2D Discrete Fourier Transform (DFT)

2

Outline

• Circular and linear convolutions

• 2D DFT

• 2D DCT

• Properties

• Other formulations

• Examples

3

Circular convolution

• Finite length signals (N0 samples) → circular or periodic convolution– the summation is over 1 period– the result is a N0 period sequence

• The circular convolution is equivalent to the linear convolution of the zero-padded equal length sequences

[ ]f m

m *

[ ]g m

m

[ ]* [ ]f m g m

m=

Length=P Length=Q Length=P+Q-1

For the convolution property to hold, M must be greater than or equal to P+Q-1.

[ ]* [ ] [ ] [ ]f m g m F k G k⇔

0 1

0[ ] [ ] [ ] [ ] [ ]

N

nc k f k g k f n g k n

−

=

= ⊗ = −∑

4

Convolution

• Zero padding[ ]* [ ] [ ] [ ]f m g m F k G k⇔

[ ]f m

m *

[ ]g m

m

[ ]* [ ]f m g m

m=

[ ]F k

4-point DFT(M=4)

[ ]G k [ ] [ ]F k G k

5

In words

• Given 2 sequences of length N and M, let y[k] be their linear convolution

• y[k] is also equal to the circular convolution of the two suitably zero padded sequences making them consist of the same number of samples

• In this way, the linear convolution between two sequences having a different length (filtering) can be computed by the DFT (which rests on the circular convolution)

– The procedure is the following• Pad f[n] with Nh-1 zeros and h[n] with Nf-1 zeros• Find Y[r] as the product of F[r] and H[r] (which are the DFTs of the corresponding zero-padded

signals)• Find the inverse DFT of Y[r]

• Allows to perform linear filtering using DFT

[ ] [ ] [ ] [ ] [ ]n

y k f k h k f n h k n+∞

=−∞

= ∗ = −∑

0 1

0

0

[ ] [ ] [ ] [ ] [ ]

+ 1: length of the zero-padded seq

N

n

f h

c k f k h k f n h k n

N N N

−

=

= ⊗ = −

= −∑

6

2D Discrete Fourier Transform

• Fourier transform of a 2D signal defined over a discrete finite 2D grid of size MxN

or equivalently

• Fourier transform of a 2D set of samples forming a bidimensionalsequence

• As in the 1D case, 2D-DFT, though a self-consistent transform, can be considered as a mean of calculating the transform of a 2D sampled signal defined over a discrete grid.

• The signal is periodized along both dimensions and the 2D-DFT can be regarded as a sampled version of the 2D DTFT

7

2D Discrete Fourier Transform (2D DFT)

• 2D Fourier (discrete time) Transform (DTFT) [Gonzalez]

2 ( )( , ) [ , ] j um vn

m n

F u v f m n e π∞ ∞

− +

=−∞ =−∞

= ∑ ∑

1 1 2

0 0

1[ , ] [ , ]k lM N j m nM N

m nF k l f m n e

MN

π ⎛ ⎞− − − +⎜ ⎟⎝ ⎠

= =

= ∑∑

• 2D Discrete Fourier Transform (DFT)

2D DFT can be regarded as a sampled version of 2D DTFT.

a-periodic signalperiodic transform

periodized signalperiodic and sampled transform

8

2D DFT: Periodicity

1 1 2

0 0

1[ , ] [ , ]k lM N j m nM N

m nF k l f m n e

MN

π ⎛ ⎞− − − +⎜ ⎟⎝ ⎠

= =

= ∑∑

1 1 2

0 0

1[ , ] [ , ]k M l NM N j m n

M N

m nF k M l N f m n e

MN

π + +⎛ ⎞− − − +⎜ ⎟⎝ ⎠

= =

+ + = ∑∑1 1 2 2

0 0

1 [ , ]k l M NM N j m n j m nM N M N

m nf m n e e

MN

π π⎛ ⎞ ⎛ ⎞− − − + − +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

= =

= ∑∑

1

• A [M,N] point DFT is periodic with period [M,N]– Proof

[ , ]F k l=

(In what follows: spatial coordinates=k,l, frequency coordinates: u,v)

9

2D DFT: Periodicity

• Periodicity

• This has important consequences on the implementation and energycompaction property– 1D

[ , ] [ , ] [ , ] [ , ]F u v F u mM v F u v nN F u mM v nN= + = + = + +

[ , ] [ , ] [ , ] [ , ]f k l f k mM l f k l nN f k mM l nN= + = + = + +

f[u]

uM/2 M0

[ ] [ ]F N u F u∗− =The two inverted periods meet here

f[k] real→F[u] is symmetricM/2 samples are enough

10

Periodicity: 1D

f[u]

uM/2 M0It is more practical to have one complete period positioned in [0, M-1]

[ ] [ ]f k F u↔

0

0

2

0

2 22

0

[ ]e [ ]

e e e ( 1)2

( 1) [ ] [ ]2

u kj

M

u k Mkj j j k kM M

k

f k F u u

Mu

Mf k F u

π

π π π

↔ −

= → = = = −

− ↔ −

changing the sign of every other sample puts F[0] at the center of the interval [0,M]

The two inverted periods meet here

11

Periodicity: 2D

DFT periods

MxN values

4 inverted periods meet here

M/2

-M/2

N/2

-N/2

F[u,v]

(0,0)

12

Periodicity: 2D

DFT periods

MxN values

4 inverted periods meet here

M/2

N/2

F[u,v]

(0,0) M-1

N-10 02 ( )

0 0

0 0

[ , ]e [ , ]

,2 2

( 1) [ ] ,2 2

u k v lj

M N

k l

f k l F u u v vM Nu v

M Nf k F u v

π +

+

↔ − −

= = →

⎡ ⎤− ↔ − −⎢ ⎥⎣ ⎦

data contain one centered complete period

13

Periodicity: 2D

M/2

N/2

F[u,v]

(0,0) M-1

N-1

4 inverted periods meet here

14

Periodicity in spatial domain

1 1 2

0 0

[ , ] [ , ]k lM N j m nM N

k l

f m n F k l eπ ⎛ ⎞− − +⎜ ⎟⎝ ⎠

= =

= ∑∑

1

• [M,N] point inverse DFT is periodic with period [M,N]

1 1 2 ( ) ( )

0 0

[ , ] [ , ]k lM N j m M n NM N

k l

f m M n N F k l eπ ⎛ ⎞− − + + +⎜ ⎟⎝ ⎠

= =

+ + = ∑∑

1 1 2 2

0 0

[ , ]k l k lM N j m n j M NM N M N

k l

F k l e eπ π⎛ ⎞ ⎛ ⎞− − + +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

= =

= ∑∑

[ , ]f m n=

15

Angle and phase spectra

[ ] [ ] [ ]

[ ] [ ]{ } [ ]{ }

[ ] [ ]{ }[ ]{ }

[ ]

,

1/ 22 2

2

, ,

, Re , Im ,

Im ,, arctan

Re ,

[ , ] ,

j u vF u v F u v e

F u v F u v F u v

F u vu v

F u v

P u v F u v

Φ=

⎡ ⎤= +⎣ ⎦⎡ ⎤

Φ = ⎢ ⎥⎣ ⎦

=

modulus (amplitude spectrum)

phase

power spectrum

For a real function

[ , ] [ , ][ , ] [ , ][ , ] [ , ]

F u v F u vF u v F u v

u v u v

∗− − =

− − =

Φ − − = −Φ

conjugate symmetric with respect to the origin

16

Translation and rotation

[ ]

[ ] [ ]

2

2

[ , ] ,

, ,

m nj k lM N

m nj k lM N

f k l e F u m v l

f k m l n F u v

π

π

⎛ ⎞+⎜ ⎟⎝ ⎠

⎛ ⎞− +⎜ ⎟⎝ ⎠

↔ − −

− − ↔

[ ] [ ]0 0

cos cossin sin

, ,

k r ul r l

f r F

ϑ ω ϕϑ ω ϕ

ϑ ϑ ω ϕ ϑ

= =⎧ ⎧⎨ ⎨= =⎩ ⎩

+ ↔ +

Rotations in spatial domain correspond equal rotations in Fourier domain

17

mean value

[ ] [ ]1 1

0 0

10,0 ,N M

n m

F f n mNM

− −

= =

= ∑∑ DC coefficient

18

Separability

• The discrete two-dimensional Fourier transform of an image array is defined in series form as

• inverse transform

• Because the transform kernels are separable and symmetric, the two dimensional transforms can be computed as sequential row and column one-dimensional transforms.

• The basis functions of the transform are complex exponentials that may be decomposed into sine and cosine components.

1 1 2

0 0

1[ , ] [ , ]k lM N j m nM N

m nF k l f m n e

MN

π ⎛ ⎞− − − +⎜ ⎟⎝ ⎠

= =

= ∑∑

1 1 2

0 0

[ , ] [ , ]k lM N j m nM N

k l

f m n F k l eπ ⎛ ⎞− − +⎜ ⎟⎝ ⎠

= =

= ∑∑

19

2D DFT: summary

20

2D DFT: summary

21

2D DFT: summary

22

2D DFT: summary

other formulations

24

2D Discrete Fourier Transform

• Inverse DFT

1 1 2

0 0

1[ , ] [ , ]k lM N j m nM N

m nF k l f m n e

MN

π ⎛ ⎞− − − +⎜ ⎟⎝ ⎠

= =

= ∑∑

• 2D Discrete Fourier Transform (DFT)

1 1 2

0 0

[ , ] [ , ]k lM N j m nM N

k l

f m n F k l eπ ⎛ ⎞− − +⎜ ⎟⎝ ⎠

= =

= ∑∑

where0,1,..., 1k M= −0,1,..., 1l N= −

25

2D Discrete Fourier Transform

• Inverse DFT

1 1 2

0 0

1[ , ] [ , ]k lM N j m nM N

m nF k l f m n e

MN

π ⎛ ⎞− − − +⎜ ⎟⎝ ⎠

= =

= ∑∑

• It is also possible to define DFT as follows

1 1 2

0 0

1[ , ] [ , ]k lM N j m nM N

k l

f m n F k l eMN

π ⎛ ⎞− − +⎜ ⎟⎝ ⎠

= =

= ∑∑

where 0,1,..., 1k M= −0,1,..., 1l N= −

26

2D Discrete Fourier Transform

• Inverse DFT

1 1 2

0 0[ , ] [ , ]

k lM N j m nM N

m nF k l f m n e

π ⎛ ⎞− − − +⎜ ⎟⎝ ⎠

= =

= ∑∑

• Or, as follows

1 1 2

0 0

1[ , ] [ , ]k lM N j m nM N

k l

f m n F k l eMN

π ⎛ ⎞− − +⎜ ⎟⎝ ⎠

= =

= ∑∑

where and 0,1,..., 1k M= − 0,1,..., 1l N= −

27

2D DFT

• The discrete two-dimensional Fourier transform of an image array is defined in series form as

• inverse transform

2D DCT

Discrete Cosine Transform

29

2D DCT

• based on most common form for 1D DCT

u,x=0,1,…, N-1

“mean” value

30

1D basis functions

Cosine basis functions are orthogonal

Figure 1

31

2D DCT

• Corresponding 2D formulation

u,v=0,1,…., N-1

direct

inverse

32

2D basis functions

• The 2-D basis functions can be generated by multiplying the horizontally oriented 1-D basis functions (shown in Figure 1) with vertically oriented set of the same functions.

• The basis functions for N = 8 are shown in Figure 2. – The basis functions exhibit a progressive increase in frequency both in

the vertical and horizontal direction. – The top left basis function assumes a constant value and is referred to

as the DC coefficient.

33

2D DCT basis functionsFigure 2

34

Separability

The inverse of a multi-dimensional DCT is just a separable product of the inverse(s) of the corresponding one-dimensional DCT , e.g. the one-dimensional inverses applied along one dimension at a time

35

Separability

• Symmetry– Another look at the row and column operations reveals that these

operations are functionally identical. Such a transformation is called a symmetric transformation.

– A separable and symmetric transform can be expressed in the form

– where A is a NxN symmetric transformation matrix which entries a(i,j) are given by

• This is an extremely useful property since it implies that the transformation matrix can be pre computed offline and then applied to the image thereby providing orders of magnitude improvement in computation efficiency.

T AfA=

36

Computational efficiency

• Computational efficiency– Inverse transform– DCT basis functions are orthogonal. Thus, the inverse transformation

matrix of A is equal to its transpose i.e. A-1= AT. This property renders some reduction in the pre-computation complexity.

37

Block-based implementation

The source data (8x8) is transformed to a linear combination of these 64 frequency squares.

Block sizeN=M=8

Block-based transform

Basis function

38

Energy compaction

39

Energy compaction

40

Appendix

• Eulero’s formula