Download presentation

Presentation is loading. Please wait.

1
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)

2
Fourier Transform F( ) is computed from f(x) by the Fourier Transform:

3
Example: Box Function

4
Box Function and Its Transform

5
Cosine and Its Transform 1 If f(x) is even, so is F( )

6
Sine and Its Transform 1 -- If f(x) is odd, so is F( )

7
Delta Function and Its Transform Fourier transform and inverse Fourier transform are qualitatively the same, so knowing one direction gives you the other

8
Shah Function and Its Transform Moving the spikes closer together in the spatial domain moves them farther apart in the frequency domain!

9
Gaussian and Its Transform

10
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

11
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

12
2D Discrete Fourier Transform

13
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

14
Qualitative Filters FG = = = H Low-pass High-pass Band-pass

15
Low-Pass Filtered Image

16
High-Pass Filtered Image

17
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

18
Convolution Example

19
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=F G –Take the inverse Fourier transform of H, to get h –Then h=f g Multiplication in the spatial domain is the same as convolution in the frequency domain

20
Sampling in Spatial Domain Sampling in the spatial domain is like multiplying by a spike function

21
Sampling in Frequency Domain Sampling in the frequency domain is like convolving with a spike function

22
Reconstruction in Frequency Domain To reconstruct, we must restore the original spectrum That can be done by multiplying by a square pulse

23
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

24
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

25
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

26
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):

Similar presentations

© 2020 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google