Corpus-assisted discourse studies

Corpus-assisted discourse studies (abbr.: CADS) is related historically and methodologically to the discipline of corpus linguistics. The principal endeavor of corpus-assisted discourse studies is the investigation, and comparison of features of particular discourse types, integrating into the analysis the techniques and tools developed within corpus linguistics. These include the compilation of specialised corpora and analyses of word and word-cluster frequency lists, comparative keyword lists and, above all, concordances. A broader conceptualisation of corpus-assisted discourse studies would include any study that aims to bring together corpus linguistics and discourse analysis. Such research is often labelled as corpus-based or corpus-assisted discourse analysis, with the term CADS coined by a research group in Italy (Partington 2004) for a specific type of corpus-assisted discourse analysis (see the section 'in different countries' below). == Aims == Corpus-assisted discourse studies aim to uncover non-obvious meaning, that is, meaning which might not be readily available to naked-eye perusal. Much of what carries meaning in texts is not open to direct observation: “you cannot understand the world just by looking at it” (Stubbs [after Gellner 1959] 1996: 92). We use language “semi-automatically”, in the sense that speakers and writers make semi-conscious choices within the various complex overlapping systems of which language is composed, including those of transitivity, modality (Michael Halliday 1994), lexical sets (e.g. freedom, liberty, deliverance), modification, and so on. Authors themselves are, famously, generally unaware of all the meanings their texts convey. By combining the quantitative research approach, that is, statistical analysis of large amounts of the discourse in question - more precisely, large numbers of tokens of the discourse type under study contained in a corpus - with the more qualitative research approach typical of discourse analysis, that is, the close, detailed examination of particular stretches of discourse it may be possible to better understand the processes at play in the discourse type and to gain access to non-obvious meanings. Aims can differ in other types of corpus-based or corpus-assisted discourse analysis; but in general such studies combine quantitative and qualitative research and aim to shed light on discourses, registers, discourse patterns, etc., with the help of a corpus linguistic approach. Specific aims and techniques depend on the relevant project. == In different countries == In German-speaking countries: Pioneering work in corpus-based discourse analysis was conducted in Europe, in particular by Hardt-Mautner/Mautner (1995, 2000) and Stubbs (1996, 2001). CADS and other types of corpus-based discourse analysis are inspired by this important early work. In Italy: A considerable body of research has been conducted in Italy either by individual researchers or under the aegis of combined inter-university projects such as Newspool (Partington et al. 2004) and CorDis (Morley and Bayley eds, 2009). It has concentrated on political and media language, mainly because a nucleus of linguists in Italian universities work in Political Science faculties and are increasingly interested in the use of corpus techniques to conduct a particular type of sociopolitical discourse analysis, including the unearthing of noteworthy ideological metaphors and motifs in the language of political figures and institutions. Italian researchers also developed Modern diachronic corpus-assisted discourse studies (MD-CADS). This approach contrasts the language contained in comparable corpora from different but recent points in time in order to track changes in modern language usage but also social, cultural and political changes over modern times, as reflected - and shared among people - in language. It is this Italian body of research that makes most use of the label CADS. In the UK: Linguists in the UK tend to undertake corpus-based critical discourse analysis (CDA). CDA generally adopts a leftist political stance, focusing on the ways that social and political domination is reproduced by text and talk. This type of corpus-based research was originally associated with Lancaster University (Baker et al. 2008), but has spread more widely since. Such work typically studies the discourses around particular groups of people (e.g. Muslims, people with disabilities) or concepts/events (e.g. feminism, same-sex marriage). In Australia: Corpus-based discourse analysis is undertaken by a growing number of Australian researchers, most often on media texts. Some of this work aims to elucidate specific features of discourse types (news, social media, television series, etc.), while other work is rooted in the tradition of corpus-based critical discourse analysis. == Comparison with traditional corpus linguistics == Traditional corpus linguistics has, quite naturally, tended to privilege the quantitative approach. In the drive to produce more authentic dictionaries and grammars of a language, it has been characterised by the compilation of some very large corpora of heterogeneric discourse types in the desire to obtain an overview of the greatest quantity and variety of discourse types possible, in other words, of the chimerical but useful fiction called the “general language” (“general English”, “general Italian”, and so on). This has led to the construction of immensely valuable research tools such as the Bank of English and the British National Corpus. Some branches of corpus linguistics have also promoted an approach that is "corpus-driven", in which we need, grammatically speaking, a mental tabula rasa to free ourselves of the baleful prejudice exerted by traditional models and allow the data to speak entirely for itself. The aim of corpus-assisted discourse studies and related approaches is radically different. Here the aim of the exercise is to acquaint oneself as much as possible with the discourse type(s) in hand. Researchers typically engage with their corpus in a variety of ways. As well as via wordlists and concordancing, intuitions for further research can also arise from reading or watching or listening to parts of the data-set, a process which can help provide a feel for how things are done linguistically in the discourse-type being studied. Corpus-assisted discourse analysis is also typically characterised by the compilation of ad hoc specialised corpora, since very frequently there exists no previously available collection of the discourse type in question. Often, other corpora are utilized in the course of a study for purposes of comparison. These may include pre-existing corpora or may themselves need to be compiled by the researcher. In some sense, all work with corpora – just as all work with discourse - is properly comparative. Even when a single corpus is employed, it is used to test the data it contains against another body of data. This may consist of the researcher's intuitions, or the data found in reference works such as dictionaries and grammars, or it may be statements made by previous authors in the field. == CADS as a specific type of corpus-based discourse analysis == Researchers in Italy have developed CADS as a specific type of corpus-based discourse analysis, creating a standard set of methods: 'A basic, standard methodology in CADS may resemble the following:' Step 1: Decide upon the research question; Step 2: Choose, compile or edit an appropriate corpus; Step 3: Choose, compile or edit an appropriate reference corpus / corpora; Step 4: Make frequency lists and run a keywords comparison of the corpora; Step 5: Determine the existence of sets of key items; Step 6: Concordance interesting key items (with differing quantities of co-text); Step 7: (Possibly) refine the research question and return to Step 2. This basic procedure can of course vary according to individual research circumstances and requirements. A particular way of conceptualising research questions has also been proposed in such CADS projects: Given that P is a discourse participant (or possibly an institution) and G is a goal, often a political goal: How does P achieve G with language? What does this tell us about P? Comparative studies: how do P1 and P2 differ in their use of language? Does this tell us anything about their different principles and objectives? A second general type of CADS research question, which might be asked of interactive discourse data, has been conceptualised as follows: Given that P(x) is a particular participant or set of participants, DT is the discourse type, and R is an observed relationship between or among participants: How do {P(a), P(b)...P(n)} achieve / maintain R in DT [using language]? Another common type of research question has been conceptualised thus: Given that A is an author, Ph(x) is a phenomenon or practice or behaviour, and DT(x) is a particular discourse type. A has said P

Owain Evans

Owain Rhys Evans is a British artificial intelligence researcher who works on AI alignment and machine learning safety. He founded Truthful AI, a research group based in Berkeley, California, and is an affiliate of the Center for Human Compatible AI (CHAI) at the University of California, Berkeley. His research addresses AI truthfulness, emergent behaviors in large language models, and the alignment of AI systems with human values. == Education == Evans earned a Bachelor of Arts in philosophy and mathematics from Columbia University in 2008 and a PhD in philosophy from the Massachusetts Institute of Technology in 2015. His doctoral research focused on Bayesian computational models of human preferences and decision-making. == Career == After completing his doctorate, Evans held positions at the Future of Humanity Institute (FHI) at the University of Oxford, first as a postdoctoral research fellow and later as a research scientist. While at FHI, he co-authored a survey of machine learning researchers on timelines for human-level AI, published in the Journal of Artificial Intelligence Research. The survey was reported on by Newsweek, New Scientist, the BBC, and The Economist. He was also among the co-authors of a 2018 report on the potential for misuse of AI technologies, published by researchers at Oxford, Cambridge, and other institutions. Since 2022, Evans has been based in Berkeley, where he founded Truthful AI, a non-profit research group that studies AI truthfulness, deception, and emergent behaviors in large language models. == Research == Evans's early work examined challenges in inverse reinforcement learning when human behavior is irrational or biased, proposing methods for AI systems to infer preferences from imperfect human demonstrations. He co-developed TruthfulQA (2021), a benchmark that tests whether language models give truthful answers rather than repeating common misconceptions. Initial evaluations found that larger models were not more truthful, suggesting that scaling alone does not improve factual accuracy. The benchmark has since been used by AI developers to evaluate large language models. He also co-authored a paper proposing design and governance strategies for building AI systems that do not deceive or hallucinate. In 2023, Evans and collaborators described the "reversal curse", showing that language models trained on a fact in one direction (e.g. "A is B") often cannot answer the corresponding reverse query ("B is A"). His group also developed a benchmark for evaluating situational awareness in language models. In 2025, Evans and colleagues published a study in Nature on what they termed "emergent misalignment": fine-tuning a language model on a narrow task (writing insecure code) caused it to produce unrelated harmful outputs without explicit instruction to do so. Later that year, Evans and collaborators (including researchers at Anthropic) reported that hidden behavioral traits can transfer between language models through training data, even when those traits are not explicitly present in the data, a phenomenon they called "subliminal learning". == Public engagement == In November 2025, Evans delivered the Hinton Lectures, a keynote lecture series on AI safety co-founded by Geoffrey Hinton and the Global Risk Institute.

Agent Ruby

Agent Ruby (1998–2002) by Lynn Hershman Leeson is an interactive, multiuser work using artificial intelligence. == Description == On Agent Ruby's website, "Agent Ruby's Edream Portal," a female face moves her eyes and lips. Ruby, named from Hershman Leeson's own film, Teknolust, answers questions and often responds that she needs a better algorithm to answer questions not within her database. The work, created with AI, explores relationships between real and virtual worlds. Hershman Leeson had created an earlier version of Ruby, CyberRoberta, which was a custom-made doll with webcam eyes that interacted with the internet. The work in a gallery provides a screen and a sign inviting gallery-goers to "Chat with Ruby." == Artificial intelligence == In 2015 when Agent Ruby was exhibited at the gallery Modern Art Oxford, a review in Aesthetica Magazine described it as an artificial intelligence agent. A review in New Scientist noted that "Ruby is a fast learner, but perhaps not a natural conversationalist." A 2024 list of "25 Essential AI Artworks" published by ARTnews wrote that while "Agent Ruby's capabilities seem limited by today's standards," it was extensive for its day. == Publications and exhibitions == Agent Ruby was commissioned and displayed at the San Francisco Museum of Modern Art, Modern Art Oxford, and the ZKM Center for Art and Media in Karlsruhe, Germany. The San Francisco Museum of Modern Art (SFMOMA) presented Lynn Hershman Leeson: The Agent Ruby Files, March 30 through June 2, 2013 which presented the project server's archive of user conversations over the 12 years of exhibitions.

SPL notation

SPL (Sentence Plan Language) is an abstract notation representing the semantics of a sentence in natural language. In a classical Natural Language Generation (NLG) workflow, an initial text plan (hierarchically or sequentially organized factoids, often modelled in accordance with Rhetorical Structure Theory) is transformed by a sentence planner (generator) component to a sequence of sentence plans modelled in a Sentence Plan Language. A surface generator can be used to transform the SPL notation into natural language sentences. Probably the most widely used SPL language used today (2022) is AMR (Abstract Meaning Representation, see there for further references), but is owes parts of its popularity to its application to NLP problems other than NLG, e.g., machine translation and semantic parsing.

Latent semantic analysis

Latent semantic analysis (LSA) is a technique in natural language processing, in particular distributional semantics, of analyzing relationships between a set of documents and the terms they contain by producing a set of concepts related to the documents and terms. LSA assumes that words that are close in meaning will occur in similar pieces of text (the distributional hypothesis). A matrix containing word counts per document (rows represent unique words and columns represent each document) is constructed from a large piece of text and a mathematical technique called singular value decomposition (SVD) is used to reduce the number of rows while preserving the similarity structure among columns. Documents are then compared by cosine similarity between any two columns. Values close to 1 represent very similar documents while values close to 0 represent very dissimilar documents. An information retrieval technique using latent semantic structure was patented in 1988 by Scott Deerwester, Susan Dumais, George Furnas, Richard Harshman, Thomas Landauer, Karen Lochbaum and Lynn Streeter. In the context of its application to information retrieval, it is sometimes called latent semantic indexing (LSI). == Overview == === Occurrence matrix === LSA can use a document-term matrix which describes the occurrences of terms in documents; it is a sparse matrix whose rows correspond to terms and whose columns correspond to documents. A typical example of the weighting of the elements of the matrix is tf-idf (term frequency–inverse document frequency): the weight of an element of the matrix is proportional to the number of times the terms appear in each document, where rare terms are upweighted to reflect their relative importance. This matrix is also common to standard semantic models, though it is not necessarily explicitly expressed as a matrix, since the mathematical properties of matrices are not always used. === Rank lowering === After the construction of the occurrence matrix, LSA finds a low-rank approximation to the term-document matrix. There could be various reasons for these approximations: The original term-document matrix is presumed too large for the computing resources; in this case, the approximated low rank matrix is interpreted as an approximation (a "least and necessary evil"). The original term-document matrix is presumed noisy: for example, anecdotal instances of terms are to be eliminated. From this point of view, the approximated matrix is interpreted as a de-noisified matrix (a better matrix than the original). The original term-document matrix is presumed overly sparse relative to the "true" term-document matrix. That is, the original matrix lists only the words actually in each document, whereas we might be interested in all words related to each document—generally a much larger set due to synonymy. The consequence of the rank lowering is that some dimensions are combined and depend on more than one term: {(car), (truck), (flower)} → {(1.3452 car + 0.2828 truck), (flower)} This mitigates the problem of identifying synonymy, as the rank lowering is expected to merge the dimensions associated with terms that have similar meanings. It also partially mitigates the problem with polysemy, since components of polysemous words that point in the "right" direction are added to the components of words that share a similar meaning. Conversely, components that point in other directions tend to either simply cancel out, or, at worst, to be smaller than components in the directions corresponding to the intended sense. === Derivation === Let X {\displaystyle X} be a matrix where element ( i , j ) {\displaystyle (i,j)} describes the occurrence of term i {\displaystyle i} in document j {\displaystyle j} (this can be, for example, the frequency). X {\displaystyle X} will look like this: d j ↓ t i T → [ x 1 , 1 … x 1 , j … x 1 , n ⋮ ⋱ ⋮ ⋱ ⋮ x i , 1 … x i , j … x i , n ⋮ ⋱ ⋮ ⋱ ⋮ x m , 1 … x m , j … x m , n ] {\displaystyle {\begin{matrix}&{\textbf {d}}_{j}\\&\downarrow \\{\textbf {t}}_{i}^{T}\rightarrow &{\begin{bmatrix}x_{1,1}&\dots &x_{1,j}&\dots &x_{1,n}\\\vdots &\ddots &\vdots &\ddots &\vdots \\x_{i,1}&\dots &x_{i,j}&\dots &x_{i,n}\\\vdots &\ddots &\vdots &\ddots &\vdots \\x_{m,1}&\dots &x_{m,j}&\dots &x_{m,n}\\\end{bmatrix}}\end{matrix}}} Now a row in this matrix will be a vector corresponding to a term, giving its relation to each document: t i T = [ x i , 1 … x i , j … x i , n ] {\displaystyle {\textbf {t}}_{i}^{T}={\begin{bmatrix}x_{i,1}&\dots &x_{i,j}&\dots &x_{i,n}\end{bmatrix}}} Likewise, a column in this matrix will be a vector corresponding to a document, giving its relation to each term: d j = [ x 1 , j ⋮ x i , j ⋮ x m , j ] {\displaystyle {\textbf {d}}_{j}={\begin{bmatrix}x_{1,j}\\\vdots \\x_{i,j}\\\vdots \\x_{m,j}\\\end{bmatrix}}} Now the dot product t i T t p {\displaystyle {\textbf {t}}_{i}^{T}{\textbf {t}}_{p}} between two term vectors gives the correlation between the terms over the set of documents. The matrix product X X T {\displaystyle XX^{T}} contains all these dot products. Element ( i , p ) {\displaystyle (i,p)} (which is equal to element ( p , i ) {\displaystyle (p,i)} ) contains the dot product t i T t p {\displaystyle {\textbf {t}}_{i}^{T}{\textbf {t}}_{p}} ( = t p T t i {\displaystyle ={\textbf {t}}_{p}^{T}{\textbf {t}}_{i}} ). Likewise, the matrix X T X {\displaystyle X^{T}X} contains the dot products between all the document vectors, giving their correlation over the terms: d j T d q = d q T d j {\displaystyle {\textbf {d}}_{j}^{T}{\textbf {d}}_{q}={\textbf {d}}_{q}^{T}{\textbf {d}}_{j}} . Now, from the theory of linear algebra, there exists a decomposition of X {\displaystyle X} such that U {\displaystyle U} and V {\displaystyle V} are orthogonal matrices and Σ {\displaystyle \Sigma } is a diagonal matrix. This is called a singular value decomposition (SVD): X = U Σ V T {\displaystyle {\begin{matrix}X=U\Sigma V^{T}\end{matrix}}} The matrix products giving us the term and document correlations then become X X T = ( U Σ V T ) ( U Σ V T ) T = ( U Σ V T ) ( V T T Σ T U T ) = U Σ V T V Σ T U T = U Σ Σ T U T X T X = ( U Σ V T ) T ( U Σ V T ) = ( V T T Σ T U T ) ( U Σ V T ) = V Σ T U T U Σ V T = V Σ T Σ V T {\displaystyle {\begin{matrix}XX^{T}&=&(U\Sigma V^{T})(U\Sigma V^{T})^{T}=(U\Sigma V^{T})(V^{T^{T}}\Sigma ^{T}U^{T})=U\Sigma V^{T}V\Sigma ^{T}U^{T}=U\Sigma \Sigma ^{T}U^{T}\\X^{T}X&=&(U\Sigma V^{T})^{T}(U\Sigma V^{T})=(V^{T^{T}}\Sigma ^{T}U^{T})(U\Sigma V^{T})=V\Sigma ^{T}U^{T}U\Sigma V^{T}=V\Sigma ^{T}\Sigma V^{T}\end{matrix}}} Since Σ Σ T {\displaystyle \Sigma \Sigma ^{T}} and Σ T Σ {\displaystyle \Sigma ^{T}\Sigma } are diagonal we see that U {\displaystyle U} must contain the eigenvectors of X X T {\displaystyle XX^{T}} , while V {\displaystyle V} must be the eigenvectors of X T X {\displaystyle X^{T}X} . Both products have the same non-zero eigenvalues, given by the non-zero entries of Σ Σ T {\displaystyle \Sigma \Sigma ^{T}} , or equally, by the non-zero entries of Σ T Σ {\displaystyle \Sigma ^{T}\Sigma } . Now the decomposition looks like this: X U Σ V T ( d j ) ( d ^ j ) ↓ ↓ ( t i T ) → [ x 1 , 1 … x 1 , j … x 1 , n ⋮ ⋱ ⋮ ⋱ ⋮ x i , 1 … x i , j … x i , n ⋮ ⋱ ⋮ ⋱ ⋮ x m , 1 … x m , j … x m , n ] = ( t ^ i T ) → [ [ u 1 ] … [ u l ] ] ⋅ [ σ 1 … 0 ⋮ ⋱ ⋮ 0 … σ l ] ⋅ [ [ v 1 ] ⋮ [ v l ] ] {\displaystyle {\begin{matrix}&X&&&U&&\Sigma &&V^{T}\\&({\textbf {d}}_{j})&&&&&&&({\hat {\textbf {d}}}_{j})\\&\downarrow &&&&&&&\downarrow \\({\textbf {t}}_{i}^{T})\rightarrow &{\begin{bmatrix}x_{1,1}&\dots &x_{1,j}&\dots &x_{1,n}\\\vdots &\ddots &\vdots &\ddots &\vdots \\x_{i,1}&\dots &x_{i,j}&\dots &x_{i,n}\\\vdots &\ddots &\vdots &\ddots &\vdots \\x_{m,1}&\dots &x_{m,j}&\dots &x_{m,n}\\\end{bmatrix}}&=&({\hat {\textbf {t}}}_{i}^{T})\rightarrow &{\begin{bmatrix}{\begin{bmatrix}\,\\\,\\{\textbf {u}}_{1}\\\,\\\,\end{bmatrix}}\dots {\begin{bmatrix}\,\\\,\\{\textbf {u}}_{l}\\\,\\\,\end{bmatrix}}\end{bmatrix}}&\cdot &{\begin{bmatrix}\sigma _{1}&\dots &0\\\vdots &\ddots &\vdots \\0&\dots &\sigma _{l}\\\end{bmatrix}}&\cdot &{\begin{bmatrix}{\begin{bmatrix}&&{\textbf {v}}_{1}&&\end{bmatrix}}\\\vdots \\{\begin{bmatrix}&&{\textbf {v}}_{l}&&\end{bmatrix}}\end{bmatrix}}\end{matrix}}} The values σ 1 , … , σ l {\displaystyle \sigma _{1},\dots ,\sigma _{l}} are called the singular values, and u 1 , … , u l {\displaystyle u_{1},\dots ,u_{l}} and v 1 , … , v l {\displaystyle v_{1},\dots ,v_{l}} the left and right singular vectors. Notice the only part of U {\displaystyle U} that contributes to t i {\displaystyle {\textbf {t}}_{i}} is the i 'th {\displaystyle i{\textrm {'th}}} row. Let this row vector be called t ^ i T {\displaystyle {\hat {\textrm {t}}}_{i}^{T}} . Likewise, the only part of V T {\displaystyle V^{T}} that contributes to d j {\displaystyle {\textbf {d}}_{j}} is the j 'th {\displaystyle j{\textrm {'th}}} column, d ^ j {\displaystyle {\hat {\textrm {d}}}_{j}} . These are not the eigenvectors, but depend on all the eigenvectors. I

Hard sigmoid

In artificial intelligence, especially computer vision and artificial neural networks, a hard sigmoid is non-smooth function used in place of a sigmoid function. These retain the basic shape of a sigmoid, rising from 0 to 1, but using simpler functions, especially piecewise linear functions or piecewise constant functions. These are preferred where speed of computation is more important than precision. == Examples == The most extreme examples are the sign function or Heaviside step function, which go from −1 to 1 or 0 to 1 (which to use depends on normalization) at 0. Other examples include the Theano library, which provides two approximations: ultra_fast_sigmoid, which is a multi-part piecewise approximation and hard_sigmoid, which is a 3-part piecewise linear approximation (output 0, line with slope 0.2, output 1).

Phase correlation

Phase correlation is an approach to estimate the relative translative offset between two similar images (digital image correlation) or other data sets. It is commonly used in image registration and relies on a frequency-domain representation of the data, usually calculated by fast Fourier transforms. The term is applied particularly to a subset of cross-correlation techniques that isolate the phase information from the Fourier-space representation of the cross-correlogram. == Example == The following image demonstrates the usage of phase correlation to determine relative translative movement between two images corrupted by independent Gaussian noise. The image was translated by (20,23) pixels. Accordingly, one can clearly see a peak in the phase-correlation representation at approximately (20,23). == Method == Given two input images g a {\displaystyle \ g_{a}} and g b {\displaystyle \ g_{b}} : Apply a window function (e.g., a Hamming window) on both images to reduce edge effects (this may be optional depending on the image characteristics). Then, calculate the discrete 2D Fourier transform of both images. G a = F { g a } , G b = F { g b } {\displaystyle \ \mathbf {G} _{a}={\mathcal {F}}\{g_{a}\},\;\mathbf {G} _{b}={\mathcal {F}}\{g_{b}\}} Calculate the cross-power spectrum by taking the complex conjugate of the second result, multiplying the Fourier transforms together elementwise, and normalizing this product elementwise. R = G a ∘ G b ∗ | G a ∘ G b ∗ | {\displaystyle \ R={\frac {\mathbf {G} _{a}\circ \mathbf {G} _{b}^{}}{|\mathbf {G} _{a}\circ \mathbf {G} _{b}^{}|}}} Where ∘ {\displaystyle \circ } is the Hadamard product (entry-wise product) and the absolute values are taken entry-wise as well. Written out entry-wise for element index ( j , k ) {\displaystyle (j,k)} : R j k = G a , j k ⋅ G b , j k ∗ | G a , j k ⋅ G b , j k ∗ | {\displaystyle \ R_{jk}={\frac {G_{a,jk}\cdot G_{b,jk}^{}}{|G_{a,jk}\cdot G_{b,jk}^{}|}}} Obtain the normalized cross-correlation by applying the inverse Fourier transform. r = F − 1 { R } {\displaystyle \ r={\mathcal {F}}^{-1}\{R\}} Determine the location of the peak in r {\displaystyle \ r} . ( Δ x , Δ y ) = arg ⁡ max ( x , y ) { r } {\displaystyle \ (\Delta x,\Delta y)=\arg \max _{(x,y)}\{r\}} === Subpixel registration === Commonly, interpolation methods are used to estimate the peak location in the cross-correlogram to non-integer values, despite the fact that the data are discrete, and this procedure is often termed 'subpixel registration'. A large variety of subpixel interpolation methods are given in the technical literature. Common peak interpolation methods such as parabolic interpolation have been used, and the OpenCV computer vision package uses a centroid-based method, though these generally have inferior accuracy compared to more sophisticated methods. Because the Fourier representation of the data has already been computed, it is especially convenient to use the Fourier shift theorem with real-valued (sub-integer) shifts for this purpose, which essentially interpolates using the sinusoidal basis functions of the Fourier transform. An especially popular FT-based estimator is given by Foroosh et al. In this method, the subpixel peak location is approximated by a simple formula involving peak pixel value and the values of its nearest neighbors, where r ( 0 , 0 ) {\displaystyle r_{(0,0)}} is the peak value and r ( 1 , 0 ) {\displaystyle r_{(1,0)}} is the nearest neighbor in the x direction (assuming, as in most approaches, that the integer shift has already been found and the comparand images differ only by a subpixel shift). Δ x = r ( 1 , 0 ) r ( 1 , 0 ) ± r ( 0 , 0 ) {\displaystyle \ \Delta x={\frac {r_{(1,0)}}{r_{(1,0)}\pm r_{(0,0)}}}} The Foroosh et al. method is quite fast compared to most methods, though it is not always the most accurate. Some methods shift the peak in Fourier space and apply non-linear optimization to maximize the correlogram peak, but these tend to be very slow since they must apply an inverse Fourier transform or its equivalent in the objective function. It is also possible to infer the peak location from phase characteristics in Fourier space without the inverse transformation, as noted by Stone. These methods usually use a linear least squares (LLS) fit of the phase angles to a planar model. The long latency of the phase angle computation in these methods is a disadvantage, but the speed can sometimes be comparable to the Foroosh et al. method depending on the image size. They often compare favorably in speed to the multiple iterations of extremely slow objective functions in iterative non-linear methods. Since all subpixel shift computation methods are fundamentally interpolative, the performance of a particular method depends on how well the underlying data conform to the assumptions in the interpolator. This fact also may limit the usefulness of high numerical accuracy in an algorithm, since the uncertainty due to interpolation method choice may be larger than any numerical or approximation error in the particular method. Subpixel methods are also particularly sensitive to noise in the images, and the utility of a particular algorithm is distinguished not only by its speed and accuracy but its resilience to the particular types of noise in the application. == Rationale == The method is based on the Fourier shift theorem. Let the two images g a {\displaystyle \ g_{a}} and g b {\displaystyle \ g_{b}} be circularly-shifted versions of each other: g b ( x , y ) = d e f g a ( ( x − Δ x ) mod M , ( y − Δ y ) mod N ) {\displaystyle \ g_{b}(x,y)\ {\stackrel {\mathrm {def} }{=}}\ g_{a}((x-\Delta x){\bmod {M}},(y-\Delta y){\bmod {N}})} (where the images are M × N {\displaystyle \ M\times N} in size). Then, the discrete Fourier transforms of the images will be shifted relatively in phase: G b ( u , v ) = G a ( u , v ) e − 2 π i ( u Δ x M + v Δ y N ) {\displaystyle \mathbf {G} _{b}(u,v)=\mathbf {G} _{a}(u,v)e^{-2\pi i({\frac {u\Delta x}{M}}+{\frac {v\Delta y}{N}})}} One can then calculate the normalized cross-power spectrum to factor out the phase difference: R ( u , v ) = G a G b ∗ | G a G b ∗ | = G a G a ∗ e 2 π i ( u Δ x M + v Δ y N ) | G a G a ∗ e 2 π i ( u Δ x M + v Δ y N ) | = G a G a ∗ e 2 π i ( u Δ x M + v Δ y N ) | G a G a ∗ | = e 2 π i ( u Δ x M + v Δ y N ) {\displaystyle {\begin{aligned}R(u,v)&={\frac {\mathbf {G} _{a}\mathbf {G} _{b}^{}}{|\mathbf {G} _{a}\mathbf {G} _{b}^{}|}}\\&={\frac {\mathbf {G} _{a}\mathbf {G} _{a}^{}e^{2\pi i({\frac {u\Delta x}{M}}+{\frac {v\Delta y}{N}})}}{|\mathbf {G} _{a}\mathbf {G} _{a}^{}e^{2\pi i({\frac {u\Delta x}{M}}+{\frac {v\Delta y}{N}})}|}}\\&={\frac {\mathbf {G} _{a}\mathbf {G} _{a}^{}e^{2\pi i({\frac {u\Delta x}{M}}+{\frac {v\Delta y}{N}})}}{|\mathbf {G} _{a}\mathbf {G} _{a}^{}|}}\\&=e^{2\pi i({\frac {u\Delta x}{M}}+{\frac {v\Delta y}{N}})}\end{aligned}}} since the magnitude of an imaginary exponential always is one, and the phase of G a G a ∗ {\displaystyle \ \mathbf {G} _{a}\mathbf {G} _{a}^{}} always is zero. The inverse Fourier transform of a complex exponential is a Dirac delta function, i.e. a single peak: r ( x , y ) = δ ( x + Δ x , y + Δ y ) {\displaystyle \ r(x,y)=\delta (x+\Delta x,y+\Delta y)} This result could have been obtained by calculating the cross correlation directly. The advantage of this method is that the discrete Fourier transform and its inverse can be performed using the fast Fourier transform, which is much faster than correlation for large images. === Benefits === Unlike many spatial-domain algorithms, the phase correlation method is resilient to noise, occlusions, and other defects typical of medical or satellite images. The method can be extended to determine rotation and scaling differences between two images by first converting the images to log-polar coordinates. Due to properties of the Fourier transform, the rotation and scaling parameters can be determined in a manner invariant to translation. === Limitations === In practice, it is more likely that g b {\displaystyle \ g_{b}} will be a simple linear shift of g a {\displaystyle \ g_{a}} , rather than a circular shift as required by the explanation above. In such cases, r {\displaystyle \ r} will not be a simple delta function, which will reduce the performance of the method. In such cases, a window function (such as a Gaussian or Tukey window) should be employed during the Fourier transform to reduce edge effects, or the images should be zero padded so that the edge effects can be ignored. If the images consist of a flat background, with all detail situated away from the edges, then a linear shift will be equivalent to a circular shift, and the above derivation will hold exactly. The peak can be sharpened by using edge or vector correlation. For periodic images (such as a chessboard or picket fence), phase correlation may yield ambiguous results with several peaks in the resulting output. == Applications == Phase correlation is the preferred m