사용자:CrMT/연습장/0

합성곱(convolution, 콘벌루션)은 실수 전체 또는 그 일부나, 정수 집합과 같은 이산적인 집합, 더 나아가 적당한 측도가 주어진 군 위에서 정의된 실- 또는 복소함수 둘 사이의 이항연산이다.

합성곱은 푸리에 변환과 매우 밀접한 관계를 가지는데, 합성곱의 푸리에 변환은 푸리에 변환의 곱과 같다. (연속과 이산 모두.) 즉,

[math]\displaystyle{ \mathcal F(f * g) = \mathcal F(f) \mathcal F(g) }[/math], [math]\displaystyle{ \mathcal F(f g) = \mathcal F(f)* \mathcal F(g) }[/math],

이다. ([math]\displaystyle{ (fg)(x) = f(x) g(x). }[/math]) 이는 합성곱을 다음과 같이 계산할 수 있게 한다:

[math]\displaystyle{ f * g = \mathcal F^{-1}(\mathcal F(f) \mathcal F(g)) }[/math]

푸리에 변환을 알고 있을 때 합성곱의 원래 정의인 적분을 이용하지 않아도 쉽게 합성곱을 계산할 수 있다.

합성곱은 수학의 확률론, 수치해석 등이나 공학의 신호처리 등에서 매우 중요한 연산이다. 머신 러닝 분야에서는 최근 합성곱 신경망이 이미지넷 등에서 매우 우수한 성적을 보여 연구 중이다.

정의

실수 전체에서 정의된 두 실- 또는 복소함수 [math]\displaystyle{ f, ~ g \to \mathbb R \text{ or }\mathbb C }[/math]의 합성곱(convolution)은 다음과 같이 정의된다:

[math]\displaystyle{ (f * g)(t) := \int_\mathbb R f(\theta) g(t-\theta) ~\mathrm d \theta = \int_\mathbb R g(\theta) f(t-\theta) ~\mathrm d \theta. }[/math]

두 번째 등호는 [math]\displaystyle{ \theta \longrightarrow t - \theta }[/math]의 치환으로 얻어지며 따라서 합성곱 연산은 가환(commutative)이다. 이때 두 함수의 공역은 곱셈과 적분이 잘 정의되는 것이면 합성곱을 정의할 수 있지만,

정수 집합에서 정의된 두 이산 신호(함수) [math]\displaystyle{ x[n],~y[n] }[/math]의 합성곱은 다음과 같이 정의된다:

[math]\displaystyle{ (x*y)[n] := \sum_{m\in \mathbb Z} x[m] y[n-m] = \sum_{m\in \mathbb Z} y[m] x[n-m] }[/math]

더 많은 정의는 밑 문단를 참고.

Derivations

Convolution describes the output (in terms of the input) of an important class of operations known as linear time-invariant (LTI). See LTI system theory for a derivation of convolution as the result of LTI constraints. In terms of the Fourier transforms 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). See Convolution theorem 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.

• 항등원: [math]\displaystyle{ \int_{-\infty}^\infty \delta(\tau)\, g(t - \tau)\, d\tau = g(t) }[/math]

원형 합성곱과 주기 합성곱

정의에 의하여, 만약 [math]\displaystyle{ \tilde g:\;\mathbb R \to \mathbb C }[/math]가 주기 [math]\displaystyle{ p }[/math]를 가지는 주기함수이면, 다른 함수 [math]\displaystyle{ f:\;\mathbb R \to \mathbb C }[/math]와의 합성곱은 (간단한 계산으로) 다음과 같음을 보일 수 있다: 임의의 [math]\displaystyle{ t, ~ \theta_0 \in \mathbb R }[/math]에 대하여,

[math]\displaystyle{ (f * \tilde g)(t) = \int_{\theta_0}^{\theta_0+p} \left[\sum_{k \in \mathbb Z} f(\theta + kp)\right] \tilde g(t - \theta)\, \mathrm d\theta. }[/math]

주기함수의 [math]\displaystyle{ [0,~p) }[/math]를 원 [math]\displaystyle{ S^1 }[/math]과 적절히 대응시킬 수 있으므로, 이러한 합성곱의 표현을 원형 합성곱(circular convolution, cyclic convolution)이라고 한다. 이때 위의 대괄호 안에 있는 식을

[math]\displaystyle{ \tilde f(\theta) := \sum_{k \in \mathbb Z} f(\theta + kp) }[/math]

로 정의하면 [math]\displaystyle{ \tilde f }[/math]가 주기 [math]\displaystyle{ p }[/math]의 주기함수가 되는 것을 알 수 있다. 즉 한 함수가 주기적이면 두 함수의 합성곱은 두 주기함수의 합성곱으로 바꿀 수 있으며, 적분 구간을 [math]\displaystyle{ [0,~p] }[/math]로 할 수 있다: ([math]\displaystyle{ \theta_0 = 0 }[/math]을 대입)

[math]\displaystyle{ (f * \tilde g)(t) = \int_{0}^{p}\tilde f(\theta) \tilde g(t - \theta)\, \mathrm d\theta. }[/math]

이를 주기 합성곱(periodic convolution)이라고 한다.

같은 방법으로, 정의역이 [math]\displaystyle{ \mathbb Z }[/math]인 이산 신호에 대해서도 잘 정의된다. 임의의 [math]\displaystyle{ n,~m_0 \in \mathbb Z }[/math]에 대하여,

[math]\displaystyle{ \begin{align*}(x * \tilde y)[n] &= \sum_{m=m_0}^{m_0 + N-1} \left[ \sum_{k\in \mathbb Z} x[m+kN] \right] \tilde g[n-m] \\ &= \sum_{m=0}^{0 + N-1} \tilde f[m] \tilde g[n-m].\end{align*} }[/math]

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 of multi-digit numbers, which can therefore be efficiently implemented with transform techniques (틀:Harvnb; 틀:Harvnb).

틀:EquationNote requires N arithmetic operations per output value and N² operations for N outputs. That can be significantly reduced with any of several fast algorithms. Digital signal processing and other applications typically use fast convolution algorithms to reduce the cost of the convolution to O(N log N) complexity.

The most common fast convolution algorithms use fast Fourier transform (FFT) algorithms via the circular convolution theorem. Specifically, the circular convolution 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 or the Mersenne transform, use fast Fourier transforms in other rings.

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 and Overlap–add method. 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 R^d is itself a complex-valued function on R^d, defined by:

[math]\displaystyle{ (f * g )(x) = \int_{\mathbf{R}^d} f(y)g(x-y)\,dy = \int_{\mathbf{R}^d} f(x-y)g(y)\,dy, }[/math]

is well-defined only if f and g 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 g at infinity can be easily offset by sufficiently rapid decay in f. The question of existence thus may involve different conditions on f and g:

Compactly supported functions

If f and g are compactly supported continuous functions, then their convolution exists, and is also compactly supported and continuous 틀:Harv. More generally, if either function (say f) is compactly supported and the other is locally integrable, then the convolution f∗g is well-defined and continuous.

Convolution of f and g is also well defined when both functions are locally square integrable on R and supported on an interval of the form [a, +∞) (or both supported on [-∞, a]).

Integrable functions

The convolution of f and g exists if f and g are both Lebesgue integrable functions in L¹(R^d), and in this case f∗g is also integrable 틀:Harv. This is a consequence of Tonelli's theorem. This is also true for functions in [math]\displaystyle{ \ell^1 }[/math], under the discrete convolution, or more generally for the convolution on any group.

Likewise, if f ∈ L¹(R^d) and g ∈ L^p(R^d) where 1 ≤ p ≤ ∞, then f∗g ∈ L^p(R^d) and

[math]\displaystyle{ \|{f}*g\|_p\le \|f\|_1\|g\|_p. \, }[/math]

In the particular case p = 1, this shows that L¹ is a Banach algebra under the convolution (and equality of the two sides holds if f and g are non-negative almost everywhere).

More generally, Young's inequality implies that the convolution is a continuous bilinear map between suitable L^p spaces. Specifically, if 1 ≤ p,q,r ≤ ∞ satisfy

[math]\displaystyle{ \frac{1}{p}+\frac{1}{q}=\frac{1}{r}+1, }[/math]

then

[math]\displaystyle{ \left\Vert f*g\right\Vert _{r}\le\left\Vert f\right\Vert _{p}\left\Vert g\right\Vert _{q},\quad f\in\mathcal{L}^{p},\ g\in\mathcal{L}^{q}, }[/math]

so that the convolution is a continuous bilinear mapping from L^p×L^q to L^r. The Young inequality for convolution is also true in other contexts (circle group, convolution on Z). The preceding inequality is not sharp on the real line: when 1 < p, q, r < ∞, there exists a constant B_{p, q} < 1 such that:

[math]\displaystyle{ \left\Vert f*g\right\Vert _{r}\le B_{p,q}\left\Vert f\right\Vert _{p}\left\Vert g\right\Vert _{q},\quad f\in\mathcal{L}^{p},\ g\in\mathcal{L}^{q}. }[/math]

The optimal value of B_{p, q} was discovered in 1975.

A stronger estimate is true provided 1 < p, q, r < ∞ :

[math]\displaystyle{ \|f*g\|_r\le C_{p,q}\|f\|_p\|g\|_{q,w} }[/math]

where [math]\displaystyle{ \|g\|_{q,w} }[/math] is the weak L^q norm. Convolution also defines a bilinear continuous map [math]\displaystyle{ L^{p,w}\times L^{q.w}\to L^{r,w} }[/math] for [math]\displaystyle{ 1\lt p,q,r\lt \infty }[/math], owing to the weak Young inequality:

[math]\displaystyle{ \|f*g\|_{r,w}\le C_{p,q}\|f\|_{p,w}\|g\|_{r,w}. }[/math]

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 틀:Harv.

Distributions

틀:Main article 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 function and g is a distribution, then f∗g is a smooth function defined by a distributional formula analogous to

[math]\displaystyle{ \int_{\mathbf{R}^d} {f}(y)g(x-y)\,dy. }[/math]

More generally, it is possible to extend the definition of the convolution in a unique way so that the associative law

[math]\displaystyle{ f*(g*\varphi) = (f*g)*\varphi\, }[/math]

remains valid in the case where f is a distribution, and g a compactly supported distribution 틀:Harv.

Measures

The convolution of any two Borel measures μ and ν of bounded variation is the measure λ defined by 틀:Harv

[math]\displaystyle{ \int_{\mathbf{R}^d} f(x)d\lambda(x) = \int_{\mathbf{R}^d}\int_{\mathbf{R}^d}f(x+y)\,d\mu(x)\,d\nu(y). }[/math]

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

[math]\displaystyle{ \|\mu*\nu\|\le \|\mu\|\|\nu\| \, }[/math]

where the norm is the total variation of a measure. Because the space of measures of bounded variation is a Banach space, convolution of measures can be treated with standard methods of functional analysis that may not apply for the convolution of distributions.

Properties

Algebraic properties

틀:See also The convolution defines a product on the linear space 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 algebra without identity 틀:Harv. 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 algebras.

Commutativity

[math]\displaystyle{ f * g = g * f \, }[/math]

Proof: By definition

[math]\displaystyle{ f * g = \int^{\infty}_{-\infty} f(\tau)g(t-\tau) d\tau }[/math]

Changing the variable of integration to [math]\displaystyle{ u=t-\tau }[/math] and the result follows.

Associativity

[math]\displaystyle{ f * (g * h) = (f * g) * h \, }[/math]

Proof: This follows from using Fubini's theorem (i.e., double integrals can be evaluated as iterated integrals in either order).

Distributivity

[math]\displaystyle{ f * (g + h) = (f * g) + (f * h) \, }[/math]

Proof: This follows from linearity of the integral.

Associativity with scalar multiplication

[math]\displaystyle{ a (f * g) = (a f) * g \, }[/math]

for any real (or complex) number [math]\displaystyle{ {a}\, }[/math].

Multiplicative identity

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 or, at the very least (as is the case of L¹) admit approximations to the identity. The linear space of compactly supported distributions does, however, admit an identity under the convolution. Specifically,

[math]\displaystyle{ f*\delta = f\, }[/math]

where δ is the delta distribution.

Inverse element

Some distributions have an inverse element for the convolution, S⁽⁻¹⁾, which is defined by

[math]\displaystyle{ S^{(-1)} * S = \delta. \, }[/math]

The set of invertible distributions forms an abelian group under the convolution.

Complex conjugation

[math]\displaystyle{ \overline{f * g} = \overline{f} * \overline{g} \!\ }[/math]

Relationship with differentiation

[math]\displaystyle{ (f*g)'=f'*g=f*g' }[/math]

Proof:

[math]\displaystyle{ (f*g)'= \frac{d}{dt} \int^{\infty}_{-\infty} f(\tau) g(t-\tau) d\tau }[/math]

[math]\displaystyle{ =\int^{\infty}_{-\infty} f(\tau) \frac{\partial}{\partial t} g(t-\tau) d\tau }[/math]

[math]\displaystyle{ =\int^{\infty}_{-\infty} f(\tau) g'(t-\tau) d\tau= f*g'. }[/math]

Relationship with integration

If [math]\displaystyle{ F(t)=\int^t_{-\infty} f(\tau) d\tau, }[/math] and [math]\displaystyle{ G(t)=\int^t_{-\infty} g(\tau) d\tau, }[/math] then

[math]\displaystyle{ (F*g)(t)=(f*G)(t)=\int^t_{-\infty}(f*g)(\tau)d\tau. }[/math]

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:

[math]\displaystyle{ \int_{\mathbf{R}^d}(f*g)(x) \, dx=\left(\int_{\mathbf{R}^d}f(x) \, dx\right)\left(\int_{\mathbf{R}^d}g(x) \, dx\right). }[/math]

This follows from Fubini's theorem. 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,

[math]\displaystyle{ \frac{d}{dx}(f * g) = \frac{df}{dx} * g = f * \frac{dg}{dx} \, }[/math]

where d/dx is the derivative. More generally, in the case of functions of several variables, an analogous formula holds with the partial derivative:

[math]\displaystyle{ \frac{\partial}{\partial x_i}(f * g) = \frac{\partial f}{\partial x_i} * g = f * \frac{\partial g}{\partial x_i}. }[/math]

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 inequality. For instance, when f is continuously differentiable with compact support, and g is an arbitrary locally integrable function,

[math]\displaystyle{ \frac{d}{dx}({f} * g) = \frac{df}{dx} * g. }[/math]

These identities also hold much more broadly in the sense of tempered distributions if one of f or g is a compactly supported 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:

[math]\displaystyle{ D(f*g) = (Df)*g = f*(Dg).\, }[/math]

Convolution theorem

The convolution theorem states that

[math]\displaystyle{ \mathcal{F}\{f * g\} = k\cdot \mathcal{F}\{f\}\cdot \mathcal{F}\{g\} }[/math]

where [math]\displaystyle{ \mathcal{F}\{f\}\, }[/math] denotes the Fourier transform of [math]\displaystyle{ f }[/math], and [math]\displaystyle{ k }[/math] is a constant that depends on the specific normalization of the Fourier transform. Versions of this theorem also hold for the Laplace transform, two-sided Laplace transform, Z-transform and Mellin transform.

See also the less trivial Titchmarsh convolution theorem.

Translation invariance

The convolution commutes with translations, meaning that

[math]\displaystyle{ \tau_x ({f}*g) = (\tau_x f)*g = {f}*(\tau_x g)\, }[/math]

where τ_xf is the translation of the function f by x defined by

[math]\displaystyle{ (\tau_x f)(y) = f(y-x).\, }[/math]

If f is a Schwartz function, then τ_xf is the convolution with a translated Dirac delta function τ_xf = 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 linear operator acting on functions which commutes with translations: S(τ_xf) = τ_x(Sf) for all x. Then S is given as convolution with a function (or distribution) g_S; that is Sf = g_S∗f.

Thus any translation invariant operation can be represented as a 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 L¹ 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 tempered distribution whose Fourier transform is bounded. To wit, they are all given by bounded Fourier multipliers.

Convolutions on groups

두 함수의 정의역을 측도가 주어진 군으로 확장할 수 있다. 위에서 [math]\displaystyle{ t-\theta }[/math]나 [math]\displaystyle{ n-m }[/math]로 보아 역원의 존재를 가정해야 하고(덧셈군에서의 뺄셈은 그 역원과의 덧셈이다.), [math]\displaystyle{ \int }[/math]를 보면 측도가 주어져야 함을 알 수 있다. [math]\displaystyle{ G }[/math]가 측도 [math]\displaystyle{ \lambda }[/math]와 함께 주어지는 군이고 [math]\displaystyle{ f }[/math]와 [math]\displaystyle{ g }[/math]가 [math]\displaystyle{ G }[/math]에서 (르베그) 적분 가능한 함수[math]\displaystyle{ : \; G \to \mathbb R \text{ or }\mathbb C }[/math]라고 하자. 그러면 이 둘의 합성곱은 다음과 같이 정의할 수 있다:

[math]\displaystyle{ (f * g)(x) = \int_G f(y) g(y^{-1}x)\,\mathrm{d}\lambda(y). }[/math]

이때 [math]\displaystyle{ y^{-1}x }[/math]인 이유는 [math]\displaystyle{ y (y^{-1}x) = x }[/math]의 관계를 만족하게 하기 위해서이다. (비가환군에서 정의할 수도 있으므로.) [math]\displaystyle{ \mathrm{d}\lambda(y) }[/math]에 대한 정보가 없으면, 당연하게도 일반적으로 합성곱은 가환이 아니다.

It is not commutative in general. In typical cases of interest G is a locally compact Hausdorff topological group and λ is a (left-) Haar measure. In that case, unless G is unimodular, the convolution defined in this way is not the same as [math]\displaystyle{ \textstyle{\int f(xy^{-1})g(y) \, d\lambda(y)} }[/math]. The preference of one over the other is made so that convolution with a fixed function g commutes with left translation in the group:

[math]\displaystyle{ L_h(f*g) = (L_hf)*g. }[/math]

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 groups, 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 L¹(T), we have the following familiar operator acting on the Hilbert space L²(T):

[math]\displaystyle{ T {f}(x) = \frac{1}{2 \pi} \int_{\mathbf{T}} {f}(y) g( x - y) \, dy. }[/math]

The operator T is compact. A direct calculation shows that its adjoint T* is convolution with

[math]\displaystyle{ \bar{g}(-y). \, }[/math]

By the commutativity property cited above, T is 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 spectral theory, there exists an orthonormal basis {h_k} that simultaneously diagonalizes S. This characterizes convolutions on the circle. Specifically, we have

[math]\displaystyle{ h_k (x) = e^{ikx}, \quad k \in \mathbb{Z},\; }[/math]

which are precisely the 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 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 L² 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 topological group. If μ and ν are finite Borel measures on G, then their convolution μ∗ν is defined by

[math]\displaystyle{ (\mu * \nu)(E) = \int\!\!\!\int 1_E(xy) \,d\mu(x) \,d\nu(y) }[/math]

for each measurable subset E of G. The convolution is also a finite measure, whose total variation satisfies

[math]\displaystyle{ \|\mu * \nu\| \le \|\mu\| \|\nu\|. \, }[/math]

In the case when G is locally compact with (left-)Haar measure λ, and μ and ν are absolutely continuous with respect to a λ, 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 (R,+), then the convolution μ∗ν is the probability distribution of the sum X + Y of two independent random variables X and Y whose respective distributions are μ and ν.

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

[math]\displaystyle{ X \xrightarrow{\Delta} X\otimes X \xrightarrow{\phi\otimes\psi} X\otimes X \xrightarrow{\nabla} X. \, }[/math]

The convolution appears notably in the definition of Hopf algebras 틀:Harv. A bialgebra is a Hopf algebra if and only if it has an antipode: an endomorphism S such that

[math]\displaystyle{ S * \operatorname{id}_X = \operatorname{id}_X * S = \eta\circ\varepsilon. }[/math]

Applications

Gaussian blur can be used in order to obtain a smooth grayscale digital image of a halftone print

Convolution and related operations are found in many applications in science, engineering and mathematics.

• In image processing

In digital image processing convolutional filtering plays an important role in many important algorithms in edge detection and related processes.

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, a weighted moving average is a convolution.

• In acoustics, reverberation is the convolution of the original sound with 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 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 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, wherever there is a linear system with a "superposition principle", a convolution operation makes an appearance. For instance, in spectroscopy line broadening due to the Doppler effect on its own gives a Gaussian spectral line shape and collision broadening alone gives a Lorentzian line shape. When both effects are operative, the line shape is a convolution of Gaussian and Lorentzian, a Voigt function.

In 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 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. 틀:Harv.

• In radiotherapy treatment planning systems, most part of all modern codes of calculation applies a convolution-superposition algorithm.틀:Clarify
• Convolutional neural networks apply multiple cascaded convolution kernels with applications in machine vision and artificial intelligence

See also

• Analog signal processing
• Circulant matrix
• Convolution for optical broad-beam responses in scattering media
• Convolution power
• Cross-correlation
• Deconvolution
• Dirichlet convolution
• Jan Mikusinski
• List of convolutions of probability distributions
• LTI system theory#Impulse response and 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)
• Multidimensional discrete convolution

References

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• 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.
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External links

• Earliest Uses: The entry on Convolution has some historical information.
• Convolution, on The Data Analysis BriefBook
• 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
• 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
• Convolution at MathWorld
• Freeverb3 Impulse Response Processor: Opensource zero latency impulse response processor with VST plugins
• Stanford University CS 178 interactive Flash demo showing how spatial convolution works.
• A video lecture on the subject of convolution given by Salman Khan