sampling and quantization - uc santa barbarasampling).pdf · jan 21, 2003 sampling 2 sampling and...

Sampling and Quantization

Lecture #5January 21, 2002

Jan 21, 2003 Sampling 2

Sampling and Quantization

� Spatial Resolution (Sampling)� Determines the smallest perceivable image

detail.� What is the best sampling rate?

� Gray-level resolution (Quantization)� Smallest discernible change in the gray level

value.� Is there an optimal quantizer?

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Image sampling and quantization

In 1-D

f(x,y)

(Continuous image)

Samplerfs(m,n)

Quantizeru(m,n)

To Computer

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1-D1−D

x(t)

Time domain

X(u)

Frequency

Ts(t)

xs(t) = x(t) s(t) = Σ x(kt) δ (t-kT)

s(t)1/T

1/TXs(f)

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2-D: Comb function

ycomb(x,y;∆x,∆y)

x∆x

∆y

Comb( , ; , ) ( , )x y x y x m x y n ynm

∆ ∆ ∆ ∆≅ − −=−∞

∞

=−∞

∞

∑∑ δ

Jan 21, 2003 Sampling 6

Sampled Image

f x y f x y x y x y

f m x n y x m x y n y

x y x y u v

x yu v

x y

s

nm

( , ) ( , ) ( , ; , )

( , ) ( , )

( , ; , ) ( , )

, ; , )

=

= − −

← → ==−∞

∞

=−∞

∞

ℑ

∑∑

comb

comb COMB

comb(

∆ ∆

∆ ∆ ∆ ∆

∆ ∆

∆ ∆ ∆ ∆

δ

1 1 1

Jan 21, 2003 Sampling 7

Sampled Spectrum

F u v F u v u v

x yF u v u k

xv l

y

x yF u k

xv l

y

s

k l

k l

( , ) ( , ) ( , )

( , ) ,

,

,

,

= ∗

= ∗ − −FHG

IKJ

= − −FHG

IKJ

∑∑

∑∑

=−∞

∞

=−∞

∞

COMB

1

1

∆ ∆ ∆ ∆

∆ ∆ ∆ ∆

δ

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Sampled Spectrum: Example

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Bandlimited Images

A function f(x,y) is said to be band limited if the Fourier transform

F(u,v) = 0 for | u | > u0, | v | > v0

u0, v0 Band width of the image in the x- and y- directions

v0

u0region of support

Jan 21, 2003 Sampling 10

Foldover Frequencies

Sampling frequencies:Let us and vs be the sampling frequencies

Then us > 2u0 ; vs > 2v0

or ∆ x <1/2u0 ; ∆ y <1/2v0

Frequencies above half the sampling frequencies are called fold over frequencies.

Jan 21, 2003 Sampling 11

Sampling Theorem

A band limited image f(x,y) with F(u,v) as its Fourier transform; and F(u,v) = 0 |u| > u0 |v| > v0 ; and sampled uniformly on a rectangular grid with spacing ∆∆∆∆x and ∆∆∆∆y, can be recovered without error from the sample values f(m ∆∆∆∆x, n ∆∆∆∆y) provided the sampling rate is greater than the nyquist rate.

i.e 1/ ∆∆∆∆x = us > 2 u0, 1/ ∆∆∆∆y = vs > 2 v0

The reconstructed image is given by the interpolation formula:

f x y f m x n y xu mxu m

y v ny v nm n

s

s

s

s

( , ) ( , ) ( )( )

( )( ),

= −−

−−∑∑

=−∞

∞

∆ ∆ sin sinππ

ππ

Jan 21, 2003 Sampling 12

Reconstruction

0

R1

R2

δR

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Reconstruction via LPF

F(u,v) can be recovered by a LPF with

H u vx y u v R

R RR

R

F u v H u v F u v F u vf x y F u v

s

( , )( , )

~( , ) ( , ) ( , ) ( , )

( , ) [ ( , )]

=∈RST

= =

= ℑ −

∆ ∆0

1

2

1

Other wise is any region whose boundary is contained

within the annular ring between the rectangles and in the figure. Reconstructed signal is

∂

Jan 21, 2003 Sampling 14

Aliasing

Note: If us and vs are below the Nyquist rate, the periodicreplications will overlap, resulting in a distorted spectrum.

This overlapping of successive periods of the spectrumcauses the foldover frequencies in the original image toappear as frequencies below us/2, vs/2 in the sampled image. This is called aliasing.

Jan 21, 2003 Sampling 15

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Example

there will be aliasing.

f x y x yF u v u v u v

u v

x y u v u vs s

( , ) ( ( ))( , ) ( , ) ( , )

,

. ,.

= += - - + + +

fi = =

= = fi = = = <

2 2 3 43 4 3 4

3 4

0 2 10 2

5 2

0 0

0 0

cos

Let ,< 2

pd d

D D

Jan 21, 2003 Sampling 17

Example:(contd.)

F

Let

Otherwise

cos

s ( , ) ( , )

[ ( , ) ( , )]

( , ). . , . .

( , ) ( , ) ( , )( , ) ( , )

~( , ) (2 (2 ))

,

,

u v F u ku v lv

u k v l u k v l

H u vu u

F u v H u v F u vu v u v

f x y x y

s sk l

k l

s

= − −

= − − − − + + − + −

=− ≤ ≤ − ≤ ≤RST

∴ == + + + − −

∴ = +

∑∑

∑∑

=−∞

∞

=−∞

∞

25

25 3 5 4 5 3 5 4 5

2 5 2 5 2 5 2 50

2 1 2 1

2

125

δ δ

δ δπ

Jan 21, 2003 Sampling 18

Examples

Original and the reconstructed image from samples.

Jan 21, 2003 Sampling 19

Another example

Sampling filter

sampled image

Jan 21, 2003 Sampling 20

Aliasing Problems (real images!)