Page 302 - Biomedical Engineering and Design Handbook Volume 2, Applications

280 DIAGNOSTIC EQUIPMENT DESIGN

10.3.3 Discrete and Finite Data Sampling
The data contained in a digitally recorded image is an ordered finite array of discrete values of inten-
sity (grayscale). To manipulate this data, the continuous integrals defining the Fourier transform and
convolution must be expressed as approximating summations. For a series of discrete samples x(nt)
of a continuous function x(t), a representation in the frequency domain can be written as
∞
Xw) = ∑ x n )exp(− iwn ) (10.44)
τ
τ
(
(
−∞
where t is the sample interval. For a discrete function x(0), x(2t), . . . , x((N − 1)t) of N elements,
Eq. (10.44) becomes
⎛ 1 ⎞ 1 N −1 ⎧ ⎡ ⎛ 1 ⎞ ⎤ ⎫
Xu = ∑ xnτ)exp ⎨ − i2π u ( nτ)⎬ (10.45)
(
⎨
⎝ Nτ ⎠ N ⎢ ⎣ ⎝ Nτ ⎠ ⎥ ⎦
n =0 ⎩ ⎭
where the substitution w = u(1/Nt) has been made, with u = 0, 1, 2, . . . , N − 1.
This in effect provides samples X , X , . . . , X of the continuous function X(w) at intervals 1/Nt
0 1 N−1
of the frequency w. Hence, for the discrete function x , x , . . . , x , we can express the Fourier
0 1 N−1
transform as
N−1
2 ⎞
1
X = ∑ x exp ⎧ ⎨ i − ⎛ π un ⎬ ⎫ (10.46)
u
N n=0 n ⎩ ⎝ N ⎠ ⎭
for u = 0, 1, 2, . . . , N − 1. Hence, a sampling interval of t in the t domain is equivalent to a sampling
interval of 1/Nt in the frequency domain. The inverse of Eq. (10.46) is given by
N−1 ⎛ π ⎫
2 ⎞
X = ∑ X exp ⎧ i ⎨ un ⎬ (10.47)
u
n
n=0 ⎩ ⎝ N ⎠ ⎭
for n = 0, 1, 2, . . . , N − 1. The finite Fourier transform Eq. (10.46) and its inverse Eq. (10.47) define
sequences that are periodically replicated, according to the expressions X = X , x = x , for
N,m+k k N,m+k k
all integer values of m.
To determine the convolution for a discrete sample we follow Eq. (10.43) and find the product of
two finite Fourier transforms Z = X Y and take the inverse of the result, according to
u u u
N−1 ⎧ ⎛ π ⎫
2 ⎞
⎬
z = ∑ exp i ⎨ un X Y (10.48)
n ⎝ N ⎠ u u
n=0 ⎩ ⎭
Substituting from Eq. (10.46) and rearranging the order of summation gives
1 N−1 N−1 N−1 ⎧ ⎛ π ⎫
2 ⎞
Z = 2 ∑ ∑ xy ∑ exp ⎨ i − un up uq ⎬ (10.49)
−
−
n p q ⎝ N ⎠
N ⎩ ⎭
p=0 q=0 u u=0
Because of the replication of sequences, the convolution [Eq. (10.49)] can be simplified to
N−1
1
z = ∑ x y
−
n n q q (10.50)
N
q=0
for n = 0, 1, 2, . . . , N − 1.
With discrete data sampling, the interval t between data points determines how well the original
signal is modeled. The choice of t is referred to the Nyquist criterion, which states that a signal must