Image Processing Reference
InDepth Information
rect(
x
) = 1 when 
x

≤
1/2
= 0 when 
x

>
1/2
1/2
1/2
rect(
x
/
b
) = 1 when 
x
/
b

≤
1/2
= 0 when 
x
/
b

>
1/2

b
/2
b
/2
rect(
xb
) = 1 when 
xb

≤
1/2
= 0 when 
xb
 > 1/2
1/(2
b
)
1/(2
b
)
rect[(
x

x
′)/
b
]
x
′
Figure A.1
Examples of rect notation.
Frequency domain
Time or spatial domain
F.T.
u
/
a
1/2
a
1/2
a
Sync function
Figure A.2
Example of Fourier transform of rect function.
Convolution and Correlation
F
=
F
{
f
} and
G
=
F
{
g
}, then
∞
∫
1.
hx
()
=
fxgx x
()()
d
is projection of
g
on
f
−∞
∞
∫
2.
hx
()
=
fx gx
()(
′
− ′
xx
)
d ′ is convolution of
g
with
f.
−∞
∞
∫
3.
hx
()
=
fx gx
()(
′
′ −
xx
)
d ′ is the correlation of
g
with
f
.
−∞
We can think of convolution as a sliding projection but notice that
g
(
x
)
becomes
g
(−
x
) in the integral and so is mirrored about the
x
origin. If
H
=
F
{
h
} =
FG
, we have case (ii) in the Fourier domain. For case (iii),
H
=
FG
*
where the asterisk (*) denotes complex conjugate. A graphical illustration of
correlation can be found in Figures A.4 and A.5.
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