wavelets

1. wavelets Introduction

2. Mathematical background • Ortho-normal basis • The vector V=(e1,…,en) is an orthonormated basis of the space S if the following conditions are accomplished: • V is basis for the space S. • The basis V is normated: • <ei, ei> = 1, for every I in 1 to n. • The basis is orthogonal: • <ei, ej> = 0 • Observation • Given an basis V’ it can be transformed into an ortonormated basis using the Gram-Schmidt procedure.

3. Mathematical background • Inner product • The inner product of two vectors: • <u, v> = • For functions • <f, g> = • For complex-valued functions: f(x) = f1(x) + j* f2(x) • <f, g> = , if f and g are continuous complex valued on interval [a, b] • Given a function f and a set of function that form a basis, there exists coefficients i.e: • f(x) = • can be computed by formula:

4. Mathematical background • Hilbert Space A Hilbert space is a vector space H with an inner product <f , g> such that the norm defined by |f|=sqrt (<f , f>) turns H into a complete metric space. If the metric defined by the norm is not complete, then H is instead known as an inner product space.

5. Hilbert spaces • Examples: • 1. The real numbers R^n with <v,u> the vector dot product of v and u. • 2. The complex numbers C^n with <v,u> the vector dot product of v and the complex conjugate of u. • 3. An example of an infinite-dimensional Hilbert space is L^2(R), the set of all functions f:R->R such that the integral of f^2 over the whole real line is finite. In this case, the inner product is • f is L^2(R) if exists and is finite.

6. Wavelets • A wave is an oscillating function of time or space and is periodic , like in the next figure: • A wavelet is similar to waves but their “energy” is concentrated in time and space:

7. Wavelets • Definition • Generally a wavelet is a wave with finite energy, so it’s considered to be part of the Hilbert space, so that , where f is the wavelet function. • p = 2 is the most widely used value for the f ’s space.

8. Wavelets • Examples • Meyer’s wavelet(complex) v is a smoothing function with properties: and v(x) + v(1-x) = 1. Example: v(x) = • Mexican hat wavelet

9. Wavelets • Examples • Morlet’s wavelet where w0 >0.8 is constant. This wavelet is a complex wavelet. • Haar’s wavelet • The Haar wavelet is the simplest possible wavelet. • The disadvantage of the Haar wavelet is that it is not continuous and therefore not differentiable.

10. Wavelets • “Child wavelets” • If the is the wavelet function then expression is the wavelet translated with and scaled with s>0. Scaling either dilates (expands) or compresses a signal. is named the “mother wavelet”. represent a “child wavelet”

11. Wavelet transform • There are two types of wavelet transform: • Continuous wavelet transform • Discrete wavelet transform

12. Continuous wavelet transform • Continuous wavelet transform is given by the equation, where x(t) is the signal to be analyzed. ψ(t) is the mother wavelet or the basis function. • The original signal could be retrieved applying the inverse transform. • is the translation parameter • relates to the location of the wavelet function as it is shifted through the signal. • Thus, it corresponds to the time information in the Wavelet Transform. • s is the scale parameter and correspond to the frequency information. • Large scales (low frequencies) dilate the signal and provide global information hidden in the signal. • small scales (high frequencies) compress the signal and provide detailed information about the signal.

13. The Wavelet Synthesis • The original signal can be retrieved from the transformed signal with formula: • is a constant that depends on the wavelet used • The inverse transform is possible only if • This condition is satisfied if • A lot of wavelets could be founded with that property • Isn’t necessary that the basis to be orthonormated

14. Continuous wavelet transform vs. Short Time Fourier transform • Remarks • Short Time Fourier Transform(STFT) can also be used to analyze non-stationary signals. • STFT use a constant resolution to all frequencies : w(t- ). • only translate parameter is present • No scaling, so the resolution is constant • the Wavelet Transform uses multi-resolution technique by which different frequencies are analyzed with different resolutions.

15. Continuous wavelet transform vs. Short Time Fourier transform • Both STFT and wavelet transform are time-frequency based transformations. • Why the wavelet transform is a better approach? • Answer: • Most of the real signals have the following properties: • high frequencies (low scales) do not last for a long duration, but instead, appear as short bursts • while low frequencies (high scales) usually last for entire duration of the signal. • wavelets provide an optimal representation for many signals containing singularities (jumps and spikes)

16. Wavelet series • The Wavelet Series is obtained by discretizing CWT. This aids in computation of CWT using computers and is obtained by sampling the time-scale plane. • The sampling rate can be changed accordingly with scale change without violating the Nyquist criterion. • At higher scales (lower frequencies), the sampling rate can be decreased • At lower scales (higher frequencies), the sampling rate can be incresed • This technique is called digital filtering technique.

17. Wavelet series • Observations • Nyquist criterion states that, the minimum sampling rate that allows reconstruction of the original signal is 2ω radians, where ω is the highest frequency in the signal. • Restrictions? • The discretization can be done in any way without any restriction as far as the analysis of the signal is concerned. • If synthesis is not required, even the Nyquist criteria does not need to be satisfied. • The restrictions on the discretization and the sampling rate become important if, and only if, the signal reconstruction is desired.

18. Wavelet series • Dyadic discretization • In function , s = , • Result: • Transform formula becomes: • If form an orthogonal basis the reconstruction is obtained with: • the most convenient and used parameters are s0 = 2, b0=1

19. Wavelet series • Application • Wavelet based approximation of the function • We have: • Because the wavelet basis is orthonormated we obtain:

20. Continuous wavelet transform. Wavelet series • Disadvantages • The main drawback of this approach is computational and resources demand.

21. Discrete wavelet transform • Similar to CWT, discrete wavelet transform use the digital filtering technique. • Filters of different frequencies are used to analyze the signal at different scales. • The signal is passed through: • a series of high pass filters to analyze the high frequencies, • and through a series of low pass filters to analyze the low frequencies. • The resolution of the signal(the amount of information it contains) • Is changed by the filters • The scale is changed through sampling

22. Discrete wavelet transform • Sampling • Sub-sampling • Reducing the number of samples, eliminating some of them. • Up-sampling • increasing the sampling rate of a signal by adding new samples to the signal. • For example, up-sampling by two refers to adding a new sample, usually a zero or an interpolated value, between every two samples of the signal.

23. Discrete wavelet transform • Remarks and notations • Let be x[n] the signal to analyze • DWT coefficients are usually sampled from the CWT on a dyadic grid, i.e., s0 = 2 and t 0 = 1, yielding s=2j and t =k • Filtering a signal corresponds to the mathematical operation of convolution of the signal with the impulse response of the filter.

24. Discrete wavelet transform • The process • The procedure starts with passing this signal (sequence) through a half band digital low pass filter with impulse response h[n]. • The convolution operation in discrete time is defined as follows • A half band low pass filter removes all frequencies that are above half of the highest frequency in the signal. • if a signal has a maximum of 1000 Hz component, then half band low pass filtering removes all the frequencies above 500 Hz. • According to the Nyquist rule, half of the samples could be eliminated. • As a result, the scale is doubled. • The sampling procedure by a factor of two:

25. Discrete wavelet transform • Remarks • Filtering affect the resolution, but doesn’t affect the scale. • Sampling affect the scale, but doesn’t affect the resolution.

26. Sub-band coding • Two operations are performed at every level of decomposition: • The original signal x[n] is first passed through • a half band high pass filter g[n] • and a low pass filter h[n]. • After filtering, half of the samples are eliminated • The high pass samples contains the high frequencies samples • The low pass samples contains the low frequencies samples

27. Sub-band coding • These two steps are expressed as follow: • yhigh[k] and ylow[k] are the outputs of the high pass and low pass filters, respectively, after sub-sampling by 2. • Yhigh signal constitute the DWT coefficients of the corresponding level • Ylow signal is passed for further decomposition

28. Pyramidal coding • Sub-band coding process is easier to understand if is expressed in pyramidal form: • The DWT of the original signal is then obtained by concatenating all coefficients starting from the last level of decomposition (remaining two samples) • The DWT will then have the same number of coefficients as the original signal.

29. Filters types • Orthogonal • The filters are of the same length and are not symmetric • The filters coefficients are real numbers • Relationship between low pass filter, Gand the high pass filter, H0 • Bio-orthogonal • filters do not have the same length • low pass filter is always symmetric • high pass filter could be either symmetric or anti-symmetric. • The coefficients of the filters are either real numbers or integers.

30. High and low filters relationship • The high pass and low pass filters are not independent of each other, and they are related by • L is the filter length (in number of points)

31. Signal synthesis • The signal is reconstructed through inverse order of the sub-band coding procedure: • Synthesis filtering and up-sampling • The synthesis filters h’ and g’ could be the same as the analysis filters h and g respectively • A good reconstruction is possible if and only if the filters form an orthonormal basis.

32. Data reduction • Analyzing the DWT coefficients, at some level the information they provide is poorly and thus these samples could be eliminated. • In general the last and the first’s levels DWT coefficients don’t any significant information. • This constitute for a method of compressing images, without losing any resolution information.

33. Perfect reconstruction • Reconstruction = synthesis of the original signal x(t) from the wavelet coefficients • Conditions: • Anti-aliasing free reconstruction • Distortion’s amplitude has amplitude of 1 • OBS. In most of the pattern recognition applications the reconstruction is not needed.

34. Limitations of wavelet transform • The wavelet transform suffers from four fundamental, intertwined shortcomings: • Oscillations • the wavelet coefficients tend to oscillate positive and negative around singularities • singularity extraction and signal modeling are required • Shift variance • Aliasing • Lack of directionality

35. Summary • Any finite energy analog signal x(t ) can be decomposed in terms of wavelets(high pass) and scaling(low pass) functions: • The scaling coefficients c(n) and wavelet coefficients d( j, n) are computed via the inner products

36. Summary • The coefficients c(n) and d( j, n) are computed in linear time complexity based on recursively applying a discrete-time low-pass filterh0(n), a high-pass filter h1(n), and up-sampling and down-sampling operations.

37. Dual-tree complex discrete wavelet transform • The Dual-tree complex wavelet transform (DTCWT) calculates the complex transform of a signal using two separate DWT decompositions (tree a and tree b).

38. Dual-tree complex wavelet transform • Complex-valued wavelets • The complex wavelet coefficients: • dc( j, n) = dr( j, n) + j di( j, n) • Magnitude: • Orientation:

39. Complex representation • The dual tree transform • h0(n), h1(n) denote the low-pass/high-pass filter pair for the upper FB; • let ψh(t ) be the wavelet associated with this transform • g0(n), g1(n) denote the low-pass/high-pass filter pair for the lower FBψg(t ) • This transform can be represented by the complex wavelet: • ψ(t ) := ψh(t ) + jψg(t )

40. Dual tree complex transform • In order to eliminate the short comings of the simple wavelet transform (anti-aliasing, shift invariance etc), the complex valued transform is chosen to be approximately an analytic function: • they are designed so that ψg(t ) is approximately the Hilbert transform of ψh(t ) [denoted ψg(t ) ≈ H{ψh(t )}].

41. 2-D DUAL-TREE CWT • Six wavelets are computed using • low-pass function φ(·) • φ(t):= φh(t) + j φg(t) • and the high-pass function ψ(·) • ψ(t ) := ψh(t ) + jψg(t ) • ψ1(x, y) = φ(x) ψ(y) (LH wavelet) • φ alongthefirstdimension and • Ψalongtheseconddimension • ψ2(x, y) = ψ(x) φ(y) (HL wavelet) • ψ3(x, y) = ψ(x) ψ(y) (HH wavelet) • ψ4(x, y) = φ(x) ψ*(y) • ψ2(x, y) = ψ(x) φ*(y) • ψ6(x, y) = ψ(x) ψ*(y), where z* isthecomplexconjugate of thecomplexnumber z • These wavelets correspond to six different directions.

42. 2-D DUAL-TREE CWT • Figure a) shows the imaginary parts, b) shows the real parts and c) shows the magnitudes of these six wavelets.

43. “Real“ 2-D DUAL-TREE CWT • Two pairs of low-pass, high pass functions are used; • two pairs of LH, HL , HH sub-coding are obtained, the first representing h0(n) and h1(n) and the second g0(n) , g1(n).

44. Applications • Compression • Example • Original image: • Compressed image: • Decomposition wavelet: • bior 4.4, level 4 • Compress ratio:30%

45. Applications • Image de-noising • Original image: • De-noised image: • Wavelet: db3, level 4

46. Applications • Image fusion • Example1 • Input images: • Output image: • Decomposition using db3 at level 3

47. Applications • Image fusion • Example2 • Input images: • Output image: • Decomposition using db1 at level 3 • Fusion method: UD_fusion

48. Applications • Signal extension/truncation • Image extension/truncation

49. Bibliography • www.wikipedia.org • http://users.rowan.edu/~polikar/WAVELETS/WTtutorial.html • The Wavelet Tutorial • http://web.media.mit.edu/~rehmi/wavelet/wavelet.html • A Gentle Introduction to Wavelets • The dual-tree complex wavelet transform, • Ivan W. Selesnick, Richard G. Baraniuk, and Nick G. Kingsbury