The frequency domain

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The frequency domain Part 2
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Instead of talking about one dimensional signals
that represent changes in amplitude in time, here
we are dealing with two dimensional signals which
represent intensity variations in space. These
signals come in the form of images.
Thus an MxN image has an MxN set of (complex)
fourier coefficients. To implement this
transform, we would like an analog of the FFT,
which will let us quickly compute the
coefficients of the transform. In fact, we can do
better. The two dimensional DFT is seperable into
two one dimensional DFTs which can be implemented
with an FFT algorithm.
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The spectra of an image
The Fourier Transform produces a complex number
valued output image which can be displayed with
two images, either with the real and imaginary
part or with magnitude and phase. In image
processing, often only the magnitude of the
Fourier Transform is displayed, as it contains
most of the information of the geometric
structure of the spatial domain image. However,
if we want to re-transform the Fourier image into
the correct spatial domain after some processing
in the frequency domain, we must make sure to
preserve both magnitude and phase of the Fourier
image.
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The spectra of an image
The result of an FFT is always a complex number.
This however, is not complicated, all it means is
that we get a pair of numbers and from this pair
we can calculate the pair of numbers we really
want from each harmonic the amplitude and phase
(often called the modulus and argument).
The result of the FFT is a complex number C a
ib illustrated as the point on the diagram. The
position of this point can also be described by
the distance from the center of the diagram A and
the angle q with the real axis. A is amplitude
(or modulus) and q is phase (or argument). Simple
algebra tells us that
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Fourier spectra play an important role

• The Fourier transform of a real function is a
  complex function

where R(u,v) and I(u,v) are, respectively, the
real and imaginary components of F(u,v).

• The magnitude function F(u,v) is called the
  frequency spectrum of image f(m,n). The
  magnitudes correspond to the amplitudes of the
  basis images in our Fourier representation. The
  array of magnitudes is termed the amplitude
  spectrum of the image

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Fourier spectra play an important role
The array of phases is termed the phase spectrum.

When the term spectrum is used on its own, the
amplitude spectrum is normally implied.
The power spectrum of an image is simply the
square of its amplitude spectrum

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Importance of phase and magnitude
Equation indicates that the Fourier transform of
an image can be complex. This is illustrated
below in Figures 1a-c. Figure 4a shows the
original image am,n, Figure 1b the magnitude in
a scaled form , and Figure 1c the phase.
Figure 1 a) b) c)
Both the magnitude and the phase functions are
necessary for the complete reconstruction of an
image from its Fourier transform. Figure 2a shows
what happens when Figure 1a is restored solely on
the basis of the magnitude information and Figure
2b shows what happens when Figure 1a is restored
solely on the basis of the phase information.
Figure 2 a)
Figure 2 b)
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Properties of the Fourier transform
Periodicity - F(u,v ) repeats itself endlessly
in both directions, with a period of N . This
means that
The N x N block of coefficients that we compute
from an N x N image with our two-dimensional FFT
algorithm is a single period from this infinite
sequence.
If f(x,y) is real, its Fourier transform is
conjugate symmetry, that is
negative frequencies are mirror images of
positive frequencies
The complex conjugate of a complex number
is defined to be
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Frequency Content Frequency Content Location
The DFT coefficients produced by the 2D DFT
equations , are arranged in a somewhat awkward
manner as shown in the diagram below. (Figure 3)
It is considered much more intuitive to have low
frequency content in the center of the image and
high frequency content on the outsides of the
image. Due to the periodicity of the content, and
the fact that we could have done our DFT over any
period of the image, we chose to modify the
frequency domain contents representation by
interchanging the 1st and 3rd quadrants and 2nd
and 4th quadrants. This layout is shown below.
(Figure4)
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Spectrum is more easily interpreted (visually) if
we shift the results
It is common practice to multiply the input image
function by (-1)xy prior to computing the
Fourier transform.
That is F(0,0) is located at u N/2 and v N/2
. Multiplying f(x,y) by (-1)xy shifts the origin
of F(u,v) to frequency coordinates (N/2,N/2),
which is the center of the N x N area occupied by
the 2-D DFT. We refer to this area of the
frequency domain as the frequency rectangle. It
extends from u0 to uN-1and v0 to vN-1 ( u and
v are integer and N should be even number.
We have the following relationships between
samples in the spatial and frequency domain
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Frequency Content Location
The Fourier Transform is used if we want to
access the geometric characteristics of a spatial
domain image. Because the image in the Fourier
domain is decomposed into its sinusoidal
components, it is easy to examine or process
certain frequencies of the image, thus
influencing the geometric structure in the
spatial domain. In most implementations the
Fourier image is shifted in such a way that the
DC-value (i.e. the image mean) F(0,0) is
displayed in the center of the image. The further
away from the center an image point is, the
higher is its corresponding frequency.
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An example of frequency domain image processing
Figure 5
Figure 6
Figure 6 is a representation of the result of
performing an FFT on Figure 5. This diagram shows
values of amplitude for each of the two
dimensional sine wave frequencies, with high
values being shown as lighter than low ones, with
black indicating a zero amplitude. In practice
the amplitude is a floating point number which
has been mapped into 256 grey levels to produce
this image. The amplitude values F(u,v) in this
image have been calculated from the real and
imaginary values produced by the FFT algorithm
and these values are stored in two additional
arrays Real(u,v) and Imag(u,v).
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How can we display the discrete Fourier transform
of an image ?
Amplitude spectra are normally visualized as
8-bit grayscale images. In order to do this we
must scale the magnitude to lie in a 0-255 range
. The obvious approach of multiplying by a
scaling factor
As u and v increase , the contribution of these
high frequencies to the image becomes less and
less important and thus the value of the
corresponding coefficients F(u,v) become smaller.
For displaying purpose , people use logarithmic
mapping of the data.
Since the logarithm is not defined for 0, many
implementations of this operator add the value 1
to the image before taking the logarithm.
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Logarithmic Operator
The logarithmic operator is a simple point
processor where the mapping function is a
logarithmic curve. In other words, each pixel
value is replaced with its logarithm.
The scaling constant c is chosen so that the
maximum output value is 255 (providing an 8-bit
format). That means if R is the value with the
maximum magnitude in the input image, c is given
by
The degree of compression (which is equivalent to
the curvature of the mapping function) can be
controlled by adjusting the range of the input
values. Since the logarithmic function becomes
more linear close to the origin, the compression
is smaller for an image containing small input
values. The mapping function is shown for two
different ranges of input values in Figure
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Logarithmic Operator
The image shows one bright spot in the center and
two darker spots on the diagonal. We can infer
from the image that these three frequencies are
the main components of the image with the
DC-value having the largest magnitude. Applying
the logarithmic transform to the Fourier image
yields
The image is the linearly scaled Fourier
Transform of
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Logarithmic Operator
The logarithmic operator enhances the low
intensity pixel values, while compressing high
intensity values into a relatively small pixel
range. Hence, if an image contains some important
high intensity information, applying the
logarithmic operator might lead to loss of
information.
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Logarithmic Operator
Here, we can see that the image contains many
more frequencies. However, it is now hard to tell
which are the dominating ones, since all high
magnitudes are compressed into a rather small
pixel value range. The magnitude of compression
is large in this case because there are extremely
high intensity values in the output of the
Fourier Transform (in this case up to ).
This image is the result of first multiplying
each pixel with 0.0001 and then taking its
logarithm. Now, we can recognize all the main
components of the Fourier image and can even see
the difference in their intensities.
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Spectra of simple periodic patterns
The image shows 2 pixel wide vertical stripes.
The Fourier transform of this image is shown in
If we look carefully, we can see that it contains
3 main values the DC-value and, since the
Fourier image is symmetrical to its center, two
points corresponding to the frequency of the
stripes in the original image. Note that the two
points lie on a horizontal line through the image
center, because the image intensity in the
spatial domain changes the most if we go along it
horizontally.
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Spectra of simple periodic patterns
The distance of the points to the center can be
explained as follows the maximum frequency which
can be represented in the spatial domain are one
pixel wide stripes.
Hence, the two pixel wide stripes in the above
image represent
Thus, the points in the Fourier image are halfway
between the center and the edge of the image,
i.e. the represented frequency is half of the
maximum.
Further investigation of the Fourier image shows
that the magnitude of other frequencies in the
image is less than 1/100 of the DC-value, i.e.
they don't make any significant contribution to
the image. The magnitudes of the two minor points
are each two-thirds of the DC-value.
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Aliasing
Aliasing is a very important concept, when using
the FFT for frequency domain image processing.
Nyquist's theorem says that we must sample a
signal at a rate which is at least twice the
highest frequency present if we are to avoid
errors.
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Aliasing
The first curve has been sampled with
approximately 10 points per wavelength, the
second with about 5, the third with around three
the last with less than two. What can be clearly
seen is that the sample points on the last curve
do not clearly define the frequency and a second
curve which could equally well fit the sampled
points has been included. If a frequency which is
higher than the Nyquist frequency is present, it
will be under-sampled like the last curve and
will be seen by the FFT as a lower frequency
whose size is difficult to predict. This is
aliasing and this is why the highest frequency
used by the FFT is equal to half the number of
sampled points in the signal - any higher
frequency would not have been properly
interpreted by the sampling process.
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Filtering of images and Fourier transform
The Fourier transform is of great use in the
calculation of image convolutions

The convolution theorem

Thus we may write
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Filtering in the Frequency Domain
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Filtering of images and Fourier transform
In a time-based signal, a low frequency signal is
one which changes slowly, whereas a high
frequency signal has a more rapid change. To
extend this concept to a spatial signal, it is
easy to see that low-frequency data occurs where
intensity values change slowly, i.e. a smooth
gradient, and high frequencies equate to a rapid
change in intensity, i.e. a sharp edge. Armed
with these concepts, we can now anticipate the
results of filtering an image.
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Frequency Filter
Frequency filtering is based on the Fourier
Transform. The operator usually takes an image
and a filter function in the Fourier domain. This
image is then multiplied with the filter function
in a pixel-by-pixel fashion
where F(k,l) is the input image in the Fourier
domain, H(k,l) the filter function and G(k,l) is
the filtered image. To obtain the resulting image
in the spatial domain, G(k,l) has to be
re-transformed using the inverse Fourier
Transform.
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The form of the filter function determines the
effects of the operator. There are basically
three different kinds of filters lowpass,
highpass and bandpass filters. A low-pass filter
attenuates high frequencies and retains low
frequencies unchanged. The result in the spatial
domain is equivalent to that of a smoothing
filter as the blocked high frequencies
correspond to sharp intensity changes, i.e. to
the fine-scale details and noise in the spatial
domain image. A highpass filter, on the other
hand, yields edge enhancement or edge detection
in the spatial domain, because edges contain many
high frequencies. Areas of rather constant
graylevel consist of mainly low frequencies and
are therefore suppressed. A bandpass attenuates
very low and very high frequencies, but retains a
middle range band of frequencies. Bandpass
filtering can be used to enhance edges
(suppressing low frequencies) while reducing the
noise at the same time (attenuating high
frequencies).
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Low pass filtering
The most simple lowpass filter is the ideal low
pass. It suppresses all frequencies higher than
the cut-off frequency and leaves smaller
frequencies unchanged
In most implementations, Do is given as a
fraction of the highest frequency represented in
the Fourier domain image.
The drawback of this filter function is a
ringing effect that occurs along the edges of the
filtered spatial domain image. This phenomenon is
illustrated in the next Figure 5, which shows the
shape of the one-dimensional filter in both the
frequency and spatial domains for two different
values of Do .
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Ideal low pass filter
We obtain the shape of the two-dimensional filter
by rotating these functions about the y-axis. As
mentioned earlier, multiplication in the Fourier
domain corresponds to a convolution in the
spatial domain. Such a kernel will have large
positive coefficients at its center, but these
will be surrounded by a ring of smaller, negative
coefficients.
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Ideal low pass filter
Top Original image. Bottom Image filtered with
ideal lowpass filter on Y axis, normalized cutoff
frequency .15. X axis is an all pass.
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Ideal low pass filter
When we try to use an rectangular lowpass filter
in the Y direction two things are illustrated.
First, an ideal rectangular filter cannot be used
because it creates "ringing" artifacts, the same
as in a one-dimensional transform. The second and
more important realization is that a filter
varying only in the Y frequency direction, and
equal across all X, has its effects only in the Y
direction of the image. We expect this from the
rotation property, and from this we can infer,
properly it turns out, that a filter is just as
seperable as the transform, and therefore the
direction of a filter will be the direction of
its effect. Notice the way the shadows ripple up
and down from horizonal lines in the original
image, whereas vertical lines such as the edge of
the car door are unaffected.
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Butterworth low pass filter
Better results can be achieved with a Gaussian
shaped filter function. The advantage is that the
Gaussian has the same shape in the spatial and
Fourier domains and therefore does not incur the
ringing effect in the spatial domain of the
filtered image. A commonly used discrete
approximation to the Gaussian is the Butterworth
filter. Applying this filter in the frequency
domain shows a similar result to the Gaussian
smoothing in the spatial domain. One difference
is that the computational cost of the spatial
filter increases with the standard deviation
(i.e. with the size of the filter kernel),
whereas the costs for a frequency filter are
independent of the filter function. Hence, the
spatial Gaussian filter is more appropriate for
narrow lowpass filters, while the Butterworth
filter is a better implementation for wide
lowpass filters.
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Smoothing Through Low Pass Filters
Top Original image. Bottom Image filtered with
5th order Butterworth lowpass filter, normalized
cutoff frequency .3
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Smoothing Through Low Pass Filters
This indicates which frequencies will be kept,
just the very lowest. This binary image which
contains 1s at the center and zeros everywhere
else is multiplied by the Real and Imag arrays.
This means that only the longest wavelength sine
waves remain in the list. In fact anything finer
than six variations per image width or height is
excluded.
This is the result of performing the low pass
filtering operation on figure 5.
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It is also possible to do much more complicated
filtering operations
This filter consists of a lot of small areas
which correspond to the peaks in the amplitude
spectrum (figure 4), which form a geometric
pattern and are, therefore caused by one periodic
source. This binary image which contains 1s in
the dots and zeros everywhere else is multiplied
by the Real and Imag arrays. This means that only
the variations caused by the forming fabric
remain in the list.
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Highpass Filters and Band Pass Filters
The same principles apply to highpass filters. We
obtain a highpass filter function by inverting
the corresponding lowpass filter, e.g. an ideal
highpass filter blocks all frequencies smaller
than Do and leaves the others unchanged.
Bandpass filters are a combination of both
lowpass and highpass filters. They attenuate all
frequencies smaller than a frequency Do and
higher than a frequency D1 , while the
frequencies between the two cut-offs remain in
the resulting output image. We obtain the filter
function of a bandpass by multiplying the filter
functions of a lowpass and of a highpass in the
frequency domain, where the cut-off frequency of
the lowpass is higher than that of the highpass.
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Sharpening Through Highpass Filters
If we call the transform of the original image A
and a fully attenuating highpass filter H, then
the transform of the highpassed image B(u,v)
A(u,v)H(u,v). Therefore we can create any linear
combination C aA bB aA b(AH) A(a bH)
and therefore we can create our sharpening filter
H'(u,v) (a bH(u,v)). By selecting a good
ratio of a to b as well as choosing the right
cutoff frequency for the filter, we can therefore
create natural looking sharpening of the photo.
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Text orientation finding - Example
Finally, we present an example (i.e. text
orientation finding) where the Fourier Transform
is used to gain information about the geometric
structure of the spatial domain image. Text
recognition using image processing techniques is
simplified if we can assume that the text lines
are in a predefined direction. Here we show how
the Fourier Transform can be used to find the
initial orientation of the text and then a
rotation can be applied to correct the error. We
illustrate this technique using
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Text orientation finding - Example
The logarithm of the magnitude of its Fourier
transform are
We can see that the main values lie on a vertical
line, indicating that the text lines in the input
image are horizontal.