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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


Functions on Two Variables2D Impulse and Sifting2D Discrete Fourier TransformDFT Properties

Outline




DFT 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]?



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)



Outline




DFT 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)



Example

⇓



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.



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


Aliasing

Anyway for the sake of explaining it,



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


Another Example



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.



Outline




DFT Properties



Spatial and Frequency Intervals

∆u =1

M∆T(13)

∆v =1

N∆Z(14)

We have seen this before.



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)



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)



Illustration & Shifting

f [m,n] (−1)m+n⇐⇒F [u−M/2, v −N/2]



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}?



Some more 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!



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



Illustration - Spectrum Centering and Log Operation



Illustration - Change e�ect in Spectrum



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.



Reconstruction Illustration - Phase Dominates!



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.



Summary

2D extension of ideas we discussed in 1D.

Concepts of sampling, aliasing in images.

2D DFT.

Properties

2D Convolution.



Questions

4.4, 4.6, 4.7, 4.12, 4.13, 4.15, 4.17(look at 4.3 as reference).


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