Transcript SYDE 575: Introduction to Image Processing

SYDE 575: Introduction to
Image Processing
Spatial-Frequency Domain:
Continuous Fourier Transform
Textbook: Chapter 4 (Spatial-Frequency)
Blackboard Notes
Fourier Transform

A signal can be expressed as a weighted sum of sines
and cosines
Source: Gonzalez and Woods
Euler’s Formula

Euler's formula
e = cos q + j sin q
jq

Based on this, summation of sines and cosines
can be expressed as a function of a complex
exponential
Signal Representation
• Any signal f(x,y) can be expressed as a
superposition of unit impulses using the
sifting property
f(x,y) =   f(s,t) d(x-s,y-t) ds dt
• Sketch
Relationship to Convolution
g(x,y) = T[ f(x,y) ]
=T[
  f( s,t ) d( x - s, y - t ) ds dt ]
=   f( s,t ) T[ d( x - s, y - t ) ] ds dt
=   f( s,t ) h( x - s, y - t ) ds dt
= f(x,y) * h(x,y)
Transformation -> Convolution
• Transforming a function f(x,y) then involves
transforming the impulses which are a
function of (x,y) which leads to convolution
• Example: 1-D continuous exponential PSF
– What is the step response? Edge blur
– Can we deblur?
Deconvolution
• We are able to smooth a signal using an
exponential smoothing filter
• If we know the blur model, how can we find
the original signal f(x,y) for a linear system
(where ‘*’ represents convolution):
g(x,y) = f(x,y) * h(x,y)
• Deconvolution is, in general, not easy to
perform
Inverse System
• As before, to undo the effects of an
undesirable affect to the image, generate an
inverse system using
h(x,y) * h1(x,y) = d(x,y)
• Still involves deconvolution! Why is
deconvolution difficult?
– Each output generally has contribution from many
inputs
Easier Approach
• We prefer to represent f(x,y) with components
that are not “smeared” by the LSI (linear, shift
invariant) system i.e.,
{ fk,l(x,y) } s.t. S fk,l fk,l(x,y) = f(x,y) and
T [ fk,l(x,y) ] = lk,l fk,l(x,y) so that
fk,l(x,y) * h(x,y) = lk,l fk,l(x,y)
• Defined as eigenfunctions of LSI systems
Complex Exponentials
• Complex exponentials, ej2pux, will not be altered by a
LTI system
• ‘u’ is the frequency of the complex exponential
– 1-d: cycles per unit time
– 2-d: cycles per unit distance or per degree of visual angle
ej2pux * h(x) =  ej2pu(x-s) h(s) ds
=[

h(s) e-j2pus ds ] ej2pux
= H(u) ej2pux
where H(u) represents the continuous Fourier
transform (FT) of the function h(x)
Continuous Fourier Transform
Forward
F(u) =  f(x) e-j2pux dx
Inverse
f(x) =  F(u) e j2pux du
• F(u) is complex i.e., F(u) = | F(u) | e jf
where | F(u) | is the magnitude and f is the
phase
Fourier: Magnitude and Phase

Magnitude
1/ 2
F (u , v ) = éëR (u ,v ) + I (u ,v )ùû
2
2
R = Re( F (u , v )), I = Im(F (u ,v ))

Phase
f(u,v) = tan-1( I(u,v) / R(u,v) )

Power Spectrum
P (u , v ) = F (u ,v )
2
2D Continuous Fourier
Transform
Forward
F(u,v) =  f(x,y) e-j2p(ux + vy) dx dy
Inverse
f(x,y) =  F(u,v) e j2p(ux + vy) du dv
Important Property
• If convolution is performed in the time (1d) or
spatial (2d) domain:
1d:
2d:
g(x) = f(x) * h(x)
g(x,y) = f(x,y) * h(x,y)
the transformed functions can be multiplied in
the frequency (1d) or spatial frequency (2d)
domain:
1d:
G(u) = F(u) H(u)
2d:
G(u,v) = F(u,v) H(u,v)
Examples
• Example 1: 1-d local average over window W
• Example 2: exponential blur
• Example 3: Gaussian blur