The frequency domain_part2.ppt

8/11/2019 The frequency domain_part2.ppt

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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 isseperableinto 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 realand imaginarypart or with magnitude

andphase. In image processing, often only the magnitudeof 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 imageThe 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 qwith the real axis. A is

amplitude(or modulus) and qis phase(or argument). Simple algebra tells us

that


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Fourier spectra play an important roleThe Fourier transform of a real function is a complex function

),(),( vuievuF 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 thefrequency

spectrumof 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 spectrumof 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 amplitudespectrum is normally implied.

The power spectrumof an image is simply the square of its amplitude

spectrum :


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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 a[m,n], Figure 1b the magnitude in ascaled form , and Figure 1c the phase.

Both the magnitude and the phase

functions are necessary for the

complete reconstruction of an image

from its Fourier transform. Figure 2ashows what happens when Figure 1a is

restored solely on the basis of the

magnitude information and Figure 2b

shows what happens when Figure 1a isrestored solely on the basis of the

Figure 2: a) Figure 2: b)

Figure 1: a) b) c)


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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

),(),(),(),( NvNuFNvuFvNuFvuF

The N x N block of coefficients that we compute from an N

x N image with our two-dimensional FFT algorithm is asingle period from this infinite sequence.

If f(x,y) is real, its Fourier transform is conjugate symmetry,

that is ),(),( vuFvuF

negative frequencies are mirror images of positive frequencies

The complex conjugateof a complex number

is defined to be


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requency on enFrequency Content Location

The DFT coefficients produced by

the 2D DFT equations , are arrangedin 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 anyperiod 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 resultsIt is common practice to multiply the input image function by (-1)x+y

prior to computing the Fourier transform.

)2/,2/(])1)(,([ NvNuFyxfTransformFourier yx

That isF(0,0) is located at u = N/2 and v = N/2 . Multiplyingf(x,y) by (-

1)x+yshifts the origin ofF(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

u=0 to u=N-1and v=0 to v=N-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:

xN

u

1

yNv 1


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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 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 practicethe 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 twoadditional arrays Real(u,v) and Imag(u,v).

Figure 5 Figure 6


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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. Inorder to do this we must scale the magnitude to lie in a 0-255 range . The

obvious approach of multiplying by a scaling factor

max),(

255

vuF

As uand v increase , the contribution of these high frequencies to the

image becomes less and less important and thus the value of the

corresponding coefficientsF(u,v)become smaller.

For displaying purpose , people use logarithmic mapping of the data.

]1),(log['),( vuFCvuF

Since the logarithm is not defined for 0, many implementations of this

operator add the value 1to the image before taking the logarithm.


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Logarithmic OperatorThe logarithmic operator is a simple point processorwhere the mapping

functionis a logarithmic curve. In other words, each pixel value is

replaced with its logarithm.The scaling constant cis chosen so that the maximum output value is 255

(providing an 8-bit format). That means ifRis the value with the

maximum magnitude in the input image, cis 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 closeto the origin, the compression is

smaller for an image containing

small input values. The mapping

functionis shown for two differentranges of input values in Figure


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Logarithmic Operator

The image shows one bright

spot in the center and twodarker 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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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 intensityinformation, applying the logarithmic operator

might lead to loss of information.


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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 intensityvalues in the output of the Fourier Transform

(in this case up to ).

This image is the result of first

multiplying each pixel with 0.0001andthen 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 2pixel 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, twopoints 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 patternsThe 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 saysthat 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 equallywell 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 morerapid 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.

F Filt


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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:

whereF(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, highpassand bandpassfilters. 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 smallerfrequencies 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 theshape 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 filterWe obtain the shape

of the two-

dimensional filter by

rotating thesefunctions 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 positivecoefficients 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, normalizedcutoff 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 theY 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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Better results can be achieved with a Gaussianshaped filterfunction. 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 smoothingin 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 areindependent 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.

Butterworth low pass filter


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Smoothing Through Low Pass Filters

Top: Original image.

Bottom: Image filteredwith 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 ismultiplied 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 orheight 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 peaksin 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 imageB(u,v) =

A(u,v)*H(u,v). Therefore we can create any

linear combination C = aA + bB = aA +b(A*H) = A(a + bH)and therefore we can

create our sharpening filterH'(u,v) = (a +

bH(u,v)). By selecting a good ratio of ato bas

well as choosing the right cutoff frequency for

the filter, we can therefore create natural

looking sharpening of the photo.

Text orientation finding Example


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Finally, we present an example (i.e.text

orientation finding) where the FourierTransform 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

Text orientation finding - Example

Text orientation finding Example


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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.

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