k The function Ψ(x) is called wavelet function and shows band-pass behavior. The wavelet coefficients d a,b

Transcription

1 Wavele Transform

2 Wavele Transform The wavele ransform corresponds o he decomposiion of a quadraic inegrable funcion sx ε L 2 R in a family of scaled and ranslaed funcions Ψ,l, ψ, l 1/2 = ψ l The funcion Ψx is called wavele funcion and shows band-pass behavior. The wavele coefficiens d a,b are derived as follows: 1 d =, l sx Ψ x - l dx where ε R +, l ε R and * denoes he complex conjugae funcion - The discree wavele ransform DWT represens a 1-D signal s in erms of shifed versions of a lowpass scaling funcion φ and shifed and dilaed versions of a prooype bandpass wavele funcion ψ. *

3 For special choices of φ and ψ, he funcions: for j and ε Z, form an orhonormal basis, and we have he represenaion: 2 2, 2 2, 2 /, j j j j j j = = φ φ ψ ψ = + = j j j j j j u z 0,, 0,, 0 ψ ω φ where, = d s j j *,, ψ ω = d s u j j *,, φ and

4 Reloo a F.T. expressions: F u = f x e j2πxu dx f x = F u e j2πxu Thus fx is represened here as a linear combinaion of he basis funcions: expjωx Wavele ransform on he oher hand, represens fx or f as a linear combinaion of: ψ = 2 / 2 ψ 2 l l where ψ is called he moher wavele. Parameers and l are inegers which generaes he basis funcions as he dilaed and shifed variaions of he moher wavele. du

5 The parameer plays he role of frequency and l plays he role of ime. Hence by varying and l, we have differen frequency and differen ime or space hence he erm muli-channel muliresoluion approach. Compare in discree case: The F.T.: The DWT: f f x 1 N N 1 u = 0 j2πux N = F u e ; x = 0,1,..., N / 2 = X, l[2 ψ 2 l] l where, DWT 1 1 l X CWT, l = x ψ d

6 = d lt a h x a l X DWT, 2 / ] [, 2 / lt a f a l X x l DWT = DWT Discree Wavele Transform: Forward: and Inverse: Tae, T = 1 and ime is coninuous. Synhesis filers: Analysis filers: 2 / a f a f = 2 / a h a h = Funcions h and f are derived by dilaion of a single filer. Thus he basis funcions are dilaed -> a - and shifed -> - la - versions of: 2 / lt a a f l = = ψ ψ * h f = Synhesis filers for perfec reconsrucion:

7 Visualize pseudo-frequency corresponding o a scale. Assume a cener frequency F c of he wavele and use he following relaionship: F a F = c a. where a is he scale. is he sampling period and F c is he cener frequency of a wavele in Hz. F a is he pseudo-frequency corresponding o he scale a, in Hz. The highpass and lowpass filers are no independen of each oher, and hey are relaed by he following expression: g[ L 1 N ] = 1. h n n

8 QMF ban and ypical magniude responses Decimaors Expanders H 0 z 2 2 G 0 z x ^ x H 1 z 2 2 G 1 z Analysis Ban Synhesis Ban 1 H 0 z H 1 z π/2 ω π

9 M-channel M-band QMF ban Decimaors Expanders H 0 z M M G 0 z H 1 z M M G 1 z x ^ x H M-1 z M M G M-1 z Analysis Ban Synhesis Ban

10 The DWT analyzes he signal a differen frequency bands wih differen resoluions by decomposing he signal ino coarse approximaion and deail informaion. DWT employs wo ses of funcions, called scaling funcions and wavele funcions, which are associaed wih low pass and highpass filers, respecively. The decomposiion of he signal ino differen frequency bands is simply obained by successive highpass and lowpass filering of he ime domain signal. The original signal x[n] is firs passed hrough a half-band highpass filer g[n] and a lowpass filer h[n]. Afer he filering, half of he samples can be eliminaed according o he Nyquis s rule since he signal now has a highes frequency of f max /2 radians insead of f max. The signal can herefore be sub-sampled by 2, simply by discarding every oher sample. This consiues one level of decomposiion and can mahemaically be expressed as follows: y hi [ ] = x[ n]. g[2 n] y lo [ ] = x[ n]. h[2 n]

11 Bloc diagram of he mehodology of 1-D DWT.

12 Frequency responses bandwidhs of he differen oupu channels of he wavele filer ban, for a = 2 and hree or more levels of decomposiion α/4 α/2 α β = 2α ω

13 Frequency Response of 2-channel Daubeschies 8-ap orhogonal wavele filers. Low-Pass High-Pass

14 Frequency Response of a 3-channel orhogonal wavele filers. Channel - I Channel - II Channel - III

15 Channel - IV Frequency Response of a 4-channel orhogonal wavele filers. Channel - I Channel - II Channel - III

16 Two-level maximally decimaed filer ban H 0 z 2 2 G 0 z x H 0 z 2 H 1 z 2 2 G 1 z 2 G 0 z ^ x H 0 z 2 2 G 0 z H 1 z 2 2 G 1 z H 1 z 2 2 G 1 z

17 Illusraions o demonsrae he difference beween: FT, STFT and WT

18 = cos2π 10 + cos2π 25 + cos2π 50 + cos2π100

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21 Noe ha he FT gives wha frequency componens specral componens exis in he signal. Nohing more, nohing less. When he ime localizaion of he specral componens are needed, a ransform giving he TIME-FREQUENCY REPRESENTATION of he signal is needed. Wha is Wavele Transform and how does i solve he problem? View WT as a plo on a 3-D graph, where ime is one axis, frequency he second and ampliude is he hird axis. This will show us wha frequencies, f, exis a which ime, T. There is an issue, called "uncerainy principle", which saes ha, we canno exacly now wha frequency exiss a wha ime insance, bu we can only now wha frequency bands exis a wha ime inervals.

22 The uncerainy principle, originally found and formulaed by Heisenberg, saes ha, he momenum and he posiion of a moving paricle canno be nown simulaneously. This applies o our subjec as follows: The frequency and ime informaion of a signal a some cerain poin in he ime-frequency plane canno be nown. In oher words: We canno now wha specral componen exiss a any given ime insan. The bes we can do is o invesigae wha specral componens exis a any given inerval of ime. This is a problem of resoluion, and i is he main reason why researchers have swiched from STFT o WT. STFT gives a fixed resoluion a all imes, whereas WT gives a variable or suiable resoluion as follows: Higher frequencies are beer resolved in ime, and lower frequencies are beer resolved in frequency. This means ha, a cerain high frequency componen can be locaed beer in ime wih less relaive error han a low frequency componen. On he conrary, a low frequency componen can be locaed beer in frequency compared o high frequency componen

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24 STFT x ω, f = [ x. ω ']exp j2πf d w = exp a* 2 2

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26 Narrow Window, w Broader Window, w

27 Sill larger window, w

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33 Time and Frequency Resoluions Ampliude Frequency Time Ampliude Fourier Frequency Scale Time STFT/Gabor Time Wavele

34 Frequency STFT/Gabor 2ω 1 ω 1 T 2T Time Wavele Scale Frequency ω 0 /4 ω 0 /2 ω 0 T 2T 4T Time

35 Two-dimensional Wavele Transform LPF 2 LL LPF 2 Image HPF 2 LH HPF 2 LPF HPF 2 2 HL HH

36 Level I wavele decomposiion of an image

37 Level II wavele decomposiion of an image

38 References: Mulirae Sysems and Filer bans, P. P. Vaidyanahan; Prenice Hall Inc., 1993.

39 Wavele based analysis of exure Images

40 Problem of Shape from 3-D Texures

41 2-D Texures 3-D Texures

42 Real world 3-D Texure image

43 REFERENCES 1. M. Clerc and S. Malla, The Texure Gradien Equaion for Recovering Shape from Texure, IEEE Transacions on Paern Analysis and Machine Inelligence, Vol. 24, No. 4, pp , April J. Garding, Surface Orienaion and Curvaure from Differenial Texure Disorion, Proceedings of he IEEE Conference on Compuer Vision ICCV 95, 1995, pp J. S. Kwon, H. K. Hong and J. S Choi, Obaining a 3-D orienaion of Projecive exures using a Morphological Mehod, Paern Recogniion, Vol. 29, No. 5, pp , T. Leung and J. Mali, On Perpendicular exures, or: Why do we see more flowers in he disance?, Proceedings of he IEEE Conference on Compuer Vision and Paern Recogniion CVPR 97, 1997, San Juan, Puero Rico, pp J. Mali and R. Rosenholz, Compuing Local Surface Orienaion and Shape from exure for Curved Surfaces, Inernaional Journal of Compuer Vision, Vol. 232, pp , E. Ribeiro and E. R. Hancoc, Shape from periodic Texure using he eigenvecors of local affine disorion, IEEE Transacions on Paern Analysis and Machine Inelligence, Vol. 23, No. 12, pp , Dec B. J. Super and A. C. Bovi, Planar surface orienaion from exure spaial frequencies, Paern Recogniion, Vol. 28, No. 5, pp , Suhendu Das and Thomas Greiner; Wavele based separable analysis of exure images for exracing orienaion of planar surfaces; Proceedings of he second IASTED Inernaional Conference on Visualizaion, Imaging and Image Processing IASTED-VIIP; Sepember 9-12, 2002, Malaga, Spain, pp Thomas Greiner and Suhendu Das; Recovering Orienaion of a exured planar surface using wavele ransform; Indian Conference on Compuer Vision, Graphics and Image Processing, 2002 ICVGIP '02, December 16-18, 2002, Space Applicaions Cenre SAC-ISRO, Ahmedabad, INDIA, pp

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