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# Signal Processing Course: Signal Processing in Ortho-bases

January 01, 2012

## Transcript

([0 , 1]).
3. ### Continuous image: f 2 L 2 ([0 , 1] 2

). Signals, Images and More Continuous signal: f 2 L 2 ([0 , 1]).
4. ### Continuous image: f 2 L 2 ([0 , 1] 2

). General setting: f : [0, 1]d ! Rs Videos: d = 3 , s = 1. Signals, Images and More Continuous signal: f 2 L 2 ([0 , 1]).
5. ### Continuous image: f 2 L 2 ([0 , 1] 2

). General setting: f : [0, 1]d ! Rs Videos: d = 3 , s = 1. Color image: d = 2 , s = 3. Signals, Images and More Continuous signal: f 2 L 2 ([0 , 1]).
6. ### Continuous image: f 2 L 2 ([0 , 1] 2

). General setting: f : [0, 1]d ! Rs Videos: d = 3 , s = 1. Color image: d = 2 , s = 3. Multi-spectral: d = 2, s 3. Signals, Images and More Continuous signal: f 2 L 2 ([0 , 1]).

8. ### Orthogonal basis { m }m of L2([0, 1]d) Continuous signal/image

f L2([0, 1]d). Orthogonal Decompositions
9. ### Orthogonal basis { m }m of L2([0, 1]d) f =

m f, m m ||f|| = |f(x)|2dx = m | f, m ⇥|2 Continuous signal/image f L2([0, 1]d). Orthogonal Decompositions
10. ### Orthogonal basis { m }m of L2([0, 1]d) f =

m f, m m ||f|| = |f(x)|2dx = m | f, m ⇥|2 Continuous signal/image f L2([0, 1]d). Orthogonal Decompositions m
11. ### 1-D Wavelet Basis Wavelets: j,n (x) = 1 2j/2 x

2jn 2j Position n, scale 2j, m = (n, j).
12. ### 1-D Wavelet Basis Wavelets: j,n (x) = 1 2j/2 x

2jn 2j Position n, scale 2j, m = (n, j).
13. ### m1,m2 Basis { m1,m2 (x1, x2 )}m1,m2 of L2([0, 1]2)

m1,m2 (x1, x2 ) = m1 (x1 ) m2 (x2 ) tensor product 2-D Fourier Basis Basis { m (x)}m of L2([0, 1]) m1 m2
14. ### m1,m2 Basis { m1,m2 (x1, x2 )}m1,m2 of L2([0, 1]2)

m1,m2 (x1, x2 ) = m1 (x1 ) m2 (x2 ) tensor product f(x) f, m1,m2 Fourier transform 2-D Fourier Basis Basis { m (x)}m of L2([0, 1]) m1 m2 x m
15. ### 3 elementary wavelets { H, V , D}. Orthogonal basis

of L2([0, 1]2): k j,n (x) = 2 j (2 jx n) k=H,V,D j<0,2j n [0,1]2 2-D Wavelet Basis V (x) H(x) D(x)
16. ### 3 elementary wavelets { H, V , D}. Orthogonal basis

of L2([0, 1]2): k j,n (x) = 2 j (2 jx n) k=H,V,D j<0,2j n [0,1]2 2-D Wavelet Basis V (x) H(x) D(x)

x (j, n, k)
18. ### Discrete Computations Discrete orthogonal basis { m } of CN

. f = m f, m m
19. ### Fast Fourier Transform (FFT), O(N log(N)) operations. Discrete Computations Discrete

orthogonal basis { m } of CN . m [n] = 1 N e2i N nm f = m f, m m
20. ### Fast Fourier Transform (FFT), O(N log(N)) operations. Fast Wavelet Transform,

O(N) operations. Discrete Wavelet basis: no closed-form expression. Discrete Computations Discrete orthogonal basis { m } of CN . m [n] = 1 N e2i N nm f = m f, m m

28. ### Best basis Fastest error decay ||f fM ||2 log(||f fM

||) log(M) Efficiency of Transforms Fourier DCT Local DCT Wavelets

38. ### Compression by Transform-coding Image f Zoom on f f forward

a[m] = ⇥f, m ⇤ R transform
39. ### Compression by Transform-coding Image f Zoom on f f forward

a[m] = ⇥f, m ⇤ R transform Quantization: q[m] = sign(a[m]) |a[m]| T ⇥ Z ˜ a[m] T T 2T 2T a[m] Quantized q[m] bin T q[m] Z
40. ### Compression by Transform-coding Image f Zoom on f f forward

a[m] = ⇥f, m ⇤ R coding transform Entropic coding: use statistical redundancy (many 0’s). Quantization: q[m] = sign(a[m]) |a[m]| T ⇥ Z ˜ a[m] T T 2T 2T a[m] Quantized q[m] bin T q[m] Z
41. ### Compression by Transform-coding Image f Zoom on f f forward

a[m] = ⇥f, m ⇤ R coding decoding q[m] Z transform Entropic coding: use statistical redundancy (many 0’s). Quantization: q[m] = sign(a[m]) |a[m]| T ⇥ Z ˜ a[m] T T 2T 2T a[m] Quantized q[m] bin T q[m] Z
42. ### Compression by Transform-coding Image f Zoom on f f forward

Dequantization: ˜ a[m] = sign(q[m]) |q[m] + 1 2 ⇥ T a[m] = ⇥f, m ⇤ R coding decoding q[m] Z ˜ a[m] dequantization transform Entropic coding: use statistical redundancy (many 0’s). Quantization: q[m] = sign(a[m]) |a[m]| T ⇥ Z ˜ a[m] T T 2T 2T a[m] Quantized q[m] bin T q[m] Z
43. ### Compression by Transform-coding Image f Zoom on f f ,

R =0.2 bit/pixel f forward Dequantization: ˜ a[m] = sign(q[m]) |q[m] + 1 2 ⇥ T a[m] = ⇥f, m ⇤ R coding decoding q[m] Z ˜ a[m] dequantization transform backward fR = m IT ˜ a[m] m transform Entropic coding: use statistical redundancy (many 0’s). Quantization: q[m] = sign(a[m]) |a[m]| T ⇥ Z ˜ a[m] T T 2T 2T a[m] Quantized q[m] bin T q[m] Z
44. ### Compression by Transform-coding Image f Zoom on f f ,

R =0.2 bit/pixel f forward Dequantization: ˜ a[m] = sign(q[m]) |q[m] + 1 2 ⇥ T a[m] = ⇥f, m ⇤ R coding decoding q[m] Z ˜ a[m] dequantization transform backward fR = m IT ˜ a[m] m transform Entropic coding: use statistical redundancy (many 0’s). Quantization: q[m] = sign(a[m]) |a[m]| T ⇥ Z ˜ a[m] T T 2T 2T a[m] Quantized q[m] bin T q[m] Z ||f fM ||2 = O(M ) =⇥ ||f fR ||2 = O(log (R)R ) Theorem:

47. ### Denoising thresh. f = N 1 m=0 f, m ⇥

m ˜ f = | f, m ⇥|>T f, m ⇥ m
48. ### Denoising thresh. f = N 1 m=0 f, m ⇥

m ˜ f = | f, m ⇥|>T f, m ⇥ m In practice: T 3 for T = 2 log(N) Theorem: if ||f0 f0,M ||2 = O(M ), E(|| ˜ f f0 ||2) = O( 2 +1 )