Computer Graphics

Recitation 6

2

Motivation – Image compression

What linear combination of 8x8 basis signals produces an 8x8 block in the image?

3

The plan today

Fourier Transform (FT). Discrete Cosine Transform (DCT).

4

What is a transform?

Function: rule that tells how to obtain result y given some input x

Transform: rule that tells how to obtain a function G(f) from another function g(t) Reveal important properties of g More compact representation of g

5

Periodic function

Definition: g(t) is periodic if there exists P such that g(t+P) = g(t)

Period of a function: smallest constant P that satisfies g(t+P) = g(t)

6

Attributes of periodic function

Amplitude: max value of g(t) in any period Period: P Frequency: 1/P, cycles per second, Hz Phase: position of the function within a period

7

Time and Frequency

example : g(t) = sin(2pf t) + (1/3)sin(2p(3f) t)

8

Time and Frequency

example : g(t) = sin(2pf t) + (1/3)sin(2p(3f) t)

= +

9

Time and Frequency

example : g(t) = sin(2pf t) + (1/3)sin(2p(3f) t)

= +

10

Time and Frequency

example : g(t) = {1, a/2 < t < a/2 0, elsewhere

11

Time and Frequency

example : g(t) = {

= +

=

1, a/2 < t < a/2 0, elsewhere

12

Time and Frequency

example : g(t) = {

= +

=

1, a/2 < t < a/2 0, elsewhere

13

Time and Frequency

example : g(t) = {

= +

=

1, a/2 < t < a/2 0, elsewhere

14

Time and Frequency

example : g(t) = {

= +

=

1, a/2 < t < a/2 0, elsewhere

15

Time and Frequency

example : g(t) = {

= +

=

1, a/2 < t < a/2 0, elsewhere

16

Time and Frequency

example : g(t) = {

=

1, a/2 < t < a/2 0, elsewhere

1

1sin(2 )

k

A ktk

17

Time and Frequency

If the shape of the function is far from regular wave its Fourier expansion will include infinite num of frequencies.

= 1

1sin(2 )

k

A ktk

18

Frequency domain

Spectrum of freq. domain : range of freq. Bandwidth of freq. domain : width of the spectrum DC component (direct current): component of zero freq. AC – all others

19

Fourier transform

Every periodic function can be represented as the sum of sine and cosine functions

Transform a function between a time and freq. domain

( ) ( )[cos(2 ) sin(2 )]

( ) ( )[cos(2 ) sin(2 )]

G f g t ft i ft dt

g t G f ft i ft df

20

Fourier transform

Discrete Fourier Transform:1

0

1

0

1 2 2( ) ( )[cos( ) sin( )] 0 1

1 2 2( ) ( )[cos( ) sin( )] 0 1

n

t

n

t

ft ftG f g t i f n

n n n

ft ftg t G f i t n

n n n

0 n-1

21

FT for digitized image

Each pixel Pxy = point in 3D (z coordinate is value of color/gray level

Each coefficient describes the 2D sinusoidal function needed to reconstruct the surface

In typical image neighboring pixels have “close” values surface is very smooth most FT coefficients small

22

Sampling theory

Image = continuous signal of intensity function I(t)

Sampling: store a finite sequence in memory I(1)…I(n)

The bigger the sample, the better the quality? – not necessarily

23

Sampling theory

We can sample an image and reconstruct it without loss of quality if we can :Transform I(t) function from to freq. DomainFind the max frequency fmax

Sample I(t) at rate > 2 fmax

Store the sampled values in a bitmap

2fmax is called Nyquist rate

24

Sampling theory

Some loss of image quality because:fmax can be infinite.

choose a value such that freq. > fmax do not contribute much (low amplitudes)

Bitmap may be too small – not enough samples

25

Fourier Transform

Periodic function can be represented as sum of sine waves that are integer multiple of fundamental (basis) frequencies

Frequency domain can be applied to a non periodic function if it is nonzero over a finite range

26

Discrete Cosine Transform

A variant of discrete Fourier transformReal numbersFast implementationSeparable (row/column)

27

Discrete Cosine Transform

Definition of 2D DCT: Input: Image I(i, j) 1 i N1, 1 j N2

Output: a new “image” B(u, v), each pixel stores the corresponding coefficient of the DCT

1 2

1 1 1 2

( , ) 4 ( , ) cos (2 1) cos (2 1)2 2

N N

i j

u vB u v I i j i j

N N

28

Using DCT in JPEG

DCT on 8x8 blocks

29

Using DCT in JPEG

DCT – basis

30

Using DCT in JPEG

Block sizesmall block

faster correlation exists between neighboring pixels

large block better compression in smooth regions

Power of 2 – for fast implementation

31

Using DCT in JPEG

For smooth, slowly changing images most coefficients of the DCT are zero

For images that oscillate – high frequency present – more coefficients will be non-zero

32

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

33

Image compression using DCT

DCT enables image compression by concentrating most image information in the low frequencies

Loose unimportant image info (high frequencies) by cutting B(u,v) at bottom right

The decoder computes the inverse DCT – IDCT

Bye-bye