HOME

TheInfoList



Click Here for Items Related To - Convolution kernel
OR:
Convolution kernel on:   [Wikipedia]   [Google]   [Amazon]

In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
(in particular,
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
), convolution is a
mathematical operation In mathematics, an operation is a Function (mathematics), function which takes zero or more input values (also called "''operands''" or "arguments") to a well-defined output value. The number of operands is the arity of the operation. The most c ...
on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' refers to both the result function and to the process of computing it. It is defined as the
integral In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
of the product of the two functions after one is reflected about the y-axis and shifted. The choice of which function is reflected and shifted before the integral does not change the integral result (see
commutativity In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
). The integral is evaluated for all values of shift, producing the convolution function. Some features of convolution are similar to
cross-correlation In signal processing, cross-correlation is a measure of similarity of two series as a function of the displacement of one relative to the other. This is also known as a ''sliding dot product'' or ''sliding inner-product''. It is commonly used fo ...
: for real-valued functions, of a continuous or discrete variable, convolution (f*g) differs from cross-correlation (f \star g) only in that either or is reflected about the y-axis in convolution; thus it is a cross-correlation of and , or and . For complex-valued functions, the cross-correlation operator is the
adjoint In mathematics, the term ''adjoint'' applies in several situations. Several of these share a similar formalism: if ''A'' is adjoint to ''B'', then there is typically some formula of the type :(''Ax'', ''y'') = (''x'', ''By''). Specifically, adjoin ...
of the convolution operator. Convolution has applications that include
probability Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...
,
statistics Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
,
acoustics Acoustics is a branch of physics that deals with the study of mechanical waves in gases, liquids, and solids including topics such as vibration, sound, ultrasound and infrasound. A scientist who works in the field of acoustics is an acoustician ...
,
spectroscopy Spectroscopy is the field of study that measures and interprets the electromagnetic spectra that result from the interaction between electromagnetic radiation and matter as a function of the wavelength or frequency of the radiation. Matter wa ...
,
signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, and scientific measurements. Signal processing techniq ...
and
image processing An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
,
geophysics Geophysics () is a subject of natural science concerned with the physical processes and physical properties of the Earth and its surrounding space environment, and the use of quantitative methods for their analysis. The term ''geophysics'' som ...
,
engineering Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad rang ...
,
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
,
computer vision Computer vision is an interdisciplinary scientific field that deals with how computers can gain high-level understanding from digital images or videos. From the perspective of engineering, it seeks to understand and automate tasks that the hum ...
and
differential equations In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
. The convolution can be defined for functions on
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
and other
groups A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
(as
algebraic structures In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set ...
). For example,
periodic function A periodic function is a function that repeats its values at regular intervals. For example, the trigonometric functions, which repeat at intervals of 2\pi radians, are periodic functions. Periodic functions are used throughout science to desc ...
s, such as the
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 values. The DTFT is often used to analyze samples of a continuous function. The term ''discrete-time'' refers to the ...
, can be defined on a
circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...
and convolved by periodic convolution. (See row 18 at .) A ''discrete convolution'' can be defined for functions on the set of
integers An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language o ...
. Generalizations of convolution have applications in the field of
numerical analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of ...
and
numerical linear algebra Numerical linear algebra, sometimes called applied linear algebra, is the study of how matrix operations can be used to create computer algorithms which efficiently and accurately provide approximate answers to questions in continuous mathematic ...
, and in the design and implementation of
finite impulse response In signal processing, a finite impulse response (FIR) filter is a filter whose impulse response (or response to any finite length input) is of ''finite'' duration, because it settles to zero in finite time. This is in contrast to infinite impulse ...
filters in signal processing. Computing the inverse of the convolution operation is known as
deconvolution In mathematics, deconvolution is the operation inverse to convolution. Both operations are used in signal processing and image processing. For example, it may be possible to recover the original signal after a filter (convolution) by using a deco ...
.


Definition

The convolution of and is written , denoting the operator with the symbol . It is defined as the integral of the product of the two functions after one is reflected about the y-axis and shifted. As such, it is a particular kind of
integral transform In mathematics, an integral transform maps a function from its original function space into another function space via integration, where some of the properties of the original function might be more easily characterized and manipulated than in ...
: :(f * g)(t) := \int_^\infty f(\tau) g(t - \tau) \, d\tau. An equivalent definition is (see
commutativity In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
): :(f * g)(t) := \int_^\infty f(t - \tau) g(\tau)\, d\tau. While the symbol is used above, it need not represent the time domain. At each ''t'', the convolution formula can be described as the area under the function weighted by the function shifted by the amount . As changes, the weighting function emphasizes different parts of the input function ; If is a positive value, then is equal to that slides or is shifted along the \tau-axis toward the right (toward ) by the amount of , while if is a negative value, then is equal to that slides or is shifted toward the left (toward ) by the amount of . For functions , supported on only (i.e., zero for negative arguments), the integration limits can be truncated, resulting in: :(f * g )(t) = \int_^ f(\tau) g(t - \tau)\, d\tau \quad \ \text f, g :
domain_of_definition_ In_mathematics,_a_partial_function__from_a_Set_(mathematics),_set__to_a_set__is_a_function_from_a_subset__of__(possibly__itself)_to_._The_subset_,_that_is,_the_Domain_of_a_function,_domain_of__viewed_as_a_function,_is_called_the_domain_of_defini_...
''_(below).


__Notation_

A_common_engineering_notational_convention_is: :_f(t)_*_g(t)_\mathrel_\underbrace_, which_has_to_be_interpreted_carefully_to_avoid_confusion._For_instance,__is_equivalent_to_,_but__is_in_fact_equivalent_to_.


__Relations_with_other_transforms_

Given_two_functions__f(t)__and__g(t)__with_Two-sided_Laplace_transform.html" ;"title="#Domain_of_definition.html" "title=", \infty) \to \mathbb. For the multi-dimensional formulation of convolution, see ''#Domain of definition">domain of definition In mathematics, a partial function from a Set (mathematics), set to a set is a function from a subset of (possibly itself) to . The subset , that is, the Domain of a function, domain of viewed as a function, is called the domain of defini ...
'' (below).


Notation

A common engineering notational convention is: : f(t) * g(t) \mathrel \underbrace_, which has to be interpreted carefully to avoid confusion. For instance, is equivalent to , but is in fact equivalent to .


Relations with other transforms

Given two functions f(t) and g(t) with Two-sided Laplace transform">bilateral Laplace transforms (two-sided Laplace transform) : F(s) = \int_^\infty e^ \ f(u) \ \textu and : G(s) = \int_^\infty e^ \ g(v) \ \textv respectively, the convolution operation f(t) * g(t) can be defined as the inverse Laplace transform of the product of F(s) and G(s) . More precisely, : \begin F(s) \cdot G(s) &= \int_^\infty e^ \ f(u) \ \textu \cdot \int_^\infty e^ \ g(v) \ \textv \\ &= \int_^\infty \int_^\infty e^ \ f(u) \ g(v) \ \textu \ \textv \end Let t = u + v such that : \begin F(s) \cdot G(s) &= \int_^\infty \int_^\infty e^ \ f(u) \ g(t - u) \ \textu \ \textt \\ &= \int_^\infty e^ \underbrace_ \ \textt \\ &= \int_^\infty e^ (f(t) * g(t)) \ \textt \end Note that F(s) \cdot G(s) is the bilateral Laplace transform of f(t) * g(t) . A similar derivation can be done using the unilateral Laplace transform (one-sided Laplace transform). The convolution operation also describes the output (in terms of the input) of an important class of operations known as ''linear time-invariant'' (LTI). See
LTI system theory LTI can refer to: * ''LTI – Lingua Tertii Imperii'', a book by Victor Klemperer * Language Technologies Institute, a division of Carnegie Mellon University * Linear time-invariant system, an engineering theory that investigates the response of ...
for a derivation of convolution as the result of LTI constraints. In terms of the
Fourier transforms A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
of the input and output of an LTI operation, no new frequency components are created. The existing ones are only modified (amplitude and/or phase). In other words, the output transform is the pointwise product of the input transform with a third transform (known as a
transfer function In engineering, a transfer function (also known as system function or network function) of a system, sub-system, or component is a function (mathematics), mathematical function that mathematical model, theoretically models the system's output for ...
). See
Convolution theorem In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two functions (or signals) is the pointwise product of their Fourier transforms. More generally, convolution in one domain (e.g. ...
for a derivation of that property of convolution. Conversely, convolution can be derived as the inverse Fourier transform of the pointwise product of two Fourier transforms.


Visual explanation


Historical developments

One of the earliest uses of the convolution integral appeared in
D'Alembert Jean-Baptiste le Rond d'Alembert (; ; 16 November 1717 – 29 October 1783) was a French mathematician, mechanician, physicist, philosopher, and music theorist. Until 1759 he was, together with Denis Diderot, a co-editor of the ''Encyclopédie ...
's derivation of
Taylor's theorem In calculus, Taylor's theorem gives an approximation of a ''k''-times differentiable function around a given point by a polynomial of degree ''k'', called the ''k''th-order Taylor polynomial. For a smooth function, the Taylor polynomial is the t ...
in ''Recherches sur différents points importants du système du monde,'' published in 1754. Also, an expression of the type: :\int f(u)\cdot g(x - u) \, du is used by
Sylvestre François Lacroix Sylvestre François Lacroix (28 April 176524 May 1843) was a French mathematician. Life He was born in Paris, and was raised in a poor family who still managed to obtain a good education for their son. Lacroix's path to mathematics started wit ...
on page 505 of his book entitled ''Treatise on differences and series'', which is the last of 3 volumes of the encyclopedic series: ''Traité du calcul différentiel et du calcul intégral'', Chez Courcier, Paris, 1797–1800. Soon thereafter, convolution operations appear in the works of
Pierre Simon Laplace Pierre-Simon, marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French scholar and polymath whose work was important to the development of engineering, mathematics, statistics, physics, astronomy, and philosophy. He summarized ...
, Jean-Baptiste Joseph Fourier,
Siméon Denis Poisson Baron Siméon Denis Poisson FRS FRSE (; 21 June 1781 – 25 April 1840) was a French mathematician and physicist who worked on statistics, complex analysis, partial differential equations, the calculus of variations, analytical mechanics, electri ...
, and others. The term itself did not come into wide use until the 1950s or 60s. Prior to that it was sometimes known as ''Faltung'' (which means ''folding'' in
German German(s) may refer to: * Germany (of or related to) ** Germania (historical use) * Germans, citizens of Germany, people of German ancestry, or native speakers of the German language ** For citizens of Germany, see also German nationality law **Ge ...
), ''composition product'', ''superposition integral'', and ''Carson's integral''. Yet it appears as early as 1903, though the definition is rather unfamiliar in older uses. The operation: :\int_0^t \varphi(s)\psi(t - s) \, ds,\quad 0 \le t < \infty, is a particular case of composition products considered by the Italian mathematician
Vito Volterra Vito Volterra (, ; 3 May 1860 – 11 October 1940) was an Italian mathematician and physicist, known for his contributions to mathematical biology and integral equations, being one of the founders of functional analysis. Biography Born in Anc ...
in 1913.


Circular convolution

When a function is periodic, with period , then for functions, , such that exists, the convolution is also periodic and identical to: :(f * g_T)(t) \equiv \int_^ \left sum_^\infty f(\tau + kT)\rightg_T(t - \tau)\, d\tau, where is an arbitrary choice. The summation is called a
periodic summation In signal processing, any periodic function s_P(t) with period ''P'' can be represented by a summation of an infinite number of instances of an aperiodic function s(t), that are offset by integer multiples of ''P''. This representation is called pe ...
of the function . When is a periodic summation of another function, , then is known as a ''circular'' or ''cyclic'' convolution of and . And if the periodic summation above is replaced by , the operation is called a ''periodic'' convolution of and .


Discrete convolution

For complex-valued functions defined on the set Z of integers, the ''discrete convolution'' of and is given by: :(f * g) = \sum_^\infty f g - m or equivalently (see
commutativity In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
) by: :(f * g) = \sum_^\infty f -mg The convolution of two finite sequences is defined by extending the sequences to finitely supported functions on the set of integers. When the sequences are the coefficients of two
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
s, then the coefficients of the ordinary product of the two polynomials are the convolution of the original two sequences. This is known as the
Cauchy product In mathematics, more specifically in mathematical analysis, the Cauchy product is the discrete convolution of two infinite series. It is named after the French mathematician Augustin-Louis Cauchy. Definitions The Cauchy product may apply to infini ...
of the coefficients of the sequences. Thus when has finite support in the set \ (representing, for instance, a
finite impulse response In signal processing, a finite impulse response (FIR) filter is a filter whose impulse response (or response to any finite length input) is of ''finite'' duration, because it settles to zero in finite time. This is in contrast to infinite impulse ...
), a finite summation may be used: :(f* g) \sum_^M f -m


Circular discrete convolution

When a function is periodic, with period , then for functions, , such that exists, the convolution is also periodic and identical to: :(f * g_N) \equiv \sum_^ \left(\sum_^\infty + kNright) g_N - m The summation on is called a
periodic summation In signal processing, any periodic function s_P(t) with period ''P'' can be represented by a summation of an infinite number of instances of an aperiodic function s(t), that are offset by integer multiples of ''P''. This representation is called pe ...
of the function . If is a periodic summation of another function, , then is known as a
circular convolution Circular convolution, also known as cyclic convolution, is a special case of periodic convolution, which is the convolution of two periodic functions that have the same period. Periodic convolution arises, for example, in the context of the discre ...
of and . When the non-zero durations of both and are limited to the interval ,  reduces to these common forms: The notation () for ''cyclic convolution'' denotes convolution over the
cyclic group In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...
of integers modulo . Circular convolution arises most often in the context of fast convolution with a fast Fourier transform (FFT) algorithm.


Fast convolution algorithms

In many situations, discrete convolutions can be converted to circular convolutions so that fast transforms with a convolution property can be used to implement the computation. For example, convolution of digit sequences is the kernel operation in
multiplication Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being additi ...
of multi-digit numbers, which can therefore be efficiently implemented with transform techniques (; ). requires arithmetic operations per output value and operations for outputs. That can be significantly reduced with any of several fast algorithms.
Digital signal processing Digital signal processing (DSP) is the use of digital processing, such as by computers or more specialized digital signal processors, to perform a wide variety of signal processing operations. The digital signals processed in this manner are ...
and other applications typically use fast convolution algorithms to reduce the cost of the convolution to O( log ) complexity. The most common fast convolution algorithms use fast Fourier transform (FFT) algorithms via the circular convolution theorem. Specifically, the
circular convolution Circular convolution, also known as cyclic convolution, is a special case of periodic convolution, which is the convolution of two periodic functions that have the same period. Periodic convolution arises, for example, in the context of the discre ...
of two finite-length sequences is found by taking an FFT of each sequence, multiplying pointwise, and then performing an inverse FFT. Convolutions of the type defined above are then efficiently implemented using that technique in conjunction with zero-extension and/or discarding portions of the output. Other fast convolution algorithms, such as the
Schönhage–Strassen algorithm The Schönhage–Strassen algorithm is an asymptotically fast multiplication algorithm for large integers. It was developed by Arnold Schönhage and Volker Strassen in 1971.A. Schönhage and V. Strassen,Schnelle Multiplikation großer Zahlen, ''C ...
or the Mersenne transform, use fast Fourier transforms in other
ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
s. If one sequence is much longer than the other, zero-extension of the shorter sequence and fast circular convolution is not the most computationally efficient method available. Instead, decomposing the longer sequence into blocks and convolving each block allows for faster algorithms such as the
overlap–save method In signal processing, ''overlap–save'' is the traditional name for an efficient way to evaluate the discrete convolution between a very long signal x /math> and a finite impulse response (FIR) filter h /math>: where for ''m'' outside the regio ...
and
overlap–add method In signal processing, the overlap–add method is an efficient way to evaluate the discrete convolution of a very long signal x /math> with a finite impulse response (FIR) filter h /math>: where for ''m'' outside the region . This article uses c ...
. A hybrid convolution method that combines block and FIR algorithms allows for a zero input-output latency that is useful for real-time convolution computations.


Domain of definition

The convolution of two complex-valued functions on is itself a complex-valued function on , defined by: :(f * g )(x) = \int_ f(y)g(x-y)\,dy = \int_ f(x-y)g(y)\,dy, and is well-defined only if and decay sufficiently rapidly at infinity in order for the integral to exist. Conditions for the existence of the convolution may be tricky, since a blow-up in at infinity can be easily offset by sufficiently rapid decay in . The question of existence thus may involve different conditions on and :


Compactly supported functions

If and are
compactly supported In mathematics, the support of a real-valued function f is the subset of the function domain containing the elements which are not mapped to zero. If the domain of f is a topological space, then the support of f is instead defined as the smalles ...
continuous function In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...
s, then their convolution exists, and is also compactly supported and continuous . More generally, if either function (say ) is compactly supported and the other is
locally integrable In mathematics, a locally integrable function (sometimes also called locally summable function) is a function which is integrable (so its integral is finite) on every compact subset of its domain of definition. The importance of such functions li ...
, then the convolution is well-defined and continuous. Convolution of and is also well defined when both functions are locally square integrable on and supported on an interval of the form (or both supported on ).


Integrable functions

The convolution of and exists if and are both Lebesgue integrable functions in (), and in this case is also integrable . This is a consequence of Tonelli's theorem. This is also true for functions in , under the discrete convolution, or more generally for the convolution on any group. Likewise, if ()  and  ()  where ,  then  (),  and :\, * g\, _p\le \, f\, _1\, g\, _p. In the particular case , this shows that is a
Banach algebra In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach ...
under the convolution (and equality of the two sides holds if and are non-negative almost everywhere). More generally, Young's inequality implies that the convolution is a continuous bilinear map between suitable spaces. Specifically, if satisfy: :\frac+\frac=\frac+1, then :\left\Vert f*g\right\Vert_r\le\left\Vert f\right\Vert_p\left\Vert g\right\Vert_q,\quad f\in\mathcal^p,\ g\in\mathcal^q, so that the convolution is a continuous bilinear mapping from to . The Young inequality for convolution is also true in other contexts (circle group, convolution on ). The preceding inequality is not sharp on the real line: when , there exists a constant such that: :\left\Vert f*g\right\Vert_r\le B_\left\Vert f\right\Vert_p\left\Vert g\right\Vert_q,\quad f\in\mathcal^p,\ g\in\mathcal^q. The optimal value of was discovered in 1975 and independently in 1976, see
Brascamp–Lieb inequality In mathematics, the Brascamp–Lieb inequality is either of two inequalities. The first is a result in geometry concerning integrable functions on ''n''-dimensional Euclidean space \mathbb^. It generalizes the Loomis–Whitney inequality and Höl ...
. A stronger estimate is true provided : :\, f* g\, _r\le C_\, f\, _p\, g\, _ where \, g\, _ is the weak norm. Convolution also defines a bilinear continuous map L^\times L^\to L^ for 1< p,q,r<\infty, owing to the weak Young inequality: :\, f* g\, _\le C_\, f\, _\, g\, _.


Functions of rapid decay

In addition to compactly supported functions and integrable functions, functions that have sufficiently rapid decay at infinity can also be convolved. An important feature of the convolution is that if ''f'' and ''g'' both decay rapidly, then ''f''∗''g'' also decays rapidly. In particular, if ''f'' and ''g'' are rapidly decreasing functions, then so is the convolution ''f''∗''g''. Combined with the fact that convolution commutes with differentiation (see #Properties), it follows that the class of Schwartz functions is closed under convolution .


Distributions

Under some circumstances, it is possible to define the convolution of a function with a distribution, or of two distributions. If ''f'' is a
compactly supported In mathematics, the support of a real-valued function f is the subset of the function domain containing the elements which are not mapped to zero. If the domain of f is a topological space, then the support of f is instead defined as the smalles ...
function and ''g'' is a distribution, then ''f''∗''g'' is a smooth function defined by a distributional formula analogous to :\int_ (y)g(x-y)\,dy. More generally, it is possible to extend the definition of the convolution in a unique way so that the associative law :f* (g* \varphi) = (f* g)* \varphi remains valid in the case where ''f'' is a distribution, and ''g'' a compactly supported distribution .


Measures

The convolution of any two Borel measures ''μ'' and ''ν'' of
bounded variation In mathematical analysis, a function of bounded variation, also known as ' function, is a real-valued function whose total variation is bounded (finite): the graph of a function having this property is well behaved in a precise sense. For a conti ...
is the measure \mu*\nu defined by :\int_ f(x) \, d(\mu*\nu)(x) = \int_\int_f(x+y)\,d\mu(x)\,d\nu(y). In particular, : (\mu*\nu)(A) = \int_1_A(x+y)\, d(\mu\times\nu)(x,y), where A\subset\mathbf R^d is a measurable set and 1_A is the
indicator function In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , one has \mathbf_(x)=1 if x\i ...
of A. This agrees with the convolution defined above when μ and ν are regarded as distributions, as well as the convolution of L1 functions when μ and ν are absolutely continuous with respect to the Lebesgue measure. The convolution of measures also satisfies the following version of Young's inequality :\, \mu* \nu\, \le \, \mu\, \, \nu\, where the norm is the
total variation In mathematics, the total variation identifies several slightly different concepts, related to the (local or global) structure of the codomain of a function or a measure. For a real-valued continuous function ''f'', defined on an interval 'a'' ...
of a measure. Because the space of measures of bounded variation is a
Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
, convolution of measures can be treated with standard methods of
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
that may not apply for the convolution of distributions.


Properties


Algebraic properties

The convolution defines a product on the
linear space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
of integrable functions. This product satisfies the following algebraic properties, which formally mean that the space of integrable functions with the product given by convolution is a commutative
associative algebra In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplic ...
without
identity Identity may refer to: * Identity document * Identity (philosophy) * Identity (social science) * Identity (mathematics) Arts and entertainment Film and television * ''Identity'' (1987 film), an Iranian film * ''Identity'' (2003 film), ...
. Other linear spaces of functions, such as the space of continuous functions of compact support, are closed under the convolution, and so also form commutative associative algebras. ;
Commutativity In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
: f * g = g * f Proof: By definition: (f * g)(t) = \int^\infty_ f(\tau)g(t - \tau)\, d\tau Changing the variable of integration to u = t - \tau the result follows. ;
Associativity In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement f ...
: f * (g * h) = (f * g) * h Proof: This follows from using
Fubini's theorem In mathematical analysis Fubini's theorem is a result that gives conditions under which it is possible to compute a double integral by using an iterated integral, introduced by Guido Fubini in 1907. One may switch the order of integration if th ...
(i.e., double integrals can be evaluated as iterated integrals in either order). ;
Distributivity In mathematics, the distributive property of binary operations generalizes the distributive law, which asserts that the equality x \cdot (y + z) = x \cdot y + x \cdot z is always true in elementary algebra. For example, in elementary arithmetic, ...
: f * (g + h) = (f * g) + (f * h) Proof: This follows from linearity of the integral. ; Associativity with scalar multiplication: a (f * g) = (a f) * g for any real (or complex) number a. ;
Multiplicative identity In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
: No algebra of functions possesses an identity for the convolution. The lack of identity is typically not a major inconvenience, since most collections of functions on which the convolution is performed can be convolved with a delta distribution (a unitary impulse, centered at zero) or, at the very least (as is the case of ''L''1) admit approximations to the identity. The linear space of compactly supported distributions does, however, admit an identity under the convolution. Specifically, f * \delta = f where ''δ'' is the delta distribution. ; Inverse element: Some distributions ''S'' have an
inverse element In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers. Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that is ...
''S''−1 for the convolution which then must satisfy S^ * S = \delta from which an explicit formula for ''S''−1 may be obtained.The set of invertible distributions forms an
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commut ...
under the convolution. ; Complex conjugation: \overline = \overline * \overline ; Relationship with differentiation: (f * g)' = f' * g = f * g' Proof: \begin (f * g)' & = \frac \int^\infty_ f(\tau) g(t - \tau) \, d\tau \\[4pt] & =\int^\infty_ f(\tau) \frac g(t - \tau) \, d\tau \\[4pt] & =\int^\infty_ f(\tau) g'(t - \tau) \, d\tau = f* g'. \end ; Relationship with integration: If F(t) = \int^t_ f(\tau) d\tau, and G(t) = \int^t_ g(\tau) \, d\tau, then (F * g)(t) = (f * G)(t) = \int^t_(f * g)(\tau)\,d\tau.


Integration

If ''f'' and ''g'' are integrable functions, then the integral of their convolution on the whole space is simply obtained as the product of their integrals: : \int_(f * g)(x) \, dx=\left(\int_f(x) \, dx\right) \left(\int_g(x) \, dx\right). This follows from
Fubini's theorem In mathematical analysis Fubini's theorem is a result that gives conditions under which it is possible to compute a double integral by using an iterated integral, introduced by Guido Fubini in 1907. One may switch the order of integration if th ...
. The same result holds if ''f'' and ''g'' are only assumed to be nonnegative measurable functions, by Tonelli's theorem.


Differentiation

In the one-variable case, : \frac(f * g) = \frac * g = f * \frac where ''d''/''dx'' is the derivative. More generally, in the case of functions of several variables, an analogous formula holds with the partial derivative: : \frac(f * g) = \frac * g = f * \frac. A particular consequence of this is that the convolution can be viewed as a "smoothing" operation: the convolution of ''f'' and ''g'' is differentiable as many times as ''f'' and ''g'' are in total. These identities hold under the precise condition that ''f'' and ''g'' are absolutely integrable and at least one of them has an absolutely integrable (L1) weak derivative, as a consequence of Young's convolution inequality. For instance, when ''f'' is continuously differentiable with compact support, and ''g'' is an arbitrary locally integrable function, : \frac(f* g) = \frac * g. These identities also hold much more broadly in the sense of tempered distributions if one of ''f'' or ''g'' is a distribution (mathematics)#Convolution versus multiplication, rapidly decreasing tempered distribution, a compactly supported tempered distribution or a Schwartz function and the other is a tempered distribution. On the other hand, two positive integrable and infinitely differentiable functions may have a nowhere continuous convolution. In the discrete case, the difference operator ''D'' ''f''(''n'') = ''f''(''n'' + 1) − ''f''(''n'') satisfies an analogous relationship: : D(f * g) = (Df) * g = f * (Dg).


Convolution theorem

The convolution theorem states that : \mathcal\ = k\cdot \mathcal\\cdot \mathcal\ where \mathcal\ denotes the Fourier transform of f, and k is a constant that depends on the specific Normalizing constant, normalization of the Fourier transform. Versions of this theorem also hold for the Laplace transform, two-sided Laplace transform, Z-transform and Mellin transform. On the other hand, if \mathcal W is the DFT matrix, Fourier transform matrix, then : \mathcal W\left(C^x \ast C^y\right) = \left(\mathcal W C^ \bull \mathcal W C^\right)(x \otimes y) = \mathcal W C^x \circ \mathcal W C^y, where \bull is Khatri–Rao product#Face-splitting product, face-splitting product, \otimes denotes Kronecker product, \circ denotes Hadamard product (matrices), Hadamard product (this result is an evolving of count sketch properties).


Translational equivariance

The convolution commutes with translations, meaning that : \tau_x (f * g) = (\tau_x f) * g = f * (\tau_x g) where τ''x''f is the translation of the function ''f'' by ''x'' defined by : (\tau_x f)(y) = f(y - x). If ''f'' is a Schwartz function, then ''τxf'' is the convolution with a translated Dirac delta function ''τ''''x''''f'' = ''f'' ∗ ''τ''''x'' ''δ''. So translation invariance of the convolution of Schwartz functions is a consequence of the associativity of convolution. Furthermore, under certain conditions, convolution is the most general translation invariant operation. Informally speaking, the following holds : Suppose that ''S'' is a bounded linear operator acting on functions which commutes with translations: ''S''(''τxf'') = ''τx''(''Sf'') for all ''x''. Then ''S'' is given as convolution with a function (or distribution) ''g''''S''; that is ''Sf'' = ''g''''S'' ∗ ''f''. Thus some translation invariant operations can be represented as convolution. Convolutions play an important role in the study of time-invariant systems, and especially LTI system theory. The representing function ''g''''S'' is the impulse response of the transformation ''S''. A more precise version of the theorem quoted above requires specifying the class of functions on which the convolution is defined, and also requires assuming in addition that ''S'' must be a continuous linear operator with respect to the appropriate topology. It is known, for instance, that every continuous translation invariant continuous linear operator on ''L''1 is the convolution with a finite Borel measure. More generally, every continuous translation invariant continuous linear operator on ''L''''p'' for 1 ≤ ''p'' < ∞ is the convolution with a Distribution (mathematics)#Tempered distributions and Fourier transform, tempered distribution whose Fourier transform is bounded. To wit, they are all given by bounded Fourier multipliers.


Convolutions on groups

If ''G'' is a suitable group (mathematics), group endowed with a measure (mathematics), measure λ, and if ''f'' and ''g'' are real or complex valued Lebesgue integral, integrable functions on ''G'', then we can define their convolution by :(f * g)(x) = \int_G f(y) g\left(y^x\right)\,d\lambda(y). It is not commutative in general. In typical cases of interest ''G'' is a locally compact Hausdorff space, Hausdorff topological group and λ is a (left-) Haar measure. In that case, unless ''G'' is unimodular group, unimodular, the convolution defined in this way is not the same as \int f\left(xy^\right)g(y) \, d\lambda(y). The preference of one over the other is made so that convolution with a fixed function ''g'' commutes with left translation in the group: :L_h(f* g) = (L_hf)* g. Furthermore, the convention is also required for consistency with the definition of the convolution of measures given below. However, with a right instead of a left Haar measure, the latter integral is preferred over the former. On locally compact
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commut ...
s, a version of the convolution theorem holds: the Fourier transform of a convolution is the pointwise product of the Fourier transforms. The circle group T with the Lebesgue measure is an immediate example. For a fixed ''g'' in ''L''1(T), we have the following familiar operator acting on the Hilbert space ''L''2(T): :T (x) = \frac \int_ (y) g( x - y) \, dy. The operator ''T'' is compact operator on Hilbert space, compact. A direct calculation shows that its adjoint ''T* '' is convolution with :\bar(-y). By the commutativity property cited above, ''T'' is normal operator, normal: ''T''* ''T'' = ''TT''* . Also, ''T'' commutes with the translation operators. Consider the family ''S'' of operators consisting of all such convolutions and the translation operators. Then ''S'' is a commuting family of normal operators. According to compact operator on Hilbert space, spectral theory, there exists an orthonormal basis that simultaneously diagonalizes ''S''. This characterizes convolutions on the circle. Specifically, we have :h_k (x) = e^, \quad k \in \mathbb,\; which are precisely the Character (mathematics), characters of T. Each convolution is a compact multiplication operator in this basis. This can be viewed as a version of the convolution theorem discussed above. A discrete example is a finite
cyclic group In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...
of order ''n''. Convolution operators are here represented by circulant matrices, and can be diagonalized by the discrete Fourier transform. A similar result holds for compact groups (not necessarily abelian): the matrix coefficients of finite-dimensional unitary representations form an orthonormal basis in ''L''2 by the Peter–Weyl theorem, and an analog of the convolution theorem continues to hold, along with many other aspects of harmonic analysis that depend on the Fourier transform.


Convolution of measures

Let ''G'' be a (multiplicatively written) topological group. If μ and ν are finite Borel measures on ''G'', then their convolution ''μ''∗''ν'' is defined as the pushforward measure of the Group action (mathematics), group action and can be written as :(\mu * \nu)(E) = \iint 1_E(x+y) \,d\mu(x) \,d\nu(y) for each measurable subset ''E'' of ''G''. The convolution is also a finite measure, whose
total variation In mathematics, the total variation identifies several slightly different concepts, related to the (local or global) structure of the codomain of a function or a measure. For a real-valued continuous function ''f'', defined on an interval 'a'' ...
satisfies :\, \mu * \nu\, \le \left\, \mu\right\, \left\, \nu\right\, . In the case when ''G'' is locally compact with (left-)Haar measure λ, and μ and ν are absolute continuity, absolutely continuous with respect to a λ, Radon–Nikodym theorem, so that each has a density function, then the convolution μ∗ν is also absolutely continuous, and its density function is just the convolution of the two separate density functions. If μ and ν are probability measures on the topological group then the convolution ''μ''∗''ν'' is the probability distribution of the sum ''X'' + ''Y'' of two statistical independence, independent random variables ''X'' and ''Y'' whose respective distributions are μ and ν.


Infimal convolution

In convex analysis, the infimal convolution of proper (not identically +\infty) convex functions f_1,\dots,f_m on \mathbb R^n is defined by: (f_1*\cdots*f_m)(x)=\inf_x \. It can be shown that the infimal convolution of convex functions is convex. Furthermore, it satisfies an identity analogous to that of the Fourier transform of a traditional convolution, with the role of the Fourier transform is played instead by the Legendre transform: \varphi^*(x) = \sup_y ( x\cdot y - \varphi(y)). We have: (f_1*\cdots *f_m)^*(x) = f_1^*(x) + \cdots + f_m^*(x).


Bialgebras

Let (''X'', Δ, ∇, ''ε'', ''η'') be a bialgebra with comultiplication Δ, multiplication ∇, unit η, and counit ''ε''. The convolution is a product defined on the endomorphism algebra End(''X'') as follows. Let ''φ'', ''ψ'' ∈ End(''X''), that is, ''φ'', ''ψ'': ''X'' → ''X'' are functions that respect all algebraic structure of ''X'', then the convolution ''φ''∗''ψ'' is defined as the composition :X \mathrel X \otimes X \mathrel X \otimes X \mathrel X. The convolution appears notably in the definition of Hopf algebras . A bialgebra is a Hopf algebra if and only if it has an antipode: an endomorphism ''S'' such that :S * \operatorname_X = \operatorname_X * S = \eta\circ\varepsilon.


Applications

Convolution and related operations are found in many applications in science, engineering and mathematics. * In
image processing An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
** In digital image processing convolutional filtering plays an important role in many important algorithms in edge detection and related processes (see Kernel (image processing)) ** In optics, an out-of-focus photograph is a convolution of the sharp image with a lens function. The photographic term for this is bokeh. ** In image processing applications such as adding blurring. * In digital data processing ** In analytical chemistry, Savitzky–Golay smoothing filters are used for the analysis of spectroscopic data. They can improve signal-to-noise ratio with minimal distortion of the spectra ** In
statistics Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
, a weighted moving average is a convolution. * In
acoustics Acoustics is a branch of physics that deals with the study of mechanical waves in gases, liquids, and solids including topics such as vibration, sound, ultrasound and infrasound. A scientist who works in the field of acoustics is an acoustician ...
, reverberation is the convolution of the original sound with echo (phenomenon), echoes from objects surrounding the sound source. ** In digital signal processing, convolution is used to map the impulse response of a real room on a digital audio signal. ** In electronic music convolution is the imposition of a Spectrum, spectral or rhythmic structure on a sound. Often this envelope or structure is taken from another sound. The convolution of two signals is the filtering of one through the other. * In electrical engineering, the convolution of one function (the Signal (electrical engineering), input signal) with a second function (the impulse response) gives the output of a linear time-invariant system (LTI). At any given moment, the output is an accumulated effect of all the prior values of the input function, with the most recent values typically having the most influence (expressed as a multiplicative factor). The impulse response function provides that factor as a function of the elapsed time since each input value occurred. * In
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, wherever there is a linear system with a "superposition principle", a convolution operation makes an appearance. For instance, in
spectroscopy Spectroscopy is the field of study that measures and interprets the electromagnetic spectra that result from the interaction between electromagnetic radiation and matter as a function of the wavelength or frequency of the radiation. Matter wa ...
line broadening due to the Doppler effect on its own gives a Normal distribution, Gaussian spectral line shape and collision broadening alone gives a Cauchy distribution, Lorentzian line shape. When both effects are operative, the line shape is a convolution of Gaussian and Lorentzian, a Voigt function. ** In Time-resolved spectroscopy#Time-resolved fluorescence spectroscopy, time-resolved fluorescence spectroscopy, the excitation signal can be treated as a chain of delta pulses, and the measured fluorescence is a sum of exponential decays from each delta pulse. ** In computational fluid dynamics, the large eddy simulation (LES) turbulence model uses the convolution operation to lower the range of length scales necessary in computation thereby reducing computational cost. * In probability theory, the probability distribution of the sum of two independent (probability), independent random variables is the convolution of their individual distributions. ** In kernel density estimation, a distribution is estimated from sample points by convolution with a kernel, such as an isotropic Gaussian. * In radiotherapy treatment planning systems, most part of all modern codes of calculation applies a convolution-superposition algorithm. * In structural reliability, the reliability index can be defined based on the convolution theorem. ** The definition of reliability index for limit state functions with nonnormal distributions can be established corresponding to the joint distribution function. In fact, the joint distribution function can be obtained using the convolution theory. * Convolutional neural networks apply multiple cascaded ''convolution'' kernels with applications in machine vision and artificial intelligence. Though these are actually cross-correlations rather than convolutions in most cases. * In Smoothed-particle hydrodynamics, simulations of fluid dynamics are calculated using particles, each with surrounding kernels. For any given particle i, some physical quantity A_i is calculated as a convolution of A_j with a weighting function, where j denotes the neighbors of particle i: those that are located within its kernel. The convolution is approximated as a summation over each neighbor.


See also

* Analog signal processing * Circulant matrix * Convolution for optical broad-beam responses in scattering media * Convolution power * Deconvolution * Dirichlet convolution * Generalized signal averaging * Jan Mikusinski * List of convolutions of probability distributions * LTI system theory#Impulse response and convolution * Multidimensional discrete convolution * Scaled correlation * Titchmarsh convolution theorem * Toeplitz matrix (convolutions can be considered a Toeplitz matrix operation where each row is a shifted copy of the convolution kernel)


Notes


References


Further reading

* . * * * Dominguez-Torres, Alejandro (Nov 2, 2010). "Origin and history of convolution". 41 pgs. http://www.slideshare.net/Alexdfar/origin-adn-history-of-convolution. Cranfield, Bedford MK43 OAL, UK. Retrieved Mar 13, 2013. * * * . * . * . * . * . * * * . * * . * . * . * . * * * .


External links


Earliest Uses: The entry on Convolution has some historical information.


o

* http://www.jhu.edu/~signals/convolve/index.html Visual convolution Java Applet * http://www.jhu.edu/~signals/discreteconv2/index.html Visual convolution Java Applet for discrete-time functions * https://get-the-solution.net/projects/discret-convolution discret-convolution online calculator *https://lpsa.swarthmore.edu/Convolution/CI.html Convolution demo and visualization in javascript *https://phiresky.github.io/convolution-demo/ Another convolution demo in javascript
Lectures on Image Processing: A collection of 18 lectures in pdf format from Vanderbilt University. Lecture 7 is on 2-D convolution.
by Alan Peters * * https://archive.org/details/Lectures_on_Image_Processing
Convolution Kernel Mask Operation Interactive tutorial


at MathWorld
Freeverb3 Impulse Response Processor
Opensource zero latency impulse response processor with VST plugins * Stanford University CS 17

showing how spatial convolution works.
A video lecture on the subject of convolution
given by Salman Khan (educator), Salman Khan
Example of FFT convolution for pattern-recognition (image processing)Intuitive Guide to Convolution
A blogpost about an intuitive interpretation of convolution. {{Differentiable computing Functional analysis Image processing Fourier analysis Bilinear maps Feature detection (computer vision)