This module introduces image processing, 2D convolution, 2D sampling and 2D FTs.
Image processing
Images are 2D functions
f
x
y
Linear shift invariant systems
H is LSI if:
H
1
f
1
x
y
2
f
2
x
y
H
f
1
x
y
H
f
2
x
y for all images
f
1 ,
f
2 and scalar.
H
f
x
x
0
y
y
0
g
x
x
0
y
y
0 LSI systems are expressed mathematically as 2D convolutions:
g
x
y
∞
∞
∞
∞
h
x
y
f
where
h
x
y is the 2D impulse response (also called the
point spread function ).
2d fourier analysis
u
v
y
∞
∞
x
∞
∞
f
x
y
u
x
v
y where
is the 2D FT
and
u and
v are frequency variables in
x
u and
y
v .
2D complex exponentials are
eigenfunctions for 2D LSI systems:
∞
∞
∞
∞
h
x
y
u
0
v
0
∞
∞
∞
∞
h
u
0
x
v
0
y
u
0
x
v
0
y
∞
∞
∞
∞
h
u
0
v
0
where
∞
∞
∞
∞
h
u
0
v
0
H
u
0
v
0
H
u
0
v
0 is the 2D Fourier transform of
h
x
y evaluated at frequencies
u
0 and
v
0 .
g
x
y
h
x
y
f
x
y
∞
∞
∞
∞
h
x
y
f
G
u
v
H
u
v
u
v
Inverse 2d ft
g
x
y
1
2
2
v
∞
∞
u
∞
∞
G
u
v
u
x
v
y
2d sampling theory
Think of the image as a 2D surface.
We can
sample the height of the surface
using a 2D impulse array.
Impulses spaced
x apart in the horizontal direction and
y in the vertical
f
s
x
y
S
x
y
f
x
y where
f
s
x
y is sampled image in frequency
2D FT of
s
x
y is a 2D impulse array in frequency
S
u
v
multiplication in timeconvolution in
frequency
F
s
u
v
S
u
v
u
v
u
v is bandlimited in both the horizontal and vertical
directions.
periodically replicated in
(
u
,
v
) frequency plane
Nyquist theorem
Assume that
f
x
y is bandlimited to
B
x ,
B
y :
If we sample
f
x
y at spacings of
x
B
x and
y
B
y , then
f
x
y can be perfectly recovered from the samples by
lowpass filtering:
ideal lowpass filter, 1 inside rectangle, 0 outside
Aliasing in 2d