Page 298 - Biomedical Engineering and Design Handbook Volume 2, Applications

276 DIAGNOSTIC EQUIPMENT DESIGN

The principle of linearity prescribes that the weighted input ag (u) will produce the output
in
(1)
(2)
ag out (u), and the sum of two weighted inputs ag (u) + bg (u) will produce the output ag (u) +
(1)
in
out
in
bg (2) (u), for any real numbers a and b. We can further propose that the linear system be space invari-
out
ant so that changing the position of the input merely changes the location of the output without alter-
ing its functional form. Hence, the image forming system can be treated as a linear superposition of
the outputs arising from each of the individual points on the object. An estimate of the effect of the
operation of the linear system on the contribution from each point will provide a measure of the
performance.
If the operation of the linear system is symbolically represented as L{}, the relation between the
input and output for independent variable u can be written as
g () = L g u()} (10.25)
{
u
out in
From the properties of the Dirac d function, which represents a unit area impulse, defined as
⎧ ∞ u = 0 ∞
u
δ() = ⎨ and ∫ δ()udu = 1
⎩ 0 otherwise −∞
∞
−
δ
(
we have the sifting property ∫ −∞ gu() ( u u ) du = gu ). Applying this property to Eq. (10.25)
0
0
gives
∞
g () = L {∫ −∞ g u ) ( − u du′ } (10.26)
u′
δ
( ′
u
)
out
in
which expresses g (u) as a linear combination of elementary d functions, each weighted by a
out
number g (u′). From the linearity principle, the system operator L{} can be applied to each of the
in
elementary functions so that Eq. (10.26) can be written as
∞
g () = ∫ −∞ g u L{ ( − u du′ (10.27)
( ′
δ
u′
)
)
}
u
out
in
The quantity L{d(u′− u)} is the response of the system, measured at point u in the output space, to
a d function located at the point u′ in the input space, and is called the impulse response.
For image forming systems the impulse response is usually referred to as the point-spread func-
tion (PSF). In this case we consider the response at vector point r in the image space, to an irradi-
ance distribution Φ (r′) located in object space. An element dr¢ located at r′ will emit a radiant flux
0
of Φ (r¢) dr′, which will be spread by the linear operation L{} over a blurred spot defined by the
0
function p(r¢; r). The flux density at the image point r, from the object point r′, will be given by
dΦ (r) = p(r′; r)Φ (r) dr′, so that the contribution from all points in the object space becomes
i 0
∞
r′
r′
Φ () = ∫ Φ ( ) p( ; ) dr′
r
r
i −∞ 0 (10.28)
where p(r′; r) is the point-spread function. If the system is space invariant, a point source moved over

r
the object space r¢ will produce the same blurred spot at the corresponding locations in the image
space. In this case the value of p(r′; r) depends only on the difference between the location in the
object space and the image space, namely (r′− r), so that p(r′; r) = p(r′− r) and Eq. (10.28) becomes
∞
r′
Φ () = ∫ Φ ( ) p( ′ − ) r dr′ (10.29)
r
r
i −∞ 0

Equation (10.29) is known as the two-dimensional [i.e., r = (x, y)] convolution integral, written as
∞ ∞
, ′
′
−
g x y) = ∫ −∞ ∫ −∞ f xy h x xy ydx dy′ (10.30)
) ′
−
, ′
( ′
(,
(
)