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Multiparametric Wavelet Transforms
E.Ostheimer2, V.G. Labunets1, D.E.Komarov1,
Yekaterinburg , AIST-2016
1Ural Federal University, pr. Mira, 19, Yekaterinburg, 620002, Russian Federation
2Capricat LLC 1340 S. Ocean Blvd., Suite 209 Pompano Beach 33062 Florida USA
WDT is characterized by two sets of coefficients
0 1 1
20 1 1
, ,...,, where L=2D
, ,...,n
L
L
h h hg g g
WDT
Structure of WDT
Relations between g- and h-coefficients:
0 1 1
20 1 1
, ,...,, ,...,n
L
L
h h hg g g
WDT
0 1 2 2 1
1 2 1 1 0
...|| || || ... || ||
...
L L
L L L
g g g g g
h h h h h
0 1 10 1 2 12 2
0 1 1
, ,...,, ,...,
, ,...,n n
LD
L
h h hh h h
g g gWDT WDT
0 1 1 0 1 12 2, , , , , ,n nL Dh h h WDT WDT
Multiparametric Wavelet transform:
0 1 12, , ,n D WDT
Let be the smallest positive integer such that 2log 2m D 12 2 2m mD
1 1
1
0 1 1 0 1 12 2 212 , ,..., [ , ,..., ] IL Ln n r n n r
r n mh h h h h h
WDT AWT
Arbitrary cyclic wavelet transform, written in stairs-like form:
Structure of WDT
1 3
2 3 1
0 1 2 3 0 1
0 2
0
0 1 2 3
0 1 2 3
0 1 2 3
2 3 0
2 3
2 3 0 1 0 1 2 3
0 1 2 3
2
1
13 0
g g g g
g g g g
g g g g
g g g g
g g
h h h h h h h h
h h h h h h h h
h h h h
h h h h
g g
g g g g
1
1
1
1
1 1
1 1
1 1
1 1
1 1
1 1
1 1
1 1
=
0 1 2 3
0 1 2 3
0 1 2 3
0 1 2 3
0 1 2 3
0 1 2 3
2 3 0 1
0 1 2 3
0 1 2 3
0 1 2 3
0 1 2 3
0 1 2 3
0 1 2 3
0 1 2 3
2 3 0 1
0 1 2 3
h h h h
h h h h
h h h h
h h h h
h h h h
h h h h
h h h h
g g g g
g g g g
g g g g
g g g g
g g g g
g g g g
g g g g
g g
h h
g
h
g
h
=
16 0 1 2 3WDT h , h , h , h =
Stairs-Like Structure of WDT16(4)
stepstep
step
Two Spans:H-span and g-span
Two Spans:H-span and g-span
Two Spans:H-span and g-span
3
0 1 2 3
0 1 2 3
0 1 2 3
0 1 2 3 0 1 2 3
2 3 0 1 2 3 0 1
0 1 2 3 0 1 2 3
2 3 0 1 0 1 2 3
0 1 2
2 3 0 1
12 8
h h h h
h h h h
h h h h
h h h h h h h h
h h h h h h h h
g g g g g g g g
g g g g g g g g
g g g g
g g g g
I I =
0 1 2 3
0 1 2 3
0 1 2 3
0 1 2 3
0 1 2 3
0 1 2 3
2 3 0 1
0 1 2 3
0 1 2 3
0 1 2 3
0 1 2 3
0 1 2 3
0 1 2 3
0 1 2 3
2 3 0 1
h h h h
h h h h
h h h h
h h h h
h h h h
h h h h
h h h h
g g g g
g g g g
g g g g
g g g g
g g g g
g g g g
g g g g
g g g g
=
Stairs-Like Structure of WDT16(4)
4 12 8 8 16 I IAWT AWT AWT
,
( ) ( )
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0( )
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
( ) ( )
0
i j
cos sin
si
1
1
1
1
1
1
1
1
n cos
i j
i
CS
j
.
Jacobi-Givens Rotations
1 1 0 0, , 1 , 0 0 1 12( ) ( ) ( ) , , ,n
k ki j k i j i j Lh h h CS CS CS AWT
Multiparametric presentation of AWT8(6)
0 1 2 3 4 5
0 1 2 3 4 5
4 5 0 1 2 3
2 3 4 5 0 18 0 1 2 3 4 5
5 4 3 2 1 0
5 4 3 2 1 0
1 0 5 4 3 2
3 2 1 0 5 4
, , , , ,
h h h h h hh h h h h h
h h h h h hh h h h h h
h h h h h h h h h h h hh h h h h h
h h h h h hh h h h h h
AWT
0,4 0 8 0 1 2 3 4 5
0 0 0 1 2 3 4 5
0 1 2 3 4 5
4 5 0 1 2 3
2 3 4 5 0 1
0 0 5 4 3 2 1 0
5 4 3 2 1 0
1 0 5 4 3 2
3 2 1 0 5 4
( ) , , , , ,
11
1
11
1
h h h h h h
c s h h h hh h h h h h
h h h h h hh h h h h h
s c h h h hh h h h h h
h h h h h hh h h h h h
CS AWTh h
h h
0 1 2 3
0 1 2 3 4 5
4 5 0 1 2 3
2 3 4 5 0 1
3 2 1 0
5 4 3 2 1 0
1 0 5 4 3 2
3 2 1 0 5 4
h h h hh h h h h h
h h h h h hh h h h h h
h h h hh h h h h h
h h h h h hh h h h h h
0 0
0 0
0 0cosc 0 0sins
0 5 0 0 0c h s h
0 0 0 5 0s h c h
3,7 0 2,6 0 1,5 0 0,4 0 8 0 1 2 3 4 5( ) ( ) ( ) ( ) , , , , ,h h h h h h CS CS CS CS AWT
0 0 0 1 2 3 4 5
0 0 0 1 2 3 4 5
0 0 4 5 0 1 2 3
0 0 2 3 4 5 0 1
0 0 5 4 3 2 1 0
0 0 5 4 3 2 1 0
0 0 1 0 5 4 3 2
0 0 3 2 1 0 5 4
c s h h h hc s h h h h
c s h h h hc s h h h h
s c h h h hs c h h h h
s c h h h hs c h h h h
h hh h
h hh h
h hh h
h hh h
0 1 2 3
0 1 2 3
0 1 2 3
2 3 0 18 0 1 2 3
3 2 1 0
3 2 1 0
1 0 3 2
3 2 1 0
, , ,
h h h hh h h h
h h h hh h h h
h h h hh h h hh h h h
h h h hh h h h
AWT
Multiparametric presentation of AWT8(6)
0,7 1 3,6 1 2,5 1 1,4 1 8 0 1 2 3 8 0 1( ) ( ) ( ) ( ) , , , ,h h h h h h CS CS CS CS AWT AWT