General Functions A non-periodic function can be represented as a sum of sin’s and cos’s of (possibly) all frequencies: F( ) is the spectrum of the function.

General Functions

• A non-periodic function can be represented as a sum of sin’s and cos’s of (possibly) all frequencies:

• F() is the spectrum of the function f(x)

deFxf xi)(

2

1)(

xixe xi sincos

Fourier Transform

• F() is computed from f(x) by the Fourier Transform:

dxexfF xi )()(

Example: Box Function

21 02

1 1)(

x

xxf

f

ff

fF

sinc2

sin

)(

Box Function and Its Transform

Cosine and Its Transform

-1.5

-1

-0.5

0

0.5

1

1.5

1-1

If f(x) is even, so is F()

Sine and Its Transform

-1.5

-1

-0.5

0

0.5

1

1.5

1

-1

-

If f(x) is odd, so is F()

Delta Function and Its Transform

Fourier transform and inverse Fourier transform arequalitatively the same, so knowing one direction gives you the other

Shah Function and Its Transform

Moving the spikes closer together in the spatial domain moves them farther apart in the frequency domain!

Gaussian and Its Transform

-0.02

0.03

0.08

0.13

0.18

-0.02

0.03

0.08

0.13

0.18

2

2

2

1 x

e

• The spectrum of a functions tells us the relative amounts of high and low frequencies– Sharp edges give high frequencies

– Smooth variations give low frequencies

• A function is bandlimited if its spectrum has no frequencies above a maximum limit– sin, cos are bandlimited

– Box, Gaussian, etc are not

Qualitative Properties

Functions to Images

• Images are 2D, discrete functions• 2D Fourier transform uses product of sin’s and

cos’s (things carry over naturally)• Fourier transform of a discrete, quantized function

will only contain discrete frequencies in quantized amounts

• Numerical algorithm: Fast Fourier Transform (FFT) computes discrete Fourier transforms

2D Discrete Fourier Transform

Filters

• A filter is something that attenuates or enhances particular frequencies

• Easiest to visualize in the frequency domain, where filtering is defined as multiplication:

• Here, F is the spectrum of the function, G is the spectrum of the filter, and H is the filtered function. Multiplication is point-wise

)()()( GFH

Qualitative Filters

F G

=

=

=

H

Low-pass

High-pass

Band-pass

Low-Pass Filtered Image

High-Pass Filtered Image

Filtering in the Spatial Domain

• Filtering the spatial domain is achieved by convolution

• Qualitatively: Slide the filter to each position, x, then sum up the function multiplied by the filter at that position

duuxgufgfxh )()()(

Convolution Example

Convolution Theorem

• Convolution in the spatial domain is the same as multiplication in the frequency domain– Take a function, f, and compute its Fourier transform, F– Take a filter, g, and compute its Fourier transform, G– Compute H=FG– Take the inverse Fourier transform of H, to get h– Then h=fg

• Multiplication in the spatial domain is the same as convolution in the frequency domain

Sampling in Spatial Domain

• Sampling in the spatial domain is like multiplying by a spike function

Sampling in Frequency Domain

• Sampling in the frequency domain is like convolving with a spike function

Reconstruction in Frequency Domain

• To reconstruct, we must restore the original spectrum

• That can be done by multiplying by a square pulse

Reconstruction in Spatial Domain

• Multiplying by a square pulse in the frequency domain is the same as convolving with a sinc function in the spatial domain

Aliasing Due to Under-sampling

• If the sampling rate is too low, high frequencies get reconstructed as lower frequencies

• High frequencies from one copy get added to low frequencies from another

Aliasing Implications

• There is a minimum frequency with which functions must be sampled – the Nyquist frequency– Twice the maximum frequency present in the signal

• Signals that are not bandlimited cannot be accurately sampled and reconstructed

• Not all sampling schemes allow reconstruction– eg: Sampling with a box

More Aliasing

• Poor reconstruction also results in aliasing• Consider a signal reconstructed with a box filter in

the spatial domain (which means using a sinc in the frequency domain):