Computer Graphics

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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(tP) g(t)
• Period of a function smallest constant P that
  satisfies g(tP) 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(2pft) (1/3)sin(2p(3f)t)

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Time and Frequency

• example g(t) sin(2pft) (1/3)sin(2p(3f)t)

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Time and Frequency

• example g(t) sin(2pft) (1/3)sin(2p(3f)t)

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Time and Frequency
1, -a/2 lt t lt a/2 0, elsewhere

• example g(t)

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Time and Frequency
1, -a/2 lt t lt a/2 0, elsewhere

• example g(t)

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Time and Frequency
1, -a/2 lt t lt a/2 0, elsewhere

• example g(t)

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Time and Frequency
1, -a/2 lt t lt a/2 0, elsewhere

• example g(t)

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Time and Frequency
1, -a/2 lt t lt a/2 0, elsewhere

• example g(t)

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Time and Frequency
1, -a/2 lt t lt a/2 0, elsewhere

• example g(t)

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Time and Frequency
1, -a/2 lt t lt a/2 0, elsewhere

• example g(t)

A (1/k)sin(2pkft)
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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(2pkft)

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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

• 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(2pft) - i sin(2pft)dt
g(t) G(f)cos(2pft) i sin(2pft)df
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Fourier transform

• Discrete

G(f) (1/n) g(t)cos(2pft/n) - i
sin(2pft/n) , 0ltfltn-1
g(t) (1/n) G(f)cos(2pft/n) i
sin(2pft/n) , 0lttltn-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 gt 2fm
• - Store the sampled values in a bitmap

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Sampling theory

• Some loss of image quality because
• - fm can be infinite choose a value s.t freq. gt
  fm do not contribute much (low amplitudes)
• - Bitmap may be too small
• 2fm is Nyquist rate

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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

f0 1 f17
G (½) C P cos((2t1)fp/16) , C
f
t
f
f

• Fourier transform on 8 points

G P cos(2pft/8) i P sin(2pft/8) ,
f0,1,,7
t
f
t
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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

• Example 8 points

f0 1 f17
G (½) C P cos((2t1)fp/16) , C
f
t
f
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

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Discrete Cosine Transform

• Example 8 points

The inverse DCT
P (½) C G cos((2t1)jp/16) ,
t0,1,,7
j
t
j
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Discrete Cosine Transform

• 2D DCT

G C C Pxy cos((2x1)ip/2n)cos((2y
1)jp/2n)
j
i
ij

• 2D Inverse DCT (IDCT)

P ¼ C C Gij cos((2x1)ip/16)
cos((2y1)jp/16)
xy
j
i
C f
f0 1 f17
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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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Using DCT in JPEG

• For almost flat surface most Gij0
• 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.

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Using DCT in JPEG

• 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

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Image compression using DCT

• 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

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See you next time