Lecture 13 The frequency Domain (1) Dr. Masri Ayob

Lecture 13Lecture 13The frequency The frequency

Domain (1)Domain (1)

Dr. Masri AyobDr. Masri Ayob

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The Twilight ZoneThe Twilight Zone

Image data can be represented in either the spatial domain or the frequency domain. The frequency domain contains the same information as the spatial domain but in a vastly different form.

Useful for data compressionMore efficient for certain image operations

Spatial Domain

For each location in the image, what is the value of the light intensity at that location?

Spatial Domain

For each location in the image, what is the value of the light intensity at that location?

Frequency Domain

For each frequency component in the image, what is power or its amplitude?

Frequency Domain

For each frequency component in the image, what is power or its amplitude?

Various frequency domain representations exist but the two predominant representations are the Fourier and Discrete Cosine representations.Frequency domain - an alternative representation of an image based on the frequencies of brightness or colour variation in the image.

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Fourier TransformFourier Transform

Spatial Domain vs Frequency Domain

ft

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Fourier TransformFourier Transform

Why?• alternative description• efficient calculation• less sensitive for disturbances• Obey the convolution thorem

More efficient, easier:• convolution• correlation• filtering• differentiating• shifting• compression

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Fourier TransformFourier Transform

Applications wide ranging and ever present in modern lifeApplications wide ranging and ever present in modern life

• TelecommsTelecomms - GSM/cellular phones,

• Electronics/ITElectronics/IT - most DSP-based applications,

• EntertainmentEntertainment - music, audio, multimedia,

• Accelerator controlAccelerator control (tune measurement for beam steering/control),

• Imaging, image processing,Imaging, image processing,

• Industry/researchIndustry/research - X-ray spectrometry, chemical analysis (FT spectrometry), PDE solution, radar design,

• MedicalMedical - (PET scanner, CAT scans & MRI interpretation for sleep disorder & heart malfunction diagnosis,

• Speech analysisSpeech analysis (voice activated “devices”, biometry, …).

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Fourier AnalysisFourier Analysis

Many different transforms are used in image processing (far too many begin with the letter H: Hilbert, Hartley, Hough, Hotelling, Hadamard, and Haar). The Fourier representation of any function is possible by determining

The fundamental frequencyThe coefficient of each harmonic

Fourier coefficients are typically Complex-valuedFundamental frequency is determined by the resolution of a discrete image

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Spatial FrequencySpatial Frequency

L = length of the cycle (period of the function).If the variation is spatial and L is a distance, then 1/L is termed the spatial frequency of the variation.

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Spatial FrequencySpatial Frequency

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Spatial FrequencySpatial Frequency

N=100, u=3, A=127

N=100, u=6, A=127

N=100, u=3, A=50

N=100, u=3, A=127, phase = 90

Variation in thex – direction (u)

Sin

Cosine

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Fourier TheoryFourier Theory

Techniques for the analysis and manipulation of spatial frequency.Developed a representation of functions based on frequency.The idea is “any periodic function can be represented as a sum of these simpler sinusoids”.

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Fourier AnalysisFourier Analysis

Any function can be represented as the sum of sine and cosine waves having different amplitudes and wavelengths.Fourier analysis is a way of determining the individual sin/cosine waves that, when added together, construct the desired signalConsider a square wave. Can it be represented as the sum of sin and cosine waves?

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Fourier TheoryFourier Theory

A set of sine and cosine functions having particular frequencies are choose for the representation. basic function

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Fourier AnalysisFourier Analysis

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Fourier TheoryFourier Theory

A weighted sum of these basic function is called a Fourier Series.

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Add a 3rd “harmonic” to the fundamental frequency. The amplitude is less than that of the base and the frequency is 3 times that of the base.

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Fourier TheoryFourier Theory

The weighting factors for each sine and cosine function are known as the Fourier coefficients.

The summation of basic function

No. of terms

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Fourier TheoryFourier Theory

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Fourier TheoryFourier Theory

Add a 3rd “harmonic” to the fundamental frequency. The amplitude is less than that of the base and the frequency is 3 times that of the base.

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Fourier TheoryFourier Theory

Add a 5th “harmonic” to the fundamental frequency. The amplitude is less than that of the base and the frequency is 5 times that of the base.

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Fourier TheoryFourier Theory

Add a 7th and 9th“harmonic” to the fundamental frequency.

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Fourier TheoryFourier Theory

Adding all harmonics up to the 100th.

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Fourier TheoryFourier Theory

Adding all harmonics up to the 200th.

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Fourier TheoryFourier Theory

L: period; u and v are the number of cycles fitting into one horizontal and vertical period, respectively of f(x,y).

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Fourier TheoryFourier Theory

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Discrete Fourier TransformDiscrete Fourier Transform

Fourier theory provides us with a means of determining the contribution made by any basic function to the representation of some function f(x).The contribution is determined by projecting f(x) onto that basis function.This procedure is described as a Fourier transform.

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Discrete Fourier TransformDiscrete Fourier Transform

When applying the procedure to images, we must deal explicitly with the fact that an image is:

Two-dimensionalSampledOf finite extent

These consideration give rise to the Discrete Fourier Transform (DFT).The DFT of an NxN image can be written:

1

0

/)(21

0

),(1

),(N

x

NvyuxjN

y

eyxfN

vuF or

1

0

1

0

)(2sin

)(2cos),(

1),(

N

x

N

y N

vyuxj

N

vyuxyxf

NvuF

(8.5)

Processing the image in frequency domainComplex number

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Discrete Fourier TransformDiscrete Fourier Transform

For any particular spatial frequency specified by u and v, evaluating equation 8.5 tell us how much of that particular frequency is present in the image. There also exist an inverse Fourier Transform that convert a set of Fourier coefficients into an image.

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0

/)(21

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),(1

),(N

x

NvyuxjN

y

evuFN

yxf

Processing the image in spatial domain

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Discrete Fourier TransformDiscrete Fourier Transform

F(u,v) is a complex number:

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Discrete Fourier TransformDiscrete Fourier Transform

The magnitudes correspond to the amplitudes of the basic images in our Fourier representation.The array of magnitudes is termed the amplitude spectrum (or sometime ‘spectrum’).The array of phases is termed the phase spectrum.The power spectrum is simply the square of its amplitude spectrum:

),(),(),(),( 222vuIvuRvuFvuP

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Discrete Fourier TransformDiscrete Fourier Transform

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Discrete Fourier TransformDiscrete Fourier Transform

If we attempt to reconstruct the image with an inverse Fourier Transform after destroying either the phase information or the amplitude information, then the reconstruction will fail.

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FFTFFT

The Fast Fourier Transform is one of the most important algorithms ever developed

Developed by Cooley and Tukey in mid 60s.Is a recursive procedure that uses some cool math tricks to combine sub-problem results into the overall solution.

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DFT vs FFTDFT vs FFT

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DFT vs FFTDFT vs FFT

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DFT vs FFTDFT vs FFT

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PeriodicityPeriodicity

The DFT assumes that an image is part of an infinitely repeated set of “tiles” in every direction. This is the same effect as “circular indexing”.

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Periodicity and WindowingPeriodicity and Windowing

Since “tiling” an image causes “fake” discontinuities, the spectrum includes “fake” high-frequency components

Spatial discontinuities

Windowing minimizes the artificial discontinuities by pre-processing pixel values prior to computing the DFT.

Pixel values are modulated so that they gradually fall to zero at the edges.

Three well-known windowing functions:•Bartlett•Hanning•Blackman

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Windowing FunctionsWindowing Functions

max

maxmax

0

)/(1)(

rr

rrrrrw

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2010

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maxmax12cos08.01cos5.042.0)(r

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Blackman

R is a distance from the centre of the image and rmax is its maximum value.

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FFT PackageFFT Package com.pearsoneduc.ip.opcom.pearsoneduc.ip.op

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FFT Package FFT Package com.pearsoneduc.ip.opcom.pearsoneduc.ip.op

Thank youThank you

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