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 Restrictions?
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 Bio-orthogonal
The filters are of the same length and are not symmetric
The filters coefficients are real numbers
Relationship between low pass filter, G and 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 filter h0(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 along the first dimension and
Ψ along the second dimension
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* is the complex conjugate of the complex number 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 The dual-tree complex wavelet transform,
The Wavelet Tutorial
A Gentle Introduction to Wavelets
The dual-tree complex wavelet transform,
Ivan W. Selesnick, Richard G. Baraniuk, and Nick G. Kingsbury