[PPT] - CS 4495 Computer Vision Frequency2: Sampling and Aliasing Aaron PowerPoint Presentation

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

Aaron Bobick School of Interactive Computing

CS 4495 Computer Vision Frequency2: Sampling and Aliasing

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

Administrivia

Project 1 is due tonight. Submit what you have at the

deadline.

Next problem set – stereo – will be out Thursday Sept. 11

and will be due Tuesday evening, Sept 23rd, 11:55pm.

It is easier…
Readings for this week: Still FP Chapter 4

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

Salvador Dali “Gala Contemplating the Mediterranean Sea, which at 30 meters becomes the portrait

f Abraham Lincoln”, 1976

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

Fourier: A nice set of basis

Teases away fast vs. slow changes in the image.

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

Fourier Transform

) ( ) ( ) ( ω ω ω iI R F + =

We want to understand the frequency ω of our signal. So, let’s reparametrize the signal by ω instead of x:

) +φ ωx Asin(

f(x) F(ω)

Fourier Transform

F(ω) f(x)

Inverse Fourier Transform For every ω from 0 to inf, (actually –inf to inf), F(ω) holds the amplitude A and phase φ of the corresponding sine

How can F hold both? Complex number trick!

2 2

) ( ) ( ω ω I R A + ± =

) ( ) ( tan 1 ω ω φ R I

−

=

And we can go back: Even Odd

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

Frequency Spectra – Even/Odd

Frequency actually goes from –inf to inf. Sinusoid example: Even (cos) ω ω Odd (sin) ω Magnitude Real Imaginary Power

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

2D Fourier Transforms

The two dimensional version: .
And the 2D Discrete FT:
Works best when you put the origin of k in the middle….

2 1 1 ( )

1 ( ) ( , ) ,

i y N x N N x y x y

x y

k x k y

F k f x k y e N

π = − = − = = −

+

= ∑ ∑

( ) ( )

2 (

)

, ,

i

ux vy

F u v f x y e dx dy

π ∞ − − ∞ −∞ ∞

+

∫ = ∫

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

Linearity of Sum

50 100 150 200 250 50 100 150 200 250 20 40 60 80 100 120 20 40 60 80 100 120 50 100 150 200 250 50 100 150 200 250 20 40 60 80 100 120 20 40 60 80 100 120 50 100 150 200 250 50 100 150 200 250 20 40 60 80 100 120 20 40 60 80 100 120

+ =

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

Extension to 2D – Complex plane

Both a Real and Im version

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

Examples

B.K. Gunturk

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

Fourier Transform and Convolution

h f g ∗ =

( ) ( )

∫

∞ ∞ − −

= dx e x g u G

ux i π 2

( ) ( )

∫ ∫

∞ ∞ − ∞ ∞ − −

− = dx d e x h f

ux i

τ τ τ

π 2

( )

[ ] (

)

( )

[ ]

∫ ∫

∞ ∞ − ∞ ∞ − − − −

− = dx e x h d e f

x u i u i τ π τ π

τ τ τ

2 2

( )

[ ]

( )

[ ]

∫ ∫

∞ ∞ − ∞ ∞ − − −

= ' '

' 2 2

dx e x h d e f

ux i u i π τ π

τ τ Let Then

( ) ( )

u H u F =

Convolution in spatial domain Multiplication in frequency domain

⇔

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

Fourier Transform and Convolution

h f g ∗ = FH G = fh g = H F G ∗ =

Spatial Domain (x) Frequency Domain (u) So, we can find g(x) by Fourier transform

g = f ∗ h G = F × H

FT FT IFT

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

Example use: Smoothing/Blurring

We want a smoothed function of f(x)

( ) ( ) ( )

x h x f x g ∗ =

H(u) attenuates high frequencies in F(u) (Low-pass Filter)!

Convolution in space is multiplication in freq:

( ) ( ) ( )

u H u F u G =

πσ 2 1 u

( )

u H

( )

     − =

2 2

2 1 exp 2 1 σ σ π x x h

Let us use a Gaussian kernel

σ

( )

x h x

Fat Gaussian in space is skinny Gaussian in

frequency. Why?

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

2D convolution theorem example

* f(x,y) h(x,y) g(x,y) |F(sx,sy)| |H(sx,sy)| |G(sx,sy)| ( or |F(u,v)| )

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

Low and High Pass filtering

Ringing

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

Fourier Transform smoothing pairs

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

Properties of Fourier Transform

Spatial Domain (x) Frequency Domain (u)

Linearity

( ) ( )

x g c x f c

2 1

+

( ) ( )

u G c u F c

2 1

+

Shifting

( )

x x f −

( )

u F e

ux i 2π −

Symmetry

( )

x F

( )

u f −

Conjugation

( )

x f ∗

( )

u F −

∗

Convolution

( ) ( )

x g x f ∗

( ) ( )

u G u F

Differentiation

( )

n n

dx x f d

( ) ( )

u F u i

n

π 2

Scaling

( )

ax f       a u F a 1 Shrink Stretch

Differentiate

Multiply by u

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

Fourier Pairs (from Szeliski)

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

Fourier Transform Sampling Pairs

FT of an “impulse train” is an impulse train

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

Sampling and Aliasing

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

Sampling and Reconstruction

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

Sampled representations

How to store and compute with continuous functions?
Common scheme for representation: samples
write down the function’s values at many points
S. Marschner

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

Reconstruction

Making samples back into a continuous function
for output (need realizable method)
for analysis or processing (need mathematical method)
amounts to “guessing” what the function did in between
S. Marschner

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

1D Example: Audio

low high frequencies

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

Sampling in digital audio

Recording: sound to analog to samples to disc
Playback: disc to samples to analog to sound again
how can we be sure we are filling in the gaps correctly?
S. Marschner

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

Sampling and Reconstruction

Simple example: a sign wave
S. Marschner

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

Undersampling

What if we “missed” things between the samples?
Simple example: undersampling a sine wave
unsurprising result: information is lost
S. Marschner

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

Undersampling

What if we “missed” things between the samples?
Simple example: undersampling a sine wave
unsurprising result: information is lost
surprising result: indistinguishable from lower frequency
S. Marschner

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

Undersampling

What if we “missed” things between the samples?
Simple example: undersampling a sine wave
unsurprising result: information is lost
surprising result: indistinguishable from lower frequency
also was always indistinguishable from higher frequencies
aliasing: signals “traveling in disguise” as other frequencies
S. Marschner

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

Aliasing in video

S. Seitz

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

Aliasing in images

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

What’s happening?

Input signal:

x = 0:.05:5; imagesc(sin((2.^x).*x))

Plot as image:

Alias! Not enough samples

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

Antialiasing

What can we do about aliasing?
Sample more often
Join the Mega-Pixel craze of the photo industry
But this can’t go on forever
Make the signal less “wiggly”
Get rid of some high frequencies
Will loose information
But it’s better than aliasing

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

Preventing aliasing

Introduce lowpass filters:
remove high frequencies leaving only safe, low frequencies
choose lowest frequency in reconstruction (disambiguate)
S. Marschner

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

(Anti)Aliasing in the Frequency Domain

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

Impulse Train

■ Define a comb function (impulse train) in 1D as follows

[ ] [ ]

M k

comb x x kM δ

∞ =−∞

= −

∑

where M is an integer

2[ ]

comb x x

1

B.K. Gunturk

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

Impulse Train in 1D

2( )

comb x x u

1

12 1 2

1 ( ) 2 comb u

1 2 2

Scaling

( )

ax f       a u F a 1 Remember:

B.K. Gunturk

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

Fourier Transform of an impulse train is also an impulse train:

Impulse Train in 2D (bed of nails)

( )

1 , ,

k l k l

k l x kM y lN u v MN M N δ δ

∞ ∞ ∞ ∞ =−∞ =−∞ =−∞ =−∞

  − − ⇔ − −    

∑ ∑ ∑ ∑

1 1 , ( , ) M N

comb u v

, ( , ) M N

comb x y

( )

, ( , )

,

M N k l

comb x y x kM y lN δ

∞ ∞ =−∞ =−∞

≡ − −

∑ ∑

As the comb samples get further apart, the spectrum samples get closer together!

B.K. Gunturk

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

Impulse Train

2[ ]

comb n n u

1

12

1 2

1 ( ) 2 comb u

1 2

B.K. Gunturk

Scaling

( )

ax f       a u F a 1 Remember:

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

Sampling low frequency signal

x

( ) f x

x

M

( )

M

comb x

u

( ) F u

u

1 M

1 ( ) M

comb u

x

( ) ( )

M

f x comb x

u

1

( )* ( )

M

F u comb u B.K. Gunturk

Multiply: Convolve:

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

Sampling low frequency signal

x

( ) f x

u

( ) F u

u

1

( )* ( )

M

F u comb u

x

( ) ( )

M

f x comb x

W W − M W 1 M

1 2W M >

No “problem” if

B.K. Gunturk

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

Sampling low frequency signal

u

1

( )* ( )

M

F u comb u

x

( ) ( )

M

f x comb x

M W 1 M

If there is no overlap, the original signal can be recovered from its samples by low-pass filtering.

1 2M

B.K. Gunturk

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

Sampling high frequency signal

u

( ) F u

W W −

u

1

( )* ( )

M

F u comb u

( ) ( )

M

f x comb x

W 1 M

Overlap: The high frequency energy is folded over into low

frequency. It is “aliasing” as lower

frequency energy. And you cannot fix it once it has happened.

x

( ) f x

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

Sampling high frequency signal

u

( ) F u

u

[ ]

( )* ( ) ( )

M

f x h x comb x

W W − 1M

Anti-aliasing filter

u

W W −

( )* ( ) f x h x

1 2M

B.K. Gunturk

x

( ) f x

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

Sampling high frequency signal

u

[ ]

( )* ( ) ( )

M

f x h x comb x

1 M

u

( ) ( )

M

f x comb x

W 1 M

■ Without anti-aliasing filter: ■ With anti-aliasing filter:

B.K. Gunturk

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

Aliasing in Images

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

Image half-sizing

This image is too big to fit on the screen. How can we reduce it? How to generate a half- sized version?

S. Seitz

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

Image sub-sampling

Throw away every other row and column to create a 1/2 size image

called image sub-sampling

1/4 1/8

S. Seitz

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

Image sub-sampling

1/4 (2x zoom) 1/8 (4x zoom) Aliasing! What do we do? 1/2

S. Seitz

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

Gaussian (lowpass) pre-filtering

G 1/4 G 1/8 Gaussian 1/2

Solution: filter the image, then subsample

Filter size should double for each ½ size reduction. Why?
S. Seitz

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

Subsampling with Gaussian pre-filtering

G 1/4 G 1/8 Gaussian 1/2

S. Seitz

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

Compare with...

1/4 (2x zoom) 1/8 (4x zoom) 1/2

S. Seitz

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

Campbell-Robson contrast sensitivity curve

The higher the frequency the less sensitive human visual system is…

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

Lossy Image Compression (JPEG)

Block-based Discrete Cosine Transform (DCT) on 8x8

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

Using DCT in JPEG

The first coefficient B(0,0) is the DC component, the

average intensity

The top-left coeffs represent low frequencies, the bottom

right – high frequencies

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

Image compression using DCT

DCT enables image compression by concentrating most

image information in the low frequencies

Lose unimportant image info (high frequencies) by cutting

B(u,v) at bottom right

The decoder computes the inverse DCT – IDCT
Quantization Table

3 5 7 9 11 13 15 17 5 7 9 11 13 15 17 19 7 9 11 13 15 17 19 21 9 11 13 15 17 19 21 23 11 13 15 17 19 21 23 25 13 15 17 19 21 23 25 27 15 17 19 21 23 25 27 29 17 19 21 23 25 27 29 31

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

JPEG compression comparison

89k 12k

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

Maybe the end?

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

Or not!!!

A teaser on pyramids…

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

Image Pyramids

Known as a Gaussian Pyramid [Burt and Adelson, 1983]

In computer graphics, a mip map [Williams, 1983]
A precursor to wavelet transform
S. Seitz

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

Band-pass filtering

Laplacian Pyramid (subband images)
Created from Gaussian pyramid by subtraction

Gaussian Pyramid (low-pass images)

These are “bandpass” images (almost).

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

Why almost bandpass?

Why are the subtracted sequential Gaussians (“difference
f Gaussians”) “almost” bandpass?

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

Laplacian Pyramid

How can we reconstruct (collapse) this

pyramid into the original image?

Need this! Original image

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

Computing the Laplacian Pyramid

Reduce Expand Need Gk to reconstruct Don’t worry about these details – YET! (PS4?)

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

What can you do with band limited imaged?

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

Apples and Oranges in bandpass

L0 L2 L4 Reconstructed Coarse Fine

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

What can you do with band limited imaged?

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

Really the end…

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

Fourier Transform: Properties

Bahadir K. Gunturk 71

■ Linearity ■ Shifting ■ Modulation ■ Convolution ■ Multiplication ■ Separable functions

( , ) ( , ) ( , ) ( , ) af x y bg x y aF u v bG u v + ⇔ +

( , )* ( , ) ( , ) ( , ) f x y g x y F u v G u v ⇔

( , ) ( , ) ( , )* ( , ) f x y g x y F u v G u v ⇔

( , ) ( ) ( ) ( , ) ( ) ( ) f x y f x f y F u v F u F v = ⇔ =

2 ( )

( , ) ( , )

j ux vy

f x x y x e F u v

π − +

− − ⇔

2 ( )

( , ) ( , )

j u x v y

e f x y F u u v v

π +

⇔ − −

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

Fourier Transform: Properties

Bahadir K. Gunturk 72

■ Separability

2 ( )

( , ) ( , )

j ux vy

F u v f x y e dxdy

π ∞ ∞ − + −∞ −∞

= ∫ ∫

2 2

( , )

j ux j vy

f x y e dx e dy

π π ∞ ∞ − − −∞ −∞

  =    

∫ ∫

2

( , )

j vy

F u y e dy

π ∞ − −∞

= ∫ 2D Fourier Transform can be implemented as a sequence of 1D Fourier Transform operations.

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

Fourier Transform: Properties

Bahadir K. Gunturk 73

■ Energy conservation

2 2

( , ) ( , ) f x y dxdy F u v dudv

∞ ∞ ∞ ∞ −∞ −∞ −∞ −∞

=

∫ ∫ ∫ ∫

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

Fourier Transform: 2D Discrete Signals

Bahadir K. Gunturk 74

■ Fourier Transform of a 2D discrete signal is defined as where

2 ( )

( , ) [ , ]

j um vn m n

F u v f m n e

π ∞ ∞ − + =−∞ =−∞

= ∑ ∑

1 1 , 2 2 u v − ≤ <

1/2 1/2 2 ( ) 1/2 1/2

[ , ] ( , )

j um vn

f m n F u v e dudv

π + − −

= ∫ ∫

■ Inverse Fourier Transform

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

Fourier Transform: Properties

Bahadir K. Gunturk 75

■ Periodicity: Fourier Transform of a discrete signal is periodic with period 1.

( )

2 ( ) ( )

( , ) [ , ]

j u k m v l n m n

F u k v l f m n e

π ∞ ∞ − + + + =−∞ =−∞

+ + = ∑ ∑

( )

2 2 2

[ , ]

j um vn j km j ln m n

f m n e e e

π π π ∞ ∞ − + − − =−∞ =−∞

= ∑ ∑

2 ( )

[ , ]

j um vn m n

f m n e

π ∞ ∞ − + =−∞ =−∞

= ∑ ∑

1 1

( , ) F u v =

Arbitrary integers

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

Fourier Transform: Properties

Bahadir K. Gunturk 76

■ Linearity, shifting, modulation, convolution, multiplication, separability, energy conservation properties also exist for the 2D Fourier Transform of discrete signals.

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

Fourier Transform: Properties

Bahadir K. Gunturk 77

■ Linearity ■ Shifting ■ Modulation ■ Convolution ■ Multiplication ■ Separable functions ■ Energy conservation

[ , ] [ , ] ( , ) ( , ) af m n bg m n aF u v bG u v + ⇔ +

2 ( )

[ , ] ( , )

j um vn

f m m n n e F u v

π − +

− − ⇔ [ , ] [ , ] ( , )* ( , ) f m n g m n F u v G u v ⇔

[ , ]* [ , ] ( , ) ( , ) f m n g m n F u v G u v ⇔

2 ( )

[ , ] ( , )

j u m v n

e f m n F u u v v

π +

⇔ − −

[ , ] [ ] [ ] ( , ) ( ) ( ) f m n f m f n F u v F u F v = ⇔ =

2 2

[ , ] ( , )

m n

f m n F u v dudv

∞ ∞ ∞ ∞ =−∞ =−∞ −∞ −∞

=

∑ ∑ ∫ ∫

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

Fourier Transform: Properties

Bahadir K. Gunturk 78

■ Define Kronecker delta function ■ Fourier Transform of the Kronecker delta function

1, for 0 and [ , ] 0, otherwise m n m n δ = =   =    

( ) ( )

2 2

( , ) [ , ] 1

j um vn j u v m n

F u v m n e e

π π

δ

∞ ∞ − + − + =−∞ =−∞

  = = =  

∑ ∑

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

Fourier Transform: Properties

Bahadir K. Gunturk 79

■ Fourier Transform of 1 To prove: Take the inverse Fourier Transform of the Dirac delta function and use the fact that the Fourier Transform has to be periodic with period 1.

( )

2

( , ) 1 ( , ) 1 ( , )

j um vn m n k l

f m n F u v e u k v l

π

δ

∞ ∞ ∞ ∞ − + =−∞ =−∞ =−∞ =−∞

  = ⇔ = = − −  

∑ ∑ ∑ ∑

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

Sampling Theorem

Continuous signal: Shah function (Impulse train):

( )

x f x

Sampled function:

( ) ( ) ( ) ( ) ( )

∑

∞ −∞ =

− = =

n s

nx x x f x s x f x f δ

( )

x s x x

( ) ( )

∑

∞ −∞ =

− =

n

nx x x s δ

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

Sampling Theorem

Sampled function:

( ) ( ) ( ) ( ) ( )

∑

∞ −∞ =

− = =

n s

nx x x f x s x f x f δ F

S u

( )= F u ( )∗S u ( )= F u ( )∗ 1

x0 δ u − n x0      

n=−∞ ∞

∑

( )

u F

max

u

A

u

( )

u FS

max

u

x A 1x

u

Only if

max

2 1 x u ≤

Sampling frequency

1 x

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Frequency and Fourier Transform CS 4495 Computer Vision – A. Bobick

Nyquist Theorem

If

max

2 1 x u >

( )

u FS

max

u

x A 1x

u

Aliasing When can we recover from ?

( )

u F

( )

u FS

Only if

max

2 1 x u ≤

(Nyquist Frequency) We can use

( )

     < =

therwise

2 1 x u x u C

Then

( ) ( ) ( )

u C u F u F

S

=

( ) ( ) [ ]

u F x f IFT =

and Sampling frequency must be greater than

max