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Procrustes Superimposition
In statistics, Procrustes analysis is a form of statistical shape analysis used to analyse the distribution of a set of shapes. The name ''Procrustes'' ( el, Προκρούστης) refers to a bandit from Greek mythology who made his victims fit his bed either by stretching their limbs or cutting them off. In mathematics: * an orthogonal Procrustes problem is a method which can be used to find out the optimal ''rotation and/or reflection'' (i.e., the optimal orthogonal linear transformation) for the Procrustes Superimposition (PS) of an object with respect to another. * a constrained orthogonal Procrustes problem, subject to det(''R'') = 1 (where ''R'' is a rotation matrix), is a method which can be used to determine the optimal ''rotation'' for the PS of an object with respect to another (reflection is not allowed). In some contexts, this method is called the Kabsch algorithm. When a shape is compared to another, or a set of shapes is compared to an arbitrarily selected refe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Procrustes Superimposition
In statistics, Procrustes analysis is a form of statistical shape analysis used to analyse the distribution of a set of shapes. The name ''Procrustes'' ( el, Προκρούστης) refers to a bandit from Greek mythology who made his victims fit his bed either by stretching their limbs or cutting them off. In mathematics: * an orthogonal Procrustes problem is a method which can be used to find out the optimal ''rotation and/or reflection'' (i.e., the optimal orthogonal linear transformation) for the Procrustes Superimposition (PS) of an object with respect to another. * a constrained orthogonal Procrustes problem, subject to det(''R'') = 1 (where ''R'' is a rotation matrix), is a method which can be used to determine the optimal ''rotation'' for the PS of an object with respect to another (reflection is not allowed). In some contexts, this method is called the Kabsch algorithm. When a shape is compared to another, or a set of shapes is compared to an arbitrarily selected refe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mean
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set. For a data set, the '' arithmetic mean'', also known as "arithmetic average", is a measure of central tendency of a finite set of numbers: specifically, the sum of the values divided by the number of values. The arithmetic mean of a set of numbers ''x''1, ''x''2, ..., x''n'' is typically denoted using an overhead bar, \bar. If the data set were based on a series of observations obtained by sampling from a statistical population, the arithmetic mean is the ''sample mean'' (\bar) to distinguish it from the mean, or expected value, of the underlying distribution, the ''population mean'' (denoted \mu or \mu_x).Underhill, L.G.; Bradfield d. (1998) ''Introstat'', Juta and Company Ltd.p. 181/ref> Outside probability and statistics, a wide range of other notions of mean ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Biometrics
Biometrics are body measurements and calculations related to human characteristics. Biometric authentication (or realistic authentication) is used in computer science as a form of identification and access control. It is also used to identify individuals in groups that are under surveillance. Biometric identifiers are the distinctive, measurable characteristics used to label and describe individuals. Biometric identifiers are often categorized as physiological characteristics which are related to the shape of the body. Examples include, but are not limited to fingerprint, palm veins, face recognition, DNA, palm print, hand geometry, iris recognition, retina, odor/scent, voice, shape of ears and gait. Behavioral characteristics are related to the pattern of behavior of a person, including but not limited to mouse movement, typing rhythm, gait, signature, behavioral profiling, and credentials. Some researchers have coined the term behaviometrics to describe the latter cl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alignments Of Random Points
Alignments of random points in a plane can be demonstrated by statistics to be counter-intuitively easy to find when a large number of random points are marked on a bounded flat surface. This has been put forward as a demonstration that ley lines and other similar mysterious alignments believed by some to be phenomena of deep significance might exist solely due to chance alone, as opposed to the supernatural or anthropological explanations put forward by their proponents. The topic has also been studied in the fields of computer vision and astronomy. A number of studies have examined the mathematics of alignment of random points on the plane. In all of these, the width of the line — the allowed displacement of the positions of the points from a perfect straight line — is important. It allows the fact that real-world features are not mathematical points, and that their positions need not line up exactly for them to be considered in alignment. Alfred Watkins, in his classic w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Active Shape Model
Active shape models (ASMs) are statistical models of the shape of objects which iteratively deform to fit to an example of the object in a new image, developed by Tim Cootes and Chris Taylor in 1995. The shapes are constrained by the PDM ( point distribution model) Statistical Shape Model to vary only in ways seen in a training set of labelled examples. The shape of an object is represented by a set of points (controlled by the shape model). The ASM algorithm aims to match the model to a new image. The ASM works by alternating the following steps: * Generate a suggested shape by looking in the image around each point for a better position for the point. This is commonly done using what is called a "profile model", which looks for strong edges or uses the Mahalanobis distance to match a model template for the point. * Conform the suggested shape to the point distribution model, commonly called a "shape model" in this context. The figure to the right shows an example. The techn ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Standing Stones
A menhir (from Brittonic languages: ''maen'' or ''men'', "stone" and ''hir'' or ''hîr'', "long"), standing stone, orthostat, or lith is a large human-made upright stone, typically dating from the European middle Bronze Age. They can be found individually as monoliths, or as part of a group of similar stones. Menhirs' size can vary considerably, but they often taper toward the top. They are widely distributed across Europe, Africa and Asia, but are most numerous in Western Europe; particularly in Ireland, Great Britain, and Brittany, where there are about 50,000 examples, and northwestern France, where there are some 1,200 further examples. Standing stones are usually difficult to date. They were constructed during many different periods across pre-history as part of the larger megalithic cultures in Europe and near areas. Some menhirs stand next to buildings that have an early or current religious significance. One example is the South Zeal Menhir in Devon, which formed the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David George Kendall
David George Kendall FRS (15 January 1918 – 23 October 2007) was an English statistician and mathematician, known for his work on probability, statistical shape analysis, ley lines and queueing theory. He spent most of his academic life in the University of Oxford (1946–1962) and the University of Cambridge (1962–1985). He worked with M. S. Bartlett during World War II, and visited Princeton University after the war. Life and career David George Kendall was born on 15 January 1918 in Ripon, West Riding of Yorkshire, and attended Ripon Grammar School before attending Queen's College, Oxford, graduating in 1939. He worked on rocketry during the World War II, before moving to Magdalen College, Oxford, in 1946. In 1962 he was appointed the first Professor of Mathematical Statistics in the Statistical Laboratory, University of Cambridge; in which post he remained until his retirement in 1985. He was elected to a professorial fellowship at Churchill College, and he ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Biological Data
Biological data refers to a compound or information derived from living organisms and their products. A medicinal compound made from living organisms, such as a serum or a vaccine, could be characterized as biological data. Biological data is highly complex when compared with other forms of data. There are many forms of biological data, including text, sequence data, protein structure, genomic data and amino acids, and links among others. Biological Data and Bioinformatics Biological data works closely with Bioinformatics, which is a recent discipline focusing on addressing the need to analyze and interpret vast amounts of genomic data. In the past few decades, leaps in genomic research have led to massive amounts of biological data. As a result, bioinformatics was created as the convergence of genomics, biotechnology, and information technology, while concentrating on biological data. Biological Data has also been difficult to define, as bioinformatics is a wide-encompassing fi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of n-dimensional Euclidean space. One-dimensional manifolds include lines and circles, but not lemniscates. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, and also the Klein bottle and real projective plane. The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows complicated structures to be described in terms of well-understood topological properties of simpler spaces. Manifolds naturally arise as solution sets of systems of equations and as graphs of functions. The concept has applications in computer-graphics given the need to associate pictures with coordinates ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and only if the matrix is invertible and the linear map represented by the matrix is an isomorphism. The determinant of a product of matrices is the product of their determinants (the preceding property is a corollary of this one). The determinant of a matrix is denoted , , or . The determinant of a matrix is :\begin a & b\\c & d \end=ad-bc, and the determinant of a matrix is : \begin a & b & c \\ d & e & f \\ g & h & i \end= aei + bfg + cdh - ceg - bdi - afh. The determinant of a matrix can be defined in several equivalent ways. Leibniz formula expresses the determinant as a sum of signed products of matrix entries such that each summand is the product of different entries, and the number of these summands is n!, the factorial of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Singular Value Decomposition
In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix. It generalizes the eigendecomposition of a square normal matrix with an orthonormal eigenbasis to any \ m \times n\ matrix. It is related to the polar decomposition. Specifically, the singular value decomposition of an \ m \times n\ complex matrix is a factorization of the form \ \mathbf = \mathbf\ , where is an \ m \times m\ complex unitary matrix, \ \mathbf\ is an \ m \times n\ rectangular diagonal matrix with non-negative real numbers on the diagonal, is an n \times n complex unitary matrix, and \ \mathbf\ is the conjugate transpose of . Such decomposition always exists for any complex matrix. If is real, then and can be guaranteed to be real orthogonal matrices; in such contexts, the SVD is often denoted \ \mathbf^\mathsf\ . The diagonal entries \ \sigma_i = \Sigma_\ of \ \mathbf\ are uniquely determined by and are known as the singular values ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rotation Matrix
In linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix :R = \begin \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end rotates points in the plane counterclockwise through an angle with respect to the positive axis about the origin of a two-dimensional Cartesian coordinate system. To perform the rotation on a plane point with standard coordinates , it should be written as a column vector, and multiplied by the matrix : : R\mathbf = \begin \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end \begin x \\ y \end = \begin x\cos\theta-y\sin\theta \\ x\sin\theta+y\cos\theta \end. If and are the endpoint coordinates of a vector, where is cosine and is sine, then the above equations become the trigonometric summation angle formulae. Indeed, a rotation matrix can be seen as the trigonometric summation angle formulae in matrix form. On ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
