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Windowed Fourier Transform



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One way to approach a joint space-wave number representation is the windowed Fourier transform. As the name says, the Fourier transform is not applied to the whole image but only to a section of the image that is formed by multiplying the image with a window function w( x ). The window function has a maximum at x 0 and decreases monotonically with x towards zero. The maximum of the window function is then put at each point x of the image to compute a windowed Fourier transform for each point:


 

gˆ ( x, k 0) =


∫                            .            Σ
g( x ')w( x ' − x ) exp − 2π i k 0 x ')  dx'2.                          (5.1)

− ∞


±                                                                                             − =
The integral in Eq. (5.1) almost looks like a convolution integral (Eq. (2.56), R4). To convert it into a convolution integral we observe that w( k )

w( k ) and rearrange the second part of Eq. (5.1):


.           Σ
w( x ' − x ) exp − 2π i k 0 x '

= w( x x ') exp.2π i k 0( x x ')Σ  exp (− 2π i k 0 x )).


 

(5.2)


128                                                                              5 Multiscale Representation

 

Then we can write Eq. (5.1) as a convolution

gˆ ( x, k 0) = (g( x ) ∗ w( x ) exp (2π i k 0 x )) exp (− 2π i k 0 x ).                            (5.3)

This means that the local Fourier transform corresponds to a convolu- tion with the complex convolution kernel w( x ) exp(2π i k 0 x ) except for a phase factor exp( 2π i k 0 x ). Using the shift theorem (Theorem 3, p. 52,

±
R4), the transfer function of the convolution kernel can be computed to be

w( x ) exp (2π i k 0 x ) ◦      • wˆ ( k k 0).                            (5.4)

This means that the convolution kernel w( x ) exp(2π i k 0 x ) is a bandpass fi lter with a peak wave number of k 0. The width of the bandpass is inversely proportional to the width of the window function. In this way, the spatial and wave number resolutions are interrelated to each other. As an example, we take a Gaussian window function

 

exp. x2 Σ .                                               (5.5)

 

− 2σ 2

Its Fourier transform (± R4, ± R5), is

1


2
x
x
√ π σ


exp.− 2π 2k2σ 2Σ  .                                        (5.6)


k
x
x
k
Consequently, the product of the standard deviations in the space and wave number domain (σ 2 = 1/(4π σ 2)) is a constant: σ 2σ 2 = 1/(4π )).

This fact establishes the classical uncertainty relation (Theorem 7, p. 55). It states that the product of the standard deviations of any Fourier trans- form pair is larger than or equal to 1/(4π ). As the Gaussian window function reaches the theoretical minimum it is an optimal choice; a bet- ter wave number resolution cannot be achieved with a given spatial res- olution.

 

5.2 Scale Space†

As we have seen with the example of the windowed Fourier transform in the previous section, the introduction of a characteristic scale adds a new coordinate to the representation of image data. Besides the spa- tial resolution, we have a new parameter that characterizes the current resolution level of the image data. The scale parameter is denoted by ξ. A data structure that consists of a sequence of images with diff erent resolutions is known as a scale space; we write g( x, ξ ) to indicate the scale space of the image g( x ).

Next, in Section 5.2.1, we discuss a physical process, diff usion, that is suitable for generating a scale space. Then we discuss the general properties of a scale space in Section 5.2.2.


5.2 Scale Space†                                                                                                    129

 

5.2.1 Scale Generation by Diff usion

The generation of a scale space requires a process that can blur images to a controllable degree. Diff usion is a transport process that tends to level out concentration diff erences [23]. In physics, diff usion processes govern the transport of heat, matter, and momentum leading to an ever increasing equalization of spatial concentration diff erences. If we iden- tify the time with the scale parameter ξ, the diff usion process establishes a scale space.

To apply a diff usion process to an image, we regard the gray value g as the concentration of a chemical species. The elementary law of diff u- sion states that the fl ux density j is directed against the concentration gradient g and proportional to it:

j = − D g                                                    (5.7)

where the constant D is known as the diff usion coeffi cient. Using the continuity equation


 

 

the diff usion equation is


∂ g

∂ t +j = 0                                                 (5.8)

 

∂ g

∂ t = (D g).                                                 (5.9)


For the case of a homogeneous diff usion process (D does not depend on the position), the equation reduces to

∂ g

∂ t = D∆ g                                                 (5.10)


where


∆ ∂ 2           ∂ 2

 

           


= ∂ x2 + ∂ y2                                             (5.11)

is the Laplacian operator. It is easy to show that the general solution to this equation is equivalent to a convolution with a smoothing mask. To this end, we perform a spatial Fourier transform which results in

∂ t
∂ gˆ ( k ) = − 4π 2D| k |2gˆ ( k )                                         (5.12)

reducing the equation to a linear fi rst-order diff erential equation with the general solution

gˆ ( k, t) = exp(− 4π 2D| k |2t)gˆ ( k, 0),                                      (5.13) where gˆ ( k, 0) is the Fourier transformed image at time zero.


130                                                                              5 Multiscale Representation

 



















A                                                                    b

C                                                                    d

Figure 5.2: Scale space of some one-dimensional signals: a edges and lines; b a periodic pattern; c a random signal; d row 10 from the image shown in Fig. 11.6a. The vertical coordinate is the scale parameter ξ.

 

Multiplication of the image in the Fourier space with the Gaussian function in Eq. (5.13) is equivalent to a convolution with the same func- tion but of reciprocal width. Thus,


 

g( x, t)


  1   exp.


| x |2 Σ

 


 

g( x, 0)               (5.14)


 

 

with


= 2π σ 2(t)


− 2σ 2(t)    ∗

 

 


σ (t) =  2Dt.                                                (5.15)

,
Equation Eq. (5.15) shows that the degree of smoothing expressed by the standard deviation σ increases only by the power half with time. Therefore we set the scale parameter ξ equal to the square of the stan- dard deviation:

ξ = 2Dt.                                                   (5.16)

It is important to note that this formulation of the scale space is valid for images of any dimension. It could also be extended to image sequences. The scale parameter is not identical to the time although we used a physical diff usion process that proceeds with time to derive it.


5.2 Scale Space†                                                                                                    131

 











A                                                                    b

C                                                                    d

Figure 5.3: Scale space of a two-dimensional image: a original image; b, c, and

d at scale parameters σ 1, 2, and 4, respectively.

 

If we compute a scale space representation of an image sequence, it is useful to scale the time coordinate with a characteristic velocity u0 so that it has the same dimension as the spatial coordinates:

t' = u0t.                                                    (5.17)

We add this coordinate to the spatial coordinates and get a new coordi- nate vector

x = [x1, x2, u0t]T or x = [x1, x2, x3, u0t]T.                              (5.18)

In the same way, we extend the wave number vector by a scaled fre- quency:

k = [k1, k2, ω /u0]T or k = [k1, k2, k3, ω /u0]T.                                  (5.19)

With Eqs. (5.18) and (5.19) all equations derived above, e. g., Eqs. (5.13) and (5.14), can also be applied to scale spaces of image sequences. For discrete spaces, of course, no such scaling is required. It is automatically fi xed by the spatial and temporal sampling intervals: u0 = ∆ x/∆ t.


132                                                                              5 Multiscale Representation

 

→ ∞
As an illustration, Fig. 5.2 shows the scale space of some character- istic one-dimensional signals: noisy edges and lines, a periodic pattern, a random signal, and a row of an image. These examples nicely demon- strate a general property of scale spaces. With increasing scale parame- ter ξ, the signals become increasingly blurred, more and more details are lost. This feature can be most easily seen by the transfer function of the scale space representation in Eq. (5.13). The transfer function is always positive and monotonically decreasing with the increasing scale parame- ter ξ for all wave numbers. This means that no structure is amplifi ed. All structures are attenuated with increasing ξ, and smaller structures always faster than coarser structures. In the limit of ξ the scale space converges to a constant image with the mean gray value. A certain feature exists only over a certain scale range. In Fig. 5.2a we can observe that edges and lines disappear and two objects merge into one.

For two-dimensional images, a continuous representation of the scale space would give a three-dimensional data structure. Therefore Fig. 5.3 shows individual images for diff erent scale parameters ξ as indicated.

 


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