Concepts inDiffusion maps for edge-aware image editing
Diffusion map
A diffusion map is a machine learning algorithm for dealing with dimensionality reduction. The algorithm was first introduced by R.R. Coifman and S. Lafon in Applied and Computational Harmonic Analysis and Diffusion Maps and Geometric Harmonics. Unlike other dimensionality reduction methods such as Principle Component Analysis (PCA) and Multi-Dimensional Scaling (MDS), diffusion mapping is a non-linear method which focuses on discovering the underlying manifold in which the data is embedded.
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Euclidean distance
In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" distance between two points that one would measure with a ruler, and is given by the Pythagorean formula. By using this formula as distance, Euclidean space (or even any inner product space) becomes a metric space. The associated norm is called the Euclidean norm. Older literature refers to the metric as Pythagorean metric.
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Interpolation
In the mathematical field of numerical analysis, interpolation is a method of constructing new data points within the range of a discrete set of known data points. In engineering and science, one often has a number of data points, obtained by sampling or experimentation, which represent the values of a function for a limited number of values of the independent variable. It is often required to interpolate (i.e.
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Codomain
In mathematics, the codomain or target set of a function is the set Y into which all of the output of the function is constrained to fall. It is the set Y in the notation f: X → Y. The codomain is also sometimes referred to as the range but that term is ambiguous as it may also refer to the image. The codomain is part of the modern definition of a function f as a triple (X, Y, F), with F a subset of the Cartesian product X × Y.
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Eigenvalues and eigenvectors
The eigenvectors of a square matrix are the non-zero vectors that, after being multiplied by the matrix, remain parallel to the original vector. For each eigenvector, the corresponding eigenvalue is the factor by which the eigenvector is scaled when multiplied by the matrix. The prefix eigen- is adopted from the German word "eigen" for "self" in the sense of a characteristic description. The eigenvectors are sometimes also called characteristic vectors.
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Spatial analysis
Spatial analysis or spatial statistics includes any of the formal techniques which study entities using their topological, geometric, or geographic properties.
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Matrix (mathematics)
In mathematics, a matrix (plural matrices, or less commonly matrixes) is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements is Matrices of the same size can be added or subtracted element by element. The rule for matrix multiplication is more complicated, and two matrices can be multiplied only when the number of columns in the first equals the number of rows in the second.
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Degree of a polynomial
The degree of a polynomial is the highest degree of its terms, when the polynomial is expressed in canonical form (i.e. as a linear combination of monomials). The degree of a term is the sum of the exponents of the variables that appear in it. The word degree is now standard, but in some older books, the word order may be used instead. For example, the polynomial has three terms. (Notice, this polynomial can also be expressed as .
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