LLH LHL LHHLLL

3.4 Dyadic or Octave Band Subband Tree Structure

Logarithmic Wavelet Transform.

The dyadic or octave b an d tree is a special irreg u lar tree structure. It

splits only the low er half of the spectrum into tw o equal b an d s at an y level of

the tree. Therefore the detail or higher-hand com ponent of the signal at any

level of the tree is decom posed no further. The dyadic tree configuration an d its

co rresponding frequency resolution for L = 3 are given in Figure 3.3.

LH

LLL LLH K K K

i T ~2

Figure 3.3: The Dyadic Subband Tree Structure - Logarithmic Wavelet.

K

A n exam ination of this analysis-synthesis stru ctu re show s th a t a half­

resolution frequency step is used at each level. Therefore it is also called the

octave-band or constant-Q subband tree structure. First, low (L) an d h ig h (H)

signal ban d s are obtained here. W hile the b an d (L) p ro v id es a coarser version of

th e original signal, b an d (H) contains the detail inform ation. If the low spectral

com ponent or b a n d (L) is interpolated by 2, the detail inform ation or the

in terp o latio n error is com pensated by the interp o lated version of b a n d (H).

C onsequently, the original is perfectly recovered in this one-step dyadic tree

com ponent of the higher-level n ode of the tree. N ote th at the total d ata rate for

the sub b an d signals at the o u tp u t of the analysis stage equals , the d a ta rate

of the source signal. Therefore the dyadic tree is also critically sam pled.

This m ultiresolution (coarse-to-fine) signal decom position idea w as first

p ro p o sed for 2D signals by Burt an d A delson for vision an d im age coding

p roblem s [34]. In fact the orthonorm al w avelet transform also em ploys this

dyadic subband tree. In th at case, the coefficients in the m u ltireso lu tio n w avelet

decom position of a continuous-tim e signal, are calculated usin g the discrete­

tim e dyadic sub b an d tree p resented here.

3.5 Discrete Wavelet Transform (DWT) Algorithm

W e have seen th at the 's, /:=0,...,2 A^- 1, constitute the im pulse

response of the QMF Fq and th at the im pulse response of the QMF is related

b y

For the m om ent, an d for ease of notatio n w e w ill restrict ourselves to sim plest,

alth o u g h the m ost localised, m em ber of the class of w avelets discovered by

D aubechies [36] [38], often called DAUB4 w hich has only four coefficients

Co,Cj ,C2,C3. In other w o rd s w e require to satisfy the ap p roxim ation condition of

o rd e r N = 2 . The nu m b er of coefficients increases by tw o each tim e N is

increased by one.

Let us consider the follow ing transform ation m atrix acting on a colum n vector

Cq C j C j C 3 C-, — Cr, C, — C,₀ C , Cr, C3 C 2 Cj Cq Cz C3 C l - C o ^ 0 ^ 2 ^3 C3 - C2 Cl - Cq C q Cl C l — Cr, (3.8)

Blank entries to the above m atrix signify zeros. The stru ctu re of the m atrix is

such that, the o d d row s generate one com ponent of the data co nvoluted w ith

the filter Co,q,C2,C3, an d the even row s perform a different convolution w ith

coefficients Cj^,-C2,c^,-Cq. The last tw o row s w ra p -aro u n d like convolutions

w ith periodic b o u n d ary conditions.

The b ehaviour of the m atrix, is to perform tw o related convolutions, th en to

decim ate each of them by half, by th row ing aw ay half the values, an d finally to

interleave the rem aining halves.

This results in the o u tp u t of Fq, reduced b y 50%, accurately rep resen tin g the

d a ta 's "approxim ation" (trend, sm ooth) inform ation. The o u tp u t of F ,, also

decim ated, is referred to as d ata's "detail" (fluctuation, difference) inform ation.

It m u st be possible to reconstruct the original d ata vector of length L from its

L/2 detail or d - com ponents. That is achieved b y req u irin g the above m atrix to

be orthogonal, so th at its inverse is just the tran sp o sed m atrix. In oth er w ords,

w e require the W avelet Filter Coefficients of this m atrix to satisfy all the

conditions th at w e have seen before in Section 2.4.

"approxim ation" vector of length L /2 , th en to the "approxim ation-

approxim ation" vector of length L /4 , an d so on u n til only a trivial n u m b er of

"approxim ation - ... - approxim ation" com ponents (usually 2) rem ain. The

p ro ced u re is som etim es called a P yram id A lgorithm , for obvious reasons. The

o u tp u t of the DWT consists of these rem aining com ponents an d all the "detail"

com ponents th at w ere accum ulated along the w ay. A d iag ram sh o u ld m ake the

pro ced u re clear:

- a , - a , ' A , ' A , ' a ; ' ^ 2 a j 0 , A a A a ' X3 f l j a , ^ 2 A 3 A , X4 d . a . _A A 4 A a ^ 5 « 3 a ; A 3 A A ^ 6 d . a s A A D a a # a . A 4 A A X g _{W FCM ^} d . Permute ^ W FC M ^ D . Permute ^ O 4 W F C M ^ A a ; d , d , d . r f . • ^ 1 0 d s d . 4 4 A X i i _{a &} d . d . 4 ^ 3 ^ 1 2 d . 4 d . d . d . • ^ 1 3 a , d s 4 d s 4 d . _{d .} d . < • ^ 1 5 a s d , d , d . A . ^ 1 6 . d i _ d s _ A . _ d g _ _ d g _

If the length of the d ata w ere a h igher p o w er of tw o, there w o u ld be m ore

stages of ap plying the w avelet coefficients m atrix an d p erm uting. A t the en d

there w ill alw ays be a vector w ith tw o A " s an d a sequence of A 's, D 's, d 's,

etc. W e can notice from the above m atrix, th a t once d 's are generated, they

The value d . , of any level is term ed a "w avelet transform detail coefficient" of

the original signal vector. The final values A,", should strictly be called

"w avelet transform m other-approxim ation coefficients", alth o u g h the term

"w avelet coefficients" is often used loosely for b o th d ' s an d final A " s . Since

the full p ro ced u re is a com position of orthogonal linear operations, the w hole

DWT is itself an orthogonal linear operator. In o rd er to in v ert the DWT, w e can

sim ply reverse the procedure, starting w ith the sm allest level of the hierarchy

a n d w orking the above pro ced u re from the rig h t to the left. The follow ing

inverse m atrix is u sed instead of the first m atrix (3.8).

C0 C3