Computer Graphics

Recitation 7

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Motivation – Image compression

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

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The plan today

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

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What is a transformation?

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

Transformation : rule that tells how to obtain a function G(f) from another function g(t)

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What do we need transformations for?

Mathematical tool to solve problems Change a quantity to another form that might

exhibit useful features Example:

XCVI x XII 96 x 12 = 1152 MCLII

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Periodic function

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

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

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Attributes of periodic function

Amplitude: max value it has in any period Period: P Frequency: 1/P, cycles per second,Hz Phase: position of function within a period

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Time and Frequency

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

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Time and Frequency

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

= +

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Time and Frequency

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

= +

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Time and Frequency

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

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Time and Frequency

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

= +

=

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Time and Frequency

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

= +

=

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Time and Frequency

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

= +

=

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Time and Frequency

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

= +

=

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Time and Frequency

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

= +

=

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Time and Frequency

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

= A (1/k)sin(2kft)

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Time and Frequency

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

A (1/k)sin(2kft) =

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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

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Fourier transform

G(f) = g(t)[cos(2ft) - i sin(2ft)]dt

g(t) = G(f)[cos(2ft) + i sin(2ft)]df

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

Transform a function between a time and freq. domain

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Fourier transform

Discrete

g(t) = (1/n) G(f)[cos(2ft/n) + i sin(2ft/n)] , 0<t<n-1

G(f) = (1/n) g(t)[cos(2ft/n) - i sin(2ft/n)] , 0<f<n-1

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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 almost flat most FT coefficients small

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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

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Sampling theory

We can sample an image and reconstruct it without loss of quality if we can :

- Transform i(t) function from time to freq. Domain

- Find the max freq. fm

- Sample i(t) at rate > 2fm

- Store the sampled values in a bitmap

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Some loss of image quality because:

- fm can be infinite: choose a value s.t freq. > fm do not contribute much (low amplitudes)

- Bitmap may be too small

2fm is Nyquist rate

Sampling theory

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Fourier Transform

Periodic function can be represented as sum of sine waves that are integer multiple of

fundamental (basis) frequencies Freq. domain can be applied to a non periodic

function if it is nonzero over a finite range

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Discrete Cosine Transform

A variant of discrete Fourier transform - Real numbers - Fast implementation-Separable (row/column)

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Discrete Cosine Transform

Example: DCT on 8 points

G = (½) C P cos((2t+1)f/16) , C = { f f t

f=0 1 f=1…7

f

Fourier transform on 8 points

G = P cos(2ft/8) – i P sin(2ft/8) ,

f=0,1,…,7

f tt

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Discrete Cosine Transform

Example 8 points:

Same meaning: the 8 numbers Gf tell what sinusoidal func. should be combined to

approximate the function described by the 8 original numbers Pt

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Discrete Cosine Transform

G = (½) C P cos((2t+1)f/16) , C = { f f t

f=0 1 f=1…7

f

G3 = contribution of sinusoidal with freq. 3tp/16 to the 8 numbers Pt

G7 = contribution of sinusoidal with freq. 7tp/16 to the 8 numbers Pt

Example 8 points:

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The inverse DCT

P = (½) C G cos((2t+1)j/16) , t=0,1,…,7 j jt

Discrete Cosine Transform

Example 8 points:

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Discrete Cosine Transform

2D DCT

G = C C Pxy cos((2x+1)i/2n)cos((2y+1)j/2n)i jij

2D Inverse DCT (IDCT)

P =¼ C C Gij cos((2x+1)i/16) cos((2y+1)j/16)ixy j

f=0 1 f=1…7

C f = {

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Using DCT in JPEG

DCT on 8x8 blocks

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Using DCT in JPEG

Block size : small block

- faster - correlation exists between neighboring pixels

large block - better compression in “flat” regions

Power of 2 – for fast implementation

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Using DCT in JPEG

DCT – basis

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For almost flat surface most Gij=0

For surface that oscillates much many Gij non zero

G00 = DC coefficient

Numbers at top left of Gij contribution of low freq. sinusoidal to the surface, bottom right – high freq.

Using DCT in JPEG

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Numbers at top left of Gij contribution of low freq. sinusoidal to the surface, bottom right – high freq.

Scan each block in zig-zag order

Using DCT in JPEG

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

• Loose unimportant image info buy cut Gij at right bottom

• Decoder computes the inverse IDCT

Image compression using DCT

See you next time