Movatterモバイル変換

[0]ホーム
URL:

Jump to content
WikipediaThe Free Encyclopedia
Search

Convolution

From Wikipedia, the free encyclopedia
Integral expressing the amount of overlap of one function as it is shifted over another
For other uses, seeConvolution (disambiguation).
Visual comparison of convolution,cross-correlation, andautocorrelation. For the operations involving functionf{\displaystyle f}, and assuming the height off{\displaystyle f} is 1.0, the value of the result at 5 different points is indicated by the shaded area below each point. The symmetry off{\displaystyle f} is the reasonf⋆g{\displaystyle f\star g} andg∗f{\displaystyle g*f} are identical in this example.

Inmathematics (in particular,functional analysis),convolution is amathematical operation on twofunctionsf{\displaystyle f} andg{\displaystyle g} that produces a third functionf∗g{\displaystyle f*g}, as theintegral of the product of the two functions after one is reflected about the y-axis and shifted. The termconvolution refers to both the resulting function and to the process of computing it. The integral is evaluated for all values of shift, producing the convolution function. The choice of which function is reflected and shifted before the integral does not change the integral result (seecommutativity). Graphically, it expresses how the 'shape' of one function is modified by the other.

Some features of convolution are similar tocross-correlation: for real-valued functions, of a continuous or discrete variable, convolutionf∗g{\displaystyle f*g} differs from cross-correlationf⋆g{\displaystyle f\star g} only in that eitherf(x){\displaystyle f(x)} org(x){\displaystyle g(x)} is reflected about the y-axis in convolution; thus it is a cross-correlation ofg(−x){\displaystyle g(-x)} andf(x){\displaystyle f(x)}, orf(−x){\displaystyle f(-x)} andg(x){\displaystyle g(x)}.[A] For complex-valued functions, the cross-correlation operator is theadjoint of the convolution operator.

Convolution has applications that includeprobability,statistics,acoustics,spectroscopy,signal processing andimage processing,geophysics,engineering,physics,computer vision anddifferential equations.[1]

The convolution can be defined for functions onEuclidean space and othergroups (asalgebraic structures).[citation needed] For example,periodic functions, such as thediscrete-time Fourier transform, can be defined on acircle and convolved byperiodic convolution. (See row 18 atDTFT § Properties.) Adiscrete convolution can be defined for functions on the set ofintegers.

Generalizations of convolution have applications in the field ofnumerical analysis andnumerical linear algebra, and in the design and implementation offinite impulse response filters in signal processing.[citation needed]

Computing theinverse of the convolution operation is known asdeconvolution.

Definition

[edit]

The convolution off{\displaystyle f} andg{\displaystyle g} is writtenf∗g{\displaystyle f*g}, denoting the operator with the symbol∗{\displaystyle *}.[B] 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 ofintegral transform:

(f∗g)(t):=∫−∞∞f(τ)g(t−τ)dτ.{\displaystyle (f*g)(t):=\int _{-\infty }^{\infty }f(\tau )g(t-\tau )\,d\tau .}

An equivalent definition is (seecommutativity):

(f∗g)(t):=∫−∞∞f(t−τ)g(τ)dτ.{\displaystyle (f*g)(t):=\int _{-\infty }^{\infty }f(t-\tau )g(\tau )\,d\tau .}

While the symbolt{\displaystyle t} is used above, it need not represent the time domain. At eacht{\displaystyle t}, the convolution formula can be described as the area under the functionf(τ){\displaystyle f(\tau )} weighted by the functiong(−τ){\displaystyle g(-\tau )} shifted by the amountt{\displaystyle t}. Ast{\displaystyle t} changes, the weighting functiong(t−τ){\displaystyle g(t-\tau )} emphasizes different parts of the input functionf(τ){\displaystyle f(\tau )}; Ift{\displaystyle t} is a positive value, theng(t−τ){\displaystyle g(t-\tau )} is equal tog(−τ){\displaystyle g(-\tau )} that slides or is shifted along theτ{\displaystyle \tau }-axis toward the right (toward+∞{\displaystyle +\infty }) by the amount oft{\displaystyle t}, while ift{\displaystyle t} is a negative value, theng(t−τ){\displaystyle g(t-\tau )} is equal tog(−τ){\displaystyle g(-\tau )} that slides or is shifted toward the left (toward−∞{\displaystyle -\infty }) by the amount of|t|{\displaystyle |t|}.

For functionsf{\displaystyle f},g{\displaystyle g}supported on only[0,∞){\displaystyle [0,\infty )} (i.e., zero for negative arguments), the integration limits can be truncated, resulting in:

(f∗g)(t)=∫0tf(τ)g(t−τ)dτ for f,g:[0,∞)→R.{\displaystyle (f*g)(t)=\int _{0}^{t}f(\tau )g(t-\tau )\,d\tau \quad \ {\text{for }}f,g:[0,\infty )\to \mathbb {R} .}

For the multi-dimensional formulation of convolution, seedomain of definition (below).

Notation

[edit]

A common engineering notational convention is:[2]

f(t)∗g(t):=∫−∞∞f(τ)g(t−τ)dτ⏟(f∗g)(t),{\displaystyle f(t)*g(t)\mathrel {:=} \underbrace {\int _{-\infty }^{\infty }f(\tau )g(t-\tau )\,d\tau } _{(f*g)(t)},}

which has to be interpreted carefully to avoid confusion. For instance,f(t)∗g(t−t0){\displaystyle f(t)*g(t-t_{0})} is equivalent to(f∗g)(t−t0){\displaystyle (f*g)(t-t_{0})}, butf(t−t0)∗g(t−t0){\displaystyle f(t-t_{0})*g(t-t_{0})} is in fact equivalent to(f∗g)(t−2t0){\displaystyle (f*g)(t-2t_{0})}.[3]

Relations with other transforms

[edit]

Given two functionsf(t){\displaystyle f(t)} andg(t){\displaystyle g(t)} withbilateral Laplace transforms (two-sided Laplace transform)

F(s)=∫−∞∞e−su f(u) du{\displaystyle F(s)=\int _{-\infty }^{\infty }e^{-su}\ f(u)\ {\text{d}}u}

and

G(s)=∫−∞∞e−sv g(v) dv{\displaystyle G(s)=\int _{-\infty }^{\infty }e^{-sv}\ g(v)\ {\text{d}}v}

respectively, the convolution operation(f∗g)(t){\displaystyle (f*g)(t)} can be defined as theinverse Laplace transform of the product ofF(s){\displaystyle F(s)} andG(s){\displaystyle G(s)}.[4][5] More precisely,

F(s)⋅G(s)=∫−∞∞e−su f(u) du⋅∫−∞∞e−sv g(v) dv=∫−∞∞∫−∞∞e−s(u+v) f(u) g(v) du dv{\displaystyle {\begin{aligned}F(s)\cdot G(s)&=\int _{-\infty }^{\infty }e^{-su}\ f(u)\ {\text{d}}u\cdot \int _{-\infty }^{\infty }e^{-sv}\ g(v)\ {\text{d}}v\\&=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }e^{-s(u+v)}\ f(u)\ g(v)\ {\text{d}}u\ {\text{d}}v\end{aligned}}}

Lett=u+v{\displaystyle t=u+v}, then

F(s)⋅G(s)=∫−∞∞∫−∞∞e−st f(u) g(t−u) du dt=∫−∞∞e−st∫−∞∞f(u) g(t−u) du⏟(f∗g)(t) dt=∫−∞∞e−st(f∗g)(t) dt.{\displaystyle {\begin{aligned}F(s)\cdot G(s)&=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }e^{-st}\ f(u)\ g(t-u)\ {\text{d}}u\ {\text{d}}t\\&=\int _{-\infty }^{\infty }e^{-st}\underbrace {\int _{-\infty }^{\infty }f(u)\ g(t-u)\ {\text{d}}u} _{(f*g)(t)}\ {\text{d}}t\\&=\int _{-\infty }^{\infty }e^{-st}(f*g)(t)\ {\text{d}}t.\end{aligned}}}

Note thatF(s)⋅G(s){\displaystyle F(s)\cdot G(s)} is the bilateral Laplace transform of(f∗g)(t){\displaystyle (f*g)(t)}. A similar derivation can be done using theunilateral 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 aslinear time-invariant (LTI). SeeLTI system theory for a derivation of convolution as the result of LTI constraints. In terms of theFourier 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 atransfer function). SeeConvolution 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.

Visual explanation

[edit]
  1. Express each function in terms of adummy variableτ.{\displaystyle \tau .}
  2. Reflect one of the functions:g(τ){\displaystyle g(\tau )} →g(−τ).{\displaystyle g(-\tau ).}
  3. Add an offset of the independent variable,t{\displaystyle t}, which allowsg(−τ){\displaystyle g(-\tau )} to slide along theτ{\displaystyle \tau }-axis. Ift is a positive value, theng(t−τ){\displaystyle g(t-\tau )} is equal tog(−τ){\displaystyle g(-\tau )} that slides or is shifted along theτ{\displaystyle \tau }-axis toward the right (toward+∞{\displaystyle +\infty }) by the amount oft{\displaystyle t}. Ift{\displaystyle t} is a negative value, theng(t−τ){\displaystyle g(t-\tau )} is equal tog(−τ){\displaystyle g(-\tau )} that slides or is shifted toward the left (toward−∞{\displaystyle -\infty }) by the amount of|t|{\displaystyle |t|}.
  4. Startt{\displaystyle t} at−∞{\displaystyle -\infty } and slide it all the way to+∞{\displaystyle +\infty }. Wherever the two functions intersect, find the integral of their product. In other words, at timet{\displaystyle t}, compute the area under the functionf(τ){\displaystyle f(\tau )} weighted by the weighting functiong(t−τ).{\displaystyle g(t-\tau ).}

The resultingwaveform (not shown here) is the convolution of functionsf{\displaystyle f} andg{\displaystyle g}.

Iff(t){\displaystyle f(t)} is aunit impulse, the result of this process is simplyg(t){\displaystyle g(t)}. Formally:

∫−∞∞δ(τ)g(t−τ)dτ=g(t){\displaystyle \int _{-\infty }^{\infty }\delta (\tau )g(t-\tau )\,d\tau =g(t)}
In this example, the red-colored "pulse", g(τ),{\displaystyle \ g(\tau ),} is aneven function( g(−τ)=g(τ) ),{\displaystyle (\ g(-\tau )=g(\tau )\ ),} so convolution is equivalent to correlation. A snapshot of this "movie" shows functionsg(t−τ){\displaystyle g(t-\tau )} andf(τ){\displaystyle f(\tau )} (in blue) for some value of parametert,{\displaystyle t,} which is arbitrarily defined as the distance along theτ{\displaystyle \tau } axis from the pointτ=0{\displaystyle \tau =0} to the center of the red pulse. The amount of yellow is the area of the productf(τ)⋅g(t−τ),{\displaystyle f(\tau )\cdot g(t-\tau ),} computed by the convolution/correlation integral. The movie is created by continuously changingt{\displaystyle t} and recomputing the integral. The result (shown in black) is a function oft,{\displaystyle t,} but is plotted on the same axis asτ,{\displaystyle \tau ,} for convenience and comparison.
In this depiction,f(τ){\displaystyle f(\tau )} could represent the response of aresistor-capacitor circuit to a narrow pulse that occurs atτ=0.{\displaystyle \tau =0.} In other words, ifg(τ)=δ(τ),{\displaystyle g(\tau )=\delta (\tau ),} the result of convolution is justf(t).{\displaystyle f(t).} But wheng(τ){\displaystyle g(\tau )} is the wider pulse (in red), the response is a "smeared" version off(t).{\displaystyle f(t).} It begins att=−0.5,{\displaystyle t=-0.5,} because we definedt{\displaystyle t} as the distance from theτ=0{\displaystyle \tau =0} axis to thecenter of the wide pulse (instead of the leading edge).

Historical developments

[edit]

One of the earliest uses of the convolution integral appeared inD'Alembert's derivation ofTaylor's theorem inRecherches sur différents points importants du système du monde, published in 1754.[6]

Also, an expression of the type:

∫f(u)⋅g(x−u)du{\displaystyle \int f(u)\cdot g(x-u)\,du}

is used bySylvestre François Lacroix on page 505 of his book entitledTreatise 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.[7] Soon thereafter, convolution operations appear in the works ofPierre Simon Laplace,Jean-Baptiste Joseph Fourier,Siméon Denis Poisson, and others. The term itself did not come into wide use until the 1950s or 1960s. Prior to that it was sometimes known asFaltung (which meansfolding inGerman),composition product,superposition integral, andCarson's integral.[8] Yet it appears as early as 1903, though the definition is rather unfamiliar in older uses.[9][10]

The operation:

∫0tφ(s)ψ(t−s)ds,0≤t<∞,{\displaystyle \int _{0}^{t}\varphi (s)\psi (t-s)\,ds,\quad 0\leq t<\infty ,}

is a particular case of composition products considered by the Italian mathematicianVito Volterra in 1913.[11]

Circular convolution

[edit]
Main article:Circular convolution

When a functiongT{\displaystyle g_{T}} is periodic, with periodT{\displaystyle T}, then for functions,f{\displaystyle f}, such thatf∗gT{\displaystyle f*g_{T}} exists, the convolution is also periodic and identical to:

(f∗gT)(t)≡∫t0t0+T[∑k=−∞∞f(τ+kT)]gT(t−τ)dτ,{\displaystyle (f*g_{T})(t)\equiv \int _{t_{0}}^{t_{0}+T}\left[\sum _{k=-\infty }^{\infty }f(\tau +kT)\right]g_{T}(t-\tau )\,d\tau ,}

wheret0{\displaystyle t_{0}} is an arbitrary choice. The summation is called aperiodic summation of the functionf{\displaystyle f}.

WhengT{\displaystyle g_{T}} is a periodic summation of another function,g{\displaystyle g}, thenf∗gT{\displaystyle f*g_{T}} is known as acircular orcyclic convolution off{\displaystyle f} andg{\displaystyle g}.

And if the periodic summation above is replaced byfT{\displaystyle f_{T}}, the operation is called aperiodic convolution offT{\displaystyle f_{T}} andgT{\displaystyle g_{T}}.

Discrete convolution

[edit]
Discrete 2D Convolution Animation

For complex-valued functionsf{\displaystyle f} andg{\displaystyle g} defined on the setZ{\displaystyle \mathbb {Z} } of integers, thediscrete convolution off{\displaystyle f} andg{\displaystyle g} is given by:[12]

(f∗g)[n]=∑m=−∞∞f[m]g[n−m],{\displaystyle (f*g)[n]=\sum _{m=-\infty }^{\infty }f[m]g[n-m],}

or equivalently (seecommutativity) by:

(f∗g)[n]=∑m=−∞∞f[n−m]g[m].{\displaystyle (f*g)[n]=\sum _{m=-\infty }^{\infty }f[n-m]g[m].}

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 twopolynomials, then the coefficients of theordinary product of the two polynomials are the convolution of the original two sequences. This is known as theCauchy product of the coefficients of the sequences.

Thus, wheng is non-zero over a finite interval [-M,+M] (representing, for instance, afinite impulse response), a finite summation may be used:[13]

(f∗g)[n]=∑m=−MMf[n−m]g[m].{\displaystyle (f*g)[n]=\sum _{m=-M}^{M}f[n-m]g[m].}

Circular discrete convolution

[edit]

When a functiongN{\displaystyle g_{_{N}}} is periodic, with periodN,{\displaystyle N,} then for functions,f,{\displaystyle f,} such thatf∗gN{\displaystyle f*g_{_{N}}} exists, the convolution is also periodic and identical to:

(f∗gN)[n]≡∑m=0N−1(∑k=−∞∞f[m+kN])gN[n−m].{\displaystyle (f*g_{_{N}})[n]\equiv \sum _{m=0}^{N-1}\left(\sum _{k=-\infty }^{\infty }{f}[m+kN]\right)g_{_{N}}[n-m].}

The summation onk{\displaystyle k} is called aperiodic summation of the functionf.{\displaystyle f.}

IfgN{\displaystyle g_{_{N}}} is a periodic summation of another function,g,{\displaystyle g,} thenf∗gN{\displaystyle f*g_{_{N}}} is known as acircular convolution off{\displaystyle f} andg.{\displaystyle g.}

When the non-zero durations of bothf{\displaystyle f} andg{\displaystyle g} are limited to the interval[0,N−1],{\displaystyle [0,N-1],} f∗gN{\displaystyle f*g_{_{N}}} reduces to these common forms:

(f∗gN)[n]=∑m=0N−1f[m]gN[n−m]=∑m=0nf[m]g[n−m]+∑m=n+1N−1f[m]g[N+n−m]=∑m=0N−1f[m]g[(n−m)modN]≜(f∗Ng)[n]{\displaystyle {\begin{aligned}\left(f*g_{N}\right)[n]&=\sum _{m=0}^{N-1}f[m]g_{N}[n-m]\\&=\sum _{m=0}^{n}f[m]g[n-m]+\sum _{m=n+1}^{N-1}f[m]g[N+n-m]\\[2pt]&=\sum _{m=0}^{N-1}f[m]g[(n-m)_{\bmod {N}}]\\[2pt]&\triangleq \left(f*_{N}g\right)[n]\end{aligned}}}        Eq.1

The notationf∗Ng{\displaystyle f*_{N}g} forcyclic convolution denotes convolution over thecyclic group ofintegers moduloN.

Circular convolution arises most often in the context of fast convolution with afast Fourier transform (FFT) algorithm.

Fast convolution algorithms

[edit]

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 inmultiplication of multi-digit numbers, which can therefore be efficiently implemented with transform techniques (Knuth 1997, §4.3.3.C;von zur Gathen & Gerhard 2003, §8.2).

Eq.1 requiresN arithmetic operations per output value andN2 operations forN 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 logN) complexity.

The most common fast convolution algorithms usefast Fourier transform (FFT) algorithms via thecircular convolution theorem. Specifically, thecircular 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 theSchönhage–Strassen algorithm or the Mersenne transform,[14] use fast Fourier transforms in otherrings. The Winograd method is used as an alternative to the FFT.[15] It significantly speeds up 1D,[16] 2D,[17] and 3D[18] convolution.

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.[19] Instead, decomposing the longer sequence into blocks and convolving each block allows for faster algorithms such as theoverlap–save method andoverlap–add method.[20] A hybrid convolution method that combines block andFIR algorithms allows for a zero input-output latency that is useful for real-time convolution computations.[21]

Domain of definition

[edit]

The convolution of two complex-valued functions onRd is itself a complex-valued function onRd, defined by:

(f∗g)(x)=∫Rdf(y)g(x−y)dy=∫Rdf(x−y)g(y)dy,{\displaystyle (f*g)(x)=\int _{\mathbf {R} ^{d}}f(y)g(x-y)\,dy=\int _{\mathbf {R} ^{d}}f(x-y)g(y)\,dy,}

and is well-defined only iff andg 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 ing at infinity can be easily offset by sufficiently rapid decay inf. The question of existence thus may involve different conditions onf andg:

Compactly supported functions

[edit]

Iff andg arecompactly supportedcontinuous functions, then their convolution exists, and is also compactly supported and continuous (Hörmander 1983, Chapter 1). More generally, if either function (sayf) is compactly supported and the other islocally integrable, then the convolutionf∗g is well-defined and continuous.

Convolution off andg is also well defined when both functions are locally square integrable onR and supported on an interval of the form[a, +∞) (or both supported on[−∞,a]).

Integrable functions

[edit]

The convolution off andg exists iff andg are bothLebesgue integrable functions inL1(Rd), and in this casef∗g is also integrable (Stein & Weiss 1971, Theorem 1.3). This is a consequence ofTonelli's theorem. This is also true for functions inL1, under the discrete convolution, or more generally for theconvolution on any group.

Likewise, iff ∈L1(Rd)  and  g ∈Lp(Rd)  where1 ≤p ≤ ∞,  then  f*g ∈Lp(Rd),  and

‖f∗g‖p≤‖f‖1‖g‖p.{\displaystyle \|{f}*g\|_{p}\leq \|f\|_{1}\|g\|_{p}.}

In the particular casep = 1, this shows thatL1 is aBanach algebra under the convolution (and equality of the two sides holds iff andg are non-negative almost everywhere).

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

1p+1q=1r+1,{\displaystyle {\frac {1}{p}}+{\frac {1}{q}}={\frac {1}{r}}+1,}

then

‖f∗g‖r≤‖f‖p‖g‖q,f∈Lp, g∈Lq,{\displaystyle \left\Vert f*g\right\Vert _{r}\leq \left\Vert f\right\Vert _{p}\left\Vert g\right\Vert _{q},\quad f\in L^{p},\ g\in L^{q},}

so that the convolution is a continuous bilinear mapping fromLp×Lq toLr.The Young inequality for convolution is also true in other contexts (circle group, convolution onZ). The preceding inequality is not sharp on the real line: when 1 <p,q,r < ∞, there exists a constantBp,q < 1 such that:

‖f∗g‖r≤Bp,q‖f‖p‖g‖q,f∈Lp, g∈Lq.{\displaystyle \left\Vert f*g\right\Vert _{r}\leq B_{p,q}\left\Vert f\right\Vert _{p}\left\Vert g\right\Vert _{q},\quad f\in L^{p},\ g\in L^{q}.}

The optimal value ofBp,q was discovered in 1975[22] and independently in 1976,[23] seeBrascamp–Lieb inequality.

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

‖f∗g‖r≤Cp,q‖f‖p‖g‖q,w{\displaystyle \|f*g\|_{r}\leq C_{p,q}\|f\|_{p}\|g\|_{q,w}}

where‖g‖q,w{\displaystyle \|g\|_{q,w}} is theweakLq norm. Convolution also defines a bilinear continuous mapLp,w×Lq,w→Lr,w{\displaystyle L^{p,w}\times L^{q,w}\to L^{r,w}} for1<p,q,r<∞{\displaystyle 1<p,q,r<\infty }, owing to the weak Young inequality:[24]

‖f∗g‖r,w≤Cp,q‖f‖p,w‖g‖r,w.{\displaystyle \|f*g\|_{r,w}\leq C_{p,q}\|f\|_{p,w}\|g\|_{r,w}.}

Functions of rapid decay

[edit]

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 iff andg both decay rapidly, thenf∗g also decays rapidly. In particular, iff andg arerapidly decreasing functions, then so is the convolutionf∗g. Combined with the fact that convolution commutes with differentiation (see#Properties), it follows that the class ofSchwartz functions is closed under convolution (Stein & Weiss 1971, Theorem 3.3).

Distributions

[edit]
Main article:Distribution (mathematics)

Iff is a smooth function that iscompactly supported andg is a distribution, thenf∗g is a smooth function defined by

∫Rdf(y)g(x−y)dy=(f∗g)(x)∈C∞(Rd).{\displaystyle \int _{\mathbb {R} ^{d}}{f}(y)g(x-y)\,dy=(f*g)(x)\in C^{\infty }(\mathbb {R} ^{d}).}

More generally, it is possible to extend the definition of the convolution in a unique way withφ{\displaystyle \varphi } the same asf above, so that the associative law

f∗(g∗φ)=(f∗g)∗φ{\displaystyle f*(g*\varphi )=(f*g)*\varphi }

remains valid in the case wheref is a distribution, andg a compactly supported distribution (Hörmander 1983, §4.2).

Measures

[edit]

The convolution of any twoBorel measuresμ andν ofbounded variation is the measureμ∗ν{\displaystyle \mu *\nu } defined by (Rudin 1962)

∫Rdf(x)d(μ∗ν)(x)=∫Rd∫Rdf(x+y)dμ(x)dν(y).{\displaystyle \int _{\mathbf {R} ^{d}}f(x)\,d(\mu *\nu )(x)=\int _{\mathbf {R} ^{d}}\int _{\mathbf {R} ^{d}}f(x+y)\,d\mu (x)\,d\nu (y).}

In particular,

(μ∗ν)(A)=∫Rd×Rd1A(x+y)d(μ×ν)(x,y),{\displaystyle (\mu *\nu )(A)=\int _{\mathbf {R} ^{d}\times \mathbf {R} ^{d}}1_{A}(x+y)\,d(\mu \times \nu )(x,y),}

whereA⊂Rd{\displaystyle A\subset \mathbf {R} ^{d}} is a measurable set and1A{\displaystyle 1_{A}} is theindicator function ofA{\displaystyle 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

‖μ∗ν‖≤‖μ‖‖ν‖{\displaystyle \|\mu *\nu \|\leq \|\mu \|\|\nu \|}

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

Properties

[edit]

Algebraic properties

[edit]
See also:Convolution algebra

The convolution defines a product on thelinear 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 commutativeassociative algebra withoutidentity (Strichartz 1994, §3.3). Other linear spaces of functions, such as the space of continuous functions of compact support, areclosed under the convolution, and so also form commutative associative algebras.

Commutativity
f∗g=g∗f{\displaystyle f*g=g*f} Proof: By definition:(f∗g)(t)=∫−∞∞f(τ)g(t−τ)dτ{\displaystyle (f*g)(t)=\int _{-\infty }^{\infty }f(\tau )g(t-\tau )\,d\tau } Changing the variable of integration tou=t−τ{\displaystyle u=t-\tau } the result follows.
Associativity
f∗(g∗h)=(f∗g)∗h{\displaystyle f*(g*h)=(f*g)*h} Proof: This follows from usingFubini's theorem (i.e., double integrals can be evaluated as iterated integrals in either order).
Distributivity
f∗(g+h)=(f∗g)+(f∗h){\displaystyle f*(g+h)=(f*g)+(f*h)} Proof: This follows from linearity of the integral.
Associativity with scalar multiplication
a(f∗g)=(af)∗g{\displaystyle a(f*g)=(af)*g} for any real (or complex) numbera{\displaystyle a}.
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 adelta distribution (a unitary impulse, centered at zero) or, at the very least (as is the case ofL1) admitapproximations to the identity. The linear space of compactly supported distributions does, however, admit an identity under the convolution. Specifically,f∗δ=f{\displaystyle f*\delta =f} whereδ is the delta distribution.
Inverse element
Some distributionsS have aninverse elementS−1 for the convolution which then must satisfyS−1∗S=δ{\displaystyle S^{-1}*S=\delta } from which an explicit formula forS−1 may be obtained.
The set of invertible distributions forms anabelian group under the convolution.
Complex conjugation
f∗g¯=f¯∗g¯{\displaystyle {\overline {f*g}}={\overline {f}}*{\overline {g}}}
Time reversal
If  q(t)=r(t)∗s(t),{\displaystyle q(t)=r(t)*s(t),}  then  q(−t)=r(−t)∗s(−t).{\displaystyle q(-t)=r(-t)*s(-t).}

Proof (usingconvolution theorem):

q(t) ⟺F  Q(f)=R(f)S(f){\displaystyle q(t)\ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ \ Q(f)=R(f)S(f)}

q(−t) ⟺F  Q(−f)=R(−f)S(−f){\displaystyle q(-t)\ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ \ Q(-f)=R(-f)S(-f)}

q(−t)=F−1{R(−f)S(−f)}=F−1{R(−f)}∗F−1{S(−f)}=r(−t)∗s(−t){\displaystyle {\begin{aligned}q(-t)&={\mathcal {F}}^{-1}{\bigg \{}R(-f)S(-f){\bigg \}}\\&={\mathcal {F}}^{-1}{\bigg \{}R(-f){\bigg \}}*{\mathcal {F}}^{-1}{\bigg \{}S(-f){\bigg \}}\\&=r(-t)*s(-t)\end{aligned}}}

Relationship with differentiation
(f∗g)′=f′∗g=f∗g′{\displaystyle (f*g)'=f'*g=f*g'} Proof:
(f∗g)′=ddt∫−∞∞f(τ)g(t−τ)dτ=∫−∞∞f(τ)∂∂tg(t−τ)dτ=∫−∞∞f(τ)g′(t−τ)dτ=f∗g′.{\displaystyle {\begin{aligned}(f*g)'&={\frac {d}{dt}}\int _{-\infty }^{\infty }f(\tau )g(t-\tau )\,d\tau \\&=\int _{-\infty }^{\infty }f(\tau ){\frac {\partial }{\partial t}}g(t-\tau )\,d\tau \\&=\int _{-\infty }^{\infty }f(\tau )g'(t-\tau )\,d\tau =f*g'.\end{aligned}}}
Relationship with integration
IfF(t)=∫−∞tf(τ)dτ,{\textstyle F(t)=\int _{-\infty }^{t}f(\tau )d\tau ,} andG(t)=∫−∞tg(τ)dτ,{\textstyle G(t)=\int _{-\infty }^{t}g(\tau )\,d\tau ,} then(F∗g)(t)=(f∗G)(t)=∫−∞t(f∗g)(τ)dτ.{\displaystyle (F*g)(t)=(f*G)(t)=\int _{-\infty }^{t}(f*g)(\tau )\,d\tau .}

Integration

[edit]

Iff andg are integrable functions, then the integral of their convolution on the whole space is simply obtained as the product of their integrals:[25]

∫Rd(f∗g)(x)dx=(∫Rdf(x)dx)(∫Rdg(x)dx).{\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).}

This follows fromFubini's theorem. The same result holds iff andg are only assumed to be nonnegative measurable functions, byTonelli's theorem.

Differentiation

[edit]

In the one-variable case,

ddx(f∗g)=dfdx∗g=f∗dgdx{\displaystyle {\frac {d}{dx}}(f*g)={\frac {df}{dx}}*g=f*{\frac {dg}{dx}}}

whereddx{\displaystyle {\frac {d}{dx}}} is thederivative. More generally, in the case of functions of several variables, an analogous formula holds with thepartial derivative:

∂∂xi(f∗g)=∂f∂xi∗g=f∗∂g∂xi.{\displaystyle {\frac {\partial }{\partial x_{i}}}(f*g)={\frac {\partial f}{\partial x_{i}}}*g=f*{\frac {\partial g}{\partial x_{i}}}.}

A particular consequence of this is that the convolution can be viewed as a "smoothing" operation: the convolution off andg is differentiable as many times asf andg are in total.

These identities hold for example under the condition thatf andg are absolutely integrable and at least one of them has an absolutely integrable (L1) weak derivative, as a consequence ofYoung's convolution inequality. For instance, whenf is continuously differentiable with compact support, andg is an arbitrary locally integrable function,

ddx(f∗g)=dfdx∗g.{\displaystyle {\frac {d}{dx}}(f*g)={\frac {df}{dx}}*g.}

These identities also hold much more broadly in the sense of tempered distributions if one off org is arapidly 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, thedifference operatorDf(n) =f(n + 1) −f(n) satisfies an analogous relationship:

D(f∗g)=(Df)∗g=f∗(Dg).{\displaystyle D(f*g)=(Df)*g=f*(Dg).}

Convolution theorem

[edit]

Theconvolution theorem states that[26]

F{f∗g}=F{f}⋅F{g}{\displaystyle {\mathcal {F}}\{f*g\}={\mathcal {F}}\{f\}\cdot {\mathcal {F}}\{g\}}

whereF{f}{\displaystyle {\mathcal {F}}\{f\}} denotes theFourier transform off{\displaystyle f}.

Convolution in other types of transformations

[edit]

Versions of this theorem also hold for theLaplace transform,two-sided Laplace transform,Z-transform andMellin transform.

Convolution on matrices

[edit]

IfW{\displaystyle {\mathcal {W}}} is theFourier transform matrix, then

W(C(1)x∗C(2)y)=(WC(1)∙WC(2))(x⊗y)=WC(1)x∘WC(2)y{\displaystyle {\mathcal {W}}\left(C^{(1)}x\ast C^{(2)}y\right)=\left({\mathcal {W}}C^{(1)}\bullet {\mathcal {W}}C^{(2)}\right)(x\otimes y)={\mathcal {W}}C^{(1)}x\circ {\mathcal {W}}C^{(2)}y},

where∙{\displaystyle \bullet } isface-splitting product,[27][28][29][30][31]⊗{\displaystyle \otimes } denotesKronecker product,∘{\displaystyle \circ } denotesHadamard product (this result is an evolving ofcount sketch properties[32]).

This can be generalized for appropriate matricesA,B{\displaystyle \mathbf {A} ,\mathbf {B} }:

W((Ax)∗(By))=((WA)∙(WB))(x⊗y)=(WAx)∘(WBy){\displaystyle {\mathcal {W}}\left((\mathbf {A} x)\ast (\mathbf {B} y)\right)=\left(({\mathcal {W}}\mathbf {A} )\bullet ({\mathcal {W}}\mathbf {B} )\right)(x\otimes y)=({\mathcal {W}}\mathbf {A} x)\circ ({\mathcal {W}}\mathbf {B} y)}

from the properties of theface-splitting product.

Translational equivariance

[edit]

The convolution commutes with translations, meaning that

τx(f∗g)=(τxf)∗g=f∗(τxg){\displaystyle \tau _{x}(f*g)=(\tau _{x}f)*g=f*(\tau _{x}g)}

where τxf is the translation of the functionf byx defined by

(τxf)(y)=f(y−x).{\displaystyle (\tau _{x}f)(y)=f(y-x).}

Iff is aSchwartz function, thenτxf is the convolution with a translated Dirac delta functionτxf =f ∗τxδ. 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 thatS is a boundedlinear operator acting on functions which commutes with translations:S(τxf) =τx(Sf) for allx. ThenS is given as convolution with a function (or distribution)gS; that isSf =gS ∗f.

Thus some translation invariant operations can be represented as convolution. Convolutions play an important role in the study oftime-invariant systems, and especiallyLTI system theory. The representing functiongS is theimpulse response of the transformationS.

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 thatS must be acontinuous linear operator with respect to the appropriatetopology. It is known, for instance, that every continuous translation invariant continuous linear operator onL1 is the convolution with a finiteBorel measure. More generally, every continuous translation invariant continuous linear operator onLp for 1 ≤p < ∞ is the convolution with atempered distribution whoseFourier transform is bounded. To wit, they are all given by boundedFourier multipliers.

Convolutions on groups

[edit]

IfG is a suitablegroup endowed with ameasure λ, and iff andg are real or complex valuedintegrable functions onG, then we can define their convolution by

(f∗g)(x)=∫Gf(y)g(y−1x)dλ(y).{\displaystyle (f*g)(x)=\int _{G}f(y)g\left(y^{-1}x\right)\,d\lambda (y).}

It is not commutative in general. In typical cases of interestG is alocally compactHausdorfftopological group and λ is a (left-)Haar measure. In that case, unlessG isunimodular, the convolution defined in this way is not the same as∫f(xy−1)g(y)dλ(y){\textstyle \int f\left(xy^{-1}\right)g(y)\,d\lambda (y)}. The preference of one over the other is made so that convolution with a fixed functiong commutes with left translation in the group:

Lh(f∗g)=(Lhf)∗g.{\displaystyle L_{h}(f*g)=(L_{h}f)*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.

Onlocally compact abelian groups, a version of theconvolution theorem holds: the Fourier transform of a convolution is the pointwise product of the Fourier transforms. Thecircle groupT with the Lebesgue measure is an immediate example. For a fixedg inL1(T), we have the following familiar operator acting on theHilbert spaceL2(T):

Tf(x)=12π∫Tf(y)g(x−y)dy.{\displaystyle T{f}(x)={\frac {1}{2\pi }}\int _{\mathbf {T} }{f}(y)g(x-y)\,dy.}

The operatorT iscompact. A direct calculation shows that its adjointT* is convolution with

g¯(−y).{\displaystyle {\bar {g}}(-y).}

By the commutativity property cited above,T isnormal:T*T =TT* . Also,T commutes with the translation operators. Consider the familyS of operators consisting of all such convolutions and the translation operators. ThenS is a commuting family of normal operators. According tospectral theory, there exists an orthonormal basis {hk} that simultaneously diagonalizesS. This characterizes convolutions on the circle. Specifically, we have

hk(x)=eikx,k∈Z,{\displaystyle h_{k}(x)=e^{ikx},\quad k\in \mathbb {Z} ,\;}

which are precisely thecharacters ofT. Each convolution is a compactmultiplication operator in this basis. This can be viewed as a version of the convolution theorem discussed above.

A discrete example is a finitecyclic group of ordern. Convolution operators are here represented bycirculant matrices, and can be diagonalized by thediscrete Fourier transform.

A similar result holds for compact groups (not necessarily abelian): the matrix coefficients of finite-dimensionalunitary representations form an orthonormal basis inL2 by thePeter–Weyl theorem, and an analog of the convolution theorem continues to hold, along with many other aspects ofharmonic analysis that depend on the Fourier transform.

Convolution of measures

[edit]

LetG be a (multiplicatively written) topological group.If μ and ν areRadon measures onG, then their convolutionμ∗ν is defined as thepushforward measure of thegroup action and can be written as[33]

(μ∗ν)(E)=∬1E(xy)dμ(x)dν(y){\displaystyle (\mu *\nu )(E)=\iint 1_{E}(xy)\,d\mu (x)\,d\nu (y)}

for each measurable subsetE ofG. The convolution is also a Radon measure, whosetotal variation satisfies

‖μ∗ν‖≤‖μ‖‖ν‖.{\displaystyle \|\mu *\nu \|\leq \left\|\mu \right\|\left\|\nu \right\|.}

In the case whenG islocally compact with (left-)Haar measure λ, and μ and ν areabsolutely 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. In fact, ifeither measure is absolutely continuous with respect to the Haar measure, then so is their convolution.[34]

If μ and ν areprobability measures on the topological group(R,+), then the convolutionμ∗ν is theprobability distribution of the sumX +Y of twoindependentrandom variablesX andY whose respective distributions are μ and ν.

Infimal convolution

[edit]

Inconvex analysis, theinfimal convolution of proper (not identically+∞{\displaystyle +\infty })convex functionsf1,…,fm{\displaystyle f_{1},\dots ,f_{m}} onRn{\displaystyle \mathbb {R} ^{n}} is defined by:[35](f1∗⋯∗fm)(x)=infx{f1(x1)+⋯+fm(xm)|x1+⋯+xm=x}.{\displaystyle (f_{1}*\cdots *f_{m})(x)=\inf _{x}\{f_{1}(x_{1})+\cdots +f_{m}(x_{m})|x_{1}+\cdots +x_{m}=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 theLegendre transform:φ∗(x)=supy(x⋅y−φ(y)).{\displaystyle \varphi ^{*}(x)=\sup _{y}(x\cdot y-\varphi (y)).}We have:(f1∗⋯∗fm)∗(x)=f1∗(x)+⋯+fm∗(x).{\displaystyle (f_{1}*\cdots *f_{m})^{*}(x)=f_{1}^{*}(x)+\cdots +f_{m}^{*}(x).}

Bialgebras

[edit]

Let (X, Δ, ∇,ε,η) be abialgebra with comultiplication Δ, multiplication ∇, unit η, and counitε. The convolution is a product defined on theendomorphism algebra End(X) as follows. Letφ,ψ ∈ End(X), that is,φ,ψ:X →X are functions that respect all algebraic structure ofX, then the convolutionφ∗ψ is defined as the composition

X→ΔX⊗X→ϕ⊗ψX⊗X→∇X.{\displaystyle X\mathrel {\xrightarrow {\Delta } } X\otimes X\mathrel {\xrightarrow {\phi \otimes \psi } } X\otimes X\mathrel {\xrightarrow {\nabla } } X.}

The convolution appears notably in the definition ofHopf algebras (Kassel 1995, §III.3). A bialgebra is a Hopf algebra if and only if it has an antipode: an endomorphismS such that

S∗idX=idX∗S=η∘ε.{\displaystyle S*\operatorname {id} _{X}=\operatorname {id} _{X}*S=\eta \circ \varepsilon .}

Applications

[edit]
Gaussian blur can be used to obtain a smooth grayscale digital image of ahalftone print.

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

See also

[edit]

Notes

[edit]
  1. ^Reasons for the reflection include:
  2. ^The symbolU+2217 ∗ASTERISK OPERATOR is different thanU+002A *ASTERISK, which is often used to denote complex conjugation. SeeAsterisk § Mathematical typography.

References

[edit]
  1. ^Bahri, Mawardi; Ashino, Ryuichi; Vaillancourt, Rémi (2013)."Convolution Theorems for Quaternion Fourier Transform: Properties and Applications"(PDF).Abstract and Applied Analysis.2013:1–10.doi:10.1155/2013/162769.Archived(PDF) from the original on 2020-10-21. Retrieved2022-11-11.
  2. ^Smith, Stephen W (1997)."13.Convolution".The Scientist and Engineer's Guide to Digital Signal Processing (1 ed.). California Technical Publishing.ISBN 0-9660176-3-3. Retrieved22 April 2016.
  3. ^Irwin, J. David (1997). "4.3".The Industrial Electronics Handbook (1 ed.). Boca Raton, FL: CRC Press. p. 75.ISBN 0-8493-8343-9.
  4. ^Differential Equations (Spring 2010), MIT 18.03."Lecture 21: Convolution Formula".MIT Open Courseware. MIT. Retrieved22 December 2021.{{cite web}}: CS1 maint: numeric names: authors list (link)
  5. ^"18.03SC Differential Equations Fall 2011"(PDF).Green's Formula, Laplace Transform of Convolution.Archived(PDF) from the original on 2015-09-06.
  6. ^Dominguez-Torres, p 2
  7. ^Dominguez-Torres, p 4
  8. ^R. N. Bracewell (2005),"Early work on imaging theory in radio astronomy", in W. T. Sullivan (ed.),The Early Years of Radio Astronomy: Reflections Fifty Years After Jansky's Discovery, Cambridge University Press, p. 172,ISBN 978-0-521-61602-7
  9. ^John Hilton Grace and Alfred Young (1903),The algebra of invariants, Cambridge University Press, p. 40
  10. ^Leonard Eugene Dickson (1914),Algebraic invariants, J. Wiley, p. 85,ISBN 978-1-4297-0042-9{{citation}}:ISBN / Date incompatibility (help)
  11. ^According to[Lothar von Wolfersdorf (2000), "Einige Klassen quadratischer Integralgleichungen",Sitzungsberichte der Sächsischen Akademie der Wissenschaften zu Leipzig,Mathematisch-naturwissenschaftliche Klasse, volume128, number 2, 6–7], the source is Volterra, Vito (1913),"Leçons sur les fonctions de linges". Gauthier-Villars, Paris 1913.
  12. ^Damelin & Miller 2011, p. 219
  13. ^Press, William H.; Flannery, Brian P.; Teukolsky, Saul A.; Vetterling, William T. (1989).Numerical Recipes in Pascal. Cambridge University Press. p. 450.ISBN 0-521-37516-9.
  14. ^Rader, C.M. (December 1972). "Discrete Convolutions via Mersenne Transforms".IEEE Transactions on Computers.21 (12):1269–1273.doi:10.1109/T-C.1972.223497.S2CID 1939809.
  15. ^Winograd, Shmuel (January 1980).Arithmetic Complexity of Computations. Society for Industrial and Applied Mathematics.doi:10.1137/1.9781611970364.ISBN 978-0-89871-163-9.
  16. ^Lyakhov, P. A.; Nagornov, N. N.; Semyonova, N. F.; Abdulsalyamova, A. S. (June 2023)."Reducing the Computational Complexity of Image Processing Using Wavelet Transform Based on the Winograd Method".Pattern Recognition and Image Analysis.33 (2):184–191.doi:10.1134/S1054661823020074.ISSN 1054-6618.S2CID 259310351.
  17. ^Wu, Di; Fan, Xitian; Cao, Wei; Wang, Lingli (May 2021). "SWM: A High-Performance Sparse-Winograd Matrix Multiplication CNN Accelerator".IEEE Transactions on Very Large Scale Integration (VLSI) Systems.29 (5):936–949.Bibcode:2021ITVL...29..936W.doi:10.1109/TVLSI.2021.3060041.ISSN 1063-8210.S2CID 233433757.
  18. ^Mittal, Sparsh; Vibhu (May 2021)."A survey of accelerator architectures for 3D convolution neural networks".Journal of Systems Architecture.115 102041.doi:10.1016/j.sysarc.2021.102041.S2CID 233917781.
  19. ^Selesnick, Ivan W.; Burrus, C. Sidney (1999). "Fast Convolution and Filtering". In Madisetti, Vijay K. (ed.).Digital Signal Processing Handbook. CRC Press. p. Section 8.ISBN 978-1-4200-4563-5.
  20. ^Juang, B.H."Lecture 21: Block Convolution"(PDF). EECS at the Georgia Institute of Technology.Archived(PDF) from the original on 2004-07-29. Retrieved17 May 2013.
  21. ^Gardner, William G. (November 1994)."Efficient Convolution without Input/Output Delay"(PDF).Audio Engineering Society Convention 97. Paper 3897.Archived(PDF) from the original on 2015-04-08. Retrieved17 May 2013.
  22. ^Beckner, William (1975). "Inequalities in Fourier analysis".Annals of Mathematics. Second Series.102 (1):159–182.doi:10.2307/1970980.JSTOR 1970980.
  23. ^Brascamp, Herm Jan;Lieb, Elliott H. (1976)."Best constants in Young's inequality, its converse, and its generalization to more than three functions".Advances in Mathematics.20 (2):151–173.doi:10.1016/0001-8708(76)90184-5.
  24. ^Reed & Simon 1975, IX.4
  25. ^Weisstein, Eric W."Convolution".mathworld.wolfram.com. Retrieved2021-09-22.
  26. ^Weisstein, Eric W."From MathWorld--A Wolfram Web Resource".
  27. ^Slyusar, V. I. (December 27, 1996)."End products in matrices in radar applications"(PDF).Radioelectronics and Communications Systems.41 (3):50–53.Archived(PDF) from the original on 2013-08-11.
  28. ^Slyusar, V. I. (1997-05-20)."Analytical model of the digital antenna array on a basis of face-splitting matrix products"(PDF).Proc. ICATT-97, Kyiv:108–109.Archived(PDF) from the original on 2013-08-11.
  29. ^Slyusar, V. I. (1997-09-15)."New operations of matrices product for applications of radars"(PDF).Proc. Direct and Inverse Problems of Electromagnetic and Acoustic Wave Theory (DIPED-97), Lviv.:73–74.Archived(PDF) from the original on 2013-08-11.
  30. ^Slyusar, V. I. (March 13, 1998)."A Family of Face Products of Matrices and its Properties"(PDF).Cybernetics and Systems Analysis C/C of Kibernetika I Sistemnyi Analiz.- 1999.35 (3):379–384.doi:10.1007/BF02733426.S2CID 119661450.Archived(PDF) from the original on 2013-08-11.
  31. ^Slyusar, V. I. (2003)."Generalized face-products of matrices in models of digital antenna arrays with nonidentical channels"(PDF).Radioelectronics and Communications Systems.46 (10):9–17.Archived(PDF) from the original on 2013-08-11.
  32. ^Ninh, Pham;Pagh, Rasmus (2013).Fast and scalable polynomial kernels via explicit feature maps. SIGKDD international conference on Knowledge discovery and data mining. Association for Computing Machinery.doi:10.1145/2487575.2487591.
  33. ^Hewitt and Ross (1979)Abstract harmonic analysis, volume 1, second edition, Springer-Verlag, p 266.
  34. ^Hewitt and Ross (1979), Theorem 19.18, p 272.
  35. ^R. Tyrrell Rockafellar (1970),Convex analysis, Princeton University Press
  36. ^Zhang, Yingjie; Soon, Hong Geok; Ye, Dongsen; Fuh, Jerry Ying Hsi; Zhu, Kunpeng (September 2020). "Powder-Bed Fusion Process Monitoring by Machine Vision With Hybrid Convolutional Neural Networks".IEEE Transactions on Industrial Informatics.16 (9):5769–5779.Bibcode:2020ITII...16.5769Z.doi:10.1109/TII.2019.2956078.ISSN 1941-0050.S2CID 213010088.
  37. ^Chervyakov, N.I.; Lyakhov, P.A.; Deryabin, M.A.; Nagornov, N.N.; Valueva, M.V.; Valuev, G.V. (September 2020)."Residue Number System-Based Solution for Reducing the Hardware Cost of a Convolutional Neural Network".Neurocomputing.407:439–453.doi:10.1016/j.neucom.2020.04.018.S2CID 219470398.Convolutional neural networks represent deep learning architectures that are currently used in a wide range of applications, including computer vision, speech recognition, time series analysis in finance, and many others.
  38. ^Atlas, Homma, and Marks."An Artificial Neural Network for Spatio-Temporal Bipolar Patterns: Application to Phoneme Classification"(PDF).Neural Information Processing Systems (NIPS 1987).1.Archived(PDF) from the original on 2021-04-14.{{cite journal}}: CS1 maint: multiple names: authors list (link)
  39. ^Zölzer, Udo, ed. (2002).DAFX:Digital Audio Effects, p.48–49.ISBN 0471490784.
  40. ^Diggle 1985.
  41. ^Ghasemi & Nowak 2017.
  42. ^Monaghan, J. J. (1992)."Smoothed particle hydrodynamics".Annual Review of Astronomy and Astrophysics.30:543–547.Bibcode:1992ARA&A..30..543M.doi:10.1146/annurev.aa.30.090192.002551. Retrieved16 February 2021.

Further reading

[edit]

External links

[edit]
Look upconvolution in Wiktionary, the free dictionary.
Wikimedia Commons has media related toConvolution.
Concepts
Applications
Implementations
Audio–visual
Text
Decisional
People
Architectures
Retrieved from "https://en.wikipedia.org/w/index.php?title=Convolution&oldid=1315884581"
Categories:
Hidden categories:
[8]ページ先頭
©2009-2025 Movatter.jp