Introduction to The Lifting Scheme

• Two approaches to make a wavelet transform:– Scaling function and wavelets (dilation

equation and wavelet equation) – Filter banks (low-pass filter and high-pass

filter)

• The two approaches produce same results, proved by Doubeches.

• Filter bank approach is preferable in signal processing literatures

Wavelet Transforms

|H0(w)|, |G0(w)|

/ 2/ 2Ideal low-pass filter

|H 1(w ) |, |G 1(w ) |

/ 2/ 2Ideal high- pass filter

Wavelet Transforms

2

2

2

2

+X(z) Y(z)

H0(z)

G1(z)

G0(z)

H1(z)

X1(z)

X0(z)X'0(z)

X'1(z) Y'1(z) Y1(z)

Y'0(z) Y0(z)

Practical Filter

Understanding The Lifting Scheme

signal

Splitting

)0(ks

)0(kd

)1(ks

)1(kd

signal

Merge

)1(kd

Predicting

)1(ks

Updating

)0(kd

Inverse Predicting

)0(ks

Inverse Updating

…

Transmitting

S (z )

D (z)

P (z ) U (z )

S '(z )

D '(z )

splitX (z )

-

Lifting Scheme in the Z-Transform Domain

Low band signal

High band signal

Update stage

Prediction stage

2

S '(z )

D '(z )

P (z )U (z )

S (z )

D (z )

-com-bine

X (z )

Lifting Scheme in the Z-Transform Domain

Inverse update stage

Inverse prediction stage

2

• A spatial domain construction of bi-orthogonal wavelets, consists of the following four basic operations:

• Split : sk(0)=x2i(0), dk(0)=x2i+1

(0)

• Predict : dk(r)= dk(r-1) – pj(r) sk+j(r-1)

• Update : sk(r)= sk(r-1) + uj(r) dk+j(r)

• Normalize : sk(R)=K0sk(R), dk(R)=K1dk(R)

Four Basic Stages

• Prediction and Update

d i-2(k- 1 )

p 0(k )... ...

......

d i-1(k- 1) d i

(k- 1) d i+ 1(k- 1) d i+ 2

(k- 1) d i+3(k - 1)

s i-2(k - 1 ) s i-1

(k- 1) s i(k- 1) s i+ 1

(k- 1) s i+ 2(k- 1) s i+3

(k- 1)

p 0(k )p 1

(k ) p 1(k )

d i-2(k) d i-1

(k) d i(k) d i+ 1

(k) d i+ 2(k) d i+3

(k)

u 0(k ) u 0

(k )u 1(k ) u 1

(k )

s i-2(k- 1 ) s i-1

(k - 1) s i(k - 1) s i+ 1

(k- 1) s i+ 2(k- 1) s i+3

(k- 1)

Two Main Stages

• A prediction rule : interpolation– Linear interpolation coefficients: [1,1]/2

• used in the 5/3 filter– Cubic interpolation coefficients: [-1,9,9,-1]/16

• used in the 13/7 CRF and the 13/7 SWE

xi- 1

xi+1

xi- 2

xi- 3

xi+2

xi+3xp

30- 1- 3 1

xi- 1

xi+1

xi- 2

xi- 3

xi+2

xi+3xp

30- 1- 3 1

Prediction Stage

• An update rule : preservation of average (moments) of the signal– The update coefficients in the 5/3 are [1,1]/4– The update coefficients in the 13/7 SWE are

[-1,9,9,-1]/32– The update coefficients in the 13/7 CRF are

[-1,5,5,-1]/16

Update Stage

• The 5/3 wavelet– The (2,2) lifting scheme

split

predic t

high pass

0p0p

0u 0u

2ix 1ix ix 1ix 2ix 3ix 4ix

1kd

ks

kd

0p 0p

low pass

update

Example

• We have p0 = 1/2 by linear interpolation and the detailed coefficient are given by

• In the update stage, we first assure that the average of the signal be preserved

• From an update of the form, we have

• From this, we get A=1/4 as the correct choice to maintain the average.

)(2

121 iiik xxxd

i

ik

k xs2

1

)( 1 kkik ddAxs

i

ii

ii k

kik

k xAxAdAxs 12)21(2

Example

• The coefficients of the corresponding high pass filter are {h1} = ½{-1,2,-1}

• The coefficients of the corresponding low pass filter are {h0} = ⅛{-1,2,6,2,-1}

• So, the (2,2) lifting scheme is equal to the 5/3 wavelet.

Example

• Complexity of the lifting version and the conventional version– The conventional 5/3 filter

• X_low = ( 4*x[0]+2*x[0]+2*(x[-1]+x[1])-(x[2]+x[-2]) )/8

• X_high = x[0]-(x[1]+x[-1])/2

• Number of operations per pixel = 9+3 = 12

– The (2,2) lifting• D[0] = x[0]- (x[1]+x[-1])/2

• S[0] = x[0] + (D[0]+D[1])/4

• Number of operations per pixel = 6

Example

• The lifting scheme is an alternative method of computing the wavelet coefficients

• Advantages of the lifting scheme:– Requires less computation and less memory. – Easily produces integer-to-integer wavelet

transforms for lossless compression. – Linear, nonlinear, and adaptive wavelet

transform is feasible, and the resulting transform is invertible and reversible.

Conclusions