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

17 Time and Frequency If the shape of the function is far from regular wave its Fourier expansion will include infinite num of frequencies. =

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

20 Fourier transform Discrete Fourier Transform: 0n-1

21 FT for digitized image Each pixel P xy = 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. Domain  Find the max frequency f max  Sample I(t) at rate > 2 f max  Store the sampled values in a bitmap 2f max is called Nyquist rate

24 Sampling theory Some loss of image quality because:  f max can be infinite. choose a value such that freq. > f max 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 transform  Real numbers  Fast implementation  Separable (row/column)

27 Discrete Cosine Transform Definition of 2D DCT:  Input: Image I(i, j) 1  i  N 1, 1  j  N 2  Output: a new “image” B(u, v), each pixel stores the corresponding coefficient of the DCT

28 Using DCT in JPEG DCT on 8x8 blocks

29 Using DCT in JPEG DCT – basis

30 Using DCT in JPEG Block size  small 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

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