082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

24.15 Discrete convolution and correlation 801
Figure24.23
Effect ofsetting elements to zero in the d.c.t.matrix. Moving from left to right agreater
number ofelements in the matrix are setto zero, resulting in more imageartefacts.
24.15 DISCRETECONVOLUTIONANDCORRELATION
As stated at the beginning of Section 24.8, convolution and correlation are important
techniquesinsignalandimageprocessing.Inthissectionwewilldescribetheirdiscrete
representations. Both ofthese can be implemented efficiently usingthe d.f.t.
24.15.1 Linearconvolution
The linear convolution of two real sequences f[n] andg[n] is another sequence,h[n]
say, which we denote by f ∗g, which isdefined as follows:
Linear convolution of f[n] andg[n]:
∞∑
h[n]=f∗g= f[m]g[n−m] forn=...−3,−2,−1,0,1,2,3,...
m=−∞
Notice the similaritybetween this definition and thatof convolution defined inthe context
of the continuous Fourier transform in Section 24.8. Notice also that in this formulathesequencegisfoldedand
shifted.Thiswillbeillustratedintheexample which
follows.
Frequently the sequences being considered will be finite.
Assume now that f[n] is a finite sequence ofN 1
terms, so that all terms other than
f[0], f[1],..., f[N 1
−1] arezero.
Suppose also that g[n] is a finite sequence of N 2
terms, with all terms other than
g[0],g[1],...,g[N 2
−1] being zero.
Itcanbeshownthatthesequenceh[n]willhavelengthN 1
+N 2
−1,andtheconvolution
sum simplifies tothe following:
The linear convolution of the two finite sequences f[n] andg[n] isdefined as
n∑
h[n]=f∗g= f[m]g[n−m] forn=0,1,2,3,...,(N 1
+N 2
−2)
m=0