engi 4559 signal processing for software engineers dr. richard khoury fall 2009

ENGI 4559 Signal Processing for Software Engineers

Dr. Richard KhouryFall 2009

Fourier Analysis in 2D: Overview2D Extensions of Fourier TransformAliasing and Moiré PatternsFourier Transform of ImagesMatlab

2 ENGI 4559 © Dr. Richard Khoury, 2009

2D Extensions: CTFT

ENGI 4559 © Dr. Richard Khoury, 20093

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2D Extensions: DTFT


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2D Extensions: DFT


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2D Extensions: Sampling Theorem1D sampling theorem:

If the maximum frequency contained in an analog signal is Ωmax = B, then it can be perfectly reconstructed from samples taken at the sampling frequency Ωs = 2B (or more).

2D sampling theorem:If the maximum frequency contained in an analog signal is Ω1 = B1 in one direction and Ω2 = B2 in the other, then it can be perfectly reconstructed from samples taken at the sampling frequencies Ωs1 = 2B1 in the first direction and Ωs2 = 2B2 in the other.


2D Extensions: Aliasing


Aliasing in ImagesImagine a perfect digital camera taking a

picture of a checkerboard where each square is one pixel in size

If we sample it (i.e. take the picture) at a resolution of several pixels per square, it works

16 pixels per square:

6 pixels per square:


Aliasing in ImagesAt resolutions less than one pixel per square,

severe aliasing occurs0.9 pixels per square:


Aliasing in ImagesIn DIP, modifying the sampling rate is not always possible

Downsampling (shrinking) a picture would require access to the original scene to resample it

Aliasing is unavoidableOften covered up by blurring the image slightly before

resizing


Aliasing in ImagesAliasing is very visible in diagonal edges


Moiré Patterns


Another common type of artefact

Result of sampling an image that already has a sampled periodic element

Typically visible in Two overlapping insect

window screensStriped material on TVDigital scan of

newspaper pictures

Moiré PatternsTake two patterns that are

individuallyClean and devoid of

interferencePeriodic or nearly periodicAt different angles from each

otherSuperimposing one on the

other creates a new pattern with frequencies that do not exist in the originals


Fourier Transform of ImagesThe Fourier transform of an image is

generally a complex number

where


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Fourier Transform of ImagesThe magnitude r12 is the Fourier spectrum or

frequency spectrumr12² is the power spectrum

The exponent ω12 is the phase angleWe can display the resulting matrices as

images!


Fourier Transform of ImagesLet’s run an experiment!We’ll use this imageWe compute the DFT and display

the spectrumAll the highest values are in the

cornersThey correspond to the zero

frequenciesIt’s more convenient (and

conventional) to shift the graphic to display them in the middle

Still too dark to make out the detailsSo we normalize and display


)(log1 1210 r

Fourier Transform of ImagesThe phase angle can be displayed directlyWhat does it mean?

The spectrum represents the amplitude of the sinusoids that form the signal

The phase is the displacement of each sinusoid to the origin


Fourier Transform of Images


)( 1212

jerifft )( 12rifft )( 12jeifftLenna 12r 12

Fourier Transform of ImagesThe Fourier Spectrum represents intensitiesTheir positions in the spectrum has meaning

too


Center value (origin, since we shifted the graph) is the average intensity value of the image

Low frequency values (near the origin) are slow-varying intensity regions

High frequency values (far from the origin) are regions with fast variations in intensity

MatlabMatlab already implements the 1D and 2D

DFTF = fft2(M) computes the 2D DFT of a matrix

MM = ifft2(F) computes the 2D inverse DFT of

a Fourier matrix, recovering the original matrix M

F = fftshift(M) performs a quadrant shift of the matrix, which is convenient to display the Fourier spectrum


Matlab>> z = 4 + 3i;

>> r = abs(z)

r = 5

>> r = sqrt(real(z)^2 + imag(z)^2)

r = 5

>> theta = atan2(imag(z),real(z))

theta = 0.6435

>> z = r *exp(i *theta)

z = 4.0000 + 3.0000i


)/arctan(

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MatlabComputing the Fourier spectrum and phase angle%Read the image using the method we knowimagereal = imread('lenna.png');%Convert from uint8 to double for mathimagereal = double(imagereal);%Fourier transformF = fft2(imagereal);%Fourier spectrumSpectrum = sqrt( real(F).^2 + imag(F).^2 );%Shift the spectrum, as per conventionSpectrumS = fftshift(Spectrum);%Phase anglePhase = atan2( imag(F), real(F) );


MatlabDisplaying the spectrum and phase

figure;

colormap(gray)

imagesc(1+log10(SpectrumS))

title('Fourier Spectrum')

figure;

colormap(gray)

imagesc(Phase)

title('Phase angle')


Don’t forget to normalize the spectrum display!

MatlabReconstructing the image%Shift the spectrum backSpectrumSS = fftshift(SpectrumS);%Compute the values of Fourier TransformF2 = SpectrumSS .* exp(sqrt(-1)* Phase );%Perform the inverse DFT%Note that rounding errors can leave a small % imaginary part (10-14) which we must discardreconstructed = real(ifft2(F2));%Display the imagefigure;colormap(gray)imagesc(reconstructed)title('Reconstructed image')


SummaryWe learned the 2D equations of the

continuous-time, discrete-time, and discrete Fourier transform, and of the sampling theorem

We considered the effect of aliasing and Moiré patterns, and the difference between these two phenomena

We applied our knowledge by computing the Fourier transform of images, and splitting the result into the frequency spectrum and phase angle componentsAnd we learned the meaning of these two

components


engi 4559 signal processing for software engineers dr. richard khoury fall 2009

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