Image Processing Reference
In-Depth 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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