Wave Number Space and Fourier Transform

Vector Spaces

In Section 2.2, the discussion centered around the spatial representation of digital images. Without mentioning it explicitly, we thought of an image as composed of individual pixels (Fig. 2.10). Thus we can compose each image with basis images where just one pixel has a value of one while all other pixels are zero. We denote such a basis image with a one at row m, column n by

 

m, n

 

m, n

 

m, n

.1  m = m' ∧ n = n'

P :        p ' ' =

0  otherwise.                                (2.10)

 

Any arbitrary scalar image can then be composed from the basis images in Eq. (2.10) by

M− 1N− 1

G =  .. g_{m, n} m, n P,                                        (2.11)

m=0n=0

where G_{m, n} denotes the gray value at the position (m, n).

It is easy to convince ourselves that the basis images m, n P form an orthonormal base. To that end we require an inner product (also known as scalar product) which can be defi ned similarly to the scalar product for vectors. The inner product of two images G and H is defi ned as

M− 1N− 1

• G | H )=.. g_{m, n}h_{m, n},                                       (2.12)

m=0n=0

40                                                                                        2 Image Representation

Figure 2.11: The fi rst 56 periodic patterns, the basis images of the Fourier trans- form, from which the image in Fig. 2.10 is composed.

 

where the notation for the inner product from quantum mechanics is used in order to distinguish it from matrix multiplication, which is de- noted by GH.

From Eq. (2.12), we can immediately derive the orthonormality rela- tion for the basis images m, n P:

M− 1N− 1

.. m', n' pm, nm'', n'' pm, n = δ m'− m'' δ n'− n''.                                  (2.13)

m=0n=0

×

This says that the inner product between two base images is zero if two diff erent basis images are taken. The scalar product of a basis image with itself is one. The MN basis images thus span an M N-dimensional vector space over the set of real numbers.

×                                                      ×

The analogy to the well-known two- and three-dimensional vector spaces R2 and R3 helps us to understand how other representations for images can be gained. An M N image represents a point in the M N vector space. If we change the coordinate system, the image remains the same but its coordinates change. This means that we just observe the same piece of information from a diff erent point of view. We can draw two important conclusions from this elementary fact. First, all representations are equivalent to each other. Each gives a complete rep- resentation of the image. Secondly, suitable coordinate transformations lead us from one representation to the other and back again.

From the manifold of other possible representations beside the spa- tial representation, only one has gained prime importance for image processing. Its base images are periodic patterns and the “coordinate transform” that leads to it is known as the Fourier transform. Figure 2.11

2.3 Wave Number Space and Fourier Transform                               41

 

Figure 2.12: Description of a 2-D periodic pattern by the wavelength λ, wave number k, and phase ϕ.

 

shows how the same image that has been composed by individual pixels in Fig. 2.10 is composed of periodic patterns.

A periodic pattern is fi rst characterized by the distance between two maxima or the repetition length, the wavelength λ (Fig. 2.12). The direc- tion of the pattern is best described by a vector normal to the lines of constant gray values. If we give this vector k the length 1/λ

| k |= 1/λ,                                                   (2.14)

=

the wavelength and direction can be expressed by one vector, the wave number k. The components of k [k₁, k₂]^T directly give the number of wavelengths per unit length in the corresponding direction. The wave number k can be used for the description of periodic patterns in any dimension.

=           =    ·

In order to complete the description of a periodic pattern, two more quantities are required: the amplitude r and the relative position of the pattern at the origin (Fig. 2.12). The position is given as the distance ∆ x of the fi rst maximum from the origin. Because this distance is at most a wavelength, it is best given as a phase angle ϕ 2π ∆ x/λ 2π k ∆ x (Fig. 2.12) and the complete description of a periodic pattern is given by

 

r cos(2π k ^T x − ϕ ).                                               (2.15)

=       +

=      −

This description is, however, mathematically quite awkward. We rather want a simple factor by which the base patterns have to be multiplied, in order to achieve a simple decomposition in periodic patterns. This is only possible by using complex numbers gˆ  r exp(  iϕ ) and the com- plex exponential function exp(iϕ ) cos ϕ i sin ϕ. The real part of gˆ exp(2π i k ^T x ) gives the periodic pattern in Eq. (2.15):

≡ (gˆ exp(2π i k ^T x )) = r cos(2π k ^T x − ϕ ).                                 (2.16)

42                                                                                        2 Image Representation

 

In this way the decomposition into periodic patterns requires the extension of real numbers to complex numbers. A real-valued image is thus considered as a complex-valued image with a zero imaginary part. The subject of the remainder of this chapter is rather mathemati- cal, but it forms the base for image representation and low-level image processing. After introducing both the continuous and discrete Fourier transform in Sections 2.3.2 and 2.3.3, we will discuss all properties of the Fourier transform that are of relevance to image processing in Sec- tion 2.3.5. We will take advantage of the fact that we are dealing with images, which makes it easy to illustrate some complex mathematical

relations.

 

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