Download spatial frequency domain

The imaging problem
object
imaging optics
(lenses, etc.)
image
MIT 2.71/2.710 Optics
10/27/04 wk8-b-1
The imaging problem
Illumination
(coherent vs
incoherent)
object
imaging optics
(lenses,
etc.)
free space
MIT 2.71/2.710 Optics
10/27/04 wk8-b-2
free space
image
The imaging problem
Illumination
(coherent vs
incoherent)
object
image
(spatial) linear shift-invariant system
MIT 2.71/2.710 Optics
10/27/04 wk8-b-3
The imaging problem
object
image
(spatial) linear shift-invariant system
MIT 2.71/2.710 Optics
10/27/04 wk8-b-4
Our approach
• Today:
– linear shift invariant (LSI) systems in the space/spatial
frequency domains
– mathematical properties of Fourier transforms
• Monday:
– free space propagation: Fresnel and Fraunhofer
diffraction
• Wednesday:
– examples of Fraunhofer diffraction: amplitude and
phase diffraction gratings
– wave description of light propagation through a lens
– Fourier transformation and imaging using lenses
MIT 2.71/2.710 Optics
10/27/04 wk8-b-5
Spatial filtering
MIT 2.71/2.710 Optics
10/27/04 wk8-b-6
Spatial frequency representation
space domain
3 sinusoids
MIT 2.71/2.710 Optics
10/27/04 wk8-b-7
Fourier domain
(aka spatial frequency domain)
Spatial frequency removal
space domain
2 sinusoids (1 removed)
MIT 2.71/2.710 Optics
10/27/04 wk8-b-8
Fourier domain
(aka spatial frequency domain)
From space to spatial frequency:
2D Fourier analysis
Can I express an arbitrary g(x,y)
as a superposition of
sinusoids?
MIT 2.71/2.710 Optics
10/27/04 wk8-b-9
... etc. ...
Spatial frequency representation
space domain
g(x,y)
MIT 2.71/2.710 Optics
10/27/04 wk8-b-10
Fourier domain
(aka spatial frequency domain)
Low-pass filtering
removed high-frequency content
space domain
MIT 2.71/2.710 Optics
10/27/04 wk8-b-11
Fourier domain
(aka spatial frequency domain)
Band-pass filtering
removed high-and low-frequency content
space domain
MIT 2.71/2.710 Optics
10/27/04 wk8-b-12
Fourier domain
(aka spatial frequency domain)
Example: optical lithography
Original nested Ls
original pattern
(“nested
L’s”)
MIT 2.71/2.710 Optics
10/27/04 wk8-b-13
mild
low-pass filtering
Notice:
(i) blurring at the edges
(ii) ringing
Example: optical lithography
Original nested Ls
original pattern
(“nested
L’s”)
MIT 2.71/2.710 Optics
10/27/04 wk8-b-14
severe
low-pass filtering
Notice:
(i) blurring at the edges
(ii) ringing
The 2D Fourier integral
(aka inverse Fourier transform)
superposition
sinusoids
complex weight,
expresses relative amplitude
(magnitude & phase)
of superposed sinusoids
MIT 2.71/2.710 Optics
10/27/04 wk8-b-15
The 2D Fourier integral
The complex weight coefficients G(u,v),
Aka Fourier transform of g(x,y)
are calculated from the integral
(1D so we can draw it easily ... )
MIT 2.71/2.710 Optics
10/27/04 wk8-b-16
2D Fourier transform pairs
Image removed due to copyright concerns
MIT 2.71/2.710 Optics
10/27/04 wk8-b-17
(from Goodman,
Introduction to
Fourier Optics,
page 14)
Space and spatial frequency
representations
SPACE DOMAIN
2D Fourier transform
2D Fourier integral
aka
SPATIAL FREQUENCY
DOMAIN
MIT 2.71/2.710 Optics
10/27/04 wk8-b-18
inverse 2D Fourier transform
Fourier transform properties /1
•Fourier transforms and the delta function
•Linearity of Fourier transforms
if
and
then
for any pair of complex numbers
MIT 2.71/2.710 Optics
10/27/04 wk8-b-19
Fourier transform properties /2
Let
Shift theorem (space →frequency)
Shift theorem (frequency →space)
Scaling theorem
MIT 2.71/2.710 Optics
10/27/04 wk8-b-20
Fourier transform properties /3
Let
and
Let
Convolution theorem (space →frequency)
Let
Convolution theorem (frequency →space)
MIT 2.71/2.710 Optics
10/27/04 wk8-b-21
Fourier transform properties /4
Let
and
Let
Correlation theorem (space →frequency)
Let
Correlation theorem (frequency →space)
MIT 2.71/2.710 Optics
10/27/04 wk8-b-22
2D linear shift invariant systems
output
input
transform
multiplication with transfer function
MIT 2.71/2.710 Optics
10/27/04 wk8-b-23
Inverse Fourier
transform
Fourier
convolution with impulse response
2D linear shift invariant systems
SPACE DOMAIN
convolution with impulse response
transform
SPATIAL FREQUENCY DOMAIN
Inverse Fourier
multiplication with transfer function
MIT 2.71/2.710 Optics
10/27/04 wk8-b-24
output
transform
Fourier
input
2D linear shift invariant systems
convolution with impulse response
transform
multiplication with transfer function
pair
Inverse Fourier
are
MIT 2.71/2.710 Optics
10/27/04 wk8-b-25
output
transform
Fourier
input
Sampling space and frequency
pixel size
space
domain
field size
Nyquist
relationships:
MIT 2.71/2.710 Optics
10/27/04 wk8-b-26
frequency
resolution
spatial
frequency
domain
The Space–Bandwidth Product
Nyquist relationships:
from space → spatial frequency domain:
from spatial frequency → space domain:
: 1D Space–Bandwidth Product (SBP)
aka number of pixels in the space domain
MIT 2.71/2.710 Optics
10/27/04 wk8-b-27
SBP: example
space domain
MIT 2.71/2.710 Optics
10/27/04 wk8-b-28
Fourier domain
(aka spatial frequency domain)