2D Discrete Fourier Transform (DFT)

Transcription

1 2D Discrete Fourier Transform (DFT)

2 Outline Circular and linear convolutions 2D DFT 2D DCT Properties Other formulations Examples 2

3 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 bidimensional sequence 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 6

4 2D Discrete Fourier Transform (2D DFT) 2D Fourier (discrete time) Transform (DTFT) [Gonzalez] Fuv (, ) fmne [, ] = m= n= j2 π ( um+ vn) a-periodic signal periodic transform 2D Discrete Fourier Transform (DFT) M 1N 1 k l j2π m+ n M N 1 Fkl [, ] = f[ mne, ] MN m= n= periodized signal periodic and sampled transform 2D DFT can be regarded as a sampled version of 2D DTFT. 7

5 2D DFT: Periodicity A [M,N] point DFT is periodic with period [M,N] Proof M 1N 1 k l j2π m+ n M N 1 Fkl [, ] = f[ mne, ] MN m= n= M 1N 1 k+ M l+ N j2π m+ n M N 1 Fk [ + Ml, + N] = f[ mne, ] MN 1 MN m= n= = = Fkl [, ] M 1 N 1 j k l M N 2π m n j 2 M + π N M m+ N n m= n= f[ m, n] e e 1 (In what follows: spatial coordinates=k,l, frequency coordinates: u,v) 8

6 2D DFT: Periodicity Periodicity Fuv [, ] = Fu [ + mmv, ] = Fuv [, + nn] = Fu [ + mmv, + nn] f[ k, l] = f[ k+ mm, l] = f[ k, l+ nn] = f[ k + mm, l+ nn] This has important consequences on the implementation and energy compaction property 1D FN [ u] = F[ u] f[k] real F[u] is symmetric M/2 samples are enough F[u] The two inverted periods meet here M/2 M u 9

7 f[ k] F[ u] uk j2π M f[ k]e F[ u u ] uk Mk j2π j2π M 2 M F[u] Periodicity: 1D M jπ k u = e = e = e = ( 1) 2 k M ( 1) f[ k] F[ u ] 2 k changing the sign of every other sample of the DFT puts F[] at the center of the interval [,M] The two inverted periods meet here M/2 M It is more practical to have one complete period positioned in [, M-1] u 1

8 Periodicity in 2D 1/N 128 1/M 128 I 4 semiperiodi si incontrano ai vertici I 4 semiperiodi si incontrano al centro 11

9 Periodicity, fftshift(fft2) fft2,,127=1/m,1/n 12

10 Periodicity: 2D DFT periods N/2 MxN values 4 inverted periods meet here -M/2 (,) -N/2 F[u,v] M/2 13

11 Periodicity: 2D F[u,v] N-1 uk vl j2 π ( + ) M N f[ k, l]e F[ u u, v v ] M N u =, v = 2 2 k+ l M N ( 1) f[ k] F u, v 2 2 N/2 data contain one centered complete period DFT periods MxN values M/2 (,) M-1 4 inverted periods meet here 14

12 F[u,v] Periodicity: 2D N-1 M/2 (,) M-1 N/2 4 inverted periods meet here 15

13 Periodicity in spatial domain [M,N] point inverse DFT is periodic with period [M,N] M 1N 1 k l j2 M m+ n N f[ m, n] F[ k, l] e π = k= l= M 1N 1 k l j2 M ( m+ M) + ( n+ N) N f[ m M, n N] F[ k, l] e π + + = k= l= 1 M 1 N 1 j k l k l 2π m n j 2 M + π N M M+ N N = k= l= Fkle [, ] e = f[ m, n] 16

14 Angle and phase spectra jφ[ u, v] [, ] = F[ u, v] e F u v [, ] = Re { [, ]} + Im { [, ]} [ ] F u v F u v F u v { F[ u v] } { F[ u v] } Im, Φ [ uv, ] = arctan Re, Puv [, ] = F uv, 1/2 modulus (amplitude spectrum) phase power spectrum For a real function F[ u, v] = F [ u, v] F[ u, v] = F[ u, v] Φ [ u, v] = Φ[ u, v] conjugate symmetric with respect to the origin 17

15 Translation and rotation m n j2π k+ l M N [, ] [, ] [ ] f[ k, l] e F u m, v l f k m l n F u v m n j2π k+ l M N k = rcosϑ u = ωcosϕ l = rsinϑ l = ωsinϕ f r [, ϑ+ ϑ ] F[ ω, ϕ+ ϑ ] Rotations in spatial domain correspond equal rotations in Fourier domain 18

16 mean value F 1 NM N 1M 1 [, ] f [ n, m] = DC coefficient n= m= 19

17 Separability The discrete two-dimensional Fourier transform of an image array is defined in series form as inverse transform M 1N 1 k l j2π m+ n M N 1 Fkl [, ] = f[ mne, ] MN m= n= M 1N 1 k l j2 M m+ n N f[ m, n] F[ k, l] e π = k= l= 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. 2

18 2D DFT: summary 21

19 2D DFT: summary 22

20 2D DFT: summary 23

21 2D DFT: summary 24

22 other formulations

23 2D Discrete Fourier Transform 2D Discrete Fourier Transform (DFT) M 1N 1 k l j2π m+ n M N 1 Fkl [, ] = f[ mne, ] MN m= n= where l =,1,..., N 1 k =,1,..., M 1 Inverse DFT M 1N 1 k l j2 M m+ n N f[ m, n] F[ k, l] e π = k= l= 26

24 2D Discrete Fourier Transform It is also possible to define DFT as follows M 1N 1 k l j2π m+ n M N 1 Fkl [, ] = f[ mne, ] MN m= n= where k =,1,..., M 1 l =,1,..., N 1 Inverse DFT M 1N 1 k l j2π m+ n M N 1 f[ m, n] = F[ k, l] e MN k= l= 27

25 2D Discrete Fourier Transform Or, as follows M 1N 1 k l j2π m+ n M N Fkl [, ] f[ mne, ] = m= n= where k =,1,..., M 1 and l =,1,..., N 1 Inverse DFT M 1N 1 k l j2π m+ n M N 1 f[ m, n] = F[ k, l] e MN k= l= 28

26 2D DFT The discrete two-dimensional Fourier transform of an image array is defined in series form as inverse transform 29

27 2D DCT Discrete Cosine Transform

28 2D DCT based on most common form for 1D DCT u,x=,1,, N-1 mean value 31

29 1D basis functions Figure 1 Cosine basis functions are orthogonal 32

30 2D DCT Corresponding 2D formulation direct u,v=,1,., N-1 inverse 33

31 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. 34

32 2D DCT basis functions Figure 2 35

33 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 36

34 Block-based transform Block size N=M=8 Block-based implementation The source data (8x8) is transformed to a linear combination of these 64 frequency squares. Basis function 39

35 Energy compaction 4

36 Energy compaction 41

37 Appendix Eulero s formula 42