4 Nonl Imnge Pvrssn

1.2 IMAGE SAMPLING dinerete npatial

lowation.
AcContiuoma
only at a

Sampling is the pmesw ofmeasuring the brightnews tntormatofnanplina pointa n the pine
linerete pridd
C n
benanmpled using a
each and every value
val of
1.2.1 Theory of 2D Sampling in delined at
hia ia mathemat
netion f(. ) ivatanta. Thie
that
Let f(. ) represent the amlag iage, It y)at speoltio
neana

and y. The diserete version of / )is

obtahned by dofining /(N.
(.
eally represented as
/m, n)/(mAr, nAP)
defined lor all valuesa
not
noted that /'(m,
n) is
constants. It is to be
real of his analu
c e a r a n d Ar are positivesampling interval On taking the lourier
transform

Ar are known as
mand a. Here Ar and analog signal /t.
P). wlhere
with the
The 2D sampling
starts
whieh is
denoted by P21, 82,)
Sgnal, we get the spectrum off(r.,)
(.2)

represent the Fourier

basis.
In
and e
Eq. (1.2) e4 Fourier transtorm, we get
On taking inverse
(1.3)
4r -
specificd only at specific instants
the values are
of the analog signal f(x, p),
When we perform sampling
mathematicaly as
which is represented (4
Sm, n) f(mAr, nAy)
=

transform of this sampled signal, we get

On taking the inverse Fourier

F,2,)eMmarnA" a,d,
Sm, n)=
define
For a discrete signal, we

and

where w and w are expressed in radians.

From Eq. (1.6), we h a v e = . Differentiating S, We g

 Introduction to Image-Processing
System 5

du= dS
Ar
Similarly from Eq. (1.7), we get
(1.8)

duw-dSl
Ay (1.9)
Substituting Eq. (1.8) and (1.9) in Eq. (1.5), we get

Sm, n)=- emjan duj du

0-00 Ar Ay (1.10)

Sm, n)= 2jmi" d duz

-00-O0 (1.11)
The signalf (m, n) is discretised, hence the integral can be
the entire (wj, w2) plane can be broken replaced by summation. The double integral over
into an infinite series of
integrals, each of which is over a square of
area 4T. The range of w and w is given by T+2tk
Incorporating this concept in Eq. (1.11), we get S<T+2tk, -T +2tk2 w2 <T+ 2mtk

Som,n) LF 2ejim u du du2 (1.12)

SQk.k2)
In order to change the limits of
the integral so as to remove the dependence of the limits of integration on
k and k w is replaced by wj -2tkj and w, is replaced by w -2Tk2. On including this modification in
Eq. (1.12), we get

fm, n)=2J J w-2Tk w-27Ttkzj2nk)m 27k,)J" du dw

AxAy Ay (1.13)

(1.14)
2

The exponential term in Eq. (1.14) is equal to one for al values ofthe integer variables m, k, n and k. Hence
Eq. (1.14) reduces to

w-2Tk w-2Tk2|j ej du, du (1.15)

-T
Ay
The above equation represents the inverse Fourier transform of F(w, w,) where

u-2mk -2k, (1.16)

Ay
 F(,)AxAy 2nTk 2Tk (1.17)
Ay
The concepts discussed so far can be approached in another way, i.e., by multiplying the analog image with
a two-dimensional comb function. The 2D comb function is a rectangular grid of points and 1s illustrated in
Fig. .5. The spaces between successive grid points in thex and y direction are x and ay respectively. The
three-dimensional view ofthe comb function is shown in Fig. 1.6. The 2D comb function, which 1s otherwise
known as bed-of-nail function, is defined as

combxr, y, Ar, Ay) = 2 6-kax,y-k,4) (1.18)

k=-00 k2=-00

Ayt...

Fig. 1.5 2D view of a comb function Fig. 1.6 Three-dimensional view ofacomb function

After multiplying the analog image f(x, y) with the 2D comb function, we get the discrete version of the
analog image which is givén by

S(m, n) =fx,y) x
comb(x, y, Ax, Ay) (1.19)

fm, n) =
2 shA, k,Ay)8 (x-kAx, y-k,Ay) (1.20)
kcok-0o
We know that convolution in spatial domain is
For the purpose of analysis in the
equal to multiplication in the frequency domain and vice versa.
frequency domain, let us take the Fourier transform of the input analog
image and the 2D comb function.
The Fourier transform
of the signalf(r, y) is f(21, 22).
The Fourier transform of 2D comb
function is another comb function which is represented by
comb(1, N,) =
FT(comb(x, y Ax, Ay) (1.21)

comba,2,) = p=-00 q=-o

s 9 (1.22)
 Introduction to
Image-Processing System 7

comb(,22)= comb|,2 1
(1.23)
Ay
Now the spectrum of the 2D comb function is convolved with the
given by F1, D2) ® comb (1, 2). spectrum of the analog image which is

F(1, )= F{24, 2) ® comb (Q4, 92) (1.24)

Substituting Eq. (1.22) in Eq. (1.24), we get

FC ) F (9,.,) 2 2sn,-9 (1.25)

p=-0q=-o0
Upon convolving the spectrum of the signal with the spectrum of the comb
function, we get

Fly,) Ë F%-k,-1)
Ax Ay ke-ol=-00
p=-0 q-00
k1 (1.26)

As summation is a linear operator, interchanging the order of summation, we get

F(, w)= T
Ar Ay p=-0q=-0 k
F-k, 22 -1)6 (1.27)

(1.28)
Ay
Equation (1.28) resembles Eq. (1:17).

1.2.2 Retrieving the image from its Samples

We know that discreteness in one domain leads to periodicity in another domain. Hence sampling in the spa-
tial domain leads to periodic spectrum in the fequency domain, which is illustrated in Fig. 1.7.
In order to retrieve the original image from the sampled spectrum, the following conditions have to be satisfied.

s2w0 (1.29)
where w,, = 1 and 2wo is the bandwidth ofthe spectrum in the w direction.
Ar
Similarly,
wys2uo (1.30)

where = a n d 2w,o is the bandwidth of the spectrum in the u direction. The condition given in
where wys
Ay
Eqs. (1.29) and (1.30) implies that the sampling frequency should be greater than twice the maximum signal
frequency, which is generally termed the sampling theorem. Here, 2w,0 and 2w, are called Nyquist rates.
8 Digital mage Processng

The traaler huneton

he
low-pass filter is normaly employed in orderto oxtrnet the dkonired wpectrun, 0
low-pass filter is given as tollows

H(w, w,)
( j ,w,) roglon of support
G (1.31)
otherwiso

can bo obtained
trom tne Nnngpled
ne
of support is indicated as
region 1.7,The continuous image
in Fi, IN gIvoi us
pectrum by multiplying the sampled spectrum with the low pusa filter wlhich
(1.32)
w)
P(w,w)nU(1, wz)x P(j,
as
By takmg inverse Fourier transtorm, wo got the continuous imuge
(.33)

AX

L wys
AY

2wyo

w
wxs
x0

Fig.1.7 Periodicspectrum ofthe sampled image

Criterion
1.2.3 Violation of Sampling Violation of
of violation of the sampling criterion.
let us discuss the consequences due to
In this section, leads to aliasing. Aliasing basically occurs
criterion givenin Eq. (1.29) and (1.30)
sampling
direction-
under-sampling.
Violation sampling criterion given in Eq. (1.29) leads to overlapping ofthe spectrum in thew
of
which is illustrated 1.8. Here, w <
in Fig.
whereas
20 w,s 2u,o