Multidimensional Fourier transform

The Fourier transform can easily be extended to multidimensional sig- nals.

⊆ →

Defi nition 3 (Multidimensional FT) If  g( x ): RW                   C is a square inte- grable function, that is,

2                                                                                             2

− ∞

.

.

2

∞

∫ .g( x ). d^W x =.g( x ).g( x ). =  g( x ) < ∞                  (2.31)

 

then the Fourier transform of g( x ), gˆ ( k ) is given by

− ∞

∞

gˆ ( k ) =  ∫   g( x ) exp.− 2π i k ^T x Σ  d^W x =.w ^x T k  . g( x ).                 (2.32)

46                                                                                        2 Image Representation

 

and the inverse Fourier transform by

∞

g( x ) =  ∫   gˆ ( k ) exp.2π i k ^T x Σ  d^W k =.w^{− x} T k  . gˆ ( k ). .                         (2.33)

 

− ∞

The scalar product in the exponent of the kernel x T k makes the kernel of the Fourier transform separable, that is, it can be written as

.

W

w ^x T k =   w^kpxp.                                          (2.34)

p=1

The discrete Fourier transform is discussed here for two dimensions.

The extension to higher dimensions is straightforward.

Defi nition 4 (2-D DFT) The 2-D DFT maps an M × N complex-valued ma- trices on M × N complex-valued matrices:

M− 1N− 1

1

gˆ _{u, v}  =    √

.  . gm, n exp .−  2π imu Σ  exp .−  2π inv Σ

MN m=0n=0                                                M

m=0

n=0

1  M− 1  N− 1                                 

 

                                                                                         

N

(2.35)

=            √ MN .  . gm, nw− nv  w− mu.

×

In the second line, the abbreviation defi ned in Eq. (2.27) is used. As in the one-dimensional case, the DFT expands a matrix into a set of NM basis matrices which spans the N M-dimensional vector space over the fi eld of complex numbers. The basis matrices are of the form



B  = √

u, v       1 

 

w

w

0



M

u

w

M

2u        Σ w⁰, w^v, w^2v,..., w^{(N− 1)v}Σ  .                    (2.36)

N    N                  N



M˛ × ¸ Nr             .

 w(M− 1)u 

In this equation, the basis matrices are expressed as an outer product of the column and the row vector that form the basis vectors of the one- dimensional DFT (Eq. (2.28)). This refl ects the separability of the kernel of the 2-D DFT.

Then the 2-D DFT can be written again as an inner product

.       .

gˆ _{u, v} =   B _{u, v} | G  ,                                                 (2.37) where the inner product of two complex-valued matrices is given by

M− 1N− 1

• G | H )=  .  . g_m∗  , nh_{m, n}.                                     (2.38)

m=0n=0

2.3 Wave Number Space and Fourier Transform                               47

The inverse 2-D DFT is given by

1 M− 1N− 1

g_mn = √ MN  .  . gˆ _{u, v}w^muw^nv =. B _{− m, − n}. G ˆ.  .                                  (2.39)

M     N

u=0 v=0

 

2.3.4 Alternative Defi nitions‡

 

=

In the literature several variations of the Fourier transform exist, which can lead to a lot of confusions and errors. This has to do with the defi nition of the wave number. The defi nition of the wave number as a reciprocal wavelength k 1/λ is the most useful for signal processing, because k directly gives the number of wavelengths per unit length. In physics and electrical engineering, however,

a defi nition including the factor 2π  is more common:  k˘  =  2π /λ.  With this

notation, two forms of the Fourier transform can be defi ned: the asymmetric form

˘  .     ˘ .    .                    .        ˘ . ˘ .

1

gˆ (k) =  exp(ikx)  g(x), g(x) = 2π  exp(− ikx)  gˆ (k)                                  (2.40)

and the symmetric form

.

1                                                       1

gˆ (k˘ ) = √ 2π  .exp(ik˘ x).g(x). , g(x) = √ 2π  .exp(− ik˘ x)  gˆ (k˘ ). .                           (2.41)

Because all three versions of the Fourier transform are in common use, it is likely that wrong factors in Fourier transform pairs will be obtained. The rules for conversion of Fourier transform pairs between the three versions can directly be inferred from the defi nitions and are summarized here:

k = 1/λ, Eq. (2.22) g(x) ◦ • k˘  with 2π, Eq. (2.40)  g(x)  ◦  • k˘  with 2π, Eq. (2.41)  g(x)  ◦  •

gˆ (k) gˆ (k˘ /2π )

gˆ (k˘ /√ 2π )/√ 2π.

 

(2.42)

 

⇐ Предыдущая78910111213141516Следующая ⇒

Поделиться: