Fundamentals of Digital Image Processing - Control of Color Imaging Systems: Analysis and Design - page 68

There exist two approaches to sampling rate conversion. The first technique is

continuous domain processing. In this approach, rst

reconstructed and then it is resampled at the new desired rate. This approach is
theoretically possible but is not practical. The second approach is digital processing,
which implies that the image is processed in the digital domain without conversion to
analog or continuous domain. In discrete domain, the sampling rate can be changed
by a factor of

the continuous image is

a only if it is a rational number (i.e., ratio of two integers), that is

P
Q

a ¼ ( : )

2 104

In this case, we first upsample (interpolate) the image by a factor of P and then

downsample (decimate) by a factor of Q.We first consider the upsampling process.

Here, this process will be presented for 1-D signals. Extension to 2-D signals is
straightforward.

2.7.2 U PSAMPLING BY F ACTOR OF P
To upsample a 1-D signal by a factor of P, we insert P 1 zeros between the signal

samples and LPF the resulting signal. The block diagram of an upsampler is shown
in Figure 2.31.
The output signal before low-pass

filtering is related to the input by

P n ¼

n

f 0, P, 2P, ...

y ( n ) ¼ ( 2 : 105 )

0 otherwise

The DTFT of y ( n ) is given by

e jn v ¼

1 1 1

n
P

y ( n ) e jn v ¼ f ( k ) e jkP v ¼ F ( P v) Y (v) ¼ f ( 2 : 106 )

n ¼1 n ¼1 1

is obtained by sampling a truly band-limited analog signal f a ( x )

Assume that f ( n ) at

the Nyquist rate of f s ¼

1

D x , that is

1

1
D x

j v

2 p k

F ( j v) ¼ F a ( 2 : 107 )

D x

k ¼1

P

If we now sample f a ( x ) at the rate of Pf s ¼

D x , to obtain g ( n )

, then the discrete time

Fourier transform of g ( n ) will be

y ( n )

g ( n )

f ( n ) H ( j ω)

P

FIGURE 2.31 Upsampling by a factor of P.