia's (temporary drops) The baby's temperature is normal The baby is crying Which of these bits of information should be included in the generated texts? == Issues == There are three general issues which almost always impact the content determination task, and can be illustrated with the above example. Perhaps the most fundamental issue is the communicative goal of the text, i.e. its purpose and reader. In the above example, for instance, a doctor who wants to make a decision about medical treatment would probably be most interested in the heart rate bradycardias, while a parent who wanted to know how her child was doing would probably be more interested in the fact that the baby was being given morphine and was crying. The second issue is the size and level of detail of the generated text. For instance, a short summary which was sent to a doctor as a 160 character SMS text message might only mention the heart rate bradycardias, while a longer summary which was printed out as a multipage document might also mention the fact that the baby is on a morphine IV. The final issue is how unusual and unexpected the information is. For example, neither doctors nor parents would place a high priority on being told that the baby's temperature was normal, if they expected this to be the case. Regardless, content determination is very important to users, indeed in many cases the quality of content determination is the most important factor (from the user's perspective) in determining the overall quality of the generated text. == Techniques == There are three basic approaches to document structuring: schemas (content templates), statistical approaches, and explicit reasoning. Schemas are templates which explicitly specify the content of a generated text (as well as document structuring information). Typically, they are constructed by manually analysing a corpus of human-written texts in the target genre, and extracting a content template from these texts. Schemas work well in practice in domains where content is somewhat standardised, but work less well in domains where content is more fluid (such as the medical example above). Statistical techniques use statistical corpus analysis techniques to automatically determine the content of the generated texts. Such work is in its infancy, and has mostly been applied to contexts where the communicative goal, reader, size, and level of detail are fixed. For example, generation of newswire summaries of sporting events. Explicit reasoning approaches have probably attracted the most attention from researchers. The basic idea is to use AI reasoning techniques (such as knowledge-based rules, planning, pattern detection, case-based reasoning, etc.) to examine the information available to be communicated (including how unusual/unexpected it is), the communicative goal and reader, and the characteristics of the generated text (including target size), and decide on the optimal content for the generated text. A very wide range of techniques has been explored, but there is no consensus as to which is most effective.
Chatbot
A chatbot (originally chatterbot) is a software application or web interface designed to converse through text or speech. Modern chatbots are typically online and use generative artificial intelligence systems that are capable of maintaining a conversation with a user in natural language and simulating the way a human would behave as a conversational partner. Such chatbots often use deep learning and natural language processing. Simpler chatbots have existed for decades. Chatbots have gained popularity during the AI boom of the 2020s, with the releases of generative AI chatbots such as ChatGPT, Gemini, Claude, and Grok. These chatbots typically use fine-tuned large language models to generate text. A major area where chatbots have long been used is customer service and support, with various sorts of virtual assistants. == History == === Turing test === In 1950, Alan Turing published an article entitled "Computing Machinery and Intelligence" in which he proposed what is now called the Turing test as a criterion of intelligence. This criterion depends on the ability of a computer program to impersonate a human in a real-time written conversation with a human judge, to the extent that the judge is incapable of reliably distinguishing, on the basis of the conversational content alone, between the program and a real human. === Early chatbots === Joseph Weizenbaum's program ELIZA was first published in 1966. Weizenbaum did not claim that ELIZA was genuinely intelligent, and the introduction to his paper presented it more as a debunking exercise:In artificial intelligence, machines are made to behave in wondrous ways, often sufficient to dazzle even the most experienced observer. But once a particular program is unmasked, once its inner workings are explained, its magic crumbles away; it stands revealed as a mere collection of procedures. The observer says to himself "I could have written that". With that thought, he moves the program in question from the shelf marked "intelligent", to that reserved for curios. The object of this paper is to cause just such a re-evaluation of the program about to be "explained". Few programs ever needed it more. ELIZA's key method of operation involves the recognition of clue words or phrases in the input, and the output of the corresponding pre-prepared or pre-programmed responses that can move the conversation forward in an apparently meaningful way (e.g. by responding to any input that contains the word 'MOTHER' with 'TELL ME MORE ABOUT YOUR FAMILY'). Thus an illusion of understanding is generated, even though the processing involved has been merely superficial. ELIZA showed that such an illusion is surprisingly easy to generate because human judges are ready to give the benefit of the doubt when conversational responses are capable of being interpreted as "intelligent". Following ELIZA, psychiatrist Kenneth Colby developed PARRY in 1972. From 1978 to some time after 1983, the CYRUS project led by Janet Kolodner constructed a chatbot simulating Cyrus Vance (57th United States Secretary of State). It used case-based reasoning, and updated its database daily by parsing wire news from United Press International. The program was unable to process the news items subsequent to the surprise resignation of Cyrus Vance in April 1980, and the team constructed another chatbot simulating his successor, Edmund Muskie. In 1984, an interactive version of the program Racter was released which acted as a chatbot. A.L.I.C.E. was released in 1995. This uses a markup language called AIML, which is specific to its function as a conversational agent, and has since been adopted by various other developers of, so-called, Alicebots. A.L.I.C.E. is a weak AI without any reasoning capabilities. It is based on a similar pattern matching technique as ELIZA in 1966. This is not strong AI, which would require sapience and logical reasoning abilities. Jabberwacky, released in 1997, learns new responses and context based on real-time user interactions, rather than being driven from a static database. Chatbot competitions focus on the Turing test or more specific goals. Two such annual contests are the Loebner Prize and The Chatterbox Challenge (the latter has been offline since 2015, however, materials can still be found from web archives). Pre-dating the current generation of large language models, Gavagai, a Swedish language technology startup, created a Twitter-based bot in 2015 and DBpedia created a chatbot during the 2017 Google Summer of Code that communicated through Facebook Messenger. === Modern chatbots based on large language models === Modern chatbots like ChatGPT are often based on foundational large language models called generative pre-trained transformers (GPT). They are based on a deep learning architecture called the transformer, which contains artificial neural networks. They generate text after being trained on a large text corpus, and have emergent abilities that they are not specifically trained for. Chatbots integrated into apps and websites can call image-generation models or search the web. Some platforms also enable users to interact with conversational interfaces directly through web-based chat environments, allowing real-time assistance, content generation, and task automation without requiring software installation. == Application == === Messaging apps === Many companies' chatbots run on messaging apps or simply via SMS. They are used for B2C customer service, sales and marketing. In 2016, Facebook Messenger allowed developers to place chatbots on their platform. There were 30,000 bots created for Messenger in the first six months, rising to 100,000 by September 2017. Since September 2017, this has also been as part of a pilot program on WhatsApp. Airlines KLM and Aeroméxico both announced their participation in the testing; both airlines had previously launched customer services on the Facebook Messenger platform. The bots usually appear as one of the user's contacts, but can sometimes act as participants in a group chat. Many banks, insurers, media companies, e-commerce companies, airlines, hotel chains, retailers, health care providers, government entities, and restaurant chains have used chatbots to answer simple questions, increase customer engagement, for promotion, and to offer additional ways to order from them. Chatbots are also used in market research to collect short survey responses. A 2017 study showed 4% of companies used chatbots. In a 2016 study, 80% of businesses said they intended to have one by 2020. ==== As part of company apps and websites ==== Previous generations of chatbots were present on company websites, e.g. Ask Jenn from Alaska Airlines which debuted in 2008 or Expedia's virtual customer service agent which launched in 2011. The newer generation of chatbots includes IBM Watson-powered "Rocky", introduced in February 2017 by the New York City-based e-commerce company Rare Carat to provide information to prospective diamond buyers. ==== Chatbot sequences ==== Used by marketers to script sequences of messages, very similar to an autoresponder sequence. Such sequences can be triggered by user opt-in or the use of keywords within user interactions. After a trigger occurs a sequence of messages is delivered until the next anticipated user response. Each user response is used in the decision tree to help the chatbot navigate the response sequences to deliver the correct response message. === Company internal platforms === Companies have used chatbots for customer support, human resources, or in Internet-of-Things (IoT) projects. Overstock.com, for one, has reportedly launched a chatbot named Mila to attempt to automate certain processes when customer service employees request sick leave. Other large companies such as Lloyds Banking Group, Royal Bank of Scotland, Renault and Citroën are now using chatbots instead of call centres with humans to provide a first point of contact. In large companies, like in hospitals and aviation organizations, chatbots are also used to share information within organizations, and to assist and replace service desks. === Customer service === Chatbots have been proposed as a replacement for customer service departments. In 2026, The Financial Times reported on agentic chatbots that could do shopping for customers once given instructions. In 2016, Russia-based Tochka Bank launched a chatbot on Facebook for a range of financial services, including a possibility of making payments. In July 2016, Barclays Africa also launched a Facebook chatbot. === Healthcare === Chatbots are also appearing in the healthcare industry. A study suggested that physicians in the United States believed that chatbots would be most beneficial for scheduling doctor appointments, locating health clinics, or providing medication information. A 2025 review found that participants often rated chatbot responses as more empathic than those from clinicians. In 2020, WhatsApp worked with th
Eigenface
An eigenface ( EYE-gən-) is the name given to a set of eigenvectors when used in the computer vision problem of human face recognition. The approach of using eigenfaces for recognition was developed by Sirovich and Kirby and used by Matthew Turk and Alex Pentland in face classification. The eigenvectors are derived from the covariance matrix of the probability distribution over the high-dimensional vector space of face images. The eigenfaces themselves form a basis set of all images used to construct the covariance matrix. This produces dimension reduction by allowing the smaller set of basis images to represent the original training images. Classification can be achieved by comparing how faces are represented by the basis set. == History == The eigenface approach began with a search for a low-dimensional representation of face images. Sirovich and Kirby showed that principal component analysis could be used on a collection of face images to form a set of basis features. These basis images, known as eigenpictures, could be linearly combined to reconstruct images in the original training set. If the training set consists of M images, principal component analysis could form a basis set of N images, where N < M. The reconstruction error is reduced by increasing the number of eigenpictures; however, the number needed is always chosen less than M. For example, if you need to generate a number of N eigenfaces for a training set of M face images, you can say that each face image can be made up of "proportions" of all the K "features" or eigenfaces: Face image1 = (23% of E1) + (2% of E2) + (51% of E3) + ... + (1% En). In 1991 M. Turk and A. Pentland expanded these results and presented the eigenface method of face recognition. In addition to designing a system for automated face recognition using eigenfaces, they showed a way of calculating the eigenvectors of a covariance matrix such that computers of the time could perform eigen-decomposition on a large number of face images. Face images usually occupy a high-dimensional space and conventional principal component analysis was intractable on such data sets. Turk and Pentland's paper demonstrated ways to extract the eigenvectors based on matrices sized by the number of images rather than the number of pixels. Once established, the eigenface method was expanded to include methods of preprocessing to improve accuracy. Multiple manifold approaches were also used to build sets of eigenfaces for different subjects and different features, such as the eyes. == Generation == A set of eigenfaces can be generated by performing a mathematical process called principal component analysis (PCA) on a large set of images depicting different human faces. Informally, eigenfaces can be considered a set of "standardized face ingredients", derived from statistical analysis of many pictures of faces. Any human face can be considered to be a combination of these standard faces. For example, one's face might be composed of the average face plus 10% from eigenface 1, 55% from eigenface 2, and even −3% from eigenface 3. Remarkably, it does not take many eigenfaces combined together to achieve a fair approximation of most faces. Also, because a person's face is not recorded by a digital photograph, but instead as just a list of values (one value for each eigenface in the database used), much less space is taken for each person's face. The eigenfaces that are created will appear as light and dark areas that are arranged in a specific pattern. This pattern is how different features of a face are singled out to be evaluated and scored. There will be a pattern to evaluate symmetry, whether there is any style of facial hair, where the hairline is, or an evaluation of the size of the nose or mouth. Other eigenfaces have patterns that are less simple to identify, and the image of the eigenface may look very little like a face. The technique used in creating eigenfaces and using them for recognition is also used outside of face recognition: handwriting recognition, lip reading, voice recognition, sign language/hand gestures interpretation and medical imaging analysis. Therefore, some do not use the term eigenface, but prefer to use 'eigenimage'. === Practical implementation === To create a set of eigenfaces, one must: Prepare a training set of face images. The pictures constituting the training set should have been taken under the same lighting conditions, and must be normalized to have the eyes and mouths aligned across all images. They must also be all resampled to a common pixel resolution (r × c). Each image is treated as one vector, simply by concatenating the rows of pixels in the original image, resulting in a single column with r × c elements. For this implementation, it is assumed that all images of the training set are stored in a single matrix T, where each column of the matrix is an image. Subtract the mean. The average image a has to be calculated and then subtracted from each original image in T. Calculate the eigenvectors and eigenvalues of the covariance matrix S. Each eigenvector has the same dimensionality (number of components) as the original images, and thus can itself be seen as an image. The eigenvectors of this covariance matrix are therefore called eigenfaces. They are the directions in which the images differ from the mean image. Usually this will be a computationally expensive step (if at all possible), but the practical applicability of eigenfaces stems from the possibility to compute the eigenvectors of S efficiently, without ever computing S explicitly, as detailed below. Choose the principal components. Sort the eigenvalues in descending order and arrange eigenvectors accordingly. The number of principal components k is determined arbitrarily by setting a threshold ε on the total variance. Total variance v = ( λ 1 + λ 2 + . . . + λ n ) {\displaystyle v=(\lambda _{1}+\lambda _{2}+...+\lambda _{n})} , n = number of components, and λ {\displaystyle \lambda } represents component eigenvalue. k is the smallest number that satisfies ( λ 1 + λ 2 + . . . + λ k ) v > ϵ {\displaystyle {\frac {(\lambda _{1}+\lambda _{2}+...+\lambda _{k})}{v}}>\epsilon } These eigenfaces can now be used to represent both existing and new faces: we can project a new (mean-subtracted) image on the eigenfaces and thereby record how that new face differs from the mean face. The eigenvalues associated with each eigenface represent how much the images in the training set vary from the mean image in that direction. Information is lost by projecting the image on a subset of the eigenvectors, but losses are minimized by keeping those eigenfaces with the largest eigenvalues. For instance, working with a 100 × 100 image will produce 10,000 eigenvectors. In practical applications, most faces can typically be identified using a projection on between 100 and 150 eigenfaces, so that most of the 10,000 eigenvectors can be discarded. === Matlab example code === Here is an example of calculating eigenfaces with Extended Yale Face Database B. To evade computational and storage bottleneck, the face images are sampled down by a factor 4×4=16. Note that although the covariance matrix S generates many eigenfaces, only a fraction of those are needed to represent the majority of the faces. For example, to represent 95% of the total variation of all face images, only the first 43 eigenfaces are needed. To calculate this result, implement the following code: === Computing the eigenvectors === Performing PCA directly on the covariance matrix of the images is often computationally infeasible. If small images are used, say 100 × 100 pixels, each image is a point in a 10,000-dimensional space and the covariance matrix S is a matrix of 10,000 × 10,000 = 108 elements. However the rank of the covariance matrix is limited by the number of training examples: if there are N training examples, there will be at most N − 1 eigenvectors with non-zero eigenvalues. If the number of training examples is smaller than the dimensionality of the images, the principal components can be computed more easily as follows. Let T be the matrix of preprocessed training examples, where each column contains one mean-subtracted image. The covariance matrix can then be computed as S = TTT and the eigenvector decomposition of S is given by S v i = T T T v i = λ i v i {\displaystyle \mathbf {Sv} _{i}=\mathbf {T} \mathbf {T} ^{T}\mathbf {v} _{i}=\lambda _{i}\mathbf {v} _{i}} However TTT is a large matrix, and if instead we take the eigenvalue decomposition of T T T u i = λ i u i {\displaystyle \mathbf {T} ^{T}\mathbf {T} \mathbf {u} _{i}=\lambda _{i}\mathbf {u} _{i}} then we notice that by pre-multiplying both sides of the equation with T, we obtain T T T T u i = λ i T u i {\displaystyle \mathbf {T} \mathbf {T} ^{T}\mathbf {T} \mathbf {u} _{i}=\lambda _{i}\mathbf {T} \mathbf {u} _{i}} Meaning that, if ui is an eigenvector of TTT, then vi = Tui is an eigenvector of S. If we have
Brownout (software engineering)
Brownout in software engineering is a technique that involves disabling certain features of an application. == Description == Brownout is used to increase the robustness of an application to computing capacity shortage. If too many users are simultaneously accessing an application hosted online, the underlying computing infrastructure may become overloaded, rendering the application unresponsive. Users are likely to abandon the application and switch to competing alternatives, hence incurring long-term revenue loss. To better deal with such a situation, the application can be given brownout capabilities: The application will disable certain features – e.g., an online shop will no longer display recommendations of related products – to avoid overload. Although reducing features generally has a negative impact on the short-term revenue of the application owner, long-term revenue loss can be avoided. The technique is inspired by brownouts in power grids, which consists in reducing the power grid's voltage in case electricity demand exceeds production. Some consumers, such as incandescent light bulbs, will dim – hence originating the term – and draw less power, thus helping match demand with production. Similarly, a brownout application helps match its computing capacity requirements to what is available on the target infrastructure. Brownout complements elasticity. The former can help the application withstand short-term capacity shortage, but does so without changing the capacity available to the application. In contrast, elasticity consists of adding (or removing) capacity to the application, preferably in advance, so as to avoid capacity shortage altogether. The two techniques can be combined; e.g., brownout is triggered when the number of users increases unexpectedly until elasticity can be triggered, the latter usually requiring minutes to show an effect. Brownout is relatively non-intrusive for the developer, for example, it can be implemented as an advice in aspect-oriented programming. However, surrounding components, such as load-balancers, need to be made brownout-aware to distinguish between cases where an application is running normally and cases where the application maintains a low response time by triggering brownout. == Usage in phased deprecation == A related use of the brownout concept in software engineering is the deliberate introduction of temporary outages to a system, API or feature that is being phased out. This is sometimes also called a "scream test" when it is used to discover unknown dependents of a system or API. The intention is to allow detection of downstream consumers of an API or service who may otherwise have missed deprecation announcements or to uncover hidden side-effects of the deprecation that may have been overlooked. The intention is that developers of dependent systems will notice their own system failures caused by the upstream brownout. Such brownouts are typically pre-announced scheduled outages or probabilistic in nature (such as artificially failing a percentage of requests). As a brownout is only a temporary or partial outage, it provides downstream consumers of an API or service time to remove any discovered dependencies on the deprecated API before it is fully retired. For consumers that have already prepared for the deprecation, a brownout provides valuable testing that the final removal of the service won't cause any unexpected problems.
Phase congruency
Phase congruency is a measure of feature significance in computer images, a method of edge detection that is particularly robust against changes in illumination and contrast. == Foundations == Phase congruency reflects the behaviour of the image in the frequency domain. It has been noted that edgelike features have many of their frequency components in the same phase. The concept is similar to coherence, except that it applies to functions of different wavelength. For example, the Fourier decomposition of a square wave consists of sine functions, whose frequencies are odd multiples of the fundamental frequency. At the rising edges of the square wave, each sinusoidal component has a rising phase; the phases have maximal congruency at the edges. This corresponds to the human-perceived edges in an image where there are sharp changes between light and dark. == Definition == Phase congruency compares the weighted alignment of the Fourier components of a signal A n {\displaystyle A_{\rm {n}}} with the sum of the Fourier components. P C ( t ) = max ϕ ¯ ∑ n A n cos ( ϕ n ( t ) − ϕ ¯ ) ∑ n A n {\displaystyle PC(t)=\max _{\bar {\phi }}{\frac {\sum _{\rm {n}}A_{\rm {n}}\cos(\phi _{\rm {n}}(t)-{\bar {\phi }})}{\sum _{\rm {n}}A_{n}}}} where ϕ n {\displaystyle \phi _{\rm {n}}} is the local or instantaneous phase as can be calculated using the Hilbert transform and A n {\displaystyle A_{\rm {n}}} are the local amplitude, or energy, of the signal. When all the phases are aligned, this is equal to 1. Several ways of implementing phase congruency have been developed, of which two versions are available in open source, one written for MATLAB and the other written in Java as a plugin for the ImageJ software. Given the different notations used for its formulation, a unified version has been recently presented, where a methodology for the parameter tuning is also presented. == Advantages == The square-wave example is naive in that most edge detection methods deal with it equally well. For example, the first derivative has a maximal magnitude at the edges. However, there are cases where the perceived edge does not have a sharp step or a large derivative. The method of phase congruency applies to many cases where other methods fail. A notable example is an image feature consisting of a single line, such as the letter "l". Many edge-detection algorithms will pick up two adjacent edges: the transitions from white to black, and black to white. On the other hand, the phase congruency map has a single line. A simple Fourier analogy of this case is a triangle wave. In each of its crests there is a congruency of crests from different sinusoidal functions. == Disadvantages == Calculating the phase congruency map of an image is very computationally intensive, and sensitive to image noise. Techniques of noise reduction are usually applied prior to the calculation.