Transcript ECE_427_2D_Fourier_a.. - University of Illinois at Chicago

Separable functions
A function of more than one variable is separable with respect to a certain
coordinate system if it can be written as a product of functions, where each
function depends on only one variable.
Cartesian coordinates:
g ( x y )  g x ( x) g y ( y )
Polar coordinates:
g (r )  gr (r ) g ( )
Separable functions are easier to deal with and, in particular, the Fourier transform
may be written as:

F[ g ( x y)]   g ( x y)e

 j 2  f x x  f y y 
dx dy 


 j 2 f y y
 j 2 f x x



  g x ( x )e
dx  g y ( y)e
dy  
 
  

 Fx [ g x ( x)]Fy [ g y ( y)]
(1.22)
University of Illinois at Chicago
ECE 427, Dr. D. Erricolo
In polar coordinates, the expression of the Fourier transform of a separable
function is not as straightforward as it is for rectangular cartesian coordinates.
However, it is possible to show that:
G(   )  F[ g (r )]  k  ck ( j )k e jk H k ( gr (r )) (1.23)

where:
ck 
1
2

2
0
g ( )e jk d
(1.24)
and H k () is the Hankel transform operator of order k, defined by:
H k ( gr (r ))  2 
2
0
gr (r ) J k (2 r  )r dr
(1.25)
University of Illinois at Chicago
ECE 427, Dr. D. Erricolo
Functions with circular simmetry
These functions represent a special class of separable functions in polar
coordinates that depend only one the variable r, i.e.
g (r   )  g r (r )
(1.26)
Many problems in optics have this kind of simmetry, hence we consider more in
detail the expression of the Fourier transform of functions with circular simmetry.
We start from:
G( f x  f y )   


g ( x y)e
 j 2 ( f x x  f y y )
dx dy
(1.27)
then we consider the polar representation of both the (x,y) and the (fx,fy) planes:
r  x2  y 2 

f x2  f y2 
  arctan xy 
  arctan f 
fy
x
x  r cos  
y  r sin 
(1.28)
f x   cos  
f y   sin 
University of Illinois at Chicago
ECE 427, Dr. D. Erricolo
Introducing the change of variables into our original transform, we obtain:
G0 (    )  
2
0
d 

0

rgr (r )e j 2 r  [cos cos sin sin ] dr (1.29)
G0 (    )   dr rgr (r ) 
0
0
J 0 (a) 
Bessel function identity:
And:
G0 (    )  2 

0
2
1
2

2
0
de j 2 r  cos(  )
e ja cos(  ) d
dr rgr (r ) J 0 (2 r  )  G0 (  )
(1.30)
(1.31)
Fourier-Bessel
transform OR Hankel
transform of zero
order
The Fourier transform itself is circularly simmetric.
It is also possible to show that the inverse Fourier transform is:
(1.32)

gr (r )  2  d  G0 (  ) J 0 (2 r  )
0
(1.33)
University of Illinois at Chicago
ECE 427, Dr. D. Erricolo
We observe that:
1) There is no difference between direct and inverse Fourier-Bessel transform
operation for circularly symmetric functions.
2) If B is the symbol that denotes the transform operation, then:
B B 1 ( g r (r ))   B 1 B ( g r (r ))   B B ( g r (r ))   g r (r ) (1.34)
3) The similarity theorem becomes:
B  g r ( r )   12 G0
 


(1.35)
University of Illinois at Chicago
ECE 427, Dr. D. Erricolo
Useful functions
We will frequently use the following functions:
 x  12
1

rect ( x)   12
 x  12
0 otherwise

1 x0

sgn( x)   0 x  0
1 x  0

comb  comb( x)  n  ( x  n)
sinc( x)  sin x x
 x  1
1  x 
triangle  ( x)  
otherwise
 0
1


2
2
circle  circ ( x  y )   12
0

x2  y 2  1
x2  y 2  1
otherwise

The Fourier transform of two-dimensional functions that are separable and include
some of the functions listed above are given in table 2.1 of the textbook.
University of Illinois at Chicago
ECE 427, Dr. D. Erricolo
Spatial frequency
One interpretation of the inverse Fourier transform considers the superposition of
the elementary waves of the form: exp( j 2 ( f x x  f y y ))
that are defined over all the (x,y) plane.
However, in some practical cases we observe images that contain parallel lines and
certain fixed spacing. Recalling our previous interpretation of the direction of an
elementary wave and the spacing between wavefront lines, we are tempted to
associate the values of (fx,fy) to a certain region of an image. This association
occurs with the idea of local spatial frequencies.
Consider a complex-valued function of the form:
g ( x y)  a( x y )e j ( x y )
(1.36)
where a(x,y) is a real non-negative amplitude and x,y) is a real phase
distribution.
University of Illinois at Chicago
ECE 427, Dr. D. Erricolo
We assume that a(x,y) is a slowly varying function and define local spatial
frequency the pair (flx,fly) given by
f lx 
1  x  y
2 x
fly  21
 x y
y
(1.37)
For example, if we apply this definition to an elementary wave
g ( x y )  exp  j 2 ( f x x  f y y ) 
we obtain f lx  f x 
(1.38)
fly  f y
As a second example, we consider a space-limited quadratic-phase exponential
function
g ( x y)  exp  j ( x 2  y 2 )  rect
defined all over the
(x,y) plane
 
x
2 Lx
rect
 
y
2 Ly
(1.39)
Rectangle of
dimensions 2 Lx  2 Ly
University of Illinois at Chicago
ECE 427, Dr. D. Erricolo
In this case, the local spatial frequencies depend on the location. In fact, they are
different from zero only inside a rectangle of dimensions 2 Lx  2 Ly
and correspond to:
flx   xrect 2 xLx 
fly   yrect 2 Ly y
(1.40)
If we computed the exact spectrum of g(x,y), we would find that the spectrum is
almost flat inside the rectangle ( Lx  Lx )  ( Ly  Ly ) and almost zero outside. So, in
this case, the local spatial frequencies provide a good approximation of where
significant values of the spectrum are.
In general:
1) good agreement is found if f(x,y) is sufficiently slow-varying, i.e. it can be
well approximated with only the first three terms of its Taylor series
(constant value plus the two first partial derivatives).
2) local spatial frequencies of a coherent optical wavefront correspond to the
ray directions of the geometrical optics description of that wavefront.
University of Illinois at Chicago
ECE 427, Dr. D. Erricolo