Transcript מצגת של PowerPoint - igl | Interactive Geometry Lab

Computer Graphics
Recitation 6
Motivation – Image compression
What linear combination of 8x8 basis signals produces an 8x8 block in the image?
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The plan today
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
Fourier Transform (FT).
Discrete Cosine Transform (DCT).
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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
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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)
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Attributes of periodic function
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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
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Time and Frequency

example : g(t) = sin(2pf t) + (1/3)sin(2p(3f) t)
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Time and Frequency

example : g(t) = sin(2pf t) + (1/3)sin(2p(3f) t)
=
+
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Time and Frequency

example : g(t) = sin(2pf t) + (1/3)sin(2p(3f) t)
=
+
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Time and Frequency
1, a/2 < t < a/2
 example : g(t) = {
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

1
= A sin(2 kt )
k 1 k
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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 frequencies.

1
= A sin(2 kt )
k 1 k
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Frequency domain
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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


Every periodic function can be represented as
the sum of sine and cosine functions
Transform a function between a time and freq.
domain

G( f ) 
 g (t )[cos(2 ft )  i sin(2 ft )] dt


g (t ) 
 G( f )[cos(2 ft )  i sin(2 ft )] df

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Fourier transform
Discrete Fourier Transform:
1 n 1
2 ft
2 ft
G ( f )   g (t )[cos(
)  i sin(
)] 0  f  n  1
n t 0
n
n
1 n 1
2 ft
2 ft
g (t )   G ( f )[cos(
)  i sin(
)] 0  t  n  1
n t 0
n
n
0
n-1
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FT for digitized image
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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
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Sampling theory
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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 to freq. Domain
 Find the max frequency fmax
 Sample I(t) at rate > 2 fmax
 Store the sampled values in a bitmap
2fmax is called Nyquist rate
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Sampling theory
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Some loss of image quality because:
 fmax can be infinite.

choose a value such that freq. > fmax do not
contribute much (low amplitudes)
may be too small – not enough
samples
 Bitmap
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Fourier Transform
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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
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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

Definition of 2D DCT:
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
 Input:
 u

 v

B(u, v)   4 I (i, j ) cos 
(2i  1)  cos 
(2 j  1) 
i 1 j 1
 2 N1

 2 N2

N1
N2
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Using DCT in JPEG
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DCT on 8x8 blocks
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Using DCT in JPEG
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DCT – basis
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Using DCT in JPEG

Block size
 small block
faster
 correlation exists between neighboring pixels
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 large
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
block
better compression in smooth regions
Power of 2 – for fast implementation
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Using DCT in JPEG
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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
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Using DCT in JPEG
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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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Image compression using DCT
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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
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Bye-bye