MULTIPARAMETRIC WAVELET TRANSFORMS. PART 2. THIRD AND FOURTH CANONICAL FORMS

Labunets .V.G., Lebedev S.A.1, Egiazarian K., Astola J. 2, Ostheimer E.V.3

1 Urals State Technical University-UPI, (Russia), 2 Tampere International Center for Signal Process-ing, Tampere University of Technology, P.O.Box 553, FIN-33101 Tampere, FINLAND

3 EnerSoft Inc. Fort Lauderdale, Florida, (USA)

ABSTRACT

The main goal of the paper is to show that wavelet transforms and packets have the multiparametric repre-sentations in the form of a product of the Jacobi rotation matrices. These representations we call the third and the fourth canonical multiparametric forms. Each multi-parameric wavelet transform (MPWT) depends on sev-eral free Jacobi parameters (angles). When parameters are changed multiparametric transform is changed too taking form of all known and unknown orthogonal wavelet transforms. It gives unified approach to describ-ing a wide set of cyclic orthogonal wavelet transforms and endows with adaptive properties of those trans-forms.

1. INTRODUCTION

The wide class of orthogonal wavelet transforms WT can be defined by two sets of coefficients 0, 0: 0h , 1h , …, 1Lh − and 0g , 1g , …, 1Lg − , where 2L D= is an even number. In fact WT is determined only by a set of h -coefficients 0h , 1h , …,. 1Lh − since the second set of coefficients is usually assigned according to the rule 0 1Lg h −= , 1 2Lg h −= − , …, 1 0Lg h− = − . For this reason we will designate wavelet transform as

0 1 12, , ,n Lh h h − …WT .

Let ] [2log 2m D= be the smallest positive integer such that The wavelet transform

2nWT is factorized into a product of sparse matrixes, named stairs-like atomic wavelet transforms 0 1 12

, , ,n Lh h h − …AWT :

1 1

1

0 1 1 0 1 12 2 22 , ,..., [ , ,..., ] IL Ln

n

r n rr m

h h h h h h− −

−

+ +−=

= ⊕ = ∏WT AWT

1

2 2 1

0 2 2 1

12

12 22

[ , ]I

[ , ]

Dk k

k k k

r

r

kn

n rkr m

C h h

C g g

−+

= +

−

+−=

⊗ = ⊕ ⊗ ∑∏ ,

where 2rC is the matrix of cyclic shift on one position

modulo 2r , ⊗ is the tensor product. For example, for the Daubechies-4 wavelet transform we have

16 0 1 2 3 4 12 8 8 16, , ,h h h h

= ⊕ ⊕ =I IWDT AWT AWT AWT

3

0 1 2 3

0 1 2 3

0 1 2 3 0 1 2 3

2 3 0 1 2 3 0 112 8

0 1 2 3 0 1 2 3

2 3 0 1 0 1 2 3

0 1 2

2 3 0 1

0 1 2 3

0

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 h h

g g g g g g g gg g g g g g g g

g g g g

g g g g

⊕ ⋅ =

⋅

= ⊕I I

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

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

Coefficients 0h , 1h , …, 1Lh − depend on each other. The coefficients, which we can change independently of one another, staying wavelet transform WT in the class of orthogonal wavelet transform class, will be called angle-parameters in this paper. We will prove that multi-parametric presentation of wavelet transform exists and that any orthogonal wavelet transform depends on D angle-parameters: 0 1 1 0 1 12 2

, ,..., , ,...,n nL Dh h h ϕ ϕ ϕ− − = WT WT .

2. THIRD CANONICAL FORM OF MPWT

2.1. Multiparametric presentation of atomic wavelet transforms

In order to find multiparametric form of wavelet trans-

form we use the rotation-reflection Jacobi (2 2 )n n× -matrix in that rotation is performed on an angle ϕ in the plane spanned on i and j basis vectors:

,,

1 0 0 0

0 0

0 0

0 0 0 1

( )

i j

i

j

i j

s

c s

c

=

−

CS

L L LM O M M M

L L LM M O M M

L L LM M M O M

L L L

ϕ

where ( )cosc ϕ= and ( )sins ϕ= . For cs -matrices we

have , ,( ) ( )i j i jφ φ I=CS CS and 1, ,( ) ( ).i j i jφ φ−=CS CS

We will multiply the atomic wavelet matrix

0 1 12, , ,n Lh h h − AWT … by , ( )i j ϕCS matrix sequen-

tially with such choice of angles ϕ that product

0 0 1 10 0, , 2( ) ( ) , , ,k Lk kni j i j h h hϕ ϕ − ⋅ ⋅ ⋅ CS CS AWT… … (1)

will be permutation matrix or unit matrix. As an exam-ple, we have taken the atomic Daubechies-6 ( )8 8× -

matrix: 8 0 1 2 3 4 5, , , , ,h h h h h h = AWT

0 1 2 3 4 5

0 1 2 3 4 5

4 5 0 1 2 3

2 3 4 5 0 1

5 4 3 2 1 0

5 4 3 2 1 0

1 0 5 4 3 2

3 2 1 0 5 4

2 2 2 1

0 2 2 1

4

4

[ , ]

[ , ]k k

k k k

k

k

h h h h h hh h h h h h

h h h h h hh h h h h hh h h h h h

h h h h h hh h h h h hh h h h h h

C h h

C g g+

= +

= = = − − − − − − − − − − − −

⊗

⊗∑

2 3 4 50 1

0 1 2 3 4 5

1 24 441 2

4 4 4

[ , ] [ , ][ , ].

[ , ] [ , ] [ , ]

C h h C h hI h hI g g C g g C g g

⊗ ⊗ ⊗ = + + ⊗ ⊗ ⊗

(2)

The angle 0ϕ can be chosen such a way that the coeffi-cient 5 0h = in the zeroth and fourth rows by

0,4 0( )ϕCS . In this case coefficient 4h will be zero in the same rows too. That is, coefficients are zeroed by cou-ples. Therefore,

0,4 0 8 0 1 2 3 4 5

0 0

0 0

( ) , , , , ,

11

1

11

1

h h h h h h

c s

s c

⋅ = = ⋅ −

CS ϕ AWT

0 1 2 3 4 5

0 1 2 3 4 5

4 5 0 1 2 3

2 3 4 5 0 1

5 4 3 2 1 0

5 4 3 2 1 0

1 0 5 4 3 2

3 2 1 0 5 4

0 1 2 3

0 1 2 3 4 5

4 5 0 1 2 3

2 3 4 5 0 1

3 2

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

h h h hh h h h h h

h h h h h hh h h h h h

h h

= − − −

− − − − − − − − −

′ ′ ′ ′

= ′ ′ ′−

0 0

0 0

i

h h

h h

1 0

5 4 3 2 1 0

1 0 5 4 3 2

3 2 1 0 5 4

,h hh h h h h h

h h h h h hh h h h h h

′−

− − − − − − − − −

if the angle is chosen such that 0 5 0 0 0c h s h− = , where

( )0 0cosc ϕ= and ( )0 0sins ϕ= . Hence,

3,7 0 2,6 0 1,5 0 0,4 0

8 0 1 2 3 4 5

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

( ) ( ) ( ) ( )

, , , , ,h h h h h h

c sc s

c sc s

s cs c

s cs c

ϕ ϕ ϕ ϕ⋅ ⋅ ⋅ ⋅

⋅ = = ⋅ −

− − −

CS CS CS CS

AWT

0 1 2 3 4 5

0 1 2 3 4 5

4 5 0 1 2 3

2 3 4 5 0 1

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 hh h h h

h h h hh h h h

h h h hh h h h

h h h hh 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 1

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 h

h h h hh h h h

′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ = = ′ ′ ′ ′− −

′ ′ ′ ′− − ′ ′ ′ ′− − ′ ′ ′ ′− −

' ' ' '0 1 2 3

8 0 1 2 3' ' ' '0 1 2 3

14 41 24 4

[ , ] [ , ], , , .

[ , ] [ , ]

I h h C h hh h h h

C g g C g g

⊗ ⊗′ ′ ′ ′ = + = ⊗ ⊗

AWT

As a result we get a new atomic matrix

8 0 1 2 3, , ,h h h h′ ′ ′ ′ AWT with four coefficients. To get the

atomic matrix with two coefficients we should iterate foregoing procedure

0,7 1 3,6 1 2,5 1 1,4 1 8 0 1 2 3

'' ''0 1

8 0 1'' ''0 1

0424

( ) ( ) ( ) ( ) , , ,

[ , ], .

[ , ]

h h h h

C h hh h

C g g

′ ′ ′ ′ ⋅ = ⊗

′′ ′′ = = ⊗

CS CS CS CS AWT

AWT

ϕ ϕ ϕ ϕ

(3)

Reiteration of this procedure on matrix 8 0 1,h h′′ ′′ AWT

results in:

1,7 2 0,6 2 3,5 2 2,4 2

8 0 1 8

( ) ( ) ( ) ( )

, ,h h

⋅ ⋅ ⋅ ⋅

′′ ′′ ⋅ =

CS CS CS CS

P

ϕ ϕ ϕ ϕ

AWT (4)

where 8P is a quasipermutation matrix (there are only 1+ or 1− in every row and in every column of it). As

the final result we get:

1,7 2 0,6 2 3,5 2 2,4 2

0,7 1 3,6 1 2,5 1 1,4 1

3,7 0 2,6 0 1,5 0 0,4 0

8 0 1 2 3 4 5 8

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

, , , , , .h h h h h h

ϕ ϕ ϕ ϕ

ϕ ϕ ϕ ϕ

ϕ ϕ ϕ ϕ

⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅

⋅ =

CS CS CS CS

CS CS CS CS

CS CS CS CS

PAWT

(5)

From here we obtain the multiparametric representation of the atomic wavelet transform matrix:

8 0 1 2 3 4 5

0,4 0 1,5 0 2,6 0 3,7 0

1,4 1 2,5 1 3,6 1 0,7 1

2,4 2 3,5 2 0,6 2 1,7 2 8

0 0

0 0

0 0

0 0

0 0

0 0

0 0

, , , , ,

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

h h h h h h

c sc s

c sc s

s cs c

s c

ϕ ϕ ϕ ϕ

ϕ ϕ ϕ ϕ

ϕ ϕ ϕ ϕ

−

−

−

= = ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ =

=

CS CS CS CS

CS CS CS CS

CS CS CS CS P

AWT

0 0

1 1

1 1

1 1

1 1

1 1

1 1

1 1

1 1

s c

c sc s

c sc s

s cs c

s cs c

−

−

−

−

−

⋅

⋅ ⋅

( ) ( ) ( )

2 2

2 2

2 2

2 28

2 2

2 2

2 2

2 2

0 1 28 0 8 1 8 2 8= ,

c sc s

c sc s

s cs c

s cs c

ϕ ϕ ϕ

−

−

−

−

⋅ =

− ⋅ − ⋅ − ⋅

P

T T T P

i (6)

where ( )cosi ic ϕ= , ( )sini is ϕ= , 0,1,2i = and every

matrix ( )8 iϕT is the product of the following rotation

sin/cos – matrixes:

( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )

08 0 0,4 0 1,5 0 2,6 0 3,7 018 0 1,4 1 2,5 1 3,6 1 0,7 12

8 0 2,4 2 3,5 2 0,6 2 1,7 2

, ,

.

===

T CS CS CS CST CS CS CS CST CS CS CS CS

ϕ ϕ ϕ ϕ ϕϕ ϕ ϕ ϕ ϕϕ ϕ ϕ ϕ ϕ

(7)

Let us clarify regularity in the sequences of index’s couples. We have

( ) ( )0

00, 41, 52, 63, 7

4

i

k T k

=

+↗ ↖

( ) ( )14

11, 42, 53, 60, 7

1 4

i

k T k

=

⊕ +

↗ ↖

( ) ( )24

22, 43, 50, 61, 7

2 4

i

k T k

=

⊕ +

↗ ↖

If i is a number of an iteration 12riT + 0,1, 2, ..., 1;( Di = −

0,1,2,..., 1)nr = − and ,p q are indexes of the matrix

( ),p q iϕCS , then ( ) ( )2, ,2

ssp q i k k= ⊕ + , where

0,1,2,...,2 1rk = − . We will get the same results if ( )16 16× -matrix

16 0 1 2 3 4 5, , , , ,h h h h h h AWT is chosen as the source

atomic transform matrix with the identical set of coeffi-cients. In order to zero the coefficients let us apply fore-going procedure to this matrix (compare the result with (6)):

( ) ( ) ( )2 1 016 2 16 1 16 0 16 0 1 2 3 4 5 16, , , , , ,h h h h h hϕ ϕ ϕ = T T T PAWT (8)

( ) ( ) ( )0 1 216 0 1 2 3 4 5 16 0 16 1 16 2 16, , , , , ,h h h h h h = T T T Pϕ ϕ ϕAWT (9)

where T –matrixes are the products of CS -matrixes. This result is general and valid for any ( )1 12 2r r+ +×

atomic matrix:

( )1 1 1

1

0 1 2 120

2 2 , , ,r r r

D

i Di

i h h h+ + +

← −

−=

= ⋅ ∏P T …ϕ AWT (10)

and, therefore,

( )1 11

1

0 1 2 12 20

2, , ,r rr

D

D ii

ih h h+ ++

→ −

−=

= = ∏ T P… ϕAWT

1

2 11

20 0

,22

( ) ,r

rD

r ii k r

i k k +

−→ −

= =⊕ +

= ∏ ∏CS Pϕ (11)

where 2r⊕ is addition modulo 2r and

( )1

2 1

2 ,20 2

( ).r

r

rr

ii ii k k

k+

−

⊕ +=

=∏T CSϕ ϕ

Here 1

0 1 1

0T T T T

ni n

i

→ −−

=

= ⋅⋅⋅∏ and 1

1 1 0

0

T T T Tn

i n

i

← −−

=

= ⋅⋅⋅∏

are the right and left multiplications, respectively. The

symbol 1

0 1 1 1 1

0T T T T T T T

ni n n n

i

−− −

=

= ⋅⋅⋅ = ⋅⋅⋅∏ we use for

commutative multipliers.

2.2. Multiparametric representations of wavelet transforms and wavelet packets

The classical wavelet transform with coefficients 0 1 2 1, ,..., Dh h h − is constructed from atomic wavelet trans-

forms according to the following rule:

0 1 2 1

1

1 0 1 2 1 1

2

2 2 2

[ , ,..., ]

[ , ,..., ] .

n D

n

r D n rr m

h h h

h h h

−

→ −

+ − +=

−

=

= ⊕ ∏ I

WT

AWT (12)

Substituting (11) in (12) we obtain the multiparametric presentation of wavelet transforms:

0 1 2 1

1 1

10

2

1 1 2 22 2

[ , ,..., ]

= ( )

n D

n D

i n rr m i

r ri

h h h −

→ − → −

+= =

+ + −

=

⊕ =

∏ ∏ T P Iϕ

WT

1 1

2 11 1

2 2 20 0

,22

( ) .r

r n r

n D

r ir m i k r

i k k + +

−→ − → −

−= = =

⊕ +

= ⊕

∏ ∏ ∏CS P Iϕ (13)

We will call this presentation the third canonical form. For example consider ( )16 16× Daubechies-4 wavelet

transform:

[ ][ ][ ]

( ) ( ) ( ) ( )( ) ( )

16 0 1 2 3 4 12 8 8 16

0 1 0 14 0 4 1 4 12 8 0 8 1 8 8

0 116 0 16 1 16

, , ,h h h h = = ⊕ ⊕ = ⊕ ⋅ ⊕ ⋅

⋅ =

I I

T T P I T T P I

T T P

ϕ ϕ ϕ ϕ

ϕ ϕ

WDT AWT AWT AWT

1 1

4 4 12 8 8 80 0

1

16 160

( ) ( )

( ) ,

i ii i

i i

ii

i

→ →

= =

→

=

= ⊕ ⋅ ⊕ ⋅

⋅

∏ ∏

∏

T P I T P I

T P

ϕ ϕ

ϕ

(14)

where

( ) ( ) ( ) ( )( ) ( )

0 1 0 14 4 0 4 1 4 8 8 0 8 1 8

0 116 16 0 16 1 16

, ,

.

= =

=

T T P T T P

T T P

ϕ ϕ ϕ ϕ

ϕ ϕ

AWT AWT

AWT

It is possible to obtain all the transforms of

16 0 1 2 3, , ,h h h h WT -type by changing the angles 0ϕ

and 1ϕ . All the atomic matrices in multiparametric rep-resentation of wavelet transform are characterized by the same set of angle-parameters. And all the angles have equal values in each atomic matrix and have to be chosen synchronously. Of course, it is possible to use different angles sets in different atomic matrixes and to change them not synchronously, but in this case we will get heterogeneous wavelet transforms.

The classical wavelet transform with coefficients 0 1 2 1, ,..., Dh h h − is constructed from atomic wavelet trans-

forms according to (12). The atomic transform is used only once within each iteration in (12). In fact, the atomic transform could be repeated not more then

1 12 / 2 2n r n r+ − −= times. Let 1 2 12( , ,..., ,..., )r r r r r

t n rs s s s − −=s

be a binary 12n r− − -digital integer. Every binary digit rts controls the tht position of the matrix 12r +AWT in

the thr iteration matrix:

1

11

2

22

, 1,

, 0.r

r

rt

rt

rtr

s s

I s+

++

=

==

AWTAWT

All such matrices form a packet of atomic matrices

1

11 21 1 1

1

1

2 2

2 2 22

2

... .

rr tn r

rr r n rr r r

n r

ts

ss s+

− −+ + +

− −

==

= ⊕ ⊕ ⊕

= ⊕sAWP AWT

AWT AWT AWT

(15)

Multiplying atomic packets 12rtr

s+AWT , we obtain dis-

crete controlled third multiparametric representation of wavelet packet:

1

1

0 1 2 1

1

2 12 1

1 0 0

1 1

11 21 1 1

( ) 2 ,(2 1)22

, ,...,22

2 2 2

2 2

2

[ , ,..., ]

...

( )n r r

n

Dr m

n

r m

D

r it i k r

m m n rn

rr r n rr r r

rt

r

n

ni k t k

rt

t

s

ss s

h h h

ϕ− −

→ −

−=

→ −

=

−→ −

= = =

+ −

− −+ + +

⊕ + + +

= =

= ⊕ ⊕ ⊕ =

=

∏

∏

∏ ∏ ∏

s s s s

sCS P

WP AWP

AWT AWT AWT

1

,n

r m

−

=

∏

(16)

with discrete binary parameters ( )11 2 2, ,..., ,m m m m

n ms s s − −=s

( )1 1 1 11 2 2 2, ,...,m m m m

n ms s s+ + + +− −=s ,…, ( )2 2 2

2 2,n n ns s− − −=s , ( )1 11

n ns− −=s ,

where 1 12 2 2.n n r r− − + = ⊗ P I P Multiparametric wavelet

packet represent is a generalization of multiresolution

decomposition and comprise the entire family of sub-band (tree) decomposition.

Let us introduce completed atomic matrix using in

1

0 1 2 10

1 1 12 2 2[ , ,..., ] ( )

D

D ii

r r rih h h

→ −

−=

+ + +

= = ∏ T PϕAWT

1

20 0

2 1

,22

( ) ,r

rD

r ii k r

i k k

→ −

= =

−

⊕ +

= ∏ ∏CS Pϕ (17)

maximum number of multipliers ( )2rD =

0 10

1 1 1 1

2 1

2 2 1 2 2[ , ,..., ] ( )i

i

r

r r r rih h h

→

=+ + + +

−

−

=

∏ T PϕAWT .

For example

0 1 2 3 4 5

0 1 2 3 4 5

4 5 0 1 2 3

2 3 4 5 0 1

5 4 3 2 1 0

5 4 3 2 1 0

1 0 5 4 3 2

3 2 1 0 5 4

58 0 1 2 3 4, , , , ,

h h h h h hh h h h h h

h h h h h hh h h h h hh h h h h h

h h h h h hh h h h h hh h h h h h

h h h h h h

= − − −

− − − − − − − − −

AWT

is not a completed atomic matrix, but

0 1 2 3 4 5 6 7

6 7 0 1 2 3 4 5

4 5 6 7 0 1 2 3

2 3 4 5 6 7 0 1

7 6 5 4 3 2 1 0

1 0 7 6 5 4 3 2

3 2 1 0 7 6 5 4

5 4 3 2 1 0 7 6

78 0 1 2 6, , ,..., ,

h h h h h h h hh h h h h h h hh h h h h h h hh h h h h h h hh h h h h h h hh h h h h h h hh h h h h h h hh h h h h h h h

h h h h h

= − − − −

− − − − − − − − − − − −

AWT

is a completed atomic matrix. Completed matrix de-pends on 12 2rD += h-coefficients. Substituting complete atomic matrices in (13) and (16) we obtain the com-pleted wavelet transform and packet, respectively:

1

10 0

1 12 2 2

2 1

2 2= ( )n

n

i n rr i

r

r ri

→ − →

+= =

+ + −

− ⊕

∏ ∏ T P IϕCWT , (18)

1 2 1 2 11 2

0 1 0 0( ) 2 ,(2 1)2

22 2 2

.r rn rn

rr t i k r

rn ni k t kt iϕ− − → − −−

= = = =⊕ + + +

=

∏ ∏ ∏ ∏CS PCWP (19)

2.3. The inverse multiparametric wavelet transform

The direct multiparametric wavelet transform (MPWT) is defined by expression:

1 1

1 1

0 1 2 1 2 2 2 20

2 [ , ,..., ] ( )r r n rn

n Di

D ir m i

h h h ϕ + +

→ − → −

− −= =

= ⊕

∏ ∏T P IWT , (20)

where ] [2log 2m D= . This is the orthogonal matrix and

its inverse matrix coincides with its transpose one. For this reason

10 1 2 1 0 1 2 12 2

1 1

1 1 10

1 1 2 1

1 1, 20 0 2

2 2 2 2

2 2 2

[ , , ..., ] [ , , ..., ]

( )

( ) ,

n nt

D D

n Di

r r i n rr m i

rn D

r r i n rk i kr m i k r

t

t

h h h h h h−− −

← − ← −

+ + += =

← − ← − −

+ +⊕ += = =

−

−

= =

= ⊕ =

= ⊕

∏ ∏

∏ ∏ ∏

P T I

P CS I

ϕ

ϕ

WT WT

(21)

where 2r⊕ is addition modulo 2r . Obviously inverse

wavelet packet is: 1

)0 1 2 1

1

1 1

11 21 1 1

1

11 1

1;( , ,..., ;22

2 2 22;; ;

[ , ,..., ]

...

n

Dr m

n

r m

m m n rn

rr r n rr r r

n

ss s

h h h← −

−=

← −

=

+ −

− −+ + +

−

−− −

− = =

= ⊕ ⊕ ⊕ =

∏

∏

s s s sWP AWP

AWT AWT AWT

1 12 11 2

1 0 0( ) 2 , (2 1) 2

2

1,2 2 ( ) .

n r rDn

r ir m t i k r

rt

rn i k t k

rt

st

s

ϕ− − ← − −−

−

= = = =⊕ + + +

=

∏ ∏ ∏∏P CS (22)

3. FOURTH CANONICAL FORM OF MPWT

3.1. The direct multiparametric wavelet transform

The atomic matrixes, which we took up below, were re-corded with the “normal” order of rows. That means the averaging h -rows is situated before the differencing g -rows within the atomic matrix. The fourth canonical form of MPWT can be found with using the cyclic pres-entation of atomic matrix:

[ ]

[ ]

8 0 1 5

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

4 5 0 1 2 3

4 5 0 1 2 3

2 3 4 5 0 1

2 3 4 5 0 1

2

0 1 52 2 1

80 2 2 1

4

, , ,

, , , ,k k

k k k

k

h h h

h h h h h hg g g g g g

h h h h h hg g g g g g

h h h h h hg g g g g gh h h h h hg g g g g g

h hC h h h

g g+

= +

=

= =

= ⊗ =

∑ P

…

…8

CAT

AWT

(23)

where 2nP is the permutation matrix of ideal 2-adic mixing, which swaps the rows of atomic matrix in stairs-like form [ ]0 1 52

, , ,n h h hAWT … according to the rule

1 1 1

0 1 2 3 4 2 2 2 10 2 1 2 1 2 2 1 2 1

r r

r r r r− − −

− − + − −

LL

.

In order to find third canonical form of multiparametric wavelet transforms we will multiply

[ ]0 1 2 12, , ,n Dh h h −CAT … sequentially with matrices

( )01 0CS ϕ , ( )23 0CS ϕ , … , ( )2 2,2 1 0D D− −CS ϕ choosing angles ϕ in such way as to product will be the matrix

[ ]0 1 2 32, , ,n Dh h h −′ ′ ′ ′CAT … with the new set of coefficients,

which quantity less by two then in the source matrix. For example,

( ) ( ) ( ) ( ) [ ]67 0 45 0 23 0 01 0 8 0 5,...,h h

c ss c

c ss c

c ss c

c ss c

ϕ ϕ ϕ ϕ

= ⋅

⋅ =

−

−

− −

CS CS CS CS CAT

0 1 2 3

2 3 4 5

0 1 2 3

2 3 4 5

0 1 2 3

4 5 2 3

2 3 0 1

2 3 4 5

h h h hg g g gh h h h

g g g gh h h h

g g g gh h h hg g g g

⋅ =

4 5

0 1

4 5

0 1

4 5

0 1

4 5

0 1

h hg g

h hg g

h hg g

h hg g

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

=

g g g gh h h hg g g g

h h h hg g g g

h h h hg g g gh h h h

′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ = ′ ′ ′ ′

′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′

[ ]' '1

1 2 2 18 8 0 1 2 3' '

0 2 2 14 , , , .k k

k k k

k h hC C h h h h

g g− +

= +

′ ′ ′ ′ ′= ⊗ =

∑ CAT

Let us to iterate foregoing procedure on the just got-ten new atomic matrix:

( ) ( ) ( ) ( )70 1 56 1 34 1 12 1 8 0 1 2 3, , ,h h h hϕ ϕ ϕ ϕ ′ ′ ′ ′ ′ = CS CS CS CS CAT

c sc ss c

c ss c

c ss c

s c

− = ⋅ − − −

2 3

0 1

2 3

0 1

2 3

2 3 0 1

2 3 0

0 1 2 3

0 1

0 1

0 1

0 1

0 1

0 1

0 1

0 1

g gh h

g gh h

g gh h

g gh h

h hg g

h hg g

h hg g

h hg g

′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′

′ ′ ′ ′ ⋅ = ′ ′ ′ ′

′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′

′′ ′′ ′′ ′′ ′′ ′′ ′′ ′′

= ′′ ′′′′ ′′

′′ ′′′′ ′′

0 1

2 3

0 1

2 3

0 1

2 3

0 1

2 3

g gh h

g gh h

g gh h

g gh h

[ ]' '2 2 1

8 0 1' '2 2 1

14 , .k k

k k

h hC h h

g g+

+

=

′′ ′′ ′′= ⊗ = CAT

As a result we get a block-permutation matrix with the orthogonal ( )2 2× blocks. If we will use appropriate ro-tation-reflection matrixes, we could transform this ma-trix to permutation one:

( ) ( ) ( ) ( ) [ ]01 2 67 2 45 2 23 2 8 0 1,R R R R h hϕ ϕ ϕ ϕ ′′ ′′ ′′ =CS CS CS CS CAT

0 1

0 1

0 1

0 1

0 1

0 1

0 1

0 1

28

11

11

,1

11

1

c ss c

c ss c

c ss c

c ss c

h hg g

h hg g

h hg g

h hg g

−

− = ⋅

− −

′′ ′′ ′′ ′′ ′′ ′′

′′ ′′ ⋅ = ′′ ′′

′′ ′′ ′′ ′′ ′′ ′′

− − −

− = = − −

− − −

C

where 28C is the matrix of cyclic shift modulo 8 on two

positions. Thus, we can put it down in the following way

( ) ( ) ( ) [ ]2 1 0 28 2 8 1 8 0 8 0 1 5 8, , ,h h hϕ ϕ ϕ ⋅ = −T T T CCAT …

where

( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )

28 2 23 2 45 2 67 2 01 2

18 1 12 1 34 1 56 1 70 1

08 0 01 0 23 0 45 0 67 0

,

,

.

=

=

=

T CS CS CS CS

T CS CS CS CS

T CS CS CS CS

ϕ ϕ ϕ ϕ ϕ

ϕ ϕ ϕ ϕ ϕ

ϕ ϕ ϕ ϕ ϕ

Since matrixes ( )2ni

iϕT are both symmetric and or-

thogonal, then ( ) ( )1

2 2n ni i

i iϕ ϕ−

= T T . Therefore,

[ ] [ ]( ) ( ) ( ) ( )

8 0 1 5 8 0 1 2

0 1 2 28 0 8 1 8 2 8

, , , , ,

1 ,

h h h ϕ ϕ ϕ

ϕ ϕ ϕ

= =

= − ⋅ ⋅T T T C

CAT CAT… (24)

and the atomic matrix can be represented up to multi-plicative constant (-1) as the following product:

[ ] [ ]

( ) ( ) ( )8 0 1 5 8 0 1 2

0 1 2 28 8 0 8 1 8 2 8

, , , , ,

.

h h h = =

= ⋅ ⋅ P T T T C

… ϕ ϕ ϕ

ϕ ϕ ϕ

AWT AWT (25)

Hence,

[ ] [ ] [ ][ ] ( ) ( ) ( )

( ) ( ) ( )

16 0 5 8 0 5 8 16 0 5

0 1 2 216 0 1 2 8 8 0 8 1 8 2 8 8

0 1 2 216 16 0 16 1 16 2 16

,..., ,..., ,...,

, ,

.

h h h h h h

ϕ ϕ ϕ ϕ ϕ ϕ

ϕ ϕ ϕ

= ⊕ ⋅ = = = ⋅ ⋅ ⊕ ⋅

⋅ ⋅ ⋅

I

P T T T C I

P T T T C

WT AWT AWT

WT

The flow chart of this algorithm for wavelet transform [ ]16 0 1 3, ,ϕ ϕ ϕWT is presented on Figure 1.

Figure 1. Flow chart of the algorithm for para-

metric presentation of [ ]16 0 1 2, ,WT ϕ ϕ ϕ

This result is general and valid for any ( )1 12 2r r+ +×

atomic matrix [ ]0 1 2 112 , , , Dr h h h −+ …AWT :

[ ] [ ]10 1 2 1 0 1 11 22, , ..., , ,...,rD Dr h h h ϕ ϕ ϕ+− −+ = =AWT AWT

( )1

1

01 1 12 2 2

DD

ii

r r ri ϕ

→ −−

=+ + +

= =

∏P T C

( ) ( )1 1

11

0 0 1 12 2

2 1

2 , 2 12 2 .r

r r

DD

ii k r r

k i k i+ +

→ −−

= = + +

−

⊕ + ⊕

=

∏ ∏P CS Cϕ (26)

Substituting (26) in (12) we get the multiparameer pres-entation of cyclic 2-adic orthogonal wavelet transform, which we call the fourth canonical form:

( )

0 1 12

1 11

01 1 1 12 2 22 2

[ , ,..., ]n D

n DD

ir m i

r r r n ri

ϕ ϕ ϕ

ϕ

−

→ − → −−

= =+ + + +−

=

= ⊕ =

∏ ∏P T C I

WT

(27)

( ) ( )1 1

1

0 01 1

1 12 2

2 1

12 , 2 1 2 2 22 .rn D

Di

r m i kr r

r rn rk i k i ϕ

→ − → −−

= = =+ +

+ +

−

+⊕ + ⊕ −

= ⊕

∏ ∏ ∏P CS C I

Recently, several authors [3],[4] have proposed Jcobi parametrization of wavelet transforms. E. Fosstgaard [3] proofed that all orthogonal cyclic 2-adic wavelet trans-forms can be factored into a sequence of Jacobi rotation and have the fourth canonical form in our terminology. K. Knothe [3] repeated this result and proofed it for 3-adic wavelet transforms.

3.2. The inverse multiparametric wavelet transform

For inverse matrix we have

[ ] [ ]

( )

1 1

11 1 1

10 1 1 0 1 12 2

1

02 2 2

, , , , , ,r r

Dr r r

tD D

tD

ii

i

+ +

−+ + +

−− −

→ −

=

= =

= =

∏P T C

… …ϕ ϕ ϕ ϕ ϕ ϕ

ϕ

AWT AWT

( ) ( )1 1

; 11 1 1 1 1 1

; 12 2 2 2 2 2

0 0.

D Dt D

r r r r r r

ti t t D i t

i ii i

→ − ← −−

+ + + + + +−

= =

= =

∏ ∏C T P C T Pϕ ϕ

Hence,

( )

( ) ( )

10 1 12

1 1

1 1

0

1 1 1 1

1 1 11 12 2

; 12 2 2

0

2 1; 1

2 , 2 1 2 20

2 2

2 2

[ , ,..., ]

.r

n D

n D

r m

n D

ir m k

r r r n r

r r n rr r

t D i ti

i

t D tk i k i

i

−−

← − ← −

=

← − ← −

= =

+ + + +

+ + ++ +

−

=

−−

⊕ + ⊕=

−

−

=

= ⊕ =

= ⊕

∏ ∏

∏ ∏ ∏

C T P I

C CS P I

ϕ ϕ ϕ

ϕ

ϕ

WT

where ( ) ( ) ( )11 12 2

2 , 2 1 20

2 1i

i irr r

k i k ik

r

++ +

⊕ + ⊕=

−=∏T CSϕ ϕ . In the same

manner we get the expression for inverse wavelet packet:

( )

( ) ( )

10 1 1

12 1; 1

0

12 1; 1

1 0 0

1

1 12 2

1

1

1 2 1

2 , 2 1

1 1

1 1

2

2 2

2

2

2

[ , ,..., ]

.

D

n rD

t D

r m i

n rD

t Ditr m i k

r

r r

n

n

t

n

k i k i

r r

r

r r

i ti

t

−−

− −← ← −

−

= =

− −→ −

−

== = =

+

+ +

−

=

→ − −

⊕ + ⊕

+ +

+ +

=

= = =

∏ ∏

∏ ∏ ∏

⊕

⊕

C T P

C CS P

ϕ ϕ ϕ

ϕ

ϕ

WP

4. IS AN ORTHOGONAL MATRIX A WAVELET COMPLETE PACKET?

4.1. Jacobi method

In this section, we briefly describe the Jacobi Cyclic Row Algorithm (JCRA) [1,8]. The JCRA attempts to reduce the ( )N N× orthogonal matrix U or ( )N N× symmetric matrix A to diagonal form by applying or-thogonal rotations to right of U , ( )pqϕ=Q UJ or to

left and right of A , ( ) ( )tpq pqϕ ϕ= ⋅ ⋅B J A J , where or-

thonormal Jacobi rotation with reflection

, ,

1 0 0 0

0 0

0 0

0 0 0 1

( )p q

p q

p

q

c s

s c

ϕ =

−

J

L L LM O M M M

L L LM M O M M

L L LM M M O M

L L L

is used to reduce the element pqU or pqA to zero.

4.2. Jacobi method for orthogonal matrix

Jacobi rotation ( )pqϕJ operates on p-th and q-th ele-

ment of the p-th row of U , ppU and pqU :

1 0 0 0

0 0

0 0

0 0 0 1

pp pq

qp qq

p q

p

q

Q

Q

Q

Q

= =

Q

L L LM O M M M

L L L

M M O M ML L L

M M M O ML L L

1 0 0 0 1 0 0 0

0 0 0 0,

0 0 0 0

0 0 0 10 0 0 1

pp pq

qp qq

U U c s

U U s c=

⋅ −

L L L L L LM O M M M M O M M M

L L L L L LM M O M M M M O M M

L L L L L LM M M O MM M M O M

L L LL L L

such that pqQ becomes zero. For 0pqQ = it must be re-quired: 0pp pqU c U s− + = . Hence, the expression for

( )pqtg ϕ become ( )pq pq pptg U Uϕ = . This is equivalent

to

2 2 2 2

( , ) ,pp pq

pp pq pp pp

U Uc s

U U U U

= + +

.

For example,

(1)12( )N N φ= ⋅ =Q U J

, ■ , , , ,, , , , , ,, , , , , ,, , , , , ,, , , , , ,, , , , , ,

,

where white boxes are nonzero elements and black box is the zero element. Further,

(2)12 13( ) ( )N N φ φ= ⋅ =Q U J J

, ■ ■ , , ,, , , , , ,, , , , , ,, , , , , ,, , , , , ,, , , , , ,

.

After 1N − Jacobi rotations we obtain

12 13 1( 1) ( ) ( )... ( )NN

N N φ φ φ− = ⋅ =Q U J J J

.

=

, ■ ■ ■ ■ ■, , , , , ,, , , , , ,, , , , , ,, , , , , ,, , , , , ,

But ( )1NN

−Q is an orthogonal matrix as the product of or-thogonal matrices. For this reason it can have only the following form:

12 13 1 1,

2

( 1) ( )( ) ( )... ( )N

N qq

NN N N φφ φ φ

→

=

− = ⋅ = ⋅ =∏Q U J J J U J

1

1

,N −

±

=

Q

■ ■ ■ ■ ■■ , , , , ,■ , , , , ,■ , , , ,■ , , , , ,■ , , , , ,

where 1N −Q is ( ) ( )1 1N N− × − orthogonal matrix op-

posite to 1N −=Q Q that is ( )N N× orthogonal matrix. Obviously,

( 1) 12

,1 1

( )N N N N

N N p qp q p

φ− → − →

= = +

= =∏ ∏Q JU

11

1.

11

1

± ± ±

± ± ±

■ ■ ■ ■ ■■ ■ ■ ■ ■■ ■ ■ ■ ■■ ■ ■ ■ ■■ ■ ■ ■ ■■ ■ ■ ■ ■

Hence, an orthogonal matrix U is composed of series of Jacobi rotations:

1

,1 1

( ).N N

p qp q p

φ← − ←

= = +

= ∏ ∏U J (28)

4.3. Jacobi method for symmetric matrix

The Jacobi algorithm composes a method for computing the eigen value decomposition square and symmetric matrix N N×∈A R , and was original proposed by Jacobi in 1846 [7]. Itarative applaying Jcobi rotations

( 1) ( ) (0), ,( ) ( ) i t ipq pqφ φ+ = ⋅ ⋅ =B J B J B A (29)

symmetric real matrix A is reduced to diagonal form. For each iteration step ( )pqϕJ is selected such that the

off-diagonal norm

2

1 1( ) =

N N

iji i j

off A= ≠ =

∑ ∑A (30)

is minimized. This is achieved if ( ) ( ) 0i ipq qpB B= = which

is fulfilled if ipqφ satisfies: ( ) ( ) ( ) ( )( ) 2 /( )i i i i

pq pq pp qqtg φ B B B= − . In original proposal each iteration is selected as the in-dices for which ( ) ( )i i

pq qpB B= are maximum. The algorithm converges quadratically, meaning

( ) (0)1( ) 1 ( )k

koff offN

≤ −

B B .

However, for each update ( )2NO comparisons are

required in order to locate the largest off-diagonal ele-ment. To overcome this excessive amount of searching the so called the Jacobi Cyclic Row Algorithm (JCRA) was proposed. In this case the rotation index pairs ( ),p q is chosen cyclically, e.g.

11,2 1,3 1,4 1, 1 1,

22,3 2,4 2, 1 2,

22, 1 2,

11,

... ,

... ,...,

.

Kr N N

Kr N N

NKr N N N N

NKr N N

−

−

−− − −

−−

=

=

=

=

J J J J J J

J J J J J

J J J

J J

(31)

The application of (31) for all ( 1) / 2N N + index pairs is called the Krilov sweep

1 1

,1 1 1

( ) ( ).N N N

Kr pN Kr p q

p p q p

φ φ→ − → − →

= = = +

= =∏ ∏ ∏S J Jr

where 1,2 1,3 1,( , ,..., )N Nφ φ φ φ −=r is ( 1) / 2N N + -dimension

vector of so-called the Krilov angles pqϕ and

,1

( )N

pKr p q

q pφ

→

= +

= ∏J J is the Krilov micro-sweep. According

to (28) arbitrary orthogonal matrix can be represented as Krilov sweep.

Unfortunately, subsequent rotations of sweep destroy already existing zeros in ( 1)i+B . Therefore several sweeps ( )Kr

N φS r have to be performed until the desired degree of diagonallity is achieved:

( 1) , ( ) (0) .( ) ( ), i t Kr i Krφ φ+ = ⋅ ⋅ =B S B S B Ar r (32)

This algorithm has quadratic converges, too. There are alternative orderings

11,2 2,3 3,4 2, 1 1,

21,2 2,3 3,4 2, 1

21,2 2,3

11,2

,..., ,

,..., ,...

,

,

Eu N N N N

Eu N N

NEu

NEu

− − −

− −

−

−

=

=

=

=

J J J J J J

J J J J J

J J J

J J

for the Euler sweep:

11 1

,1 1 1

( ) ( ),N pN N

Eu pN Eu p q

p p q

φ φ→ − +→ − → −

= = =

= =∏ ∏ ∏S J Jr (33)

where 1

,1

( ),N p

pEu p q

qφ

→ − +

=

= ∏J J is the Euler micro-sweep. It is

possible to proof that an arbitarary orthogonal marix can be represented as Euler sweep:

1

,1 1

( ).( )N N

p qp q p

EuN φφ

← − ←

= = +

= = ∏ ∏U S Jr (34)

Both this schemes are not amenable to parallel process-ing as they (appear) inherent sequential. For this reason numerous other schemes have been suggested [5,6].

We propose a novel scheme which allows / 2N commutative rotations:

1

2

2 12 1, ,

2 ( ) 2 2 ,(2 1)20 0

n rr

n r rn r

r i r ii k j j k

k jφ

− −

−

−→ −

⊕ + + += =

=

∏ ∏J J ,

where 0,1,..., 1r n= − , 10,1,..., 2 1n rk − −= − . e.g. 8N = :

}}}}

}}}}

}}}}

}}}}

0 1 2 32,0

0 ,4 1,5 2 ,6 3,7

0

0 1 2 32,1

1,4 2,5 3,6 0,7

0

0 1 2 32,2

2,4 3,5 0 ,6 1,7

0

0 1 2 32,3

3,4 0,5 1,6 2,7

0

,

,

2

,

.

k k k k

j

k k k k

j

k k k k

j

k k k k

j

r

= = = =

=

= = = =

=

= = = =

=

= = = =

=

=

== = =

J J J J J

J J J J J

J J J J J

J J J J J

1442443

1442443

1442443

1442443

}} }}

}} }}

0 1 0 11,0

0,2 1,3 4,6 5,7

0 1

0 1 0 11,1

1,2 0,3 5,6 4,7

0 1

,

1

.

k k k k

j j

k k k k

j j

r

= = = =

= =

= = = =

= =

= ⊕

= = ⊕

J J J J J

J J J J J

14243 14243

14243 14243

{ { { {

00,0

0,1 2,3 4,5 6,7

0 1 2 3

0 .k

j j j j

r=

= = = =

= = ⊕ ⊕ ⊕

J J J J J64444744448

A complete sweep hence requires ( 1)N − so called com-pound rotations each consisting of / 2N commutative rotations performed simultaneously:

1 2 1,

0 0

11 2 1

0 0 0 0 2

2

2 1,

( ) 2 2 ,(2 1)2

2 1

2 .

n

nr i

r i

n rn

kr i i j

r

r r

n r rn r

r ii k j jφ

→ − −

= =

− −→ − − →

+= = = = −

−

⊕ + +

−

= =

=

∏ ∏

∏ ∏ ∏ ∏

S J

J

Up to permutation matrices this sweep is completed wavelet packet

1 2 1 2 11 2

1 0 0( ) 2 ,(2 1)2

22 2 2( ) .

n r r rn

r ir m t i k r

rt

rn ni k t k

rt

st

s

ϕ− − → − −−

= = = =⊕ + + +

=

∏ ∏ ∏ ∏CS PCWP

This sweep contains the same number Jacobi matrices as the Krilov or Euler sweeps. For this reason we can suppose that an arbitrary orthogonal matrix can be rep-resented as new sweep:

1 2 1,

0 0

11 2 1

0 0 0 0 2

2 1,

( ) 2 2 ,(2 1)2

2

2 1

2 .

nr i

r i

n rn

kr i i j

r

n

r r

n r rn r

r ii k j jφ

→ − −

= =

− −→ − − →

+= = = = −

−

⊕ + +

−

= =

=

∏ ∏

∏ ∏ ∏ ∏

U J

J

Some of orthogonal transforms can be represented in this form [8-14]. Our Jacobi-like algorithm is very well suited for parallel implementations, like vector- or mul-tiprocessors, systolic and other mech-connected compu-tational (VLSI) arrays.

5. MPWT COMPRESSION PROPERTIES ESTIMATION

The technique of finding the optimal discrete orthogo-nal cyclic wavelet transform, i.e. those that would pro-vide the best reconstruction, is presented in this section. The common way to analyze and describe complicated signals and images is to represent them as a superposi-tion of simple signal or images. Most of the currently available approaches to signal/image representations are based on “Single Integral Transform Concept”. The in-tegral transforms and signal representations associated with them are the most important of any representation. The best known examples are the Fourier and Wavelet transforms. However, there are many other transforms (representations) which can be interesting for image and signal processing. An important aspect of these repre-sentations is the possibility to extract relevant informa-tion (which is actually available, but hidden) from sig-nals and images in their complex initial form. The ex-tracting of relevant information is the most important part of the signal/image analysis and interpretation. However, the diversity of problems that face science is so great that there is no available single universal method which would be well suited to all problems si-

multaneously. We hope that this difficulty can be over-come using so-called “Best Transform Concept”, based on multiparametric fast wavelet transform (13) and packet (16). One of the most important challenges is to find the best representation for the signals or images. Our approach to the signal is based on the orthogonal and multiparameteric fast wavelet transforms and pack-ets. This approach consists of three main steps:

1. Select a best transform (optimal basis) for the problem at hand from continuous family of transforms belonging to the multiparameteric wavelet transform (13) or packet (16). Each MPWT (or MPWP) depends on one or more free parameters. When parameters are changed in some way, the type and form of transform are changed as well. For example, MPWP may be the Daubechies transform for one set of values, Walsh transform for another value set and some other trans-forms for other values of parameters. This property en-ables us to calculate the spectra of signals and images for a continuous number of transforms. When parame-ters are changed, MPWP calculates the spectra of an image for all transforms belonging to MPWP and finds the best possible transform by optimizing certain crite-ria. While in the classic signal/image processing sys-tems (based on “Single Integral Transform Concept”) we observe a single “photo” of spectrum (for example, the spectrum of an image in Daubechies basis), then in MPWP we observe a “movie” consisting of image spec-tra (transformed images) for different transforms be-longing to MPWP. We can then select the best trans-form (the best coordinate system). The main reason to use a continuous family of transforms instead of a single transform (for example, Daubechies, Walsh or Haar-wavelet) is that we may lose our flexibility for handling various changes in images or manifolds.

2. Sort the components of transformed domain by “importance” for a specific task and discard “unimpor-tant” components. What is best and “important” clearly depends on the problem. Usually, a best transform for representing a signal or image should give large magni-tudes along a few axes and negligible magnitudes along most axes when signal or image is expended into the best basis associated with best transform. For image compression, a basis which provides only a few large components in the transformed domain should be used since we can discard the other components without large signal degradation. A basis through which we can “view” classes as maximally-separated point clouds in the N -dimensional space (where N is the number of pixels in each image) is an option. In this case, the “dis-tances” between classes should be used as a measure of the transform efficiency.

3. Use the surviving coordinates to solve the problem at hand using generalized non-harmonic analysis. The output of a computation can be memorized to be used in subsequent numerical computations: compression, in-terpolation, segmentation, edge detection, filtering, de-noising, etc.

Implementation of a best transform selection proce-dure for a prescribed signal/image (or a family of sig-nals/images) requires the introduction of an acceptable cost function which translates best into a minimization process. In this section we use entropy of wavelet coeffi-cient for cost function.

In order to estimate compression properties of multi-parametric orthogonal wavelet transform we have con-ducted experiments for revealing dependency of spec-tra’s coefficients entropy ( )0 1, , ,D

DE ϕ ϕ ϕ… on quantity of angle-parameters D and values of angle-parameters

iϕ . We use the entropy of spectra’s coefficients, quan-tized to integer values, as the cost function. The form of the dependency ( )2

0 1,E ϕ ϕ (case of two-parametric transform) is shown on Figure 2.

Figure 2. Entropy of spectra 2E relative to pa-rameters 0ϕ and 1ϕ for [ ]8 0 12

,ϕ ϕWDT . Test image is “Lena”

Figure 3. Minimal attainable spectra’s entropy

relative to D

The experimental gotten result for minimal attain-able spectra’s entropy ( )min 0 1, , ,D

DE ϕ ϕ ϕ… relative to D for three different images “Baboon”, “Lena” and “Light-

house” is shown on Figure 3. Only the minimal attain-able spectra’s entropy varied from image to image.

Dependency ( )0 1, , ,DDE ϕ ϕ ϕ… are multi-

dimensional if 2D > . To visualize such dependency we divide the turn-down of each parameter [ )0, 2iϕ π∈

into 2m parts. And then we treat the vector ( )0 1, , , Dϕ ϕ ϕ… as on the D -digit number in Radix- 2m

number system 1

20

(2 )m

Dm i

D ii

P p−

−=

= ∑ . This number we

can rewrite in Radix-10 number system 10

P . Thus, we

will put this number 10

P on X-axis and on Y-axis we will put the entropy of obtained spectra (see Figure 4). The entropy of quantized spectra relative to the angle-parameters values ( )0 1, , , Dϕ ϕ ϕ… in

10P -

representation is shown on the Figure 4. Figure 2 and Figure 4a ( 6m = , 2D = ) show, that

researched dependency has four minimums. Moreover, the form of dependencies and the location of minimums were the same for different images.

In order to compare the classical Daubechies wavelet transforms (for 4,6,8L = ) with multiparametrical wavelet transforms (for 2,3,4,5,6D = ) we investigated the quality of reconstructed image PSNR relative to the percent of not zeroed spectra’s coefficients. The results are in Figure 5. The values of the angle-parameters, used in this comparison, conform with the minimums of the function ( )0 1, , ,D

DE ϕ ϕ ϕ… for proper D . The test image «Lena» was used.

As shown on Figure 5, the transforms, found with assistance of multiparametric presentation of wavelet transforms, excels the classical ones with the same car-rier length with respect of compression properties.

In this chapter we researched compression properties of orthogonal wavelet transforms. In more general case, the using of multiparametric presentation of wavelet transform gives us the opportunity of finding the best wavelet basis to solve various particular problems. For example it could be the image filtering problem or spe-cial image analysing in medicine.

6. CONCLUSION

In this paper we defined the new representation of or-thogonal wavelet transform, named multiparametric form of cyclic orthogonal wavelet transform (13,27). This form is the product of sparse rotation matrixes and it describes fast algorithm for cyclic wavelet transforms. Defined representation of wavelet transform depends on finite set of free parameters, which could be changed independently of one another. For each set of parame-ters values we get the unique cyclic orthogonal wavelet transform.

(a)

(b)

(c)

Figure 4. Entropy of spectra relative to 10

P for different values of parameters:

a) 6m = , 2D = ; b) 6m = , 3D = ;c) 6m = , 4D =

Figure 5. Reconstruction of image “lena” with

different percent of preserved wavelet spectra co-efficients

7. ACKNOWLEDGMENTS

This work was supported by RFBR N 06-01-08082 and RFBR N 07-01-00409-a

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