This module introduces image processing, 2D convolution, 2D sampling and 2D FTs.
Image processing
Linear shift invariant systems
is LSI if:
-
for all images
,
and scalar.
-
LSI systems are expressed mathematically as 2D convolutions:
where
is the 2D impulse response (also called the
point spread function ).
2d fourier analysis
where
is the 2D FT
and
and
are frequency variables in
and
.
2D complex exponentials are
eigenfunctions for 2D LSI systems:
where
is the 2D Fourier transform of
evaluated at frequencies
and
.
Inverse 2d ft
2d sampling theory
We can
sample the height of the surface
using a 2D impulse array.
where
is sampled image in frequency
2D FT of
is a 2D impulse array in frequency
Nyquist theorem
Assume that
is bandlimited to
,
:
If we sample
at spacings of
and
, then
can be perfectly recovered from the samples by
lowpass filtering: