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CS443: Digital Imaging and Multimedia

Introduction to Spectral Techniques

Spring 2008

Ahmed Elgammal

Dept. of Computer Science

Rutgers University

Outlines

yFourier Series and Fourier integral yFourier Transform (FT) yDiscrete Fourier Transform (DFT) yAliasing and Nyquest Theorem y2D FT and 2D DFT yApplication of 2D-DFT in imaging yInverse Convolution yDiscrete Cosine Transform (DCT)

Sources:

yBurger and Burge "Digital Image Processing" Chapter 13, 14, 15 yFourier transform images from Prof. John M. Brayer @ UNM 2 yRepresentation and Analysis of Signals in the frequency domain yAudio: 1D temporal signal yImages: 2D spatial signal yVideo: 2D spatial signal + 1D temporal signal yHow to decompose a signal into sine and cosine function. Also known as harmonic functions. yFourier Transform, Discrete Fourier Transform,

Discrete Cosine Transform

Basics

ySine and Cosine functions are periodic yAngular Frequency: number of oscillations over the distance 2π

T: the time for a complete cycle

3

Basics

yAngular Frequency (ω) and Amplitude (a) yAngular Frequency: number of oscillations over the distance 2π

T: the time for a complete cycle

yCommon Frequency f: number of oscillation in a unit time

Basics

yPhase: Shifting a cosine function along the x axis by a distance ϕ change the phase of the cosine wave. ϕ denotes the phase angle 4 yAdding cosines and sines with the same frequency results in another sinusoid

Fourier Series and Fourier integral

yWe can represent any periodic function as sum of pairs of sinusoidal functions- using a basic (fundamental) frequency yFourier Integral: any function can be represented as combination of sinusoidal functions with many frequencies 5 yFourier Integral yHow much of each frequency contributes to a given function

Fourier Transform

6 yFourier transform yInverse Fourier transform

Frequency domain

FT

Temporal or spatial domain

Fourier Transform

yThe forward and inverse transformation are almost similar (only the sign in the exponent is different) yany signal is represented in the frequency space by its frequency "spectrum" yThe Fourier spectrum is uniquely defined for a given function. The opposite is also true. yFourier transform pairs 7 8 9 ySince of FT of a real function is generally complex, we use magnitude and phase

Lower frequencies ⇒ narrower power spectrum

Higher frequencies ⇒ wider power spectrum

u |F(u)| 2 x f(x)

Power Spectrum

10

Properties

ySymmetry: for real-valued functions yLinearity ySimilarity yShift Property 11

Important Properties:

yFT and Convolution yConvolving two signals is equivalent to multiplying their

Fourier spectra

y Multiplying two signals is equivalent to convolving their

Fourier spectra

yFT of a Gaussian is a Gaussian

Discrete Fourier Transform

yIf we discretize f(x) using uniformly spaced samples f(0), f(1),...,f(N-1), we can obtain FT of the sampled function yImportant Property:

Periodicity F(m)=F(m+N)

One period

12 Image from Computer Graphics: Principles and Practice by Foley, van Dam, Feiner, and Hughes

Impulse function

13 14

Sampling and Aliasing

yDifferences between continuous and discrete images yImages are sampled version of a continuous brightness function. successful sampling unsuccessful sampling 15

Sampling and Aliasing

ySampling involves loss of information yAliasing: high spatial frequency components appear as low spatial frequency components in the sampled signal successful sampling unsuccessful sampling Java applet from: http://www.dsptutor.freeuk.com/aliasing/AD102.html

Aliasing

yNyquist theorem: The sampling frequency must be at least twice the highest frequency present for a signal to be reconstructed from a sampled version. (Nyquist frequency) 16

Temporal Aliasing

A AA A A

The wheel appears to be moving backwards at about

angular frequency 17

Sampling, aliasing, and DFT

yDFT consists of a sum of copies of the FT of the original signal shifted by the sampling frequency: yIf shifted copies do not intersect: reconstruction is possible. yIf shifted copies do intersect: incorrect reconstruction, high frequencies are lost (Aliasing)

2-dimension

In two dimension

•These terms are sinusoids on the x,y plane whose orientation and frequency are defined by u,v 18

DFT in 2D

yFor a 2D periodic function of size MxN, DFT is defined as: yInverse transform 19 20

Visualizing 2D-DFT

yThe FT tries to represent all images as a summation of cosine-like images

Images of pure cosines

FT •Center of the image:the origin of the frequency coordinate system •u-axis: (left to right) the horizontal component of frequency •v-axis: (bottom-top) the vertical component of frequency •Center dot (0,0) frequency : image average •high frequencies in the vertical direction will cause bright dots away from the center in the vertical direction. •high frequencies in the horizontal direction will cause bright dots away from the center in the horizontal direction. ySince images are real numbers (not complex) FT image is symmetric around the origin.

FT: symmetry

FT is shift invariant

21
yIn general, rotation of the image results in equivalent rotation of its FT

Why it is not the case ?

•Edge effect ! •FT always treats an image as if it were part of a periodically replicated array of identical images extending horizontally and vertically to infinity •Solution: "windowing" the image

Edge effect

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•notice a bright band going to high frequencies perpendicular to the strong edges in the image •Anytime an image has a strong- contrast, sharp edge the gray values must change very rapidly.

It takes lots of high frequency

power to follow such an edge so there is usually such a line in its magnitude spectrum. 23

ScalingPeriodic image patterns

24

RotationOriented, elongated structures

25

Natural Images

•notice a bright band going to high frequencies perpendicular to the strong edges in the image •Anytime an image has a strong- contrast, sharp edge the gray values must change very rapidly.

It takes lots of high frequency

power to follow such an edge so there is usually such a line in its magnitude spectrum. 26

Print patternsLinear Filters in Frequency space

27

Inverse Filters - De-convolution

yHow can we remove the effect of a filter ?

Inverse Filters - De-convolution

yHow can we remove the effect of a filter ? 28
yWhat happens if we swap the magnitude spectra ? yPhase spectrum holds the spatial information (where things are), yPhase spectrum is more important for perception than magnitude spectrum.

The Discrete Cosine Transform (DCT)

yFT and DFT are designed for processing complex- valued signal and always produce a complex- valued spectrum.quotesdbs_dbs4.pdfusesText_8