This is a list of transforms in

mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

Integral transforms

Abel transform In mathematics, the Abel transform,N. H. Abel, Journal für die reine und angewandte Mathematik, 1, pp. 153–157 (1826). named for Niels Henrik Abel, is an integral transform often used in the analysis of spherically symmetric or axially symmetri ...

* Aboodh transform * Bateman transform *

Fourier transform In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...

** Short-time Fourier transform **

Gabor transform The Gabor transform, named after Dennis Gabor, is a special case of the short-time Fourier transform. It is used to determine the Sine wave, sinusoidal frequency and phase (waves), phase content of local sections of a signal as it changes over time ...

* Hankel transform * Hartley transform * Hermite transform *

Hilbert transform In mathematics and signal processing, the Hilbert transform is a specific singular integral that takes a function, of a real variable and produces another function of a real variable . The Hilbert transform is given by the Cauchy principal value ...

** Hilbert–Schmidt integral operator * Jacobi transform * Laguerre transform *

Laplace transform In mathematics, the Laplace transform, named after Pierre-Simon Laplace (), is an integral transform that converts a Function (mathematics), function of a Real number, real Variable (mathematics), variable (usually t, in the ''time domain'') to a f ...

** Inverse Laplace transform **

Two-sided Laplace transform In mathematics, the two-sided Laplace transform or bilateral Laplace transform is an integral transform equivalent to probability's moment-generating function. Two-sided Laplace transforms are closely related to the Fourier transform, the Melli ...

** Inverse two-sided Laplace transform * Laplace–Carson transform * Laplace–Stieltjes transform * Legendre transform * Linear canonical transform * Mellin transform ** Inverse Mellin transform ** Poisson–Mellin–Newton cycle * N-transform * Radon transform * Shehu transform * Stieltjes transformation * Sumudu transform * Wavelet transform (integral) * Weierstrass transform

Discrete transforms

* Binomial transform *

Discrete Fourier transform In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced Sampling (signal processing), samples of a function (mathematics), function into a same-length sequence of equally-spaced samples of the discre ...

, DFT **

Fast Fourier transform A fast Fourier transform (FFT) is an algorithm that computes the discrete Fourier transform (DFT) of a sequence, or its inverse (IDFT). A Fourier transform converts a signal from its original domain (often time or space) to a representation in ...

, a popular implementation of the DFT *

Discrete cosine transform A discrete cosine transform (DCT) expresses a finite sequence of data points in terms of a sum of cosine functions oscillating at different frequency, frequencies. The DCT, first proposed by Nasir Ahmed (engineer), Nasir Ahmed in 1972, is a widely ...

Modified discrete cosine transform The modified discrete cosine transform (MDCT) is a transform based on the type-IV discrete cosine transform (DCT-IV), with the additional property of being lapped: it is designed to be performed on consecutive blocks of a larger dataset, where s ...

* Discrete Hartley transform * Discrete sine transform * Discrete wavelet transform *

Hadamard transform The Hadamard transform (also known as the Walsh–Hadamard transform, Hadamard–Rademacher–Walsh transform, Walsh transform, or Walsh–Fourier transform) is an example of a generalized class of Fourier transforms. It performs an orthogonal ...

(or, Walsh–Hadamard transform) ** Fast wavelet transform *''Hankel transform'', the

determinant In mathematics, the determinant is a Scalar (mathematics), scalar-valued function (mathematics), function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the ...

of the Hankel matrix * Discrete Chebyshev transform **Equivalent, up to a diagonal scaling, to a

discrete cosine transform A discrete cosine transform (DCT) expresses a finite sequence of data points in terms of a sum of cosine functions oscillating at different frequency, frequencies. The DCT, first proposed by Nasir Ahmed (engineer), Nasir Ahmed in 1972, is a widely ...

* Finite Legendre transform * Spherical Harmonic transform * Irrational base discrete weighted transform * Number-theoretic transform * Stirling transform

Discrete-time transforms

These transforms have a continuous frequency domain: *

Discrete-time Fourier transform In mathematics, the discrete-time Fourier transform (DTFT) is a form of Fourier analysis that is applicable to a sequence of discrete values. The DTFT is often used to analyze samples of a continuous function. The term ''discrete-time'' refers ...

* Z-transform

Data-dependent transforms

* Karhunen–Loève transform

Other transforms

Affine transformation In Euclidean geometry, an affine transformation or affinity (from the Latin, '' affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles. More general ...

(

Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...

) * Bäcklund transform *

Bilinear transform The bilinear transform (also known as Tustin's method, after Arnold Tustin) is used in digital signal processing and discrete-time control theory to transform continuous-time system representations to discrete-time and vice versa. The bilinear t ...

* Box–Muller transform *

Burrows–Wheeler transform The Burrows–Wheeler transform (BWT) rearranges a character string into runs of similar characters, in a manner that can be reversed to recover the original string. Since compression techniques such as move-to-front transform and run-length enc ...

(

data compression In information theory, data compression, source coding, or bit-rate reduction is the process of encoding information using fewer bits than the original representation. Any particular compression is either lossy or lossless. Lossless compressi ...

) *

Chirplet transform In signal processing, the chirplet transform is an inner product of an input signal with a family of analysis primitives called chirplets.S. Mann and S. Haykin,The Chirplet transform: A generalization of Gabor's logon transform, ''Proc. Vision In ...

* Distance transform * Fractal transform * Gelfand transform *

* Hough transform (

digital image processing Digital image processing is the use of a digital computer to process digital images through an algorithm. As a subcategory or field of digital signal processing, digital image processing has many advantages over analog image processing. It allo ...

) * Inverse scattering transform * Legendre transformation *

Möbius transformation In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form f(z) = \frac of one complex number, complex variable ; here the coefficients , , , are complex numbers satisfying . Geometrically ...

* Perspective transform (

computer graphics Computer graphics deals with generating images and art with the aid of computers. Computer graphics is a core technology in digital photography, film, video games, digital art, cell phone and computer displays, and many specialized applications. ...

) * Sequence transform * Watershed transform (

) * Wavelet transform (orthonormal) * Y-Δ transform (

electrical circuit An electrical network is an interconnection of electrical components (e.g., battery (electricity), batteries, resistors, inductors, capacitors, switches, transistors) or a model of such an interconnection, consisting of electrical elements (e. ...

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