Functions on Two Variables
Extension to Two Variables
Dr. Praveen Sankaran
Department of ECE
NIT Calicut
January 16, 2013
Dr. Praveen Sankaran DIP Winter 2013
Functions on Two Variables
Outline
1 Functions on Two Variables
2D Impulse and Sifting
2D Discrete Fourier Transform
DFT Properties
Dr. Praveen Sankaran DIP Winter 2013
Functions on Two Variables2D Impulse and Sifting2D Discrete Fourier TransformDFT Properties
Outline
1 Functions on Two Variables
2D Impulse and Sifting
2D Discrete Fourier Transform
DFT Properties
Dr. Praveen Sankaran DIP Winter 2013
Functions on Two Variables2D Impulse and Sifting2D Discrete Fourier TransformDFT Properties
2D Impulse
δ (t,z) =
{∞ if t = z = 0
0 otherwise(1)
∫∞
−∞
∫∞
−∞
δ (t,z)dt dz = 1 (2)
Discrete Equivalent
δ [x ,y ] =
{1 x = y = 0
0 otherwise(3)
δ [x− x0,y − y0]?
Dr. Praveen Sankaran DIP Winter 2013
Functions on Two Variables2D Impulse and Sifting2D Discrete Fourier TransformDFT Properties
Sifting Property
Sifting Property∫∞
−∞
∫∞
−∞
f (t,z)δ (t− t0,z− z0)dt dz = f (t0,z0) (4)
∞
∑x=−∞
∞
∑y=−∞
f [x ,y ]δ [x− x0,y − y0] = f [x0,y0] (5)
Dr. Praveen Sankaran DIP Winter 2013
Functions on Two Variables2D Impulse and Sifting2D Discrete Fourier TransformDFT Properties
Outline
1 Functions on Two Variables
2D Impulse and Sifting
2D Discrete Fourier Transform
DFT Properties
Dr. Praveen Sankaran DIP Winter 2013
Functions on Two Variables2D Impulse and Sifting2D Discrete Fourier TransformDFT Properties
2D Continuous Fourier Transform
F (µ,ν) =∫
∞
−∞
∫∞
−∞
f (t,z)e−j2π(µt+νz)dt dz (6)
f (t,z) =∫
∞
−∞
∫∞
−∞
F (µ,ν)e j2π(µt+νz)dµ dν (7)
Dr. Praveen Sankaran DIP Winter 2013
Functions on Two Variables2D Impulse and Sifting2D Discrete Fourier TransformDFT Properties
Example
⇓
Dr. Praveen Sankaran DIP Winter 2013
Functions on Two Variables2D Impulse and Sifting2D Discrete Fourier TransformDFT Properties
2D Sampling
2D Impulse Train
s [t,z ] =∞
∑m=−∞
∞
∑n=−∞
δ [t−m∆T , z−n∆Z ] (8)
f (t,z)s [t,z ]⇒ sampled function.
Dr. Praveen Sankaran DIP Winter 2013
Functions on Two Variables2D Impulse and Sifting2D Discrete Fourier TransformDFT Properties
2D Sampling Theorem
1
∆T <1
2µmax(9)
and,2
∆Z <1
2νmax(10)
We have a problem here. A camera would have a �xed sampling
system that is trying to match all sorts of visual frequencies. Also
we want to limit the input signal to a speci�c spatial band. This
involves multiplying the signal with another function such as the
box we discussed earlier. But the box has wide frequency and
product in space equates to convolution in frequency. That means
the space limited signal would have wide frequency components in
practice.Dr. Praveen Sankaran DIP Winter 2013
Functions on Two Variables2D Impulse and Sifting2D Discrete Fourier TransformDFT Properties
Aliasing
Anyway for the sake of explaining it,
Dr. Praveen Sankaran DIP Winter 2013
Functions on Two Variables2D Impulse and Sifting2D Discrete Fourier TransformDFT Properties
Spatial Aliasing in Images
What we can do is to device ways to reduce the visual e�ect of
violating the sampling theorem.
Sampling is kept constant. The image taken is changed with
square size reduced from `a' through `d'. `c', `d' have square
size < one pixel and aliasing is visually evident.Dr. Praveen Sankaran DIP Winter 2013
Functions on Two Variables2D Impulse and Sifting2D Discrete Fourier TransformDFT Properties
Another Example
Dr. Praveen Sankaran DIP Winter 2013
Functions on Two Variables2D Impulse and Sifting2D Discrete Fourier TransformDFT Properties
2D DFT
F [u,v ] =M−1
∑m=0
N−1
∑n=0
f [m,n]e−j2π(um/M+vn/N) (11)
f [.] is a digital image of size M×N.
u = 0,1, · · ·M−1 and v = 0,1, · · ·N−1.
Inverse DFT
f [m,n] =1
MN
M−1
∑u=0
N−1
∑v=0
F [u,v ]e j2π(um/M+vn/N) (12)
m = 0,1, · · ·M−1 and n = 0,1, · · ·N−1.
Dr. Praveen Sankaran DIP Winter 2013
Functions on Two Variables2D Impulse and Sifting2D Discrete Fourier TransformDFT Properties
Outline
1 Functions on Two Variables
2D Impulse and Sifting
2D Discrete Fourier Transform
DFT Properties
Dr. Praveen Sankaran DIP Winter 2013
Functions on Two Variables2D Impulse and Sifting2D Discrete Fourier TransformDFT Properties
Spatial and Frequency Intervals
∆u =1
M∆T(13)
∆v =1
N∆Z(14)
We have seen this before.
Dr. Praveen Sankaran DIP Winter 2013
Functions on Two Variables2D Impulse and Sifting2D Discrete Fourier TransformDFT Properties
Periodicity
F [u,v ] = F [u+k1M,v ] = F [u,v +k2N] = F [u+k1M,v +k2N](15)
f [m,n] = f [m+k1M,n] = f [m,n+k2N] = f [m+k1M,n+k2N](16)
Dr. Praveen Sankaran DIP Winter 2013
Functions on Two Variables2D Impulse and Sifting2D Discrete Fourier TransformDFT Properties
Shifting
F [u−u0,v − v0]⇐⇒ f [m,n]e−j2π(u0m/M+v0n/N)
Multiplying f [m,n] by the modi�ed exponential term shifts the
data so that the origin is now at F [u−u0,v − v0]. Hence,
f [m,n] (−1)m+n⇐⇒ F [u−M/2, v −N/2]
Also,
f [m−m0,n−n0]⇐⇒ F [u,v ]e j2π(um0/M+vn0/N)
Dr. Praveen Sankaran DIP Winter 2013
Functions on Two Variables2D Impulse and Sifting2D Discrete Fourier TransformDFT Properties
Illustration & Shifting
f [m,n] (−1)m+n⇐⇒F [u−M/2, v −N/2]
Dr. Praveen Sankaran DIP Winter 2013
Functions on Two Variables2D Impulse and Sifting2D Discrete Fourier TransformDFT Properties
Even/Odd Functions
Any function w (x ,y) = we (x ,y) +wo (x ,y),
we (x ,y) =w (x ,y) +w (−x ,−y)
2(17)
wo (x ,y) =w (x ,y)−w (−x ,−y)
2(18)
Even
we (x ,y) = we (−x ,−y)
Symmetric
we (x ,y) =we (M− x ,N− y)
Odd
wo (x ,y) =−wo (−x ,−y)
Antisymmetric
wo (x ,y) =−wo (M− x ,N− y)
What about f = {2 1 1 1}?
Dr. Praveen Sankaran DIP Winter 2013
Functions on Two Variables2D Impulse and Sifting2D Discrete Fourier TransformDFT Properties
Some more properties
Dr. Praveen Sankaran DIP Winter 2013
Functions on Two Variables2D Impulse and Sifting2D Discrete Fourier TransformDFT Properties
How do the values shape up?
F [0,0] =M−1
∑m=0
N−1
∑n=0
f [m,n]e0⇒
F [0,0] = MN1
MN
M−1
∑m=0
N−1
∑n=0
f [m,n]⇒
F [0,0] = MNf̃ [m,n] (19)
A scaled average intensity value of the image gives the origin
point in the spectrum!
Dr. Praveen Sankaran DIP Winter 2013
Functions on Two Variables2D Impulse and Sifting2D Discrete Fourier TransformDFT Properties
Spectrum and Phase Angle
F [u,v ] = |F [u,v ]|e jφ(u,v)
Fourier Spectrum,
|F [u,v ]|=[R [u,v ]2 + I [u,v ]2
]1/2(20)
even
Phase angle,
φ (u,v) = arctan
([I [u,v ]
R [u,v ]
])(21)
odd
Dr. Praveen Sankaran DIP Winter 2013
Functions on Two Variables2D Impulse and Sifting2D Discrete Fourier TransformDFT Properties
Illustration - Spectrum Centering and Log Operation
Dr. Praveen Sankaran DIP Winter 2013
Functions on Two Variables2D Impulse and Sifting2D Discrete Fourier TransformDFT Properties
Illustration - Change e�ect in Spectrum
Dr. Praveen Sankaran DIP Winter 2013
Functions on Two Variables2D Impulse and Sifting2D Discrete Fourier TransformDFT Properties
Illustration - Phase changes
Though seemingly �lled with varying values that give no information
about the underlying image, phase is really important. The Fourier
spectrum and phase angle can be obtained by taking DFT of an
image. Now consider the case where two images are involved and
you mix the spectrum and phase information of the two for IDFT.
Dr. Praveen Sankaran DIP Winter 2013
Functions on Two Variables2D Impulse and Sifting2D Discrete Fourier TransformDFT Properties
Reconstruction Illustration - Phase Dominates!
Dr. Praveen Sankaran DIP Winter 2013
Functions on Two Variables2D Impulse and Sifting2D Discrete Fourier TransformDFT Properties
2D Convolution Theorem
f [m,n]?h [m,n] =M−1
∑j=0
N−1
∑k=0
f [j ,k]h [m− j ,n−k] (22)
f [m,n]?h [m,n]⇐⇒ F [u,v ]H [u,v ] (23)
f [m,n]h [m,n]⇐⇒ F [u,v ]?H [u,v ] (24)
Equation 23 is of particular importance to us. A spatial convolution
can be expressed as a product in frequency domain. This means
that all the linear �ltering we discussed before as convolutions can
now be easily understood as products.
Dr. Praveen Sankaran DIP Winter 2013
Functions on Two Variables2D Impulse and Sifting2D Discrete Fourier TransformDFT Properties
Summary
2D extension of ideas we discussed in 1D.
Concepts of sampling, aliasing in images.
2D DFT.
Properties
2D Convolution.
Dr. Praveen Sankaran DIP Winter 2013
Functions on Two Variables2D Impulse and Sifting2D Discrete Fourier TransformDFT Properties
Questions
4.4, 4.6, 4.7, 4.12, 4.13, 4.15, 4.17(look at 4.3 as reference).
Dr. Praveen Sankaran DIP Winter 2013