First order statistic Wavelet vs. Fourier Analogy with music score. Introduction

Transcription

1 First order statistic Wavelet vs. Fourier Analogy with music score Introduction

2 Wavelets Burt - Adelson pyramid (1983)

3 Decomposition

4 1-d signal: Discrete signal Approx Detail Wavelet: filter banks

5 Decomposition 2: images

6 8-coeff. Daubechies (orthonormal) What about filters?

7 Example: one level of decomposition (Db-8)

8 Example of more than 1 level of decomposition Nasce il concetto di Multirisoluzione!

9 Wavelet: two basic functions Base e scaling function I filtri passa basso e passa-alto sono legati a due funzioni: scaling function wavelet madre (x) (x) La funzione di scaling viene traslata e scalata per dare origine a un insieme di basi per insiemi inclusi Vj f ( x) k k ( x) k j / 2 j j, k ( x) 2 (2 x k) j / 2 j j, k ( x) 2 (2 x k)

10 Nested spaces (x) Base for Vj

11 The simplest Wavelet: Haar

12 In nested spaces: Haar

13 Haar expansion

14 Continuous Wavelet

15 FAST WAVELET TRANSFORM Analysis Similar to filter banks (but for -1 inversion) Synthesis

16 Fast Wavelet and filter banks

17 1-D multiresolution Esempio: Haar on 4 element vector

18 Reconstruction

19 Example Only for Haar: no problems on the border!

20 Time frequency decomposition Original signal (space_time domain) Fourier (frequency, costant pace) Wavelet (both with variable pace)

21 2D images

22 Three levels

23 Wavelet: relationship filters- functions

24 SEPARABILITY Function in 2D?

25 3-level FWT

26 Wavelet-packets Idea: the complete tree for wavelet-packet

27 Wavelet-packets

28 Application: fingerprints

29 Problem: search of optimum

30 Biorthogonal Wavelet

31 1. transform 2. Zeros low scale 3. Transform -1 Edge with possible directionality Edge detection

32 de-noising Denoising n. 1 Original 1. Transform 2. Thresholding 3. Transform -1 Denoising n. 2 Difference (2) Denoising n. 3 Difference(3)

33 Compression Standard for image compression: The past: JPEG (Cosine tr.) present and future: JPEG2000 (Wavelets)

34 Compression Questions: Which transform? Which quantizer? Which coder? Answers: Biorthogonal Deadzone Thresholding EZW, arithm coding

35 Comrpession Haar, Daubechies, Symlet, Biort.

36 Thresholding vs multiresolution Wavelet Coeff. Filtro Coeff. Azzerati (T=1,5) Haar % Daubechies 8 ortonormale % Symlet (4 ordine) % Biortogonali (CDF) % Livelli Immagine banda base Coeff. Troncati (T = 25) Errore (Rms) X % 1, X % 2, X 64 97% 3, X 32 98% 3, X 16 98% 3,27

37 Image Retrieval RGB Color Space Y(IQ) Color Space HSV Color Space YUV Color Space DB immagini Estrazione Signature Indici Immagini

38 Image Retrieval signature levels autocorrelation (log 2 N) -2 k MAX = (log 2 N) -1

39 Image Retrieval

40 Pattern Wavelet decomp level choice Signature comp correlations Indici Ricerca lineare (con o senza cluster) Distanza Euclidea Immagini Target images Error evaluation

41 RGB Vs Y(IQ) Gaussian noise mean: 10% - 20% - 50% Variance : 1% Image Retrieval Wavelet vs noise Scanner noise Lossy compression (BMP JPEG compresison factor20~30) (scala/spostamento/rotazione/acquisizione) I o I o + G(0.1) I o + G(0.5) I o Scanned

42 Commenti Good for texture. Good for gaussian noise Good performanc e Image Retrieval Wavelet vs noise

43 Wavelet vs. noise ORIGINALE GN m: 50% v: 1% GN m: 10% v: 1% SCANNER GN m: 20% v: 1%

44 Wavelet vs noise

45 Wavelet vs noise

46 Wavelets vs. noise

47 Wavelets and robustness vs compression

48 Image Retrieval Not only Texture!

49 Image Retrieval Not only Texture!

50 Conclusions Le Wavelet are perfect to represent in approximate way discrete signals Use instead of Fourier Plus: : FAST, multiresolution, locality preservation, adaptivity Minus: parameter dependency, too much degree of freedom, high math complexity.

51 Bibliography 1. S. Mallat (1989) A theory for multiresolution signal decomposition: the wavelet representation. IEEE Transaction on Pattern Analysis and Machine Intelligent. 11, L. Cohen (1989) Time-Frequency Distributions - A Review. In: Proceedings of the IEEE, 77(7), S. G. Mallat (1989) Multifrequency Channel Decompositions of Images and Wavelet Models. IEEE Transaction on Acoustic, Speech and Signal Processing 37(12), Ingrid Daubechies (1990) The Wavelet Transform, Time-Frequency Localization and Signal Analysis. IEEE Transaction on Information Theory 36(5), M. Vetterli, J. Kovaĉević (1995) WAVELETS, subband coding. Prentice Hall PTR. 5. G. Strang, T. Nguyen (1997) Wavelet and Filter Banks. Wellesley, Cambridge Press. 6. Pankaj N. Topiwala (1998) Wavelet Image and Video Compression. Kluwer Academic Publishers.