Waveinfo
Information on wavelets from MATLAB.
Wavelet Categories
1. Crude Wavelets.
Wavelets: Gaussian wavelets (gaus), Morlet, Mexican hat .
Properties
- phi does not exist.
- Analysis is not orthogonal.
- psi is not compactly supported.
- Reconstruction property is not insured.
Possible analysis:
- Continuous decomposition.
- Main nice properties: symmetry, psi has explicit expression.
- Main difficulties: fast algorithm and reconstruction unavailable.
2. Infinitely regular wavelets.
Wavelets: meyer (meyr).
Properties:
- phi exists and the analysis is orthogonal.
- psi and phi are indefinitely derivable.
- psi and phi are not compactly supported.
Possible analysis:
- continuous transform.
- discrete transform but with non FIR filters.
Main nice properties: symmetry, infinite regularity.
Main difficulties: fast algorithm unavailable.
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Wavelets: discrete Meyer wavelet (dmey).
Properties:
- FIR approximation of the Meyer wavelet
Possible analysis:
- continuous transform.
- discrete transform.
3. Orthogonal and compactly supported wavelets.
Wavelets: Daubechies (dbN), symlets (symN), coiflets (coifN).
General properties:
- phi exists and the analysis is orthogonal.
- psi and phi are compactly supported.
- psi has a given number of vanishing moments.
Possible analysis:
- continuous transform.
- discrete transform using FWT.
Main nice properties: support, vanishing moments, FIR filters.
Main difficulties: poor regularity.
Specific properties:
For dbN : asymmetry
For symN : near symmetry
For coifN: near symmetry and
phi as psi, has also vanishing moments.
4. Biorthogonal and compactly supported wavelet pairs.
Wavelets: B-splines biorthogonal wavelets (biorNr.Nd and
rbioNr.Nd).
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Properties:
- phi functions exist and the analysis is biorthogonal.
- psi and phi both for decomposition and reconstruction are compactly
supported.
- phi and psi for decomposition have vanishing moments.
- psi and phi for reconstruction have known regularity.
Possible analysis:
- continuous transform.
- discrete transform using FWT.
Main nice properties: symmetry with FIR filters, desirable properties
for decomposition and reconstruction are split and nice allocation is
possible.
Main difficulties: orthogonality is lost.
5. waveinfo('rbio')
RBIOINFO Information on reverse biorthogonal spline wavelets.
Reverse Biorthogonal Wavelets
General characteristics: Compactly supported biorthogonal spline
wavelets for which symmetry and exact reconstruction are possible
with FIR filters (in orthogonal case it is
impossible except for Haar).
Family
Biorthogonal
Short name
rbio
Order Nd,Nr
Nd = 1 , Nr = 1, 3, 5
r for reconstruction Nd = 2 , Nr = 2, 4, 6, 8
d for decomposition Nd = 3 , Nr = 1, 3, 5, 7, 9
Nd = 4 , Nr = 4
Nd = 5 , Nr = 5
Nd = 6 , Nr = 8
Examples
rbio3.1, rbio5.5
Orthogonal
Biorthogonal
no
yes
3
Compact support
yes
Analysis:
DWT
CWT
Support width
Filters length
rbio Nd.Nr
2Nd+1 for rec., 2Nr+1 for dec.
max(2Nd,2Nr)+2 but essentially
lr
ld
effective length
of HiF_D
rbio 1.1
rbio 1.3
rbio 1.5
rbio 2.2
rbio 2.4
rbio 2.6
rbio 2.8
rbio 3.1
rbio 3.3
rbio 3.5
rbio 3.7
rbio 3.9
rbio 4.4
rbio 5.5
rbio 6.8
possible
possible
effective length
of LoF_D
2
5
10
5
9
13
17
4
8
11
16
20
8
9
17
2
2
2
3
3
3
3
4
4
4
4
4
7
11
11
Regularity for
psi rec
Nd-1 and Nd-2 at the knots
Symmetry
yes
Number of vanishing moments
for psi dec. Nd-1
Remark: rbio 4.4 , 5.5 and 6.8 are such that reconstruction and
decomposition functions and filters are close in value.
Type: waveinfo('db')
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DBINFO Information on Daubechies wavelets.
6. Daubechies Wavelets
General characteristics: Compactly supported wavelets with
extremal phase and highest number of vanishing moments for a given
support width. Associated scaling filters are minimum-phase filters.
Family
Daubechies
Short name
Order N
Examples
db
N strictly positive integer
db1 or haar, db4, db15
Orthogonal
Biorthogonal
Compact support
DWT
CWT
yes
yes in db1
yes
possible
possible
Support width
2N-1
Filters length
2N
Regularity
about 0.2 N for large N
Symmetry
far from
Number of vanishing moments for psi
N
Reference: I. Daubechies,
Ten lectures on wavelets,
CBMS, SIAM, 61, 1994, 194-202.
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